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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I WORKS OF PROF. M. A. HOWE PUBLISHED BY JOHN WILEY & SONS. The Design of Simple Roof-trusses In Wood and Steel. With an Introduction to the Elements of Graphic Statics. Third edition, revised and enlarged. 8vo, viii + 179 pages, 87 figures and 3 folding plates. Cloth, $2.00. Retalnlng-walls for Earth. Including the Theory of Earth-pressure as Developed from the Ellipse of Stress. With a Short Treatise on Foundations. Illustrated with Examples from Prac- tice. Fifth edition, revised and enlarged, xamo, cloth, $1.25. A Treatise on Arches. Designed for the use of Engineers and Students in Technical Schools. Second edition, revised and en- larged. Svo, xxv -f 369 pages, 74 figures. Cloth, $4.00. Symmetrical Masonry Arches. Including Natural Stone, Plain-concrete, and Rein- lorced-concrete Arches. For the use of Technical Schools, Engineers, and Computers in Designing Arches according to the Elastic Theory. Svo, X + X70 pages, many illustrations. Cloth, $2.50^ iwvAi r::t.'::':".'. co.jpany THE DESIGN OF SIMPLE ROOF -TRUSSES IN WOOD AND STEEL. JV/Tff AN IN TROD UCTION TO THE ELEMENTS. OF GRAPHIC STATICS. ■BY MALVERD A. HOWE, C.E., Professor of O'lnl Engineering, Rose Fofytechnic Institute; Member of American Society of CivU Engineers. •» • •T. • •' • • • - ■*< -••.-•.:•.:: •.•:-. THIRD EDITION, REVISED AND ENLARGED. FIRST THOUSAND. NEW YORK : JOHN WILEY & SONS. London: CHAPMAN & HALL, Limited. ior2. •>-■*- • T^l«« .•»<•. / \ Copyright: I902» 19x2* BY MALVERD A. HOW& • •• THC SCISNTiFiC PRttS DWUMMOWa AWO OOOIMUW ■ROOKLVN. N^ V. PREFACE TO FIRST EDITION. Very little, if anything, new will be found in the follow- ing pages. The object in writing them has been to bring together in a small compass all the essentials required In properly designing ordinary roof-trusses in wood and steel* : At present this matter is widely scattered in the various comprehensive treatises on designing and in manufacturers* pocket-books. The student who desires to master the ele- ments of designing simple structures i^ thus compelled to procure and refer to several more or less expensive books. Students in mechanical and electrical engineering, as a rule, learn but little of the methods of designing em- ployed by students in civil engineering. For this reason the writer has been called upon for several years to give a short course in roof-truss design to all students in the Junior class of the Rose Polytechnic Institute, and in order to do so he has been compelled to collect the data he has given in this book. The tables giving the properties of standard shapes are based upon sections rolled by the Cambria Steel Company. Standard sections rolled by other manufacturers have practically the same dimensions. Malverd a. Howe. Terre Haute, Ind., September, 1902. iii 263609 PREFACE TO THE THIRD EDITION. „^.- - : ' The design ^ of details, in wood has been revised, using tiie - standard or actual sizes of lumber instead of the nominal sizes. The imit stresses for wood as given in Table XVI have been used without increasing them, although some designers use from thirty to fifty per cent larger values. If selected limibfir were always obtainable^ liie larger values could be safely employed. Considerable new matter will be foimd in the body of the text and iit the Appendix. -The author is indebted to Prof. H. A. Thomas for a careful reading of the ! text. Terre Haute, Ind., August, 1QI3. M. A. H. IV V ' ., ' -> CONTENTS. CHAFfER L GENERAL PRINCIPLES AND METHODS. ABT. PAOB «1. Equilibrium ; 1 2. The Force Polygon 1 3. Forces not iij Equilibrium — Force Required to Produce Equilibrium as far as Motion of Translation is Concerned 2 4. Perfect Equilibrium 3 •5. The Equilibrium Polygon 3 6. Application of the Equilibrium Polygon in Finding Reactions 5 7. Parallel Forces 7 8. The Direction of One Reaction Given, to Find the Magnitude and Direction of the Other 7 9. Application of the Equilibrium Polygon in Finding Centers of Gravity 8 K). Application of the Equilibrium Polygon in Finding Moments of Forces 9 11. Graphical Multiplication 12 12. To Draw an Equilibrium Polygon' through Three Given Points. ... 12 CHAPTER n. BEAMS AND TRUSSES. • 13. Vertical Loads on a Horizontal Beam, Reactions and Moments of the Outside Forces 14 14. Vertical Loads on a Simple Roof-truss — Structure considered as a Whole 15 15. Inclined Loads on a Simple Roof-truss — Structure considered as a Whole 16 16 Inclined Loads on a Simple Roof-truss, One Reaction Given in Direction — Structure considered as a Whole 16 17. Relation between the Values of R^ in Arts. 15 and 16. 17 18. Internal Equilibrium and Stresses. 18 V VI CONTENTS. AST. ?Aaa 19. Inside Forces Treated as Outside Forces 20 20. More than Two Unknown Forces Meeting at a Point 20 CHAPTER III. STRENGTH OF MATEBULS. 21. Wood in Compression — Columns or Strutd 22 22. Metal'* ** '* *' '' 27 23. End Bearing of Wood 29 23a. Bearing of Wood for Surfaces Inclined to the Fibers 30 236. End Bearing of Wood against Round Metal Pins 31 23c. Splitting Effect of Round Pins 32 23(2. Cross Bearing of Wood against Round Pins 32 24. Bearing of Steel 33 25. Bearing across the Fibers of Wood 34 26. '' '' '' '' *' Steel 34 27. Longitudinal Shear of Wood 34 28. '' '' ''Steel 35 29. Transverse Strength of Wood 36 30. '' '' ''Steel Beams 39 31. Special Case of the Bending Strength of Metal Pins 43 32. Shearing Across the Grain of Bolts, Rivets, and Pins 43 33. Shearing Across the Grain of Wood 45 34. Wood in Direct Tension 45 35. Steel and Wrought Iron in Direct Tension 45 CHAPTER IV. R00F-TBI7SSES AND THEIB DESIGN. 36. Preliminary Remarks 46 37. Roof Coverings , . 46 38. Wind Loads 47 39. Pitch of Roof 47 40. Transmission of Loads to Roof-trusses 48 41. Sizes of Timber 48 42. Steel Shapes 49 43. Round Rods 49 44. Bolts 49 45. Rivets 50 46. Local Conditions 50 CONTENTS, Vll CHAPTER V DESIGN OF A WOODEN ROOF-TRUSS. ▲BT. 47. 48. 49. 50. 51. 52. 63. 54. 55. 66. 56a. 57. 58. 59. 60, 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. Data Allowable Unit Stresses Rafters Purlins Loads at Truss Apexes Stresses in Truss Members Sizes of Compression Members of Wood Sizes of Tension Members of Wood Sizes of Steel Tension Members Design of Joint Lo with IJ" Bolts Bolts and Metal Plates . . . Nearly all Wood Steel Stirrup *' and Pin Plate Stirrup and Pin .... Steel Angle Block Cast-iron Angle Block .... Special Plank Members Soft Steel Plates and Bolts Design of Wall Bearing Design of Joint Ut '' '* '' Ui '* '' '' L. " * ' * * L» and Hook Splice * * * * * * Lit Fish-plate Splice of Wood . *' '* '* 1/8, Fish-plate of Metal Metal Splices for Tension Members of Wood . General Remarks Concerning Splice Design of Joint Ui The Attachment of Purlins The Complete Design ti tt ft it tt tt tt it tt tt tt tt tt it tt tt it it it tt PAOS 51 52 52 53 54 55 57 59 60 61 66 68 68 70 72 73 73 74 76 76 77 78 81 82 85 86 88 90 90 91 91 92 CHAPTER IV. DESIGN OF A STEEL ROOF-TRUSS. 78. Data 96 79. Allowable Stresses per Square Inch , 96 80. Sizes of Compression Members 96 81. *' '' Tension Members 99 viw' CONTENTS. ▲BT. PAOa 82. Design of Joint Lo 100 83. " " *' Ui 101 84. :" '* " L,.. 101 85. .'.'..'* .*' C/i , 102 86. Splices. 102 87. End. Supports 102 88. E^ansion 103 89. Frame Lines and Rivet Lines 103 90. Drawings 103 91. Connections for Angles ; 104 92. Purlins 104 93. End Cuts of Angles— Shape of Gugset Plates. 107 TABLES. I. Weights of Various Substances 109 II. Roof Coverings — ^Weights of Ill III. Rivets^-Standard Spacing and Sizes 115 IV. Rivets — Areas to be Deducted for 117 V. Round-headed Rivets and Bolts — ^VJeights 118 VI. Bolt Heads and Nuts — ^Weights and Dimensions 119 VII. Upset Screw Ends for Round Bars — Dimensions 120 VIII. Ri^t and Left Nuts — Dimensions and Weights 121 IX. Properties of Standard I Beams 122 X. Properties of Standard Channels 124 XI. Properties of Standard Angles with Equal Legs . 126 • XII. Properties of Standard Angles with Unequal Legs 128 XIII. Least Radii of Gyration for Two Angles Back to Back 134 XIV. Properties of T Bars 135 XV. Standard Sizes of Yellow Pine Lumber and Corresponding Areas and Section Moduli 137 XVI. Average Safe Allowable Working Unit Stresses for Wood 139 XVII. Cast-iron Washers— Weights of. 140 XVIU. Safe Shearing and Tensile Strength of Bolts 141 APPENDIX. AKT. 1. Length of Keys, Spacing of Notches and Spacing of Bolts 143 2. Plate Washers and Metal Hooks for Trusses of Wood 145 3. A Graphical Solution of the Knee-brace Problem 148 4. Trusses which may have Inclined Reactions 151 5. Tests of Joints in Wooden Trusses 155 6. Examples of Details Employed in Practice 155 7 Abstracts from General Specifications for Steel Roofs and Buildings. . 163 •< • V • •• c • • o • • GRAPHICS. CHAPTERS . GENERAL PRINCIPLES AND METHODS. 1. Equilibrium. — Forces acting upon a rigid body are in equilibrium when the body has neither motion of trans- lation nor rotation. • For forces which lie in the same plane the above condi- tions may be stated as follows : (a) There will be no motion of translation when the algebraic sums of the components of the forces resolved parallel to any two coordinate axes are zero. For conve- nience the axes are usually taken vertical and horizontal, then the vertical components equal zero and the horizontal components equal zero. {b) There will be no motion of rotation when the algebraic sum of the moments of the forces about' any center of moments is zero. 2. The Force Polygon. -Let AB, BC, CD, and DA, Fig. I, be any number of forces in equilibrium. If these forces are laid off to a common scale in succession, par- allel to the directions in Fig. i, a closed figure will be formed as shown in Fig. la. This must be true if the algebraic sums of the vertical and horizontal components respect- ively equal zero and there is no motion of translation. Such a figure is called a force polygon. -» •" *. GRAPHICS. Conversely, if any number of forces are laid off as ex- plained above and a closed figure is formed, the forces are Fig. I. in equilibrium as far as motion of translation is concerned. Motion of rotation may exist, however, when the above condition obtains. 3. Forces Not in Equilibrium. — In case a number of Fig. 2. forces, not in equilibrium, are known in direction and magni- tude, the principle of the force polygon (Art. 2) makes it pos- sible to at once determine the magnitude and direction of the force necessary to produce equi- librium. Let AB^ BC,,..,DE be forces not in equilibrium, Fig, 2. According to Art. 2, lay them off on some convenient scale, as shown in Fig. 2a. Now in Fig. 2b. order that the sum of the verti- cal components shall equal zero a force must be introduced Or -Q^ GENERAL PRINCIPLES AND METHODS. 3 having a vertical component equal to the vertical distance between E and A, and in order that the horizontal com- ponents may equal zero the horizontal component of -this force must equal the horizontal distance between E and A. These conditions are satisfied by the force EA. If this force acts in the direction shown by the arrow-head in Fig. 2a, it will keep the given forces in equiHbrixim (Art. 2) . If it acts in the opposite direction, its effect will be the same as the given forces, and hence when so acting it is called the resultant. Fig. 26 shows the force polygon for the above forces drawn in a different order. The magnitude and direction of R is the same as found in Fig. 2a. 4* Perfect Eqtiilibrium. — Let the forces AB, BC, . . . , DE, Fig. 2, act upon a rigid body. Evidently the force R, found above (Art. 3), will prevent motion, either vertically or horizontally, wherever it may be applied to the body. This ftdfills condition (a) (Art. i). For perfect equilibritmi condition {b) (Art. i) must also be satisfied. Hence there must be found a point through which R may act so that the algebraic sum of the moments of the forces given and i?, may be zero. This point is found by means of the equilibrium polygon. 5. The Equilibrium Polygon. — Draw the force polygon .(Art. 2) ABODE, Fig. 3a, and from any convenient point P draw the lines 5^, Sj, . . . , Sj. If 5^ and 52 be measured with the scale of the force polygon, they represent the mag- nitudes and directions of two forces which would keep AB in equilibrium as far as translation is concerned, for they form a closed figure with AB (Art. 2). Likewise S^ and 5, would keep BC in equilibrium, etc. Now in Fig. 3 draw GRAPHICS. S^ parallel to 5^ in Fig. 3a, 5, parallel to Sj in Fig. 3a, etc., as shown. If forces be assumed to act along these lines having the magnitudes shown in Fig. 3a, respectively, the points I, 2, 3, and 4 will be without motion, since the forces Fig. 3. Fig. 3tf. meeting at each point are in equilibrium against translation by construction, and, since they meet in a point, there carl be no rotation. In Fig. 3a, 5j and S^ form a closed figure with R ; there- fore if, in Fig. 3, S^ and S^ be prolonged until they intersect in the point r, this point will be free of all motion uiider the action of the forces 5^, 5^, and R, Since the points i, 2, 3, 4, and r in Fig. 3 have neither motion of translation nor rotation, if the forces AB, BC, CD, and DE and the force R be appHed to a rigid body in the relative positions shown in Fig. 3, this body will have no GENERAL PRINCIPLES AND METHODS. 5 motion under their action. The forces S^ and Sg keep the system A BCD in equilibrium and can be replaced by R. The lines S^^ S^y etc., in Fig. 3a are for convenience called strings, and the polygon 5^, 52, 5,, etc., in Fig. 3 is called the equilibrium polygon. The point P in Fig. ^a is called the pole. 6. Application of the Equilibrium Polygon in Finding Reactions.— Let ^ rigid body be supported at K and K\ Fig. 4, and acted upon by the forces AB, BC, CD, and Fig. 4. ___jir_. Fig. 4a. DE. Then, if equilibrium exists, it i$ clear -that two forces, one at each support, must keep the forces -/IB, BC, etc., in equilibrium. These two forces are called reactions. For convenience designate th^ one upon the left as R^, and the one upon the right as R^, The magnitudes of R^ and R^ can be found in the following manner: Construct the force 6 GRAPHICS. polygon and draw the strings 5^, Sj, etc., as shown in Fig. 4a, and then construct the equilibrixam polygon (Art. 5) as shown in Fig. 4. Unless some special condition is intro- duced the reactions R^ and i?, will be parallel to EAy Fig. 4a, and their sum equal the magnitude of EA, or the re- sultant of the forces AB, BC, CD, DE. Draw through K and iC' lines parallel to i?, and, if necessary, prolong the line 5i until it cuts oK, Fig. 4, and S^ until it cuts 5/C'. Connect o and 5, and in Fig. 4a, draw the string 5^ parallel to 05, Fig. 4, until it cuts EA in L. Now, since 5^, 5^,, and AL form a closed figure in Fig. 4a, the point o in Fig. 4 will be in equilibritun tmder the action of these three forces. For a like reason the point 5 will be in equi- libritun under the action of the three forces S^y S^y and EL, Therefore the reaction R^ =* AL and R^ = LE, and the body M will be in equilibrium under the action of the forces AB, BC, CD, DE, R^ and i?,. It may not be perfectly clear that no rotation can take place from the above demonstration, though there can be no translation since jR^ + ^2 ~ ^^* ^^^ force necessary to prevent translation under the action of the forces AB, BCy CD, and DE. To prove that rotation cannot take place let the forces AB, BC, etc., be replaced by their resultant R, acting down- ward, as shown in Fig. 4. If no rotation takes place (Art. i)> R(bK') « R,(aK') or R, « ^R. From the similar triangles 0^5, Fig. 4, and PAL, Fig. 4a, ds:aK'::R,:H or R,aK' ^ H(ds). GENERAL PRINCIPLES AND METHODS^ 7 From the similar triangles cd$, Fig. 4, and PAE, Fig. 4a, dsibK' ::R:H or R{bK') = H(ds). .\ R,{aK')^RbK' or R.^^R, or the value of R^^ by the above construction fulfills the con« dition that no rotation takes place. 7. Parallel Forces. — In case the forces AB, BC, etc., had been parallel the force polygon would become a straight line and the line A BCD , . , E would coincide with £^. All of the constructions and conclusions given above apply to such an arrangement of forces. See Figs. 9 and 9a. 8, The Direction of One Reaction Given, to Find the Magnitude and Direction of the Other. — Let the direction of J?2 be assumed as vertical, then the horizontal compo- ^' 8s Ji'" Polea^i:;^._ 80 "^ 8 Fig 5. nent, if any, of all the forces acting must be appHed at K. The force polygon (Art. 2) becomes ABC D EX, as shown in Fig. 5 a. Asstmie any pole P, and draw the strings 5j, 5„ etc. In Fig. 5, construct the equilibrium polygon (Art, 5) as shown, starting with 5^, passing through K, tJ'te only point on R^ which is known. Draw the closing line5<,', and in « GRAPHICS. Fig, ^a the string PU parallel to S^ of Fig. 5. Then EU is the magnitude of the vertical reaction R2, and L'A the mag- nitude and direction of the reaction jR^. To show that there will be no rotation under the action of the above forces, draw AE, EC, AC, and DE in Fig. 6, B parallel to S^ 5^, PF, and AE respectively in Fig. 5a. Then the point £ is in equilib- rium tmder the action of 5,, Sg, and /?, since these forces form a closed figure in Fig. sa. In Fig. 6, draw AB, CB, and BE parallel to 2?i, i?2, and AE of Fig. 5a. Then point B is in equilibrium tmder the action of jRj, i?2» and J?, and BE is parallel to ED, But J?i, 5i, and /?, and i^j, 5^, and R must form closed figures in Fig. 6, as they meet in a point in Fig. 5a respectively. Therefore BE prolonged coincides with DE, and there can be no rotation, since i?^, J?2» and R meet in a point. 9. Application of the Equilibrium Polygon in Finding Centers of Gravity. — Let abc . , . k he an tmsymmetrical body having the dimension normal to the paper equal unity. Divide the area into rectangles or triangles whose centers of gravity are readily determined. Compute the area of each small figure, and assume that this area multiplied by the weight of a unit mass is concentrated at the center of gravity of its respective area. These weights may now be considered as parallel forces P^, P, and P„ acting as shown in Fig. 7. The resultant of these forces must pass through the center of gravity of the entire mass, and hence lies in the lines R and R' formed by constructing two equilibrium GENERAL PRINCIPLES AND METHODS. 9 polygons for the forces P^, P^, and Pg, first acting vertically and then horizontally. The intersection of the lines R and /?' is the center of gravity of the mass. The load lines in Fig. 8 and Fig. 8a are not necessarily at right angles, but such an arrangement determines the point of intersection of R and R' with a maximum degree, of accuracy, since they intersect at right angles. Fig. 7. • " P N \ » .-^ <^^ 8, 9 8i\\ '. / X. ^V8« N \ , / Fig. 8a. Fig. 8. In the above constructions the weight of a unit mass is a common factor, and hence may be omitted and the areas alone of the small figures be used as the values of P^, P„ and P3. 10. Application of the Equilibrium Polygon in Finding Moments of Parallel Forces, — Let AB, BC EF be any number of parallel forces, and M' and iV' two points through which R^ and R^ pass (Fig. 9) . Construct the force ixy GRAPHICS. polygon Fig. 9a, and select some point P as a pole, so that the perpendicular distance H from the load line is looov 1 0000, or some similar quantity. Construct the equilibrium polygon Fig. 9 as explained in previous articles. Suppose the moment of AB, BC^ and CD about M' as a center of moments is desired. The moment equals A5(a,) + JBC{aj) + CZ)(a,) = M^. Prolong the lines 5^. Fig. 9. J A... - B 1 B*,^ \ \^ I 8 i T" / i ^^^ 89, / t « « 1 t / .t-^:. — „ >! Fig. 9a. 5,, and 5^ until they cut a line through M' parallel to AB, BCy etc. From the triangles Mai, Fig. 9, and ABP, Fig. 9a, aM:a^::AB:H or i4B(aJ = //(aM). Prom the triangles ab2., Fig. 9, and BCP, Fig. 9a, ab:a^:: BC : H or BC{a^ = //(a6) . GENERAL PRINCIPLES AND METHODS. " From the triangles 6^3, Fig. 9, and CDP, Fig. 9a, be: a,:: CD: H or CD(a,) = H(bc). Or AB(a,)+BC(a,)+CD(a,) =M« =H(aM + ab + bc)=H(Mc). From this it is seen that the moment of aily force equals the ordinate meastired on a line passing through the center of moments, and parallel to the given force, which is cut off between the two sides of the equilibrium polygon which are parallel to the two strings drawn from the pole P (pro- longed if necessary until they cut this line) to the ex- tremities of the load in Fig. ga; multiplied by the pole distance H. For a combination of loads the ordinate to be multiplied by H is the algebraic sum of the ordinates for each load; the loads acting downward having ordi- nates of one kind, and those acting upward of the opposite kind. To illustrate, let the moment of i?j, AB, BC, and CD about g be required. In Fig. ga the strings 5^ and S^ are drawn from the extremities of R^, hence in Fig. 9 the or- dinate gg' multiplied by H is the moment of R about g as a center of moments. The strings 5^ and 5^ are the extreme strings for AB, BC, CD, and hence the ordinate g'4 multiplied by H is the moment of these forces. Now since the reaction acts up- ward and the forces AB, BC, and CD act downward, the ordinate g4 multiplied by H is the moment of the com- bination. The above property of the equilibrium polygon is very convenient in finding the rnoments of unequal loads spaced at unequal intervals, as is the case where a locomotive stands upon a girder bridge. 12 GRAPHICS. II. Graphical Multiplication. — Let the sum of the products a^ftj, ajb^, etc., be required. The method of the previous article can be readily applied in the solution of this problem. Let 6^, fcj, etc., be taken as loads and a^, a^, etc., as the lever-arms of these loads about any convenient point as shown in Fig. lo. Then H(ab) = afi^, H{bc) = a,, ^1- H' 86 e\X ^^ Pole Fig. 10. Fig. loa. fcj, etc., and finally H{ae) = -2(a6), or the algebraic sum of the products a^fe^, ap^, etc. In case 2{ba^) is desired, the ordinates ah, be, etc., can be taken as loads replacing 6^, fej, etc., in Fig. lo. For con- venience take a pole distance H' equal to that used before and draw the polygon 5/, 5/, etc., then (ee^)H^ = 2(ba^). 12. To Draw an Equilibrium Polygon through Three Given Points. — Given the forces AB, BC, CD, and DE, it is required to pass an equilibrium polygon through the points X, y, and Z. Construct the force polygon Fig. iia, and through X and Y draw lines parallel to EA. Then, start- ing with Sg, passing through F, construct the equilibrium polygon Fig. ii, drawing the closing line S^, In Fig. iia there result the two reactions R^ and R^ when a line is drawn through P parallel to S^ of Fig. 1 1. Since the values GENERAL PRINCIPLES AND METHODS. 13 of i?i and i?2 remain constant for the given loads, the pole from which the strings in Fig. iia are drawn must lie upon a line drawn from L parallel to a line 5/' connecting X and Y in Fig. 11. That is, S^'^ is the position of the closing line for all polygons passing through X and F, and the pole can be taken anywhere upon the line P'L in Fig, i la. In order that the polygon, may also pass through Z take the loads upon the right of Z and find their resultant EB^ and through Z draw a line parallel to EB. Assume Z and Y to be two Fig. II. Fig. 11a. points through which it is desired to pass an equilibrium polygon. Proceeding as in the first case, the pole must lie somewhere upon the line UP', Fig. iia, drawn parallel to •aF, Fig. II. Then if a polygon with its pole in LP' passes through X and Y, and one with its pole in UP' passes through Z, the polygon with a pole at the intersection of these lines in P' will pass through the three points X, F, and Z. ROOF-TRUSSES. CHAPTER II, BEAMS AND TRUSSES. 13. Vertical Loads on a Horizontal Beam: Reactions and Moments of the Outside Forces.— Let the beam XY support the loads AB, BC. etc., Fig. 12, and let the ends of the beam rest upon supports .Y and Y. Required the reactions R^ and R^, neglecting the weight of the beam. In order that the beam remains in place free from all motion the outside forces AB, BC, etc., with i^, and R^ must fulfill the conditions of Art. i. Proceeding according to Art. 6, the force polygon ABCDEF is constructed, any point P taken as a pole, and the strings S^. . . . S^ drawn. Fig. 1 2a, Then, in Fig. 12, the equilibrium polygon is constructed, 14 BEylMS AND TRUSSES, JS the closing line 5^ drawn, and, parallel to this line, LP is drawn in Fig. 12a, cutting the line AF into two parts; LA being the value of R^, and LF the value of R^, The moment about any point in the vertical passing through any point x is readily found by Art. 10: M^^R,x-AB{x-a;)-BC{x- aj = {mn)H = the moment of the outside forces. 14. Vertical Loads on a Simple Roof -truss: Structure Considered as a Whole,-^^I;i this case the method of pro- cedure is precisely that given in Art. 10. The reactions R^ and R^ will of course be eqxial if the loads are equal and Fig. 13. Fig, 130. c ^ C' s B ^ u /iVn b' ^i aJ n 7 1 \ N a' ? y L? ^ 1 ,. ? . ^ibA y 1 n — ( ;8o 5P k< 1 c'? 1 A'i r-", <> ■< -^* 81 symmetrically placed about the center of the truss. This being known, the pole P may be taken on a horizontal line drawn through L, Fig. 13a, and then the closing line 5^ in Fig. 13 will be horizontal. The closing line may be made horizontal in any case by taking the pole P horizontally opposite Ly which divides the load line into the two reac- tions. It is evident from what precedes that the particular shape of the truss or its inside bracing has no influence i6 ROOF-TRUSSES. Upon the values of i?,, R^, and the ordinates to the equilib- ritim polygon. However, the internal bracing must have sufficient strength to resist the action of the outside forces and keep each point of the truss in equilibrium. 15. Inclined Loads on a Simple Roof-truss: Structure Considered as a Whole.— The case shown in Fig. 14 is that usually assumed for the action of wind upon a roof-truss. Fig. 14. Fig. 140. ,''» the truss being supported at X and Y. The directions of i?j and i?2 will be parallel to AD of Fig. 14a. The deter- mination of the values of R^ and R^ is easily accomplished by Art. 10, as shown in Figs. 14 and 14a. 16. Inclined Loads on a Simple Roof-truss, One Reaction Given in Direction: Structure Considered as a Whole. — Suppose the roof-truss to be supported upon rollers at V, Then the reaction R^ is vertical if the rollers are on a horizontal plane. The only point in i?^ which is known is the point of support X through which it must pass. Drawing the equilibrium polygon through this point, Sg cuts the direction of i?, in F' , and XY' is the closing line. Fig. 15. At y, which is by construction in equilibriimi. BEAMS AND TRUSSES. 17 there are three forces acting having the directions S^, 5^, and i?j, and these forces must make a closed figure ; hence, in Fig. isa, DL is the magnitude of R^, Since R^ must close the force polygon, LX is the magnitude and direction of R^. Fig. 15. Fig. 15**. If the rolters had been at X instead of Y, the method of procedure would have been quite similar. The equilibrium polygon would have passed through Y and ended upon a vertical through X, and the string S^ would have cut off the value of i?^ on a vertical drawn through X, Fig. 15a. 17. Relation between the Values of R^ in Articles 15 and 16. — In Article 15, i?2 ^^^ ^^ replaced by its vertical and horizontal components without altering the existing equilibrium. If the supports are in a horizontal plane, the horizontal component can be applied at X instead of Y without in any way changing the equilibrium of the stiiic- ture as a whole. Therefore the vertical component of .R^, as fotmd in Art. 15, is the same in value as the R^ found in J8 ROOF-TRUSSES. Art. i6. This fact makes it unnecessary to go through the constructions of Art. i6 when those of Art. 15 are at hand. The constructions necessary to determine R^ and K, of Art. 16 are shown by the dotted lines in Fig. 15a. 18. Intemal Equilibrium and Stresses.^ — As previously stated (Art. 14), although the structure as a whole may be in equilibrium, it is necessary that the intemal framework shall have sufficient strength to resist the stresses caused by the outside forces. For example, in Fig, 16, at the point X, R^ acts upward and the point is kept in equilibrium by the forces transmitted by the pieces Aa and La, parts of the frame. Suppose for the moment that these pieces be replaced by the stresses they transmit, as in Fig. 160. The angular directions of these forces are known, but their mag- nitudes and character are as yet imknown. Now, since X is in equilibrium under the action of the forces R^, Aa, and La, these forces must form a closed figure (Art. 2). Lay off Ri or LA, as shown in Fig. 166, and then through A draw a line Aa parallel to Aa, Fig. 16 or i6a, and through BEAMS AND TRUSSES. 19 L a line parallel to La, Fig. 16 or 166; then La and Aa are the magnitudes of the two stresses desired. Since in form- ing the closed figure Fig. 166 the forces are laid off in their true directions, one after the other, the directions will be as shown by the arrow-heads. If these arrow-heads be trans- ferred to Fig. 1 6a, it is seen that Aa acts toward X, and consequently the piece ^a in the frame Fig. 16 is in com- pression^ and in like manner the piece La is in tension. Passing, to point [/j, Fig, 16, and treating it in a similar manner, it appears that there are four forces acting to pro- duce equilibrium, two of which are known, namely, the outside force AB and the inside stress in Aa. Fig. t6c shows the closed polygon for finding the mag- nitudes and directions of the stresses in ab and Bb. Since Fig. 166 contains some of the lines found in "Fig. 1 6c, the two figures can be combined as shown in Fig. i6d. In finding the actual directions of the stresses, the forces acting around any given point must be considered independ- ently in their own closed polygon. Although Fig. i6d con- tains all the lines necessary for the determination of the stresses around X and the point L\, yet the stress diagram for one point is independent of that for the other, for Figs. 166 and 16c can be drawn to entirely different scales if the diagrams are not combined. The remaining points of the truss can be treated in the manner outlined above and the stress in each member found. Separate stress diagrams may be constructed for each point, or a combination diagram employed. Since, in case of the inside stresses, the forces meet in a point and there can be no revolution, there remain but two condi- tions of equilibrium, namely, the sum of the vertical com- 30 ROOF TRUSSES. ponents of all the forces must equal zero, and the same condition for the horizontal components. This being the case, if there are more than two unknowns among the forces acting at any point being considered, the problem cannot be solved by the above method. 19. Inside Forces Treated as Outside Forces. — Suppose the truss shown in Fig. 1 7 is cut into two parts along the line aa, then the left portion remains in equilibrium as long as the pieces Dd, dg, and gL transmit to the frame the stresses A B c N^AsP'l ! ^ — ' D C - ^ \ \ r ^\ ^ n>^ , S^'^^D^ ^ 1 V'V '\ v^ s V/9 . Fig. 17, Fig. 170. which actually existed before the cut was made. This condition may be represented by Fig. 17a. The stresses Dd, dg, and gL may now be considered as outside forces, and with the other outside forces they keep the structure as a whole in equilibrium, consequently the internal ar- rangement of the frame will have no influence upon the magnitudes of these forces. Equilibrium would still exist if the frame were of the shape shown in Fig. 176 and 176'. Fig. 17c shows the stress diagrams for the two cases shown, and also for the original arrangement of the pieces as shown in Fig. 17. 20. More than Two Unknown Forces Meeting at a Point. — ^Taking each point in turn, commencing with X, the stress diagrams are readily formed until point f/, of Fig. 1 7 is reached. Here three unknowns are found, and hence the BEAMS AND TRUSSES. 21 problem becomes indeterminate by the usual method. If now the method of Art. 19 is adopted, the bracing changed, Fig. 17^. Fig. l^b'. Fig. l^c. and the stresses in Dd, gd, and Lg found, the problem can be solved by working back from these stresses to the point r/,, as shown in Fig. l^c. CHAPTER III. STRENGTH OF MATERIALS. 21. Wood in . Compression : Columns or Struts. — ^When a piece of wood over ^teen diameters in length is subject to compression, the total load or stress required to produce failure depends upon the kind of wood and the ratio of the least dimension to its length. If the strut is circular in cross-section, then its least dimension is the -'diameter of this section ; if rectangular in section, then the least dimen- sion is the smaller side of the rectangular section. The above statements apply to the usual forms of timber which are imiform in cross-section from end to end. A piece of oak 6^^ X S'^ X 120^^ long requires about twice the load to produce failure that a similar piece 300'' long requires. A piece of oak 3" X 8" X 120" requires but about one third the load that a piece 6" X 8" X 120'' requires for failure. The actual ultimate strengths of the various woods used in structures have been determined experimentally and numerous formulas devised to represent these results. One of the later formulas, based upon the formula of A. L. Johnson, C.E., U. S. Department of Agriculture, Division of Forestry, is 700 4- 15^ 700 + 15c + c^' 22 STRENGTH OF MATERIALS. 23 where P = the tiltimate strength in pounds per square inch of the cross-section of a strut or column ; F =« the ultimate strength per square inch of wood in short pieces ; ^l^ le ngth of column in inches d least dimension in inches ^" A table of the values of P is given on page 24. The factor of safety to be used with this table depends upon the class of structure in which the wood is employed. The following statements are made in Bulletin No. 12, U. S. Department of Agrictdttire, Division of Forestry: "Since the strength of timber varies very greatly with the moisture contents (see Bulletin 8 of the Forestry Divi- sion), the economical designing of such structures will neces- sitate their being separated into groups according to the maximtmi moisture contents in use. MOISTURE CLASSIFICATION. "Class A (moisture contents, 18 per cent.) — Structures freely exposed to the weather, such as railway trestles, un- covered bridges, etc. "Class B (moisture contents, 15 per cent.) — Structures imder roof but without side shelter, freely exposed to out- side air, but protected from rain, such as roof-trusses of open shops and sheds, covered bridges over streams, etc. "Class C (moisture contents, 12 per cent.) — Structures in buildings unheated, but more or less protected from out- side air, such as roof -trusses or bams, enclosed shop^ and sheds, etc. "Class D (moisture contents, 10 per cent.) — Structures in buildings at all times protected from the outside air, 24 ROOF-TRUSSES. ULTIMATE STRENGTH OF COLUMNS. VALUES OF P. ULTIMATE STRENGTH IN POUNDS PER SQUARE INCH. Southern, Long- 4 « ^^ • Northern or Short-leaf Yel- leaf or Georgia Yellow Pine. Yellow Pine, Canadian (Ot- Douglas, Ore- gon and Wash- Spruce and Eastern Fir. Red Pine, d tawa) White ington Yellow Fir or Pine. Hemlock. Norway Pine, Pine, Canadian A A X^ m A# A ^1^ \^ am • California Red- Cypress, Cedar» (Ontario) Red wood, California Pine, White Oak. Spruce, White Pine. F » 6000 F « 5000 F=45oo F - 4000 F = 3750 I 5992 4993 4494' 3994 3740 a 5967 4973 4475 3978 3730 3 5928 4940 4446 3952 3700 4 5876 4897 4407 3918 3680 5 5813 4844 4359 3875 3630 6 5739 4782 4304 3826 3580 7 5656 ^ ^Vi 4242 3770 3530 8 5566 4638 4174 3710 3480 9 5469 4558 4102 3646 3420 10 5368 4474 4026 3579 3350 II 5264 4386 3948 3509 329<> 12 5156 4297 3867 3438 322a 13 5047 4206 3785 3365 3160 14 4937 4"4 3703 3291 3080 X5 4826 4022 3620 3217 302a i6 4716 3930 3537 3144 2950 , ^7 4606 3838 3455 3071 2880 > i8 4498 3748 3373 2998 2810 19 4391 3659 3293 2927 275<> 20 4286 3571 3214 2857 2680* 21 4183 3486 3137 2788 2620 22 4082 3402 3061 2721 2550^ 23 3983 3320 2988 2656 2490 24 3888 3240 2916 2592 243a 25 3794 3162 2846 2529 237<> 26 3703 3086 2777 2469 232a 27 3615 3013 2711 2410 2260 28 3529 2941 2647 2353 22X0 29 3446 2872 2585 2298 215a 30 3366 2805 2524 2244 2100* 32 3212 2677 2409 2142 2010 34 3068 2557 2301 2046 1920 36 2934 2445 2200 1956 1830 38 2808 2340 2106 1872 1750 40 2690 2241 2017 1793 1680 42 2579 2149 1934 1719 161O ^ 2476 2063 1857 1650 1550 46 2379 1982 1784 1586 1490 48 2288 1907 1716 1525 1430 50 2203 1835 1652 1468 1380 STRENGTH OF MATERIALS. 25 heated in the winter, such as roof-trusses in houses, halls» churches, etc/* Based upon the above classification of structures, the following table has been computed. SAFETY FACTORS TO BE USED WITH THE TABLE ON P. 24. Gafls. Yellow Pine All Othery Class A 0.20 0.23 0.28 . 0.31 0.20 " B 0.22 " C , . 0.24 " D 0.25 All struts considered in this article are assumed to have square ends. Example. — ^A white-pine column in a church is 12 feet long and 12 inches square ; what is the safe load per square inch? / 12 X 12 = 12, and from the table on page 24 d iz P = 3438 pounds per square inch. Churches belong to structures in Class D, and hence the factor of safety is 0.25 and the safe load per square inch 3438 X 0.25 =860 pounds. 860 X 144 = 123800 pounds is the total safe load for the colimm. The American Railway Engineering and Maintenance of Way Association adopted the foll6wing formula in 1907. For struts over 1 5 diameters long : = ^(^ - 6^)' 26 ROOF-TRUSSES. in which 5 = the safe strength in pounds per square inch, B = the safe end bearing stress (see Column 3, Table XVI), I = the length of the column, and d = the least side of the column. / and d are . expressed in the same unit. The following table gives the values of 5 for four values of B. The values of B used in the following table differ slightly from those recommended by The American Rail- way Engineering and Maintenance of Way Association, as they are based upon the values given in Table XVI- The imit stresses are essentially the same as given in the table on page 24, when a factor of safety of 4 is used. SAFE STRENGTH OF COLUMNS. VALUES OF S. * • SAFE STRENGTH IN POUNDS PER SQUARE INCH. • Red Pine. White Pine, Short-leaf Douglas, Oregon, and White Oak. d Norway Pine, Yellow Pine, Yellow Fir, Southern Cypress. Hemlock, Spruce, Long-leaf Cedar. Eastern Fir. Yellow Pine. B - 1000 B = iioo B ^ 1200 B» 1400 z toi5 1000 IIOO 1200 1400 16 730 810 880 1030 \l 720 790 860 1000 700 770 840 980 19 680 750 820 960 20 670 730 800 930 21 650 720 780 910 22 630 700 760 890 23 620 680 740 860 24 600 660 720 840 25 580 640 700 820 26 570 620 680 790 27 550 600 660 770 28 530 590 640 750 ap 520 570 620 720 30 500 550 600 700 / In the example on page 25, for t = 24, the safe load per square inch is 648 pounds with a factor of safety of 4. STRENGTH OF MATERIALS. 27 From the table on page 26 the corresponding value is found to be 660 pounds, the difference between the values being but 12 pounds. 22. Metal in Compression: Columns or Struts. — Steel is practically the only metal used in roof-trusses at the present time, and, unless they are very hea\y, angles are employed to the exclusion of other rolled shapes. The load req'dired to cripple a steel colimm depends upon several things, such as the kind of steel, the length, the value of the least radius of gyration for the shaps used (this is usually designated by the letter r, and the values are given in the manufacturers' pocket-books)," the manner in which the ends are held, etc. If a column has its end sections so fixed that they re- main parallel, the column is said to be square-ended. If both ends are held in place by pins which are parallel, the column is said to be pin-ended. A coltimn may have one square end and one pin end. The table on page 28 contains the ultimate strength per square inch of soft-steel columns or struts. To obtain the safe unit stress for medium steel: For quiescent loads, as in buildings, divide by 3.6 For moving loads, as in bridges, divide by 4.5 Safe unit stresses recommended by C. E. Fowler are tabulated on page 173. Example. — ^What load will cripple a square-ended col- umn of soft steel made of one standard 6" X 6" X i" angle if the length of the strut is 10 feet? From any of the pocket-books or the table at end of book the value of r is 1.18 inches, then ~ = — - = 8.5, r 1. 18 ^' 28 ROOF TRUSSES. STRENGTH OF STEEL COLUMNS OR STRUTS For Various Values op — in which L = Length in Feet and r r Radius of Gyration in Inches. P= ultimate strength in lbs. per square inch. P = Square Bearing. 45,000 FOR SOFT STEEL. Pin and Square Bearing. 45,000 1 + (12 L)*' P = 1 + (12 LY 24,000^ 36,000H To obtain safe unit stress: For quiescent loads, as in buildings, divide by 4. For moving loads, as in bridges, divide by 5. Pin Bearing. 45,000 \ I (12 LY' "^ 18,000r* ULTIMATE STRENGTH IN 1 POUNDS PER ULTIMATE STRENGTH IN POUNDS PER L SQUARE INCH. r J SQUARE INCH. r . Square. Pin and Square. Pin. Square. Pin and Square. Pin, 3.0 43437 42694 41978 12.0 28553 24142 2091 1 3.2 43230 42395 41593 12.2 28207 23771 20542 34 4301 1 42081 4II90 12.4 27863 23406 20179 3.6 42782 41754 40773 12.6 27522 23046 19823 3.8 42543 41412 40340 12.8 27185 22693 19474 4.0 42294 41058 39893 13.0 26850 22343 19133 4.2 42035 40693 39435 13.2 26524 22005 18797 44 41765 40317 38966 13.4 26189 21662 18469 4.6 41488 39930 38485 13.6 25864 21329 18148 4.8 41203 39534 37998 13.8 25543 21002 17833 5.0 40910 39130 37500 14.0 25224 20680 17523 5.2 40608 38807 36997 14.2 24909 20363 17221 5-4 40299 38300 36488 14.4 24598 20052 16925 5.6 39984 37874 35975 14.6 24290 19746 16634 5.8 39663 37443 35457 14.8 23985 19445 16350 6.0 39335 37006 34938 15.0 23684 I9148 16071 6.2 39003 36566 34416 15.2 23387 18858 15799 6.4 38665 36122 33894 15.4 23093 18572 15532 6.6 38323 35676 33371 15.6 22803 18288 15270 6.8 37976 35219 32849 15.8 22516 18015 15105 7.0 37616 34776 32328 16.0 22234 17744 14764 7.2 37272 34324 31809 16.2 21954 17478 14518 74 36914 33872 31292 16.4 21678 I7216 14279 7.6 36554 33419 30779 . 16.6 21406 16960 14043 7.8 36193 32966 30268 16.8 21137 16708 13812 STRENGTH OF MATERIALS, 29 STRENGTH OF STEEL COLUMNS OR ^TVLVl^^-^Gontinued, ULTIMATB STRBNGTH IN POUNDS PER ULTIMATE STRENGTH IN 1 'OUNDS PER L SQUARE INCH. L r 1 17.0 ' SQUARE INCH. r Square. Pin and Square. Pin. Square. Pin and Square. Pin. 8.0 35828 32514 29762 20872 16459 13584 8.2 35462 32064 29260 17.2 20611 16216 13366 8.4 35095 31615 28763 17.4 20353 15977 13150 8.6 34727 31169 28272 17.6 20098 15742 12938 8.8 34358 30724 27787 17.8 19847 15512 i273i 9.0 33988 30282 27306 18.0 19599 15286 ^5258 9.2 33611 29844 26832 18.2 19351 15063 12329 9.4 33249 29408 26364 18.4 19114 14845 12135 9.6 32880 28977 25903 18.6 18878 14630 1 1944 9.8 32511 28549 25448 18.8 18644 14420 1 1 757 10. 32143 28125 25000 19.0 18418 14218 11579 10.2 31776 27706 24559 19.2 18185 14010 1 1394 10. 4 31411 27290 24125 19.4 17961 13811 11219 10.6 31054 26879 23698 19.6 17740 13616 1 1 048 ID. 8 30684 26474 23279 19.8 17519 13422 10877 II. 30324 26072 22866 20.0 17308 13235 1071S II. 2 29965 25675 22460 20.2 17096 13050 10553 II. 4 29608 25285 22063 20.4 16888 12868 10434 II. 6 29247 24899 21671 20.6 16682 12690 10249 II. 8 28903 24517 21288 20.8 16480 12515 10087 and from the above table, P =* 34800 pounds per square inch. The area of the angle is 5.75 square inches^ hence the crippling load is 5.75 X 34800 = 200100 pounds. The safe load in a roof-tru^s is 200100 -^ 4 = 50025 pounds. If medium steel had been used, the safe load becomes 200100 -5- 3.6 = 55600 pounds. According to Fowler's formtila the safe load is 8250 X 5.75 = 47400 pounds. 23. End Bearing of Wood. — ^When a stress is trans- mitted to the ends of the fibers there must be a sufficient number to carry the load without too much compression or bending over. To illustrate, let a load P (Fig. 18) be 30 ROOF-TRUSSES. transmitted through a metal plate to the end of a wooden coltmm, then the area h Xd must be such that no crush- ing takes place. a \> ftmH**^^ \y Pig. i8. fn**HfM Pig. i8a. TABLE OF SAFE END BEARING VALUES. 1000 1100 1200 1400 Lbs. per Sq. In. Red Pine, Norway Pine, Cypress White Pine, Northern or Short-leaf Yel- low Pine, Cedar, Hemlock Spruce, Eastern Fir, Douglas, Oregon and Yellow Fir White Oak, Southern Long-leaf Pine or Geortoa Yellow Pine The values in this table haTe& factor of safety of 5 Example. — In Fig. 1 8 let 6 = 12 inches, cJ = 4 inches, and suppose the wood to be white oak; what is the safe load P? P = 4 X 12 X 1400 = 67200 pounds. 23a. Bearing of Wood for Surfaces Inclined to the Fibersv — In a large number of the connections in roof trusses it is necessary to cut one or both surfaces of contact between two members on an angle with the directions of the fibers. The allowable normal intensities of pressure upon such surfaces may be found from the STRENGTH OF MATERIALS. 31 following formtila, which is based upon the results of experiments : . . \( ^\^ r=q + {p-q)[-). where r = normal intensity on AC; q = normal intensity on BC; p = normal intensity on AB ; 6 «= angle of inclination of AC with direction of the- wood fibers. Pig. 186. SAFE BEARING VALUES FOR INCLINED SURFACES Pounds per square inch. Douglas, . » White Oak. Long-leaf Northern or Short-leaf Oregon, and Yellow Fir. White Pine* Red Pine, Norway - Yellow Pine. YeUow Spruce and Cedar. Pine. Pine. Eastern Fir. Cypress. 500 350 250 300 300 aoo to 511 363 260 3X2 3ZZ 3X0 20 ^ 402 292 249 344 339 do 600 467 ^ 3" 300 40 677 557 418 397 377 358 50 778 674 816 513 509 s; 447 60 900 628 644 555 V^ 104s 985 764 805 745 ?3 80 I2IZ 1 180 921 990 911 833 90 1400 1400 IIOO 1200 zzoo zooo 23b. End Bearing of Wood against Round Metal Pins. — In Fig. 1 8a the load P is transmitted to the wooden y^ ROOF-TRUSSES. strut by means of a casting and a round pin. The area of the end fibers which cany the load P \s b X d. The safe value of P for this area is given by the formula which is based on the normal intensities on inclined surfaces : P = bd{o.^tp + 0.54?) = bdF, where p = the allowable intensity of pressure against the ends of the fibers ; q = the allowable intensity of pressure across the fibers; d = diameter of pin; h = length of pin bearing against the wood; P = total force which the pin can safely transmit in a direction parallel to the fibers. APPROXIMATE VALUES OF F. goo 850 650 600 550 Lbs.perSq.In. Whit^Oak Long-leaf Yellow Pine Short-leaf Pine, Douglas, Ore- gon, Yellow Fir, Spruce, Fiastem Fir White Pine Cedar, Hemlock Red Pine, Norway Pine, Cypress I'he values in this table have a safety factor b e - tween 4 and 5 23c. splitting Effect of Round Pins Bearing against the End Fibers of Wood. — The round pin shown in Fig. i8a not only bears against the end fibers of the wood, but also tends to split the timber. Fortunately this tendency is comparatively small. 23d. Cross Bearing of Wood against Round Pins. — If the direction of the stress is normal to the fibers the bearing value of the wood on the pin may be taken, the same as on a flat surface having a width equal to the STRENGTH OF M/ITERIALS, 33 diameter of the pin. The safe values to be used are given in Art. 25. 24. Bearing of Steel. — Since soft and soft-medium steel are practically homogeneous in structure, the same bearing value is used for round and flat surfaces. The diameter of the pin or rivet multiplied by the thickness of the plate through which it passes is taken as the bear- ing area. This is an approximation but is sufficient for practical purposes. For soft or soft-medium steel the safe bearing value may be taken as 20000 pounds per square inch. TABLE OF SAFE BEARING VALUES. Diameter of Rivet. Area in Sq. Inches .1105 .1964 .3068 .4418 .6013 .7854 BKARING VALUE FOR D1FFERRNT THICKNKSSBS OF PLAT^ IN INCHES AT 20,000 POUNDS PER SQUARE INCH. \ 1875 2500 3125 3750 4375 5000 l'* I h \ 2344 2813 3125 3750 4375 5000 3Qo6 4688 546g 6250 4688 5625 6563 7500 5469 6563 7656 8750 6250 7500 8750 1 0000 7031 8438 9844 1 1 250 TABLE OF SAFE BEARING VALUES— Coniinii^i. Diameter of Rivet. Area in Sq. Ins. BEARING VALUE FOR DIFFERFNT THICKNESSES OF FLATE IS INCHES Al 20,000 lOUNDS PER SQUARE INCH. H 1 if 2 1 ^< I } .1105 i .1964 J .3068 7813 ■ i .4418 9375 10313 1 1 250 i .6013 1 0938 1 2031 13125 14219 15313 16406 I .7854 12500 13750 15000 16250 17500 18750 20000 34 ROOF-TRUSSES. 2$. Bearing Across the Fibers of Wood. — If a load P, Fig. 19, be transmitted through a wooden corbel to a col- timn, the area b X d, bearing directly upon the support, must.be sufficient to resist crushing. This is a point very often overlooked in construction. In Fig. 19a the same Fig. 19. 7 £.J u^e^ Fig. 19a. TABLE OF SAFE BEARING VALUES. 150 Hemlock, CaJifomia Redwood 200 White Pine, Red Pine, Norway Pine, Spruce, Eastern Fir, Cypress, Cedar, Douglas Fir, OregonFir, YeUow Fir 250 350 500 Northern Southern White or Short- Long-leaf Oak leaf-Yel- or Georgia low Pine, Yellow Chestnut Pine Lbs. per Sq. In. The values in this table have a factor of safe ty of 4 conditions obtain. The washer must. be of such a size that the area bearing upon the wood shall properly distribute the stress transmitted by the rod. 26. Bearing Across the Fibers of SteeL See Art. 24. 27. Longitudinal Shear of Wood. — In Fig. 20 let the piece A push against the notch in B, then the tendency is to push the portion above ba along the plane fea, or to shear lengthwise a surface b in length and t in width. STRENGTH OF MATERIALS. 35 A similar condition exists in Fig. 20a. The splice may fail by the shearing along the grain the two surfaces dbc and a'Vc'. A table of safe longitudinal shearing values is given below. Fig. 20. z JTL I 4-t n Tec- 5 »m I fe::=i h Fig. 2oa. TABLE OF SAFE LONGITUDINAL SHEARING VALUES. 100 150 200 Lbs. per Sq. In. White Pine, Northern or Short-leaf YeUow Pine, Canadian White Pine, Cana- dian Red Pine, Spruce, EaHtern Fir, Hemlock, California Redwood, Cedar Southern Long- leaf or Georgia Yellow, Pine, Chestnut White Oak The values in this table have a factor of safety of 4 130 DouglaH, Oregon, and YeUow Fir - 28. Longitudinal Shear of Steel. — ^For all structures considered in this book the longitudinal shear of steel is 36 ROOF-TRUSSES, fully provided for by the practical rules governing the spacing of rivets, etc. See Table III. 29, Transverse Strength of Wood. — When a beam sup- ported at the ends is loaded with concentrated loads, as shown in Fig. 2 1 , the max- imum moment is readily found by means of the equi- librium polygon. Let this moment be called A/, then for rectangular beams M - \Rhd\ Fig. 21. where M = the maximum moment in inch-pounds ; h = the breadth of the beam in inches ; d = the depth of the beam in inches; R = the allowable or safe stress per square inch in the extreme fiber. If M is given in foot-pounds, then the §econd member of the above equation becomes ^BJbd^. For a uniformly distributed load M « \wP = \Rhd^, where w = the load per linear inch of span ; / = the span in inches. Example. — An oak beam 6 inches deep has a span of 10 feet and carries a load of 100 poimds per linear foot. What must be the breadth of the beam to safely carry the load? M = \wl'^ = i X 100 X 10 X 10 = 1250 ft.-lbs. or 15000 in.-lbs. STRENGTH OF MATERIALS. 37 M = \Rhd!^ « isooo « J X 1200 X 6 X 6 X 6, or 15000 . . , & = = 2t inches. 7200 * Hence a 2 J" X 6" white-oak beam will safely cany the load; but the weight of the beam has been neglected, and consequently the breadth must be increased to, say, 2f inches. A second calculation should now be made with the weight of the beam included. TABLE OF SAFE VALUES OF R FOR WOOD. 600 700 750 800 1000 1200 Hemlock White Pine, California Douglas, Ore- Northern or Southern Spruce, Redwood gon, and Yellow Short-leaf Long-leaf or Eastern Fir, Red Fir, Yellow Pine Georgia Fir, Red Pine, Cy- Yellow Pine, Cedar press, Chestnut, California Spruce, Norway Pine, Washing- ton Fir or Pine White Oak (Red Fir) The above values are pounds per square inch. Factor of safety 6. See Table XVI, page 139. The transverse strength of wood as considered above assumes that the plane of the loads is parallel to the side of the timber having the dimension d. In case the plane of the loads makes the angle 6 with the axis BB^ Fig. 2ia, 6M cos 6 M cos e = \R'h

2I2- 2-2 i? = i?' + R'\ STRENGTH OF MATERIALS. 41 . . . -R'/i-i „, M cos e , Af cos 6 « — vi— or K = — r d\ \d 2/1-j Af sin e = or R" = — 1: x. X I2-2 These formtilas refer to point a. For point e replace X R ^ R' + 2?". Angles and Z Bars. M cos = ,. or A = — f— — z;4-4, V4-4 4-4 43 ROOf^TKUSSES. Af Sin = D3-3 ^ 3-3 where ^ = the distance of the " outer fiber " from the axis denoted by the subscript. The same fiber must be considered for both axes even if for one it is not the outermost fiber. The values for v are best determined from a fidl sized drawing. The particular fiber for which R " R' + R" is a maximum can be found by trial. An inspection of the full size drawing will usually eliminate all but two possible positions. „. Example. — ^A 4" X 4" X \" angle used as a beam has a span of 10 feet and is loaded with 150 pounds per foot of span, the plane of the loads being parallel to one leg of the angle. What is the maximum fiber stress? Ti-A = 3.28, fa-a = 8.84, = 45",^^ " 33500 in. -lbs. For the Fiber at A „, 32500 X 0.707 „ , /^ 8.82 '-^^ = S^°5 3 2500 X 0.707 STRENGTH OF MATERIALS 43 For the Fiber at B , 22500 X 0.707 „ ; ^ = —3:82 "-48 = 4473 , 22500 X 0.707 ^ = TT& ^-5^ = ^°535 R =• R +R' = 15008 lbs. Inspection shows that the maxunum fiber stress cannot be at C, hence 15008 is the maximum sought. 31. Special Case of the Bendiog Strength of Hetal Pins. — Where pins are used to connect several pieces, as in Fig. 2a, the moments of the outside forces can be determined in the usual way. This moment M = — - i?(o.o98d*), where d = the diameter of the pin in inches ; R " the safe stress in the outer fiber in pounds per -square inch. The table on page 36 gives the safe values of M for vari- ous sizes of bolts or pins. For wrought iron use i? = 15000,. and for Steel use R = 25000. 32. Shearing Across the Grain of Bolts, Rivets, and Pins. — For wrought-iron bolts use 7500 pounds per square inch, and for steel loooo pounds. The safe shearing values of rivets and bolts are given on page 44. See Table XVIII. 44 ROOF -TRUSSES. MAXIMUM BENDING MOMENTS ON PINS WITH EXTREME FIBER STRESSES, Varying from 15000 to 25000 Pounds per Square Inch. Diameter af Pin in Inches. I Ij Ij I| li If If Ij 3 2\ 2\ af 2i 3i 3 3* 3i 3i 3i 3l 3i 31 Area of Pin in Square Inches. .785 .994 1.227 1.485 1.767 2.074 2.405 2.761 3 142 3 547 3 976 4.430 4.909 5 4" 5 940 6.492 7.069 7.670 8.296 8.946 9.621 10.321 11.045 "793 12.566 MOMBNTS IN INCH-POUNDS FOR FIBRB 8TRBSSSS OF 15000 Lbs. per Sq In. 1470 2100 2900 3830 4970 6320 7890 9710 1 1 780 I413O 16770 19730 23010 26640 30630 34990 39730 44940 50550 56610 63140 70150 77660 85690 94250 xSooo Lbs. per Sq. In. 1770 2520 3450 4590 5960 7580 9470 1 1 650 1 41 40 16960 20130 23670 27610 31960 36750 41990 47680 55930 60660 67940 75770 84180 93190 102820 II31OO 20000 Lbs. per Sq. In. i960 2800 3830 5100 6630 8430 10520 12940 15710 18840 22370 26300 30680 35520 40830 46660 52970 59920 67400 75480 84180 93530 I03S40 I 14250 125660 S2300 Lbs. per Sq. In. 2210 3150 4310 5740 7460 9480 1 1 840 14560 17670 21200 25160 29590 34510 39960 45940 52490 59600 67410 75830 84920 94710 105320 I 16490 128530 I4137O 35000 Lbs. per Sq. In. 2450 3490 4790 6380 8280 10530 13150 16180 19630 23550 27960 32880 38350 44400 51040 58320 66220 74900 84250 94350 ZO5230 166910 129430 I42810 157080 SAFE SHEARING VALUES OF RIVETS AND BOLTS. Diam. of Rivet. Area in Sini^le Shear Double Shear Single Shear Double Shear Square Inches. at 7500 lbs. at Z5000 lbs. at zoooo lbs. at aoooolbs. i .H05 828 1657 HO5 2209 h .1964 1473 2945 1964 3927 .3068 23OZ 4602 3068 6136 } .4418 3313 6627 4418 8836 1^ .6013 4510 9020 6013 12026 X .7854 5891 II781 7854 15708 STRENGTH OF MAThRIMLS. 45 33. Shearing Across the Grain of Wood. SAFE TRANSVERSE SHEARING VALUES. 400 500 600 Lbs. per Sq. In. Cedar White Pine, Chestnut Hemlock Factor of safety 4 750 1000 1250 Lbs. per Sq. In. Spnice, Eafitern Fir White Oak, North- em or Short-leaf Yellow Pine Southern Long- leaf or Georgia Yellow Pine Factor of safety 4 34. Wood in Direct Tension. SAFE TENSION VALUES. 600 700 800 Lb6.perSq.In. Hemlock, Cypress White Pine, Cali- fornia Redwood, Cedar Spruce, Eastern Fir, Douglas Fir, Oregon Fir, Yellow Fir, Red Pine Factor of safety 10 900 1000 1200 Lb8.perSq.In. Northern or Short-leaf Yellow Pine V Washington Fir or Pine, Canadian White Pine and Red Pine White Oak, Southern Long- leaf or Georgia Pine Factor of Safety 10 35* Steel and Wrought Iron in Direct Tension. — For wrought iron use 12000 pounds per square inch, for steel use 16000 pounds per square inch. See Table XVIII. CHAPTER IV. ROOF-TRUSSES AND THEIR DESIGN. 36. Preliminary Remarks. — Primarily the ftinction of a roof -truss is to support a covering over a large floor-space which it is desirable to keep free of obstructions -in: the shape of permanent columns, partitions, etc. Train-sheds, power-houses, armories, large mill buildings, etc., are ex- amples of the class of buildings in which roof -trusses are commonly employed. The trusses span from side wall to side wall and are placed at intervals, depending to some extent upon the architectural arrangement of openings in the walls and upon the magnitude of the span. The top members of the trusses are connected by members called purlins, running usually at right angles to the planes of the trusses. The purlins support pieces called rafters, which run parallel to the trusses, and these carry the roof covering and any other loading, such as snow and the effect of wind. The trusses, purlins, and rafters may be of wood, steel, or a combination of the two materials. 37. Roof Covering. — This may be of various materials or their combinations, such as wood, slate, tin, copper, clay tiles, corrugated iron, flat iron, gravel and tar, etc. The weights given for roof coverings are usually per square, which is 100 square feet. 46 ROOF-TRUSSES AND THEIR DESIGN. 47 Tables I and II give the weights of various roof coverings. 38. Wind Loads. — The actual effect of the wind blowing against inclined surfaces is not very well known. The formulas in common use are given below: Let = angle of surface of roof with direction of wind ; F = force of wind in poimds per square foot ; A = pressure normal to roof, = F sin tf'-^* cos«-i . B = pressure perpendicular to direction of the wind = F cot ^ sin 0'-^ ^°^ * ; C = pressure parallel to the direction of the wind F sin 6'-^^ *^^ • . (Carnegie.) Angle 6 5" 10° 20° 30» 40« 50^ 60° 70° 80° 90** A^FX B^FX C^FX 0.125 0.122 O.OIO 0.24 0.24 0.04 0.45 0.42 0.15 0.66 0-57 0.33 0.83 0.64 0.53 0.95 0.61 0.73 1. 00 0.50 0.85 1.02 0.35 0.96 1. 01 0.17 0.99 1. 00 0.00 1. 00 39. Pitch of Roof. — The ratio of the rise to the span isj Fig. 23. called the pitch, Fig. 23 The following table gives the angles of roof s as commonly constructed: 48 ROOF-TRUSSES. Pitch. Angle 6. Sin $. CosG. Tan©. Sece. l/a 1/3 I 3(3 45' o' . 33^ 41' 0.707ZZ 0.55460 0.707ZZ 0.8331 a X .00000 0.66650 x.4X4ax z.aoi76 3o' o' 0.50000 0.86603 0.57735 I . 15470 1/4 1/5 1/6 26* 34' ax* 48' 18** a6' 0.44724 0.37137 0.3x620 0.8944Z 0.92849 0.94869 0.50004 0.39997 0.33330 Z.XX805 1.0770a 1.05408 40. Transmission of Loads to Roof-trusses. — Fig. 24 shows a common arrangement of trusses, purlins, and rafters, so that all loads are finally concentrated at the apexes B, C, ''""felfSSS i* d ^ d- - -M Fig. 24. D, etc., of the truss. Then the total weight of covering, rafters, and purlins included by the dotted lines mn, np, po, and om will be concentrated at the vertex B. The total wind load at the vertex B will be equal to the normal pressure of the wind upon the area mnop, 41 • Sizes of Timben — ^The nominal sizes of commercial timber are in even inches, as 2" X 4"» 2" X 6", 4" X 4"> etc., and in lengths of even feet, as 16', 18', 20', etc. The actual or standard sizes are smaller than the nominal sizes. Table XV gives the standard sizes for long-leaf pine, Cuban pine, short-leaf pine, and loblolly pine. ROOF-TRUSSES /IND THEIR DESIGN. 49 42. Steel Shapes. — Only such shapes should be em- ployed as are marked staftdard in the manufacturers' pocket- books. These are readily obtained and cost less per pound than the ** special'' shapes. Ordinarily all members of steel roof -trusses are com- posed of two angles placed back to back, sufficient space being left between them to admit a plate for making con- nections at the joints. See Tables IX-XII. 43. Round Rods. — In wooden trusses the vertical ten- sion members, and diagonals when in tension, are made of round rods. These rods should be upset * at the ends so that when threads are cut for the nuts, the diameter of the rod at the root of the thread is a little greater than the diameter of the body of the rod. It is common practice to buy stub ends — that is, short pieces upset — and weld these to the rods. Unless an extra-good blacksmith does the work the upsets should be made upon the rod used, without welds of any kind. Very long rods should not be spliced by welding, but connected with sleeve-nuts or tumbuckles. Upset ends, tumbuckles, and sleeve-nuts are manu- factured in standard sizes and can be purchased in the open market. See Table VII. 44. Bolts. — The sizes of bolts commonly used in wooden roof-trusses are J'' and f'^ in diameter. Larger sizes are sometimes more economical if readily obtained, f'' and l'^ bolts can be purchased almost anywhere. Care shotdd be taken to have as many bolts as possible of the same size, * Upsetfi should not be made on steel rods unless they are annealed after- wards. 50 ROOF TRUSSES, as the use of several sizes in the same structure usually causes trouble or delay. See Tables V and VI. 45. Sivets. — The rivets in steel structures should be of luiiform diameter if possible. The practical sizes for different shapes are given in the manufacturers' pocket- books. See Tables III, IV. and V. 46. Local Conditions. — In making a design local mar- kets should be considered. If material can be purchased from local dealers, although not of the sizes desired, it will often happen that even when a greater amount of the local material is used than required by the design, the total cost will be less than if special material, less in quan-^ tity, had been purchased elsewhere. This is especially true for small structures of wood. CHAPTER V. DESIGN OF A WOODEN ROOF-TRUSS. 47. Data. Wind load = 40 pounds per square foot of vertical projection of roof. Snow load =* 20 pounds per square foot of roof. Covering — slate 1^" long, i'' thick =9.2 pounds per square foot of roof. Sheathing = long-leaf Southern pine, lY thick = 4.22 pounds per square foot of root « long-leaf Southern pine, if" thick, ~ long-leaf Southern pine. - long-leaf Southern pine, for all mem- bers except verticals in tension, which will be of soft steel. Distance c. to c. of trusses = 10 feet. Pitch of roof = J. Form of truss as shown in Fig. 25. Rafters _ » Purlins Truss StMineO' Rise 90' Fig. 25. 51 52 ROOF-TRUSSES. 48, Allowable Stresses per Square Inch. SOUTHERN LONG-LEAP PINE. Tension with the grain Art. 34, 1200 lbs. End bearing Art. 23, i4oolbs. End bearing against bolts Art. 236, 850 lbs. Compression across the grain Art. 25, 350 lbs. Transverse stress — extreme fiber stress, Art. 29, 1200 lbs. Shearing with the grain Art. 27, 150 lbs. Shearing across the grain Art. 33, 1250 lbs. Coltunns and Struts. Values given in Art. 21. STEEL. Tension with the grain Art. 35, 16000 lbs. Bearing for rivets and bolts Art. 24, 20000 lbs. Transverse stress — extreme fiber stress, Art. 30, 16000 lbs. Shearing across the grain Art. 32, loooo lbs. Extreme fiber stress in bending (pins). Art. 31, 25000 lbs. 49. Rafters. — The length of each rafter c. to c. of pxu'lins is 10 X sec tf = 10 X 1.2 = 12 feet, and hence the area nmopy Fig. 24, is 12 X 10 =» 120 square feet. VERTICAL LOADS. Snow = 20.00 X 120 = 2400 lbs. Slate = 9.20 X 120 = II 04 lbs. Sheathing = 4.22 X 120 = 506 lbs. 3342 X 120 ss 4010 lbs. The normal component of this load is 4010 X cos 5, or 4010 X 0.832 = 3336 pounds. DESIGN OF A WOODEN ROOF TRUSS. $$ The normal component of the wind is (Art. 38) about 40 X 0.70 = 281bs. per square foot, and the total, 28 X 120 ■« 3360 lbs. The total normal load supported by the rafters, ex- clusive of their own weight, = 3336 + 3360 = 6696 lbs. 6696 -J- 12 = 558 lbs. per linear foot of span of the rafters. Since the thickness of the rafters has been taken as if", either the number of the rafters or their depth must be assumed. Assuming the depth as 7§", the load per linear foot which each rafter can safely carry is (Art. 29), (Table XV), wl -g- X 12 X 12 = 1200 X 15.23 = 18276; .'. zc; = 85 pounds. 558 -5- 85 = 6.56 = number of if" X 7J" rafters required. To allow for the weight of the rafters and the compo- nent of the vertical load which acts along the rafter, eight rafters will be used. If a rafter is placed immediately over each truss, the spacing of the rafters will beioXi2-5-8 = 15 inches c. to c. The weight of the rafters is 12.2 X 8 X 3.75 = 366 lbs. 50. Purlins. — The total load normal to the roof carried by one purlin, exclusive of its own weight, is 6696 + 366 X 0.832 = 7000 lbs. Although this is concentrated in loads of 7000 -s- 8 =875 lbs. spaced 15" apart, yet it may be 54 ROOF-TRUSSES. considered as uniformly distributed without serious error. The moment at the center of the purlin is J(7ooo) X lo X la = 105000 inch-pounds. The component of the vertical load parallel to the rafter is 4010 X 0.555 = 2226 pounds and the moment of this at the center of the purlin is i{2226) X 10 X 12 = 33390 inch-pounds. The purlin re- sists these two moments in the manner shown by Fig. 25a- (Since the rafters rest on top of the purlin the force parallel to the rafters pro- <^ H'*l-* . duces torsional stresses in the ''' '**■ purlin. There is also an unknown wind force parallel to the rafters which pro- duces torsional stresses of opposite character and reduces the moment 33390. Both of these effects have been neglected.) Let b = 7J" and d = 9J". The fiber stress at B, Pig. 250, will be the sum of the two fiber stresses produced by the two moments. For the force normal to the rafter R' = 6 X 105000 -s- 7i{9i)* = 932- For the force parallel to the rafter R" = 6 X 33390 -4- 9^(7^)^ = 375, R' + R" = 932 + 375 = 1307 lbs. This is a little greater than the allowable fiber stress, which is 1200 lbs. Hence the next larger size of timber must be used, or a 10" X 10" piece. The weight of the purlin is 282 pounds. 51. Loads at Truss Apexes. — Exclusive of the weight of the truss the vertical loads at each apex, Ui, Ui, Uz, Ih. and C/e. Fig- 25. is DESIGN OF A WOODEN ROOF-TRUSS. 55 Snow, slate, sheathing Art. 49, 4010 lbs. Rafters Art. 49, 366 lbs. Purlins Art. 50, 282 lbs. 4658 lbs. The weight in pounds of the truss may be found from the formula W = \dL{i + iV-L), where d is the distance in feet c. to c. of trusses, and L the span in feet. Sub- stituting for d and L, W^ = I X 10 X 60(1 + tV X 60) = 3150 lbs. The full apex load is ^^ = 525 lbs., and hence the total vertical load at each apex U1-U5, inclusive, is 4658 + 525 = 5183 lbs. In case the top chords of the end trusses are cross-braced together to provide for wind pressure, etc., this load would be increased about 75 or 100 lbs. For convenience, and since the roof assumed will re- quire light trusses, the apex loads will be increased to 6000 lbs. In an actual case it would be economy to place the trusses about 1 5 feet c. to c. The load at the supports is ^%®-^ = 3000 lbs. Wind. — The wind load for apexes U^ and [/j is 3360 lbs. {Art. 49), and at apexes Lq and U^ the load is -M/-^- = 1680 lbs. For the determination of stresses let the wind apex load be taken as 3400 lbs., and the half load as 1700 lbs. In passing, attention may be called to the fact that the weight of the truss is less than 10 per cent, of the load it lias to support exclusive of the wind; hence a slight error in assuming the truss weight will not materially affect the stresses in the several members of the truss. 52. Stresses in Truss Members. — Following the prin- ciples explained in Chapter II, the stress in each piece is xeadily determined, as indicated on Plate I. 56 ROOF-TRUSSES, Having found the stresses due to the vertical loads, the wind * loads when the wind blows from the left and when it blows from the right, these stresses must be combined in the mariner which will produce the greatest stress in the various members. The wind is assumed to blow but from one direction at the same time ; that is, the stress caused by the wind from the right cannot be combined with the stress due to the wind from the left. In localities where heavy snows may be expected it is best to determine the stresses produced by snow covering but one half of the roof as well as covering the entire roof. For convenience of reference the stresses are tabulated here. STRESSES. Vertical Loads. Wind Left. Wind Right. Maximum Stresses. UUx + 27200 + 7300 + 5600 + 34500 u,u. + 21700 + 5800 + 5600 + 27500 u,u. + 16300 + 4400 + 5600 + 20700 LoLi — 22600 — 8700 —2600 -31300 uu —22600 —8700 —2600 -31300 L,Lt — 1810O — 5600 —2600 — 23700 u,u u,u — 3000 — 2000 — 5000 U,U — 12000 —4100 -4100 — 16100 L\U + 5400 + 3700 9100 U^Lt + 7600 + 5100 12700 + signifies compression. * Some en.'];ineers consider only the lee side of the roof covered with snow when wind stresses are combined with the dead and snow load stresses. DESIGN OF A WOODEN ROOF-TRUSS. 57 53* Sizes of Compression Members of Wood. Piece LqU\. Stress = + 34500. Since the apex Ui is held in position vertically by the truss members, and horizontally by the purlins, the unsupported length of LoC/i as a column is 12 feet. To determine the size a least dimension must be assumed and a trial calculation made. This will be better explained by numerical calculations. Let the least dimension be assumed as 54", then j =* a 12 X 12 — -I = 26, and from page 24, P= 3086 lbs. per square S2 inch. The safe or allowable value is — = =771 lbs. 4 4 I)er square inch. Hence 34500 -^ 771 =44.7 = number of square inches required. If one dimension is 5 J", the other must be 9^', or a piece 55" X 9I" = 52.3 square inches, 12' long, will safely carry the stress 34500 lbs. This is the standard size of a 6" X 10" timber (Table XV). If the table on page 26 is used the safe strength per square inch, for d = 5^, is 790 pounds and the required area is is 34500 -^ 790 =43.7 square inches. Since a 6" X 8" piece has an actual area of but 41.3 square inches, the next larger size must be used, or a 6" X 10" piece, the same as found in the first trial. A piece 6" X 10" has a much greater stiffness in the 10" direction than in the 6" direction. For equal stiffness in the two directions the dimensions should be as nearly equal as possible. For example, in the above case try a piece 8" X 8", I ■¥ d = 144 -^ 7i = 19, P = 3659, and, with a safety factor of 4, the total load is 51400 poimds. This is 16900 pounds 58 ROOF-TRUSSES. greater than the stress in UoUu while the area is only 4.0 square inches greater than the area of the 6" X 10" piece. By changing the shape and adding but about 7.6 per cent to the area the safe load has been increased nearly 50 per cent. Pieces U1U2 and U2Us. Stresses + 27500 and + 20700. Letting d = s§", 27500 -5- 771 = 35.7 square inches required. Now sJ" X yi" = 41.3 square inches, hence a 6" X 8" piece can be used. However, a change in size requires a splice, and usually the cost of bolts and labor for the splice exceeds the cost of the extra material used in continuing the piece LqUi past the point U2. For this reason, and because splices are always undesirable, the top chords of roof -trusses are made imiform in size for the maximum lengths of commercial timber, and, excepting in heavy trusses, the size of the piece Lof/i is retained throughout the top chord, even when one splice is neces- sary. To illustrate the method of procedure when the size is changed, suppose U2U3 is of a different size from UiU2- To keep one dimension uniform the piece must be either 6" or 8" on one side. Try the least d as s¥\ then -1 = 26, and — = =771 lbs. 20700 -5-771 =27 square inches 4 4 required. 27 -5- 5 J indicates that a 6" X 6" piece is necessary. Commencing with LoC/i the nominal sizes composing the top chord are 6" X 10", 6" X 8", and 6" X 6". Since greater strength and stiffness can be obtained DESIGN OF A WOODEN ROOF-TRUSS. 59 without much additional expense by using the size 8" X 8" throughout, this size will be adopted. Piece f/iLi. Stress = + 9100. The unsupported length of this piece is 12 feet. Try P least d = 3f", then — = 580 and 9100 -^ 580 = 16 = the 4 number of square inches required; hence a piece 4" X 6" with an actual area of 21.1 square inches can be used. Piece U2L3. Stress = + 12700. The unsupported length = 10 X 1.6667 = 16.67 feet, / 16.67 X 12 P 1730 „ 1 = = 537 ~" = = 433 lbs. d 3.75 ^^' 4 4 ^'^^ 12700 -5- 433 = 29.3 = number of square inches re- quired, or a piece 4" X 10" must be used if d = 3.7s". ^ , t// 1 I , . P 2440 Try d = 5I", then ^r = 36 + , — = -=^^^=^ 610. a 44 12700 -^ 610 = 20.8 square inches required. The smallest size where d = si" is 5I" X si" = 30-25 square inches. In this case a 6" X 6" is more economical in material by 5.3 square inches of section, and will safely carry about 3000 lbs. greater load than the 4" X 10" piece. 54. Sizes of Tension Members of Wood. Pieces LqL^ and L^Lj. Stress = — 31300. From Art. 34 the allowable stress per square inch for Southern long-leaf pine is 1200 lbs. 6o ROOF-TRUSSES. 31300-5-1200 = 26.1 =the net number of square inches required. In order to connect the various pieces at the apexes, considerable cutting must be done for notches, bolts, etc., and where the fibres are cut off their usefulness to carry tensile stresses is destroyed. Practice indicates that in careful designing the net section must be increased by about I, or in this case the area required is 23+ 16 = 39 square inches, therefore, a piece 5 J" X 7I" = 41.3 square inches must be used. In many of the details which follow 8" X 8" pieces will be used for the bottom chord. Piece L2L3. Stress = — 23700. In a similar manner this member can be proportioned, but since splices in tension members are very undesirable, owing to the large amount of material and' labor required in making them, the best practice makes the number a minimtmi consistent with the market lengths of timber, and, consequently, in all but very large spans the bottom chord is made uniform in size from end to end. 55. Sizes of Steel Tension Members. Piece J/iLj. Stress = o. Although there is no stress in f/^L^, yet, in order that the bottom chord may be supported at Lp a round rod J^ in diameter will be used. Piece UJ^^. Stress = 7-5000. The number of square inches required is (Art. 35), 5000 -^ 16000 = 0.31 square inches. A round rod \{ inch in diam- eter is required, exclusive of the material cut away by the DESIGN OF A IVOODEN ROOF-TRUSS. 6i threads at the ends. The area at the root of the threads of dii" round rod is 0.42 square inches, hence a V round rod wiU be used. (Table VII.) Piece f/^j. Stress = *— 16100. 1 6100 -^ 16000 = 1,006 square inches. A ij" round rod has area of 1.227 square inches. This rod upset * (Table VII) to if" at the ends can be used. If the rod is not upset a diameter of if" must be used, having an area of 1.057 square inches at the root of the threads. See Table XVIII. Note that the above rods have commercial size^. 56. Design of Joint Lo.'— With i|" Bolts. — ^A common form of joint at Lo is shown in Fig. 26. The top chord rests in a notch dh in the bottom chord, and, usually, altogether too much reliance is put in the strength of this detail. The notch becomes useless when the fibers fail along dh, or when the bottom chord shears along ab. The distance ah is quite variable and depends upon the arrange- ment of rafters, gutters, cornice, etc. Let about 12" be assumed in this case, th^i it will safely resist a longitudinal shearing force ofi2X7^Xiso = 13500 lbs. (Art. 27). The area of the inclined surface due to the notch dh equals i.2(i| X 7^) = 13.5 square inches, if dc = ij". This will safely resist 13.5 X 760 = 10300 lbs. acting normal to the surface (Art. 23a), hence ,the value of the notch is but 10300 lbs., leaving 34500 — 10300 = 24200 lbs. to be held in some other manner, in this case by i|" bolts. To save cutting the bottom chord for washers, and also ^iiii ■■I I ■ . » I ■ I I I ■■! ■■ ■ ■ I ■ ■■ ■ ■ ■ mm ■ — ■■■■ — ^ ■■ ■ ■■ IM ^^^»^^^»^^^»^i^— — ^—^^^^ * Upsets on steel rods should not be used unless the entire rod is annealed after being upset at the ends. 62 ROOF-TRUSSES. to increase the bearing upon the supports of the truss it is customary to use a corbel or bolster, as shown in Fig. 26. Let a single |" holt be placed 6" from the end of the bottom chord. This will prevent the starting of a cradc at 6, and also assist in keeping the corbel in place. If it is assumed that the bolt holes are slightly larger Fig. 26. than the bolts, the instant that any motion takes place along he the bolts B will be subjected to tension. If friction along be, and between the wood and the metal- plate washer be neglected, the tension in the bolts may be determined by resolving 34500 — 10300 = 24200 into two components, one normal to the plane he, and the other in the direction of the bolts. Doing this the tension in the bolts is fotmd to be about 45000 lbs. See Fig. 26. DESIGN OF A WOODEN ROOF-TRUSS, 63 From Table XVIII a single lY' bolt will safely resist a tension of moo pounds, hence five bolts are required. Each bolt resists a tension of ^^J^^ = 9000 lbs., and hence the area of the washer bearing across the fibers of the wood must be V^ = 25.7 D" (Art. 25). As the standard cast-iron washer has an area of but 16.61 D", a single steel plate will be used for all the bolts. The total area including 5 — if" holes for bolts will be 5 (2 5.7 + 1.227) = 134.6 n", and as the top chord is 7I" wide, the plate will be assumed 7" X 20" = 140 D". The proper thickness of this plate can be determined approximately as follows: The end of the plate may be considered as an over- hanging beam fastened by the nuts or heads on the bolts and loaded with 350 lbs. per square inch of surface bear- ing against the wood. The distance from the end of the plate to the nuts is about 3", and the moment at the nuts is 350 X 7 X 3" X 3" X I = 1 1000 inch-pounds. This must equal \Rhd? = J RU^ * i X 16000 X 7 X t^ or t^ = Hm = 0.59, and hence t = 0.77" = f" about. A f" plate will be used (Art. 30). (See page 146.) The tension in the bolts must be transferred to the corbel by means of adequate washers. Where two bolts are placed side by side, steel plates will be used and for single bolts, cast-iron washers. Asstiming the steel plate washers as normal to the direction of the bolts, they bear upon a wood surface which is inclined to the direction of the fibers. The per- missible intensity of the pressure upon this surface is (Art.' 23a) for 6 =33° 41', say, 34°, about 500 potmds. 64 ROOF-TRUSSES, Since each bolt transmits a stress of 9000 pounds, the net area of the plate for two bolts is 2 (9000 -J- 500) = 36 D ". Allowing for the bolt holes the gross area is about 38.5 n". Making the corbel the same breadth as the bottom chord a plate ?§" X si" = 39.3 D" will furnish the required area. The thickness of this plate is fotmd in the manner explained for the plate in the top chord. A |" plate is suflScient. For the single bolt a bevel washer will be used. The net area bearing across the fibers of the wood must be 90oo(cos e = 0.832) -7- 350 = 21.4 n", say, 23 D", to allow for the bolt hole. A washer 5" X s" will be used. The horizontal component of the stress in the bolt is 9000(0.555) = 5000 poimds. This requires 5000 -?- 1400 -=3.6 D" for end bearing against the wood, and 5000 -^ 150 = 33.3 n" for longitudinal shear. A lug on the washer I" X f" X 5" will provide area for the end bearing, and if placed at the edge of the washer nearer the center of the truss, there will be ample shearing area provided. In the above work the washers have been designed for the stress which the bolts are assimied to take and not for the stress which the bolts can safely carry. As stated above, too much reliance should not be placed upon the shearing surface ah. Assuming this to fail the stress in the bolts becomes about 64200 pounds or 12960 pounds for each bolt, which is equivalent to a stress of 10500 poimds per square inch. The horizontal component of the tension in the bolts having been transferred to the corbel, must now be trans- ferred to the bottom chord. This is done by two white oak keys 2^" X 9" long. Each key will safely carry an DESIGN OF A IVOODEN ROOF-TRUSS. 6^ end fiber stress (Art. 23), of li X 7^ X 1400 = 13 100 lbs., and two keys 2 X 13 100, or 26200 lbs., which exceeds the total horizontal component of the stress in the bolts. The safe longitudinal shear of each key is (Art. 27), 7^" X 9 X 200 = 13500, and for both keys 2 X 13500 = 27000 lbs., a little larger than the stress to be trans- ferred. The bearing of the keys against the end fibers of the corbel and the bottom chord is safe, as the safe value for long-leaf Southern pine is the same as for white oak. The safe longitudinal shear in the end of the bottom chord is about 7I" X 12 X 150 = 13500 lbs. exclusive of the I" bolt. The safe strength at the right end of the corbel is about the same. Between the keys there is ample shearing stirface without any assistance from the bolts in both the corbel and the bottom chord. The keys have a tendency to turn and separate the corbel from the bottom chord. This will produce a small tension in the five inclined bolts if the corbel is not sufficiently stiff to hold them in place when the two end bolts are drawn up tight. One f " bolt for each key of the size used here is sufficient to prevent the keys from turning when the bolts pass through or near the keys. See Art. i, Appendix. In order to prevent bending, and also to give a large bearing surface for the vertical component of 34500 lbs., a white oak filler is placed as shown in Fig. 26, and a small oak key employed to force it tightly into place. The net area of the bottom chord must be ^^y*^ « 26.1 D'^ which inspection shows is exceeded at all sections in Fig. 26. The form of joint just considered is very common, but 66 ROOF-TRUSSES. almost always lacking in strength. In addition to the notch, usually but one or two f " bolts are used where five I J" bolts are required. The writer has even seen trusses where the bolts were omitted entirely. The joint as designed would probably fail before either the top or bottom chords gave out. If tested under a ver- tical load, the top chord would act as a lever with its ful- crum over the oak filler; this would throw an excessive tension upon the lower pair of bolts, and they would fail in the threads of the nuts. Whenever longitudinal shear of wood must be depended upon, as in Fig. 26, bolts should always be used to bring , an initial compression upon the shearing surface, thereby preventing to some extent season cracks. 56a. Design of Joint L^ — Bolts and Metal Plates. — The horizontal component of 34500 lbs. is 28700 Ibs.^ which is transferred to the bottom chord by the two metal teeth let into the chord as shown in Fig. 2 7 . Let the first plate be 7" wide and i" thick and the notch 2" deep^ then the safe moment at the point where it leaves the wood is I Rbt^ = I X 16000 X 7 X I X I = 18670 inch-potinds. A load of 18670 lbs. acting i" from the bottom of the tooth gives a moment of 18670 X i = 18670 inch-poui;^ds. This load uniformly ' distributed over the tooth = — -- — 2 /\ J = 1330 lbs. per square inch; as this is less than 1400 lbs., the safe bearing against the end fibers of the wood, the value of the tooth is 1330 X 14 = 18670 lbs. The shearing sur- face ahead of the tooth must be at least ^fH^ = 125 n"; and since the chord is 7I" thick, the length of this surface 12 c must be at least = 16.7", which is exceeded in Fig. 27* 7-5 ^ ' DESIGN GF A IVGODEU ROOF-TRUSS . 67 In like manner the value of the second tooth 7" wide and J" thick is found to be 14000 lbs., and hence the value cf both teeth is 18670 + 14000 = 32670 lbs., which exceeds the total horizontal component of 34500 lbs. or 28700 lbs. The horizontal component 28700 lbs. is transferred to the metal through the vertical plates at the end of the top chord, and these are held in place by two |" bolts as // -mi Bending 18670 * i\^ ^, J^J. .^ lyi 43^^— > !Lo -81300* — t/' — ;.* 8x8 2x7 6x8z46l0Dff -1* // -18^ »v *'i6x«Pi; -3'11- Sboltssm.wasl washers , ■ — >j< — f — » Fig. 27. shown. The bearing against the end of the top chord exceeds the allowable value about 8800 lbs. if the vertical cut is 3I" as shown. The f " plate is bolted to the bottom chord and the two bolts should be placed as near the hook as possible to prevent its drawing out of the notch. The amount of metal subject to tensile stresses and shearing stresses is greatly in excess, of that required. The net area of the bottom chord exceeds the amount required. 68 ROOF-TRUSSES. The corbel is not absolutely necessary in this detail, but it simplifies construction. To keep the ^^ plate in place two Y bolts are employed* They also keep the tooth in its proper position. The teeth should usually be about twice their thick- ness in depth, as then the bending value of the metal about equals the end bearing against the wood. This allows for a slight rounding of the comers in bending the plates. Fig. 28 shows another form of joint using one \^ plate. The bolt near the heel of the plate resists any slight lifting action of the toe of the top chord, and also assists somewhat in preventing any slipping towards the left. 57. Design of Joint L^ — Nearly all Wood. — The strength of this joint depends upon the resistance of the shearing surfaces in the bottom chord and the bearing of wood against* wood. The notches when made, as shown in Fig. 29, will safely resist the given stresses without any assistance from the bolts. A single bolt is passed through both chords to hold the parts together which might separate in hand- ling during erection. The horizontal bolts in the bottom chord are put in to prevent any tendency of the opening of season cracks, starting at the notches. The vertical bolts serve a similar purpose, as well as holding the corbel or bolster in place. 58. Design of Joint L^ — Steel Stirrup. — Fig. 30 shows one type of stirrup joint, with anotch2"deep. The safe load in bearing on the inclined surface ah is 13700 lbs., and for shearing ahead of the notch 20300 lbs. This leaves 34500 — 13700 = 20800 which must be taken by the stirrup. 20800 -4- tan 6 == — ZT~ = 31200 lbs. = stress in stirrup rod. 0.667 ^ DESIGN OF A WOODEN ROOF-TRUSS. 69 Pig. 29. 70 ROOF-TRUSSES. ^- = 1.95 = ntimber of square inches of steel re- quired, or 0.975 n" must be area of the stirrup rod. A I J" round rod will be used which has an area, at the root of the threads, of 0.893 D". To pass over the top chord the rod will be bent in the arc of a circle about 7 J" in diameter, and rest in a cast-iron yf^!-' Fig. 30. saddle, as shown in Fig. 30. The base of this saddle must have an area of HH^ = 89 D". The size of the base will be 7^' X 12". The horizontal stress 17300 transferred to the corbel will be amply provided for by the two keys which transfer it to the bottom chord. 59. Dei^ign of Joint Lq. — Steel Stirrup and Pin. — The detail shown in Fig. 31 is quite similar to that shown in Fig. 30, in the manner of resisting the stresses. In the DESIGN OF A WOODEN ROOF-TRUSS. 71 present case the tension in the steel rod is 19000 lbs., requiring a rod i|" square. Loop eyes for a 2f" pin are formed on each end of the rod as shown. Each loop Has a stress of 19000 lbs., and if this stress transmitted to the bottom chord is asstimed to act i|" from the outside stirface of the chord, the moment of this stress is i9ooo(i|" + i) = 45100 inch-pounds, reqiiiring a 2f" pin Fig. 31. (Art. 31). The pin is safe against shearing, as 5.94 X loooo = 59400 is much greater than the stress to be carried. The bearing of the pin against the end fibers of the chord is about 21000 lbs., while the permissible value is 2f X 7J X 850 = 17500 lbs. (Art. 236.) The bearing of the pin across the grain of the chord is excessive, as the vertical component of the stress in the stirrup is about 31000 lbs. It is practically impossible to use this detail with any reasonable factor of safety imless the chords are made 72 ROOF-TRUSSES. excessively large. The stirrup cannot be adjusted and will either carry the entire load or none of it. It may be well to state at this time that usually it is not possible to construct a joint so that the stress shall be divided between two different Unes of resistance. In the joints designed care has been taken to make the division of the stress such that, if the wood shears ahead of the notch, the bolts can take the entire load with a unit stress well within the elastic limit of the steel. The washers, etc., will be over-stressed in the same propor- tion as the bolts. 60. Design of Joint Lq. — Plate Stirrup and Pin. — ^Fig. 3 2 . — ^The method pursued in proportioning this type of joint 6z8x36l0Dg < 12 « tfr ^e^UU ^i^^— > -13^ Fig. 32. is the same as that followed in Art. 59. In this case the stirrup takes the entire component of 34500 lbs., the |"-bolt merely keeps the members in place. This detail has the DESIGN OF A WOODEN ROOF-TRUSS. 73 objection of excessive bearing stresses for the pin against the wood. 6i. Design of Joint Lq. — Steel Angle Block.— Fig. 33,— This joint needs no explanation. Its strength depends upon the two hooks and the shearing resistance in the bottom chord. The diagonal |" bolt is introduced to hold the block in its seat, and to reinforce the portion in direct com- -313(10* f Oast iron anffteldoed "f^xW 5 J4 metal FiQ. 33. pression. The top chord is kept in position by the top plate, and a i"-roimd steel pin driven into the end and passing through a hole in the block. 62. Design of Joint Lq. — Cast-iron Angle Block. — ^At the right, in Fig. 33, is shown a cast-iron angle block made of I" or i" metal. It is held in place by two |" bolts. The top chord is held in position by a cast-iron lug in the center of the block used to strengthen the portion of the block at its right end. In all angle block joints care must be taken to have 74 ROOr-TRUSSES. sufficient bearing surface on top of the bottom chord to safely cany the vertical component of the stress in the top chord. 63. Design of Joint Lq. — ^Special. — It sometimes happens that trusses must be introduced between walls and the truss concealed upon the outside. In this case the bottom chord rarely extends far beyond the point of intersec- •N Cast iron 4 l"thlc]t Fig. 34. tion of the center lines of the two chords. The simplest detail for this condition is a flat plate stirrup and a square pin, as shown in Fig. 34. A pin 2f " square is required. The ends are turned down to fit 2^" holes in the |" plate, and, outside of the plate the diameter is reduced for a small nut which holds a 3" plate washer in place. This detail fulfils all the conditions for bending, bearing shear, etc. If round pins are used, two will be required, each 2 J" in diameter. These should be spaced about DESIGN OF A IVOODEN ROOF-TRUSS: 75 ^'jong Fig. 35. a^zS'lPlankmierB l^bolt ■18' // I y j^i-^tt^U. wasber ^>ji'P1.7'xa4" I I ^ Fig. .^6. 76 • ROOF-TRUSSES. lo" apart and not less than 9" from the end of bottom chord. Pig. 35 shows another type of joint. This can be adjusted, but requires a heavy bottom chord and the tendency of the angles to turn creates excessive cross bear- ing stresses. 64. Design of Joint Lq. — Plank Members.— Pig. 36 shows a connection which fulfils all of the conditions of bearing, shear, bending, etc., excepting the bearing of the roimd bolts against the wood. The bearing inten- sities are about double those specified in Arts. 236 and 25. 65. Design of Joint Lo. — Steel Plates arid Bolts. — Pig. 37 shows the joint Lo composed of steel side plates, steel bearing plates, and bolts. The stresses are; trans- mitted directly to the bearing plates against the end fibers of the wood, from the bearing plates to the bolts and by the bolts to the side plates. Assuming two bearing plates on each side of the top chord, the thickness of each plate will be 34500 -^ 7i X 1400 X 4 = 0.82 or |". If six bolts are used the total bearing area for each bolt is 2d X i, and if the allowable bearing intensity is 20000 lbs., the diameter of each bolt is 34500 -^ 12 X | X 20000 =0.17 in. If the side plates are but A" thick the diameter becomes 34500 -h 12 X f X 20000 = 0.23 in. The moment to be resisted' by each bolt is 1^(34500 X 0.594) = 1708 in.-lbs. According to Art. 31 this moment requires a bolt just a little larger than |" diameter. A i" bolt permits a moment of 2450 in.-lbs., which greatly exceeds the above, hence |" bolts will be used. The shearing value of six bolts in double shear is about 72000 lbs. As is usually the case the bending values of the bolts govern the diam- DESIGN OF A WOODEN ROOF-TRUSS. 77 eters. The net distance between the bearing plates is 34500 -^ 150 X 7i X 4 = 7.6 in., say, 8", to provide for longitudinal shear of the wood. The stress in the bottom chord is not sufficiently different from that in the top chord to change any of the dimensions, so the same arrangement of plates and bolts will be used. In this detail the entire reaction should be transmitted into fimmmiwmmm^ Fig. 37. the side plates, the pin being placed as shown in Fig. 37. The pin must fulfil the conditions of bearing, shear, and tending. 66, Design of Wall Bearing. — In the designs of joint Lo ;given above, no consideration has been made of the reaction at the support. The vertical and horizontal components of the reaction are shown on Plate I, 23500 lbs. and 370c lbs. respectively. The vertical component must be provided 7? ROOF-TRUSSES. for in making the bearing area of the corbel sufficiently- large so that the allowable intensity for bearing across the grain is not exceeded. In this case 23500 t- 350 = 67.1 n" is the minimum area required. If the corbel or bolster is made of white oak only 47 D" are required. The horizontal component will usually be amply provided for by the friction between the corbel and the support, but anchor bolts should always be used in important structures. Whenever the stress in the bottom chord does not equal the horizontal component of the stress in the top chord then the difference between the two stresses must be transferred to the corbel or bolster and then to the support. In the above case 31300 — 28700 = 2600 lbs. is the excess stress to be transferred. The joints as designed amply provide for this. In all of the illustrations of the joint Lo the center lines of the top and bottom chords are shown meeting in a point over the center of the support. This is theo* retically correct but owing to the change in shape of the truss when fully loaded the top chord has a tendency to produce bending in the bottom chord which can be counter- balanced by placing the center of the support a little to the right of the intersection of the center lines of the chords. Usually the corbel will be sufficiently heavy to take care of this moment, which cannot be exactly determined. 67. Design of Joint U^. — As the rafter is continuous by this joint it will be necessary to consider only the ver- tical rod and the inclined brace. Since the stress of the rod is comparatively small, the standard size of cast-iron washer can be employed to trans- fer it to the rafter. Two forms of angle washers are shown DESIGN OF A WOODEN RCOF-TRUSS. 79 Fig. 38. Fig. 39. 3o ROOF-TRUSSES. in Figs. 38 and 40. In Figs. 39 a bent plate washer is shown which answers very well if let into the wood or made suf- ficiently heavy so that the stress in the rod cannot change the angles of the bends. Where the inclined member is so nearly at right angles with the top chord as in this case, a sqtiare bearing, as shown in Fig. 40, is all that is required if there is sufficient Pig. 40. bearing area. In this case there are 30.25 D", which has a safe bearing value of 30.25 X 350 = 10600 lbs., which is not sufficient. Fig. 38 shows a method of increasing the bearing area by means of a wfought plate, and Fig.. 39 the same end reached with a cast-iron block. In all cases the strut should be secured in place either by dowels, pins or other device. DESIGN OF A fVOODEN ROOF-lkUSS. 8i 68, Design of Joint f/i. — ^The disposition of the f rod is evident from the Figs. 41, 42, and 43 : Pig. 42. 8^ ROOF-TRUSSES. ; Fig. 41 shows the almost universal method employed by carpenters in framing inclined braces, only they seldom* take care that the center lines of all pieces meet in a point as they shotdd. If the thrust .9106 lbs. be resolved into two com- ponents respectively normal to the dotted ends, it is foimd that a notch if deep is entirely inadequate to take care of the component parallel to the rafter. The cut should be made vertical and 2f" deep. The com- FiG. 43. ponent nearly normal to the rafter is safely carried by about 22 n". Figs. 42 and 43 show the application of angle- blocks, which really make much better connections, though somewhat more expensive, than the detail first described. 69. Design of Joint L2. — Fig. 44 shows the ordinary method of connecting the pieces at this joint. The DESIGN OF A WOODEN ROOF-TRUSS. 83 horizontal component of 9100 lbs. is taken by a notch 2I" deep and 3!" long.* The brace is fastened in niain., Fig. 44. place by a |" lag-screw 8" long. The standard cast- iron washer, 3^" in diameter, gives sufficient bearing c^iag screw \^^6"long -23700* ^ g^'diam. Fig. 45. area against the bottom chord for the stress in the vertical rod. Fig. 45 shows a wooden angle-block let into the bottom chord i^". The dotted tenon on the brace need not be * The permissible bearing against a vertical cut on the brace is 760 lbs. per square inch which requires a notch 2|X3|. If the cut bisects the angle between the brace and bottom chord the notch required is 2"X3i". 84 ROOF-TRUSSES. over 2^^ thick to hold the brace in position. The principal objection to the two details just described is that the end bearing against the brace is not central, but at one cidc, thereby lowering the safe load which the brace can carry. Fig. 46 shows the application of a cast-iron angle-block. The brace is cut at the end so that an area 3 J" X4" trans- i^(5U^^>> ^ » -81300* C'xtf' •28700* ■^ ^''dlam. Fig. 46. mits the stress to the~angle-block. If the lugs on the bot- tom of the block are i J" deep, the horizontal component of the stress in Ihe brace will be safely transmitted. ^diam. Pig, 47. In Fig. 47 a f bent plate is employed. This detail requires a |" bolt passing through the brace and the bottom DESIGN OF A WOODEN ROOF-TRUSS, 8S chord to make a solid connection. The use of the bolt makes the end of the brace practically fixed, so that the stress may be assumed to be transmitted along the axis or center line. 70, Design of Joint L, and Hook Splice. — ^A very com- mon method of securing the two braces meeting at L, is shown in Fig. 48, though they are rarely dapped into the lower chord. This method does fairly well, excepting Fig. 48. when the wind blows and one brace has a much larger stress than the other. In this case the stresses are not balanced, and the struts are held in place by friction and the stiffness of the top chord. The washer for the \Y rod upset to if'' must have an area of -^fiP""45 CH'^* which is greater than the bearing area of the standard cast-iron washer, so a %" plate, (f X 8", will be used. 86 ROOF-TRUSSES. It is customary to splice the bottom chord at this joint when a splice is necessary. The net area required is V^W = 20 n". The splice shown in Fig. 48 is one commonly used in old trusses, and depends entirely upon the longitudinal shear of the wood and the end bearing of the fibers. The total end bearing required is W(Ar = 17 D ", which is obtained by two notches, each ij'' deep as shown. The total shearing area reqtdred is HiV- = 158 D". Deduct- ing bolt-holes, the area used is 2 (?§ X 12) — 2(3) = 174 n". The three bolts used simply hold the pieces in place and prevent the rotation of the hooks or tables. Fig. 49 * shows a similar splice where metal keys are used. The end-bearing area of the wood is the same as " ^V ^.g^/Metal key 2^x1^x9 Fig. 49. before, and the available area of the wood for longitudinal shear is sufficient, as shown by the dimensions given. The net area of the side pieces is 2(2 X ji) = 30 D", while but 20 n" are actually required. 71. Design of Joint L3 and Fish-plate Splice of Wood. — In this case the braces are held in position by dowels * The bearing across the grain of the wood is excessive when square metal keys are used. This is due to the tendency of the keys to rotate. DESIGN OF A WOODEN ROOF-TRUSS, 87 and a wooden angle-block. The details of the vertical rod need no explanation, as they are the same as in Art. 70. The splice is made up of two fish-plates of wood each 2i" X 7i" X 46'' long and four i^" bolts. The net area of the fish-plates is 2(2^ X 7i) - 2(2 X i^) = 27.7 D", while but 20 D'' are reqtdred. Each bolt resists in bending ^^i^(i| + if) =» 7400 fTt. .r^. -4< ^( > \ < I -—9i P1.6 X 8 j5X ^^^ JSU ^ Wx 8" i e ^j. €9 Fig. 50. inch-pounds, which is less than the safe value, or 8280 inch-pounds (Art. 31) The total end bearing of the wood fibers is 2 (4 X 2f X 1 1) = 27 D", and that required HU^ = 28 D". The longitudinal shearing area of the wood and the transverse shearing area of the bolts are evidently in excess of that required. The nuts on the bolts may be considerably smaller than the standard size, as they merely keep the pieces in place. The cast-iron washer may be replaced by the small plate 88 ROOF-TRUSSES. Washer, to make sure that no threeds are in the wood ; otherwise washers are not needed. The bolts should have a driving fit. 72. Design of Joint L, and Fish-plate Splice of Metal. — This detail, differing slightly from those previously given, requires little additional explanation. A white-oak washer Pig. 51. has been introduced so that a smaller washer can be used for the vertical rod. A small cast-iron angle block replaces the wooden block of the previous article. The splice is essentially the same, DESIGN^ QF A WOODEl^ ROOF-TRUSS, 89 with metal fish-plates. Contrary to the usual practice, plate washers have been used under the nuts. This is to make certain that the fish-plates bear against the bolt proper and not against threads. If recessed bridge jnn nuts are used, the washers can be omitted. Fig. 52 shows another metal fish-plate splice where four bolts have been replaced by one pin i^" in diameter. Sd ^{D € // rt l^rod 3-X-3- It •-23700* _/ 4x6 . " _ // 12 't ff 4x6x%L 3 l^'rod 6x8" ^^ m // •-0 Fig. 53. ^"boits // Cast }i metal ^WJ'tqGl Each casting .. has 8 lugs V>i" 41; // raqua re x riongr^tJT'^^^ ^ 23700* lJ4"rocl 6 X 8" r-M' 0- 3lJ4"rod Fig, 54. The bearing plate reduces the bending moment in the pin and increases the bearing against the wood. The struts bear against a cast-iron angle-block, with a ** pipe " 9© ROOF-TRUSSES. for the vertical rod, which transmits its stress directly to the block. Two pins in the center of the block keep the bottom chord in position laterally. 73. Metal Splices: for Tension Members of Wood. — Figs. S3 and 54 show two types of metal spUces which have the great advantage over all the splices described above in that they can be adjusted. The detail shown in Fig. 53 has one serious fault. The tension in the rods tends to CO ^ 8rl- 1>C\ m I ^ t^ 1^ Fig. 54a. rotate the angles and thereby produces excessive bearing stresses across the grain of the wood. The castings in Fig. 54 usually have roimd lugs, but square lugs are much more efficient. A very old and excellent form of splice is shown in Fig. S4a.* 74. General Remarks Concerning Splices. — ^There are a large nimiber of splices in common use which have not been considered, for the reason that most of them are faulty in design and usually very weak. In fact certain scarf -splices are almost useless, and without doubt the * See Manual for Railroad Engineers, by George L. Vose, 1872. DESIGN OF A WOODEN ROOF-TRUSS. 91 truss is only prevented from failing by the stiffness of its supports. 75. Design of Joint f/a. — ^The design of this joint is clearly shown in Figs. 55-58. No further explanation seems necessary. yiVLCtt'^ Fig. 55. [^=^(CagtIron Pig. 57. Fig. 58. 76. The Attachment of Purlins. — ^The details shown (Figs. 59-63) are self-explanatory. In all cases the adja- cent purlins should be tied together by straps as shown. This precaution may save serious damage during erection, if at no other time. ROOF-TRUSSES. The patent hangers shown in P^s. 64, 65, 66, and 67 can be employed to advantage when the purlins are placed between the top chords of the trusses. 77. The Complete Design.* — Plate I shows a complete "design for the roof-truss, with stress diagrams and bills of * The dimeusioDS and quantities ehown on Plates I aAd II are based on timber which is full size. The purlioB should be 10" X 10" instead of 6" X 10". DES/GW OF A WOODEN ROOf^TRUSS. 93 material. The weight is about 100 lbs. less than that assumed. In dimensioning the drawing a sufficient number of dimensions should be given to enable the carpenter to lay off every piece, notch, bolt hole, etc., without scaling from the drawing. To provide for settle- ment or sagging due to shrinkage and the seating of the 94 ROOF-TRUSSES. vVarious pieces when the loading comes upon the new truss, the top chord is made sowewhat longer than its com- puted length. From J" to |" for each lo' in length will be sufficient in most cases. A truss so constructed is said to be cambered. 4-12d spikes In each end Fig. 63. Fig. 64. Fig. 65. Fig. 66. Fig. 67. In computing the weights of the steel rods they have been assumed to be of uniform diameter from end to end, and increased in length an amoimt sufficient to provide metal for the upsets. See Table VII. The lengths of small bolts with heads should be given from tmder the head to the end of the bolt, and the only fraction of an inch used should be J. Plate II shows another arrangement of the web brac- ing which has some advantages. The compression mem- bers are shorter, and consequently can be made lighter. The bottom chord at the centre has a much smaller stress, DESIGN OF A WOODEN ROOF-TRUSS. 95 permitting the use of a cheap splice. On account of the increase of metal the truss is not quite as economical as that shown on Plate I. For very heavy trusses of mod- erate span the second design with the dotted diagonal is to be preferred. CHAPTER VI. DESIGN OF A STEEL ROOF-TRUSS. 78. Data. — Let the loading and arrangement of the various parts of the roof be the same as in Chapter V, and simply replace the wooden truss by a steel truss of the shape shown on Plate III. Since there is bmt little dif- ference between the weights of wooden and steel trusses of the same strength, the stresses may be taken as fotmd in Chapter V and given on Plate III. 79. Allowable Stresses per Square Inch. SOFT STEEL. Tension with the grain Art. 35, 16000 lbs. Bearing for rivets and bolts Art. 24, 20000 lbs. Transverse stress — extreme fiber stress. Art. 30, 16000 lbs. Shearing across the grain Art. 32, loooo lbs. Extreme fiber stress in bending (pins).. Art. 31, 25000 lbs. For compression use table, page 28, with a factor of safety of 4. Compare with safe values on p. 173. 80. Sizes of Compression Members. Piece LqC/j. Stress =+ 34500 lbs. The ordinary shape of the cross-section of compression members in steel is shown on Plate III. Two angles are placed back to back and separated by J" or f '^ to admit gusset-plates, by means of which all members are connected 96 DESIGN OF A STEEL ROOF -TRUSS 97 at the apexes. Generally it is more economical to employ unequal leg angles with the longer legs back to back. Let the gusset-plates be assumed V thick, then from Table XIII the least radii of gyration of angles placed as explained above can be taken. Try two 3*'^ X 2^^ X i" angles. From Table XIII the least radius of gyration (r) is 1.09. The unsupported length of thepieceL.t7>/..Hsi2,andhence^ = ^ = ii.o. From Art. 22, P = 30324 lbs. for square-ended columns when — =11.0. 30324^-4 = 7581 lbs. = the allowable stress per square inch. -\W/ = 4-5S'* number of square inches re- quired. The two angles assumed have a total area of 2.88 square inches, hence another trial must be made. An inspection of Table XIII shows that 1 .09 is also the least radius of gyration for a pair of 3^'' X 2 J" X i" angles placed f" apart, as shown; hence if any pair of 3I" X 2^" angles up to this size gives sufficient area, the pair will safely carry the load. Two T^Y X 2V X-^/ angles have an area of 2 X 2.43 = 4.86 square inches. Angles with 2i" legs do not have as much bearing for purlins as those with longer legs, and sometimes are not as economical. In this case, two 4^" X 3^^ X i\ ^ angles having an area of 4.18 square inches will safely carry 3450^^ lbs., making a better and more economical combination than that tried above. This combination will be used. Thus far it has been assumed that the two angles act as one piece. Evidently this cannot be the case unless they are firmly connected. The least radius of gyration of a single angle is about a diagonal axis as shown in 98 ROOF-TRUSSES. Table XII, and for a 4" X 3" X A" angle its value is 0.65. If the .unsupported length of a single angle is /, then in order that the single angle shall have the same strength as the combination above, — -r- must equal == 0.4, or ' 0.65 ^ 0.27 ^ ^' 7 = 6'.i. Practice makes this length not more than ^(6.1), or about 4 feet. Hence the angles will be rigidly con- nected by rivets every 4 feet. Pieces U^U^ and U^U^. Owing to the slight differences in the stresses of the top chords the entire chord is composed of the same com- bination, or two 4"X3^XtV'^ angles, having an area of 4.18 square inches. Piece U^L^. Stress = + 10 100. Although it is common practice to employ but one angle where the web stress is small, yet it is better prac- tice to use two in order that the stress may not be trans- mitted to one side of the gusset-plate. The unsupported length of this piece is 13'. 3. The least radius of g3n"ation of two 2j"X2"Xi" angles is 0.94. -L is^ -, r A T^ , . 20900 — = -^-^ = 17.0, and, from Art. 22, P«about 20900. r 0.70 4 « 5225= the allowable stress per square inch. -1.93 square mches required. Two 2i'^X2''Xi'' angles have an area of 2.12 square inches, and hence are safe according to the strut formula. For stiffness no compression member should have a dimen- sion less than -jV of its length. 13-3X12 50 = 3". 2, or the long legs of the angles should DESIGN OF A STEEL ROOF -TRUSS, 99 be 3'^.2, and the sum of the short legs not less than this amount. Hence two sVX2Y^Xi^ angles, having an area of 2.88 square inches, must be used. Tie-rivets will be used once in about every four feet. Piece Ljf/j will be the same as L^U^. Piece U^L^. Stress =+9100 lbs. Two 3" X 2y X i" angles = 2.62 square inches can evidently be used, as the dimensions and stresses are slightly less than for UiL^, ft The least radius of gyration of a single 3" X 2^" X } angle is 0.53, hence they must be riveted together every f (0.53) (12.0) = ^1.24 feet. Note that 2 J" legs can be used here, as they will receive no rivets, while in the top chord both angle legs will receive rivets as shown on Plate III. 8i. Sizes of Tension Members. Piece hj^^. Stress = — 31300 lbs. The net area required is fJrTv^^-Q^ square inches. The same general form of section is used for tension members as for compression members. In the compression members the rivets were assumed to fill the holes and transmit the stresses from one side of the holes to the other. In ten- sion members this assumption cannot be made, for the fibers- are cut off by the rivet-hole, and consequently cannot transmit any tensile stress across the rivet-holes. This being the case, the two angles employed for tension members must have an area over and above the net area required equal to the area cut out or injured by the rivet- holes. In calculating the reduction of area for rivet- holes, they are assvimed to be |" larger than the diameter I oo /?00/^- TRUSSES. of the rivet. For a |" rivet the diameter of the hole is taken as |". See Table IV. For this truss let all rivets be f'^. For a trial let the piece in hand {L^L^ be made up of two 3"X2i"Xi" angles having an area of 2.62 square inches. As shown by the arrangement of rivets on Plate III, but one rivet-hole in one leg of each angle must be deducted in getting the net area. One I" rivet-hole reduces the area of two angles 2(| X i) = 0.44 square inch, and hence the net area of two 3" X 2^" X i" angles is 2.62 — 0.44 = 2.18 square inches, which is a little greater than that required, and consequently can be safely used. Piece Ljf/g. Stress = — 1 7000 lbs. ■}-^f^ = i.o6 square inches net section required. Two 2 J" X 2" X i" angles = 2.12 square inches. 2.12 — 0.44 = 1.68 square inches net section. As this is greater than the area required, and also the smallest standard angle with i" metal which can be conveniently used with f " rivets, it will be employed. Piece LgC/g. Stress = — 16,300 lbs. Use two 2j"X2"Xi" angles having a gross area of 2.12 square inches and a net area of 1.68 square inches. 82. Design of Joint L^, Plate III. — The piece L^U^ must transfer a stress of 34500 lbs. to the gusset through a number of f" rivets. These rivets may fail in two ways. They may shear off or crush. If they shear off, two sur- faces must be sheared, and hence they are said to be in double shear. From Art. 32, a |" rivet in double shear will safely carry 8836, and hence in this case VA®u^==4 ^^ the nimiber of rivets required. DESIGN OF A STEEL ROOF -TRUSS. lOi The smallest bearing against the rivets is the f '^ gusset- plate. From Art. 24, the safe bearing value in a f " plate is 5625 lbs., showing that seven rivets must be employed to make the connection safe in bearing. It is seen that as long as the angles are at least i'^ thick, the gussets f' thick, and the rivets f^ in diameter the required number of rivets in any member equals the stress divided by the bearing value of a f " rivet in a f '^ plate, or 5625. The piece L^L^ requires -VsW-^^^ rivets. The rivets are assumed to be free from bending, as the rivet-heads clamp the pieces together firmly. The location of the rivet lines depends almost entirely upon practical considerations. The customary locations are given in Table III. 83. Design of Joint U^. — The number of rivets required in LjC/j is |^^|| = 2 rivets. The best practice uses at least three rivets, but the use of two is common. As the top chord is continuous, evidently the same number is required in it. Joint C/j will require the same treatment. 84. Design of Joint L^, L^L^ requires 6 rivets as in Art. 82. LjC/j requires 2 rivets as in Art. 83. LJJ^ requires 2 rivets as in Art. 83. Ljf/g requires ^^^^^^ = 4 rivets. L^L^' requires VAV" = 3 rivets, but the connection of L2L2' will probably be made in the field, that is, will not be made in the shop but at the building, so the number of rivets should be increased 25 per cent. Therefore 4 rivets will be provided for. ■ • » • f « * « « ■ • • • • "• • «• " •• • • *• •••'•"4 . ••••••• «• • . , • *■ 102 ROOF-TRUSSES. 85. Design of Joint l/^. U^U^ requires 7 rivets as in Art. 82. LjC^a requires 4 rivets as in Art. 83. If field- rivets are used, these numbers become 9 and 5 respectively. 86. Splices. — ^As shown on Plate III the bottom chord angles have been connected to the gusset-plate at joint L2 in the manner followed at the other joints with the addition of a plate connecting the horizontal legs of the angles. Although this connection is almost universally used, yet it is much better practice to extend L1L2 beyond the gusset- plate and then splice the angles by means of a plate between the vertical legs of the angles and a horizontal plate on the under side of the horizontal legs of the angles. See paragraph 38, page 168. 87. End Supports. — In designing joint L^ only enough rivets were placed in the bottom chord to transmit its stress to the gusset-plate. Usually a plate not less than i" thick is riveted imder the bottom-chord angles to act as a bearing plate upon the support. The entire reaction must pass through this plate and be transmitted to the gusset-plate by means of the bottom-chord angles, unless the gusset has a good bearing upon the plate. This is not the usual con- dition and is not economical. The reaction is about 24000 lbs. (Art. 65). "VeW" = 5 = "the number of f" rivets required for this purpose alone. The total number of rivets in the bottom angles is 5 H- 6 = 11 rivets. The ntmiber of rivets found by this method is in excess of the number theo- retically required. The exact number is governed by the resultant of the reaction and the stress in LoLi. DESIGN OF A STEEL ROOF-TRUSS. 103 The bearing plate should be large enotigh to distribute the load over the material upon which it bears, and to admit two anchor-bolts outside the horizontal legs of the bottom angles. 88. Expansion. — Expansion of trusses having spans less than 75 feet may be provided for by letting the bearing plate slide upon a similar plate anchored to the supports, the anchor-bolts extending through the upper plates in slotted holes. See Plate III. Trusses having spans greater than 75 feet shotdd be pro- vided with rollers at one end. In steel buildings the trusses are usually riveted to the tops of coltunns and no special provision made for ex- pansion. 89. Frame Lines and Rivet Lines. — Strictly, the rivet lines and the frame lines used in determining the stresses should coincide with the line connecting the centers of gravity of the cross-sections of the members. This is not practicable, so the rivet lines and frame lines are made to coincide. 90. Drawiiigs. — Plate III has been designed to show various details and methods of connecting the several parts of the truss and the roof members. A great many other forms of connections, purlins, roof coverings, etc., are in use, but all can be designed by the methods given above. Plate III contains all data necessary for the making of an estimate of cost, and is quite complete enough for the con- tractor to make dimensioned shop drawings from. These drawings are best made by the parties who build the truss, as their draughtsmen are familiar with the machinery and templets which will be used. I04 ROOF-TRUSSES. 91 • Connectioiis for Angles. — In designing the connec- tions of the angles, but one leg of the angle has been riveted to the gusset-plate. From a series of experiments made by Prof. F. P. McKibben {Engineering News, July s, 1906, and August 22, 1907) it appears that this connection has an efficiency of about 76 per cent based upon the net area of the angle. If short lug or hitch angles are used to connect the outstanding leg to the gusset-plate the efficiency is raised but about 10 per cent. The use of lug angles is not economical unless considerable saving can be made in the size of the gusset-pla:te. While the ordinary connection has an efficiency of but 76 per cent yet members and connections designed by this method are perfectly safe for structures of the class being con- sidered, since the stress per square inch is less than 22000 pounds. The above statements have particular reference to members in tension but are probably true for com- pression members as well, as far as efficiency is concerned. 92. Purlins. — ^When I beams or channels are used for purKns their design offers no difficulties. The loads are resolved respectively into components parallel and normal to the webs of the purlins and then the method explained in Art. 30 will determine the extreme fiber stress for the section assimied. If this exceeds or differs greatly from the allowable fiber stress, a new trial must be made. Although Art. 30 explains the method to be followed in designing purlins consisting of angles, and an example given to illustrate the method, yet it may be well to give a second example here where the loading is in two planes. From Art. 50. The moments at the center of the purlin are given for components of the loads respectively DESIGN OF A STEEL ROOF-TRUSS. 105 normal and parallel to the rafter. Let these two moments be resisted by a 6" X 4" X if" angle placed as shown in Fig. 68. Table XII gives the location of the axes i-i, Fig. 68. 2-2, 3-3, and 4-4, the axes 3-3 and 4-4 being the principal axes. Since the sum of the moments of inertia about any pair of rectangular axes is constant, Ji-i + J2-2 « /s-a + ^4-4. 1 1-1 and J2-2 are given in Table XII, io6 ROOF-TRUSSES. J3_3 = Ar^, where A and r can be found from the table. Then J4-4 = Ii-i + I2-2 - Is-s = 35-38 - SSi = 29.87. From a scale drawing or by computation the distances from the principal axes to the points a, 6, c, etc., are readily found- The two moments are resolved into components parallel to the principal axes, shown in Fig. 68. The resultant moment parallel to the axis 3-3 is 109000 in. -lbs. and that parallel to 4-4 is iiooo in. -lbs. These moments produce compression at a and 6, tension at e, and tension and compression at c and d. Inspection indicates that the maximum fiber stress wiU be at a or &. For the point a, ^ 109000 - I 7000 73 = -^y^i.25 = 2500, nence A +/3 = 13500 + 2500 = 16000 lbs., which is the fiber stress at a. The fiber stress at b is 15600 lbs. The permissible fiber stress is 16000 lbs., hence the next heavier angle must be used unless the weight of the purlin is neglected. Since the moments of inertia of the angles given in Table XII are based upon angles without fillets and roimded comers, the points a and b have been taken as shown in Fig. 68. The distances to the axes shown are values scaled from a full size drawing and are suflSciently accurate for all practical purposes. As stated in Art. 50, the planes of the loads are assumed to pass through the longitudinal gravity line of the angle. DESIGN OF A STEEL ROOF-TRUSS. 107 As the rafters axe usually placed on top of the purlin^ there is a twisting moment which has not been considered. 93. End Cuts of Angles, Shape of Gusset-plates — DimensionSi etc. — In general, it is economical to cut all angles at right angles to their length. Gusset-plates should have as few cuts as possible and in no case, where avoidable, should re-entrant cuts be made. Any frame- work which can be included in a rectangle having one side not exceeding 10 feet can be shipped by rail. This permits the riveting up of small trusses in the shop, thereby avoid- ing field riveting. Large trusses can be separated into parts which can be shipped, leaving but a few joints to be made in the field. TABLES. Table I. WEIGHTS OF VARIOUS SUBSTANCES. WOODS (seasoned). per i.u. r^oot. g^j.^ Measure. Ash, American, white 38 3.17 Cherry 42 3.50 Chestnut 41 3 .43 Ebn 35 2.96 Hemlock 25 2 .08 Hickory 53 4.42 Mahogany, Spanish 53 4.42 " Honduras 35 2 .g6 Maple 49 4.08 Oak, live $g 4.92 " white 52 4.33 Pine, white 25 2 .08 " yellow, northern 34 2 . 83 " " southern 45 3.75 Spruce . '. 25 2 .08 Sycamore 37 3 . 08 Walnut, black 38 3.17 Green timbers usually weigh from one-fifth to one-half more than dry. MASOKRT. xro».« Weight in Lbi. N*"^- per Cubic Foot. Brick-work, pressed brick 140 " ordinary 1x2 Granite or limestone, well dressed t6^ " " mortar rubble 154 " " dry 138 Sandstone, well dressed • 144 109 no TABLES. BRICK AND STONB. »-««.* Wcisrht in Lbs. "*"*• per Cubic Foot, Brick, best pressed 150 " common, hard 125 " soft, inferior 100 Cement, hydraulic loose, Rosendale 56 " LouisviUe 50 " " " English Portland 90 Granite 170 Limestones and marbles 168 " " " in small pieces 96 Quartz, common 165 Sandstones, building 151 Shales, red or black 162 Slate 175 METALS. vr Weight in Lbs. Weight in Lbs. per ^^"*' per Cubic Ft. Square Ft., 1" thick Brass (copper and zinc), cast 504 42 . 00 " rolled 524 43-66 Copper, cast 542 45 . 17 " rolled 548 45.66 Iron, cast 450 37 . 50 " wrought, purest 485 40 .42 " " average 480 40 . 00 Lead 7" 59-27 Steel 490 40.83 Tin, cast 459 38.23 Zinc 437 36.42^ TABLES. Ill Table II. WEIGHTS OF ROOF COVERINGS. CORRUGATED IRON (BLACK). Weight of corrugated iron required for one square of roof, allowing six inches lap in length and two and one-half inches in width of sheet. (Keystone.) w- (A ■ JaJi •^ ^ 3^ jS a •«4 • C J Weight in Pounds of One Square of the following Lengths. V (0 *- O' « 0^2 c Si QOu -^ is y .= ^ 5 3^3 ^^^ 5' 6' 7' 8' 9' 10' 0.065 2.61 3.28 365 358 353 350 348 346 0.049 I 97 2.48 275 270 267 264 262 261 0.035 i.40 1.76 196 192 190 188 186 185 0.028 1. 12 1. 41 156 154 152 150 149 148 0.022 0.88 I. II 123 121 119 118 117 117 0.018 0.72 0.91 lOI 99 97 97 96 95 The above table is calculated for sheets 30^ inches wide before corrugating. Purlins should not be placed over 6' apart. (Phcmix.) BLACK IRON. GALVANIZED IKON. ■r. s *= *J "" jr « J- en .J- iS*j' |g ■50 •- -18 18 •?8 1§ §^ gti. 1^ i^ i^ gtS B 0.0 Oi V 0. V a.v fus &.« £3 4-> O* t in qua oof. .S3 = 3-* tin qua >of. y •g^^os §■2^ • "S V V ei 5 i = D D C 4}!! ji S 2i 7 19.75 i§ i 15 35.0 il St 3l 15 40.0 li 5«'- H 8 11.25 \'' ' 4i i 15 45.0 2} 54 f 6 3i 8 13.75 Jfi I* 15 50.0 2l S4 14 8 16.35 I 15 55.0 A SH 44 ii6 TABLES. Tabls III — Continued, MAXIMUM SIZE OF RIVETS IN BEAMS, CHANNELS, AND ANGLES. O %, jSJS as r 3 4 5 6 7 8 9 10 12 13 I Beams. > 28 V a (A Channels. ■i 5-- 3 4 5 6 7 8 9 10 12 15 o o 4 5 6 8 9 II, o 25 50 o 75 25 13.25 15.0 20.50 33.0 > Anglet. ■si I »A If li If 2 2i 2,V ^ «^8 i i i 2i a* 3 3i 4 4i 5 5t 6 RIVET SPACING. All dimensions in Inches. Size of Minimum Pitch. Maximum Pitch at Ends of Compression Members. Minim am Pitcb in Flanges of Chords and Girders. Distance from Edge of Piece to Centre of Rivet Hole. Rivets. Minimum. Usual. i i i i I I^ l| 2i 2f 3 2i 3 3i 4 4 4 4 4 .1' li 2 TABLES. 117 Table IV. RIVETS. Tables of Areas in Square Inches, to he deducted from Riveted Plates or Shapes to Obtain Net Areas, Thick, ness Size of Hole, in Inches,* Plates ' in Inches. i .06 .08 i .09 .11 .13 .14 f .16 u .17 i .19 .20 i .22 if .23 .25 i?» i .27 A .08 .10 .12 .14 .1^ .18 .20 .21 .23 .25 .27 .29 .31 .33 { .09 .12 .14 .16 .19 .21 .23 .26 .28 .30 .33 .35 .38 .40 t'. .11 .14 .i6 .19 .22 .25 .27 .30 .33 .36 .38 .41 .44 .46 i .13 .16 .19 .22 .25 .28 .31 .34 .38 .41 .44 .47 .50 .53 tV .14 .18 .21 .25 .28 .32 .35 .39 .42 .46 .49 .53 .56 .60 t .16 .20 .23 27 .31 .35 .39 .43 .47 .51 .55 .59 .63 .66 « .17 .21 .26 .30 .34 .39 .43 .47 .52 .56 .60 .64 .69 .73 } .19 .23 .28 .33 .38 .42 .47 .52 .56 .61 .66 .70 .75 .80 !• .20 .25 .30 .36 .41 .46 .51 .56 .61 .66 .71 .76 .81 .86 .22 .27 • 33 .38 .44 .49 .55 .60 .66 .71 .77 .82 .88 .93 if .23 .29 .35 .41 .47 .53 .59 .64 .70 .76 .82 .88 .94 1. 00 I .25 .31 .38 .44 .50 .56 .63 .69 .75 .81 .88 .94 1. 00 1.06 It'« .27 .33 .40 .46 .53 .60 .66 .73 .80 .86 .93 1. 00 1. 06 1. 13 ^i .28 .35 •42 .49 .56 .63 .70 .77 .84 .91 .98I1.05 1. 13 1.20 lA .30 .37 .45 52 .59 .67 .74 .82 .89 .96 1.04 I. II 1. 19 1.26 Jl .31 .39 .47 55 .63 .70 .78 .86 .94 1.02 1.09 1.17 1.25 1.33 lA .33 .41 .49 .57 .66 .74 .81 .90 .98;i.07 1.15 1.23 1. 31 1.39 li .34 .43 .52 .60 .69 .77 .86 .95 1.03 I.I2|l.20 1.29 1.38 1.46 lA .36 .45 .54 .63 .72 .81 .90 .99 1.08 1. 17 1.26 1.35 1.44 1.53 li .38 .47 .56 .66 .75 .84 .94 1.03 1. 13 1.22 1. 31 1. 41 1.50 1.59 'f* .39 .49 .59 .68 .78 .88 .98 1.07 1. 17 1.27 1.37 1.46 1.56 1.66 i« .41 .51 .6i .71 .81 .91 1.02,1.12 I.22;i.32 1.42 I 52 1.63 1.73 ili .42 .53 .63 .74 .84 .95 1.05 1. 16 1.27 1.37 1.47 1.58 1.69 1.79 li .44 .55 .66 .77 .88 .98 1.09 1.20 1. 31 1.42 1.53 1.64 1.75 1.86 'tii .45 .57 .68 .79 • 91 1.02 1. 13 1.25 1.36:1.47 1.59 1.70 1. 81 1.93 If .47 .59 .70 .82 .94 1.05 1. 17 1.29 I.41JI.52 1.64 1.76 1.88 1.99 lit .48 .61 .73 .85 .97 1.09 1. 21 1.33 1.45 1.57 1.70 1.82 1.94 2.06 3 .50 .63 .75 .88 1.60 1. 13 1.25 1.38 1.50 1.63 1-75 1.88 2.00 2.13 * Size of hole — diameter of rivet + !"• ii8 TABLES. Table V, WEIGHTS OF ROUND-HEADED RIVETS AND ROUND-HEADED BOLTS WITHOUT NUTS PER 100. Wrought Iron. Basis: 1 cubic foot iron»480 pounds. For st^eJ add 2%. Leos^th under Head to Point. Diameter of Rivet in Inches. Inches. i i i i i 1 IJ Z 47 9.3 16.0 25.2 :7.2 52.6 71.3 xi 55 10.7 18. 1 28.3 41.3 58.0 78.2 xi 6.2 12. 1 20.2 31.3 45.5 63.5 85.1 li 7.0 13.4 22.4 34.4 49.7 68.9 92.0 2 7.8 14.8 24.5 37.5 53.9 74.4 98.9 2l 8.5 16.2 26.6 40.5 58.0 79.8 105.8 2j 93 17.5 28.8 43.6 62.2 85.3 112. 7 H 10. 1 18.9 30.9 46.7 66.4 90.7 119. 6 3 10.8 20.3 33.0 49.8 70.6 96.2 126.5 3i II. 6 21.6 35 I 52.8 74.7 10-. 6 133.4 3i 12.4 23.0 37.3 55.9 78.9 107. 1 140.3 3i 13. 1 24.3 39 4 59.0 83.1 112. 6 147.2 4 13. 9 25.7 41.5 62.0 87.3 118. 154. 1 4i 14-7 27.1 43.7 65.1 91. 4123. 5 161. Ah 15.4 28.4 45.8 68.2 95.6128.9 167.9 4i 16.2 29.8 47.9 71.2 99.8134.4 174.8 5 17.0 31.2 50.1 74.3 104.0 139.8 181. 7 5i 17.7 32.5 52.2 77.4 108 . 2 145 . 3 188.6 5 18.5 33-9 54.3 80.4 112. 3 150.7 195.6 5i 19.3 35 3 56.4 83.5 116. 5 156.2 202.5 6 20.0 36.6 58.6 86.6 120.7 161. 6 209.4 6i 20.8 38.0 60.7 89.6 124.8 167. 1 216.3 6i 21.6 39-3 62.8 92.7 129.0:172.5 223.2 6i 22.3 40.7 65.0 95.8 133.2 178.0 230.1 7 23.1 42.1 67.1 98.8 137 -4 183.5 237.0 7i 23 9 43.4 69.2 101.9 141.6I188.9 243.9 7i 24.6 44.8 71.4 105.0 145.7 194.4 250.8 7} 25 4 46.2 73.5 108.0 149.9 199 8 257.7 8 26.2 47-5 75.6 III. I 154. 1 205.3 264.6 8i 27.7 50.2 79-9 117. 2 162.2 216.2278.4 9 29.2 53.0 84.1 123.4 170.8 227.1 292.2 9i 30.8 55.7 88.4 129.5 179-1 238.0 306.0 to 33.3 58.4 92.7 135.6 187.5 248.8 319.8 loi 33.8 61.2 96.9 141. 8 195.8 259.8 333.6 II 35.4 63 9 101.2 147.9 204.2 270.7 347.4 Hi 36.9 66.6 105.4 154 I 212.5 281.6 361.2 12 38. 69.3 109.7 160.2 220.0 292.5 375 One inch in length of loo Rivets 3.07 5.45 8.52 12.27 16.70 21.82 27.61 Weight of 100 Rivet Heads 1.78 4.82 9 95 16.12 24.29 34.77 47.67 Height of i-ivet head = ^j^ diameter of rivet. TABLES. 119 Table YI. WEIGHTS AND DIMENSIONS OF BOLT HEADS. Manufacturers* Standard Sizes Basis: Hoopes & Townsend's List. Squarb. Hbxagon. AiAmftteir • of Bolt. Short Longr Thick. Weight Short Longr Thick- Weight Diameter Diameter ness. per xoo. Diameter Diameter ness. per 100. Inches. Inches. Inches. Inches. Pounds. Inches. Inches. Inches. Pounds. i {' .6x9 .707 i^ 1.0 1.7 t .505 .578 i .9 1.5 |t .840 t 2.8 !! .686 2.4 t .973 4.9 .794 i 4 3 1 i 1 1. 061 t 6.8 i .866 5-9 u 1. 193 9.9 H .974 i 8.6 ft ■ V 1.326 il 13.0 n 1.083 II. 2 \ iir 1. 591 22.0 li 1.299 f 19.0 IV 1.856 } 34.8 M^ 1. 516 1 33.1 I 2.122 ■J 54.7 ij 1.733 J 47-4 1* I 2.298 73.3 li 1.877 63.5 »i li- 2.475 !■ ■ 95-7 ij 2.021 i) 82.9 i| 2* 3.006 •^' " X56.8 2 2.309 x} 132.3 li ^f 3.359 I 215.4 'i 2.743 I§ 20^3.5 li ai 3.536 I' ' 260.3 3i 2.888 li 244.4 l| »i 3.889 ^ 341.3 'i 3.176 ij 318.4 x| 3 4.243 437.4 3 3.464 1} 408.2 3* 4.420 li 508.5 3i 3.610 469.9 Approximate rules for dimensions of finished nuts and heads for bolts (square and hexagon) * Short diameter of nut-=li diameter of bolt; Thickness of nut«l diameter of bolt; Short diameter of head ~ IJ diameter of bolt; Thickness of head=»l diameter of bolt; Long diameter of square nut or head =2 . 12 diameter of bolt; " hexagon nut or head »1 .73 diameter of of bolt. M U 120 TABLES. Table YII. tJPSET SCREW ENDa Round Ban, DIMENSIONS OP UPSBT END. DIMENSIONS AND PROPORTIONS OP BOO^ r OP BAR. 'S • «*4 o I 1 • ■2 i-' If • b m b ffi • ^5 ctcr rew. o *< I. «*4 1. b I •3 b 8 b u I C i a a '8 OQ b s. Jab '5 b 3, •0 •a < < 2 * s CQ <»« 4> 2^ Z b c •^ < b n^M B o In. 4i Sq. In .302 a 9 10 A In. < ^ I.I 6i k5J PrCt. 54 A In. b < ^ In. 4i In. Sq. In. Lbs. Sq. In.j Lbs. PrCt. .196 1.668 .249 .845 21 J 4 .420 9 * .307, 1.043 5i 37 1 I 4i .550 8 i Jl .371, 1-262 6i 48 I .4421.502 44 25 1* 4i .694 7 .519 1-763 54 34 i 1} 4} .893 7 .601' 2.044' ^\ 49 H .6902.347 44 29 if 5 1.057 6 I .785! 2.67 4i 35 ItV .8873.01 4i 19 li 5 1.295 6 ij .994 3.38 4i 30 lA 1 . 108 3 . 77 3* 17 If 5 1. 515 54 il 1.227 4.17 Ah 23 If 5J 1-744 5 It'i 1.353 4 60 5 ; 29 If 1.4855.05 4 18 li 54 2.048 5 ;f' 1.623' 5.52 4} 26 2 5i 2.302 44 1.767I 6.01 5i 30 'A 1. 9186. 52 44 20 a* 5: 2.650 44 If 2.074: 7.05 5 1 28 *i 2.2377.60 4l 18 3i Si 3.023 44 If 2.405 8.18 4J; 26 If 2.5808.77 4 17 3} 6 3.419 44 I* 2.761I 9.39 4i 24 'i 6 3.715 4 i}i 2.948 10.02 5 26 3 3.142 10.68 34 18 3t 6 4.155 4 3,V 3.341 IZ.36 4i 24 2f 3-54712.06; 4 17 3} (^ 4.619 4 2l»» 3.758 12.78 4i 23 ^i ■ 6 5.108 4 2i 3.976 13.52 5* 28 2l''l 4.200 14.28 44 22 3 6 5.428 34 2i 4.430 15 07 4} 23 3» 6i 5-957 34 2t'i 4.666 15.86 5* 28 2} 4.909 16.69 4f 21 3} 6i 6.510 34 2ft 5.157 17.53 5i. 26 21 5.412 18.40 *t 20 31 7 7.087 34 3 i 5.673 19.29 5 25 2I 5.94020.20 4i 19 3\ 7 7.548 3i 3}i 2* 6.231 21.12 4? 22 3f 7i 8. 171 3i 6.492 22.07 5i 26 »\\ 6.77723.04 4i 2Z 3i 7i 8.641 3 3 7.06924.03 6i 22 3i 74 9.305 3 34 7.670 26.08 5ii 21 4 74 9.993 3 3i 8.296 28.20 4}i 1 20 TABLES. 121 Table YIII. RIGHT AND LEFT NUTS. I ! r-W-*! i*-T-i Dimensions of Nuts from Edge Moor Bridge Works' Standard. Diam- eter Lencfth Upset. Diameter of Bar. Side of Square Bar. Length of Nut. Length oi Thread. Diam- eter of Hex. Weight ov of Screw. One Nut One Nut. and Two B O A A L T W Screw Ends. Inches. Inches. Inches. Inches. Inches. Inches. Inches. Pounds. Lbs. » 4i t A 6 lA i» 1} 4i 4 i and } t and \\ 6 'f» If If 4i l} 4 i f 6J »J 3 3 7* l} 4^ }• " ,. 6i if 3 3 7i If 5 I " iiV 7 If 2f 4} 11} li 5 »f " lA I 7 i\ »f 4i 11} 5i li xA " 1* 74 aA 2i 6i l6i l} 5i *A , Jf " .A 74 3A 2} 6v 16} ij 5i lA " H 8 *A 3i 9 23t 2 5i If " 4* ** „ , 8 2A 31 : 9 23t 2* 5i lA " iJ 84 »i 3- 12i 3i4 2j 5i iJ' " iH 84 2.- 3'- "i 3x4 2j 6 i» 9 ai' 3i: i6i 4ii 2} 6 111 " a ^•' 9 »'.■ 3j 16} 4X* 2f 6 2A " a* ;ti '• 94 3}J 4i 2li 53t 2} 6 aA 94 4i 2li 53} 2i 6 2\ " 3A 3 " 2^ 10 3A 4f 36i 66i 3 6 »« . 2i 10 If 4f 26i 66i 3i t aA " a* 'i! io4 5 32 8i 3i 7 » 1 IZ ii 5f 38i 97} 3} 7i 3 5J» ii4 3ki Si 45 ii6 4 74 3i 12 4A 6i 53i X38 4 122 TABLES. Table IX. PROPERTIES OF STANDARD I BEAMa a- n ^ ii ^ ^ \ V a 9 z d o B B B B B B B 5 5 9 9 9 9 Bi3 Bi3 Bi3 B 17 B 17 B 17 B21 B21 B21 B25 B25 B25 B 25 a & O a At as Si 3 3 3 4 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 8 8 U3 ttt e 3 o 5-5 6.5 7.5 7.5 8.5 9 5 10.5 9. 75 12.25 14.75 12.25 14.75 17.25 15.0 17.5 20.0 17.75 20.25 22.75 25 25 a o •^» «j u V V c 1.63 1. 91 2.21 2.21 2.50 2.79 3.09 2.87 3.60 4 34 3.61 4 34 5.07 4.42 5.15 5.88 5.33 5 96 6.69 7 43 .0 V I a M o u a .17 .26 .36 .19 .26 .34 .41 .21 .36 .50 .23 .35 .47 .25 .35 .46 .27 .35 .44 .53 8> e o as U s 2.33 2.42 2.52 2.66 2.73 2.81 2.88 3.00 3.15 3.29 3.33 3.45 3-57 3.66 3.76 3.87 4.00 4.08 4.17 4.26 c o « a o 8 u c 2.5 2.7 2.9 6.0 6.4 6.7 7.1 12. 1 13.6 15. 1 21.8 24.0 26.2 36.2 39.2 42.2 56. p 60.2 64.1 68.0 3 O I 8 s u a 1.7 1.8 1.9 30 3.2 3 4 3.6 4.8 5.4 6.1 7.3 8.0 8.7 10.4 II. 2 12. 1 14.2 15.0 16.0 17.0 a o >» 7 O - O K 1 10 8 JS 1.23 1. 19 1. 15 1.64 I 59 1.54 1.52 2.05 1.94 1.87 2.46 2.35 2.27 2.86 2.76 2.68 3.27 3.18 3.10 3.03 t V •-* 7 O m ** 'h s o u a .46 .53 .60 .77 .85 .93 1. 01 1.23 1.45 1.70 1.85 2.09 2.36 2.67 2.94 3.24 3.78 4.04 4.36 4 71 II a o (« O h 3 < •5 e6 u e 53 52 52 59 58 58 57 65 .63 63 .72 69 .68 .78 .76 .74 .84 .82 .81 .80 Table IX — Continited. PROPERTIES OF STANDARD I BEABIS. B29 Bag B29 Big B33 i B33 ' B33 ' B33 > B4I 1 B41 1 B41 ' B53 ' BS3 ■- B5/ 15 B65 ; B6s B65 B73 ; B73 ■■ B73 ■■ B89 : B8g : BS9 : B89 : B89 : 7 8 9 10 11 h 1 T I, !. ?: s - Is "0 m 2.= F 14 3^ f H i 3 1 1 1 1 8 ~~ V r' '% 1 1 1 T 8 a - 4.33 84-9 "ITs 3.67 T^t -90 4-45 91 -9 Zt 3.54 !.6S .88 4.61 101.9 3.40 6.4J .85 4.77 III. 8 n.s 3-30 7.31 .84 4.66 133. I H 4 4.07 6.89 .97 4.80 134. 1 a6,8| 3.90 8.5a .93 4-9S 146.4 29 3 3 77 ■9> 510 158.7 31.7. 3.67 9.50 .90 5.00 3IS-8 3«-o! 4.S3 9.5. l.OI 509 338.3 3S.0: 4,71 10.07 -99 S.ai '45- 9 4t-0, 4.57 10. 95 .96 5-50 441.8 g:l{ \t 14.61 1.08 S-5S 4SS.8 IS <«> 1.07 5.65 483.4 64.5 S 73 16.04 1.04 5 75 511.0 68. .1 S-63 17.06 1.03 584 S38.6 71.8, 5-52 18.17 l.Ol 6.00 795.6 88. J 7.07 11.19 1.15 6.10 84.. 8 93 5 6.91 33.38 1.13 6.18 881.5 97.9 6,79 23-47 34.62 6.J6 931.1 101. 4: 6.69 1.09 6.JS 1169. s 117. 0' 7.83 17.86 1. 11 6-33 13.9.8 111,0, 7.70 39.04 1.19 6.40 1268.8 136.9 7.58 30.35 1.17 7.00 3087.3 173.9 9.46 43. 86 1.36 7.07 1167.8 180.7' 9.31 44.35 1.33 7-'3 1338.4 186.5 9.10 45.70 i.3» 7.ig 3309. 193.4 9.09 77.10 1.30 7-25 2379.6 198.3 S.99 48.55 1.38 124 TABLES. Table X. PROPERTIES OF STANDARD CHANNEIA ^ I ± & i2 z ■3. 3 4 5 6 7 8 9 10 II 12 J3 ^ V "u • .0 • M . e d a CT3 1 55 1 8 u 1 d Ins. 3 1 -a e ^0 *4 u ■ « Ct 2 -< I a M s £ «^ •0 Moment of Inerti Axis i-x. Section Modulus Axis i-i. Radius of Gyratio Axis i-i. Moment of Inerti Axis 8-a. Section Modulus Axis a-a. Radius of Gyratic Axis a-a. Distance of Centre Gravity from Outs of Web. A t b I S r I' S' r' X Lbs. Sq. In. Incrics Inches Ins.* Ins.* Inches Ins.* Ins.s Inches Inches C 5 4.00 1. 19 .17 1. 41 X.6 I.I 1. 17 .20 .21 .41 .44 C 5 3 5.00 1.47 .26 1.50 1.8 1.2 1. 12 .25 .24 .41 •^ C 5 3 6.00 1.76 .36 1.60 2.X 1.4 1.08 .3X .27 .42 .46 C 9 4 5.25 1.55 .18 1.58 3.8 1.9 1.56 .32 .29 .45 .46 C 9 4 6.25 1.84 .25 1.65 4.2 2.1 1. 51 .38 .32 .45 .46 C 9 4 7.25 2.13 .33 I 73 4.6 2.3 1.46 .44 .35 .46 .46 Ci3 5 6.50 I 95 .19 1.75 7.4 3.0 1.95 .48 .38 .50 .49 C 13 5 9.00 2.65 .33 1.89 8.9 3 5 1.83 .64 .45 .49 .48 Ci3 5 11.50 3.38 .48 2.04 XO.4 4.2 1.75 .82 .54 .49 .51 Ci7 6 8.00 2.38 .20 X.92 13.0 4.3 2.34 .70 .50 .54 .52 C 17 6 10.50 3.09 .32 2.04 15 I 50 2.21 .88 .57 .53 .50 C 17 6 13.00 3.82 .44 2.16 17.3 5.8 2.13 1.07 .65 .53 .52 Ci7 6 15.50 456 .56 2.28 19.5 6.5 2.07 1.28 .74 .53 .55 C2I 9.75 2.85 .21 2.09 21. 1 6.0 2.72 .98 .63 .59 55 C2I 12.25 3.60 .32 2.20 24.2 6.9 2.59 1.19 .71 .57 .53 C2I 14.75 4.34 .42 2.30 27.2 7.8 2.50 1.40 .79 .57 .53 C 21 17.25 5.07 .53 2.41 30.2 8.6 2.44 1.62 .87 .56 .55 C2I 19. 75 5.81 .63 2,51 33.2 9.5 2.39 1.85 .96 .56 .58 C25 8 XX. 25 3.35 .22 2.26 32.3 8.1 3.10 1.33 .79 .63 .58 C25 8 13.75 4.04 .31 2.35 36.0 9.0 2.98 1.55 .87 .62 .56 C2S 8 16.25 4.78 .40 2.44 39 9 10. 2.89 1.78 .95 .61 .56 C25 8 18.75 5.51 .49 2.53 43.8 II. 2.82 2.01 1.02 .60 .57 C25 8 2X.25 6.25 .58 2.62 47.8 XI. 9 2.76 2.25 I. II .60 .59 TABLES, 125 /^ Table X — Continued. PROPERTIES OF STANDARD CHANNELS 1 ^- T^ & i^2 ^ a 9 iz: c o u 2 3 m V c c efl U « v« h j3 u «i V a a .V Q JS te V ^ a Ins. Lbs. c '■5 u ft V Sq. In. 53 13.35 3.89 15*00 4.41 20.00 25.00 15.00 20.00 25.00 30.00 3500 20.50 25.00 30.00 35.00 5.88 7.35 4.46 5.88 7.35 8.82 10.29 6.03 7.35 8.82 10.29 40.00 11.76 33.00 35.00 40.00 45.00 50.00 55.00 9.90 10.29 11.76 13.24 14.71 16.18 .0 « a M o H Inches .23 .29 .45 .61 .24 .38 .53 .68 .82 .28 .39 .51 .64 .76 .40 .43 .52 .62 .72 .82 O Inchcb 2.43 2.49 2.65 2.81 2.60 2.74 2.89 3.04 3.18 c« V . C M l-t I O (A •- K a o Ins.* 47.3 50.9 60.8 70.7 66.9 78.7 91.0 103.2 "55 2.94 128. 1 3.05144.0 3.17 161. 6 3.30,179.3 3.42196.9 3.40312.6 3 43319.9 3.52347.5 3.62375.1 3.72402.7 3.82430.2 8 «0 9 S.2 c< u S a O H 9 Ins.' 'Inches 10.5 II. 3 13.5 15.7 13.4 15.7 18.2 20.6 23.x 21.4 24.0 26.9 29 9 32.8 41.7 42.7 46.3 50.0 53.7 57.4 3.49 3.40 3.21 3.10 3.87 3.66 3.52 3.42 3.35 4.61 4.43 4.28 4.17 4.09 5.62 5.57 5.44 5.32 5.23 5.16 10 t O n a o Ins.* II 1.77 1.95 2.45 2.98 2.30 2.85 3.40 3.99 4.66 3.91 4.53 5.21 5.90 6.63 12 •a ? Ji S' Ins.* .97 1.03 1. 19 1.36 1.17 1.34 1.50 1.67 1.87 1.75 1. 91 2.09 2.27 2.46 e o cQ. u # >»« c « .5 •3 cs 13 e 3 o c!> .2 2 GO Inches 8.23 3.16 8.48 3.22 9.39; 3.43 10.29 3.63 11.22 3.85 12.19 4.07 .67 .66 .65 .64 .72 .70 .68 .67 .67 .81 .78 .77 .76 .75 .91 .91 .89 .88 .87 .87 i .82 Inches .61 .59 .58 .62 .64 .61 .62 .65 .69 .70 .68 .68 .69 .72 .79 .79 .78 .79 .80 Table XI. PROPERTIES OF STANDARD ANGLES. I 3 _3_ 9 10 II 13 Jl ■S m 3 i i 3 •Bis 3i 1 u S.-- dB2 g.S -Sn a Siv 11 i «9 1 H "3 3< m 1^ 1 ' III -3.3 1 r OX« T A « I 8 ~T' I" B" »" Inches IS7. TS. S^n lneh<-9 "t;^ TiiTT Ins.' A 5 1^1 i .58 .17 .23 .009 .017 -33 -004 .14 A s iV M -as ■as .oia .024 .22 .36 .005 .014 .14 A 7 I XI 1 .So .23 -30 .oaa .031 ■ 30 .42 .009 021 .19 A 7 I XI A i.i6 .34 -3a .030 .044 .30 .45 .013 .038 .19 A 7 I XI i 1-49 .44 ■34 .037 .056 .29 ■48 .016 ■ 034 -19 A 9 iJXi i i.oa .30 .36 ■°x .049 .38 .51 .018 -035 .a4 A 9 llxl A 1-47 -43 .38 .071 .38 ■54 .025 .047 -24 A 9 1 1. 91 .56 .40 .077 .091 -37 .57 -033 .057 ■34 A 9 ilXi A 3.3a .68 -4a .090 .109 .36 .60 .040 .066 -34 All 1 XI iV 1.79 :U ■ 44 .11 -104 .46 .63 .045 -07a ■ 39 A II 1 XI i a. 34 ■ 47 -14 ■ 134 .45 .66 .058 .0S8 -39 All t a.S6 M -49 .16 .i6a ■44 .69 .070 ■39 All I XI 3.3s .98 ■51 -19 .188 .44 :7a .oSa .114 .39 A 13 I XI ^ 2. II -6a .51 .18 .14 .54 .7* .073 -10 ■34 A.J I XI a. 77 .81 -53 -33 ■ 19 ■ 53 .75 -094 ^13 .34 A 13 t XI 3.39 .55 -a? .23 .52 .78 .113 -15 .34 A. 3 I XI 3.91 1. 17 .57 .31 -26 ■ 51 .81 ■ 133 .16 -34 A 13 1 XI A 4.56 1.34 ■59 ■35 .30 .51 .84 .15a .18 .34 A IS 2 X a A a. 43 ■ 71 ■57 ■ >7 .19 -61 .80 .11 -14 ■ 39 A IS 1 3.19 .94 .59 .35 .35 .61 .84 '14 -17 -39 A IS a X3 A 3. 92' 1.15 .61 -4a .30 .60 .87 -17 ■39 A IS i 4.6ai.36 .64 .48 -35 .59 .90 '2 A IS a Xa A 5.30:1 se .66 .54 ■ 40 .59 93 '■'3 -25 .38 A 17 a Xa } 4.0 I. 19 ■ 7a ■ 70 ■3? ■ 77 1. 01 ■ 29 .28 -49 A 17 a Xa A 5.0 1,1.46 ■ 74 .85 -48 .76 I 05 .35 .33 -49 A 17 a Xa f 5-9 ii.73 .76 ■98 ■ 57 -75 1.08 -41 -38 -48 A 17 a X2 a Xi A 6.8 2-00 .781,11 .65 ■ 7! I. II .46 -42 .48 A 17 _^ 7.7 'a.2s 8ii.a3 -J?- I.lV 52 -46 -48 TABLES, 127 Table XI — Contintied. PROPERTIES OF STANDARD ANGLES. X 3 3 • V a M u 2 H 4 • 1 5 • e (/) t < 6 7 8 9 10 II 12 S 9 si 13 • a 9 a 1 • M C « c Q c « - *♦ V <-• .S20 u ^ r en 9 1 • ^ en Cm 1 c ** I. •0 (« Qi • fj . 23 c ■ M Hit w j«a |2« C m 55 c >* c 7 M CI 32 •0 M V t Ins. Lbs. A Sq.In Q Q^^ •J axa X I S r X" I" S" r" Inches Inches Ins.* Ins.* Inches Inches Ins.* Ins.« Inches A 19 3 X3 } 4.9 1.44 .84 I 24 .58 .93 1. 19 .50 .42 .59 A 19 3 X3 A 6.0 1.78 .87 1. 51 .71 .92 1.22 .61 .50 .59 A 19 3 X3 f 72 2. II .89 1.76 .83 .91 1.26 .72 .57 .58 A 19 3 X3 A 8.3 2.43 .91 1.99 .95 .91 1.29 .82 .64 .58 A 19 3 X3 i 9.4 2.75 .93 2.22 1.07 .90 1.32 92 .70 •55 A 19 3 X3 W 10.4 3.06 .95 2.43 1. 19 .89 1.35 1.02 .76 •55 A 19 3 X3 f II. 4 3.36 .98 2.62 1.30 .88 1.38 1. 12 .81 .58 A 21 3iX3i f 8.4 2.48 1. 01 2.87 1. 15 1.07 1.43 1. 16 .81 .6a A 21 3 X34 A 9.8 2.87 1.04 3.26 1.32 1.07 1.46 1.33 .91 .68 A 21 3 X3i i II. I 3.25 1.06 3.64 1.49 1.06 1.50 I. SO 1. 00 .68 A 21 3 X3i A 12.3 3.62 1.08 3.99 1.65 I. OS 1.53 1.66 1.09 .68 A 21 3 X3i i ,13. S 3.98 1. 10 4.33 1. 81 1.04 1.56 1.82 1. 17 .68 A 21 3 X3i H 14.8 4.34 1. 12 4.65 1.96 1.04 1-59 1.97 1.24 .67 A 21 3iX3i i 159 4.69 1. 15 4.96 2. II 1.03 1.62 2.13 1. 31 .67 A 21 3iX34 if 17. 1 5.03 1. 17 5.25 2. 2 J 1.02 1.65 2.28 1.38 .67 A 23 4 X4 ^ 8.2 2.40 1. 12 3.71 1.29 1.24 1.58 1.50 .95 .79 A 23 4 X4 i 9.7 2.86 1. 14 4.36 1.52 1.23 1. 61 1.77 1. 10 .79 A 23 4 X4 A II. 2 3.31 1. 16 4.97 1.75 1.23 1.64 2.02 1.23 .78 A 23 4 X4 i 12.8 3.75 1. 18; 5.56 1.97 1.22 1.67 2.28 1.36 •75 A 23 4 X4 <» 14.2 4.18 1. 21 6.12 2.19 1. 21 1. 71 2.52 1.48 .78 A 23 4 X4 f 15.7 4.61 1.23 6.66 2.40 1.20 1.74 2.76 1.59 .77 A 23 4 X4 U 17.1 5.03 I 25 7.17 2.61 1. 19 1.77 3.00 1.70 .77 A 23 4 X4 i 18.55.44 1.27 7.66 2.81 1. 19 1.80 3.23 1.80 .77 A 23 4 X4 a 19.9 5.84 1.29 8.14 3.01 1. 18 1.83 3.46 1.89 .77 A 27 6 X6 A 17.2 5. 06 1.66 17.68 4.07 1.87 2.34 7.13 3.04 1. 19 A 27 6 X6 i 19.6:5.75 1.68 19.91 4.61 1.86 2.38 8.04 3-37 1. 18 A 27 6 X6 ♦ 21.96.43 1. 71 22.07 5.14 1.85 2.41 8.94 3.70 1. 18 A 27 6 X6 i 24.2I7.11 1.73 24.16 5.66 1.84 2.45 9.81 4.01 1. 17 A 27 6 X6 ¥ 26. 417. 78 1.75 26.19 6.17 1.83 2.48 10.67 4.31 1. 17 A 27 6 X6 i 28.78.44 1.78 28.15 6.66 1.83 2.51 11.52 4 59 1.17 A 27 6 X6 ? 30.99.09 1.80 30.06 7.15 1.82 2.54 12.35 4.86 1.17 A 27 6 X6 33.1 9.73 1.82 31.92 7.63 1. 81 2.57 13.17 5-12 1. 16 Column 9 contains the least radii of gyration for two angles back to back for all thicknesses of gusset plates. 128 TABLES. Table XII. PROPERTIES OF STANDARD ANGLES in Section Number. A A A A A A A A A A A A A A A A A A A A 91 91 91 91 91 91 93 93 93 93 93 93 95 95 95 95 95 95 95 95 A 97 A 97 A 97 A 97 A 97 Dimen- sions. bz a Inches. 2iX2 2iX3 2iX2 2iX3 2iX2 2iX2 3 3 3 3 3 3 X2j X2j X2i X2j X2i X2i 3iX2i 34x24 34X2i 34x24 34X2i 34X2i 34X2i 34x24 34x3 34x3 34x3 34x3 34x3 3 4 5 Tiiickness. Weight Ftwt. Area of of Section. t A Inches. Pounds. Sq. In A 2.8 .81 i 3.6 1.06 A 4.5 1. 31 f 5.3 1.55 iV 6.0 1.78 i 6.8 2.00 i 4.5 1. 31 A 5-5 1.62 f 6.5 1.92 A 7.5 2.21 i 8.5 2.50 A 9.4 2.78 } 4.9 1.44 ♦ 6.0 1.78 i 7. a 2. II A 8.3 2.43 J 9.4 2.75 A 10.4 3.06 f II. 4 3.36 ii 12.4 3.65 ^ 6.6 1.93 f 7.8 2.30 <' 9.0 2.65 i 10.2 3.00 A II. 4 3.34 Distanceof Centre of Gravity from Back of Longer Flange. Moment of Inertia Axis X'Z. Inches. .51 .54 .56 .58 .60 .63 .66 .68 .71 .73 .75 .77 .61 .64 .66 .68 .70 .73 .75 .77 .81 .83 .85 .88 .90 Inches.* .29 .37 .45 .51 .58 .64 .74 .90 1.04 1. 18 1.30 1.42 .78 .94 1.09 1.23 1.36 1.49 1. 61 1.72 1.58 1.85 2.09 2.33 2.55 8 Section Modulus. Axis i-i. S Inches'. .20 .25 .31 .36 .41 .46 .40 .49 .58 .66 .74 .82 .41 .50 .59 .68 .76 .84 .92 .99 .72 .85 .98 1. 10 1. 21 TABLES, 129 Table XII — Continued. PROPERTIES OF STANDARD ANGLES. 9 10 II 12 13 14 15 I Distance Radius of Gyration of Centre of Gravity from Back Moment of Inertia Section Modulus Radius of Gyration Least Radius of Gyration Axis i-i. of Shorter Flange. Axis 2-2. Axis 2-2. Axis 2-2. Tangent of Angle a Axis 3-3. Section Number. r X' I S' r' r" Inches. Inches. Inches.* Inches.' Inches. Inches. .60 .76 .51 .29 .79 .632 .43 A 91 .59 .79 .65 .38 .78 .626 .42 A 91 .58 .81 .79 .47 .78 .620 .42 A 91 .58 .83 .91 .55 .77 .614 .42 A 91 .57 .85 1.03 .62 .76 .607 .42 A 91 .56 .88 1. 14 .70 .75 .600 .42 A 91 .75 .91 1. 17 .56 .95 .684 .53 A 93 .74 .93 1.42 .69 .94 .680 .53 A 93 .74 .96 1.66 .81 .93 .676 .52 A 93 .73 .98 1.88 .93 .92 .672 .52 A 93 .72 1. 00 2.08 1.04 .91 .666 .52 A 93 .72 1.02 2.28 1. 15 .91 .661 .52 A 93 .74 I. II 1.80 .75 1. 12 .506 .54 A 95 .73 1. 14 2.19 .93 I. II .501 .54 A 95 .72 1. 16 2.56 1.09 1. 10 .496 .54 A 95 .71 1. 18 2.91 1.26 1.09 .491 .54 A 95 .70 1.20 3.24 1. 41 1.09 .486 .53 A 95 .70 1.23 3.55 1.56 1.08 .480 .53 A 95 .69 1.25 3.85 1. 71 1.07 .472 .53 A 95 .69 1.27 4.13 1.85 1.06 .468 .53 A 95 .90 1.06 2.33 .95 1. 10 .724 .63 A 97 .90 1.08 2.72 1. 13 1.09 .721 .62 A 97 .89 1. 10 3.10 1.29 1.08 .718 .62 A 97 .88 1. 13 3.45 1.45 1.07 .714 .62 A 97 .87 1. 15 3.79 1. 61 1.07 .711 .62 A 97 Column 9 contains the least radii of gyration for two angles with short legs^ back to back for all thicknesses of gusset plates. 13° TABLES. Table XII — Continued. PROPERTIES OF STANDARD ANGLES, I 2 3 4 5 6 7 8 Distance Dimen- sions. Thickness. WeiRht Font. Area of Section. of Centre of Gravity from Back of Longer Moment of Inertia Axis z-z. Section Modulus. Axis z-z. Section Number. Flange. bxa t A » I 8 Inches. Inches. Pounds. Sq. In. Inches. Inches.* Inches*. A 97 34X3 * 12.5 3.67 .92 3.76 1.33 A 97 34X3 H 13.6 4.00 .94 2.96 1.44 A 97 34X3 } 14.7 4. 31 •^! 3.15 1.54 A 97 34X3 if 15.7 4.62 .98 3.33 1. 65 A 99 4 X3 ^ 7.1 2.09 .76 1.65 .73 A 99 4 X3 A 8.5 2.48 .78 I 92 .87 A 99 4 X3 9.8 2.87 .80 2.18 .99 A 99 4 X3 J II. I 3.25 .83 2.42 1. 12 A 99 4 X3 •ft 12.3 3.62 •fs •2.66 1.23 A 99 4 X3 13.6 3.98 .87 2.87 1.35 A 99 4 X3 *> 14.8 4.34 .89 3.08 1.46 A 99 4 X3 } 15 9 4.69 .92 3.28 1.57 A 99 4 X3 il 17.1 503 .94 3.47 1.68 A loi 5 X3 t 8.2 2.40 .68 1.75 .75 A loi 5 X3 9.7 2.86 .70 2.04 .89 A loi 5 X3 ^t II. 3 3.31 .73 2.32 1.02 A loi 5 X3 i 12.8 3.75 .75 2.58 1. 15 A loi 5 X3 t 14.2 4.18 .77 2.83 1.27 A loz 5 X3 15-7 4.61 .80 3.06 1.39 A loi 5 X3 H 17.1 5.03 .82 3.29 1. 51 A loi 5 X3 i 18.5 5.44 .84 3.51 1.62 A loi 5 X3 if 19.9 5.84 .86 3.71 1.74 A 103 5 X34 f 10.4 3.05 .86 3.18 1. 21 A 103 5 X34 ■rV 12.0 3.53 .88 3.63 1.39 A 103 5 X34 i 13.6 4.00 .91 4.05 1.56 A 103 5 X34 t 15.2 4.46 .93 4. 45 1.73 A 103 5 X34 16.7 4.92 .95 4.83 1.90 A 103 5 X34 ^ 18.3 5-37 .97 5.20 2.06 A 103 5 +34 i 19.8 5.81 z.oo 5-55 2.22 A 103 5 X34 V 21.2 6.25 1.02 5.89 2.37 A 103 5 X34 22.7 6.67 1.04 6.21 2.52 TABLES. 131 Table XII — Continued. PROPERTIES OF STANDARD ANGLES. -! > «> ■*?■ 1-j' jT^ i •* X i ^ ^^ 1 — 1 ^^"""""■^"»r* • ^^ F=aj^ 1. ^-^ ' ■ i w 1 ■ ^ to 9 10 IZ 12 13 14 15 I Distance • Radius of Gyration Axis z-x. of Centre of Gravity from Back of Shorter Flanji^e. Moment of Inertia. Axis 2-2. Section Modulus Axis 2-2. Radius of Gyration Axis 2-3. Tangent of Angle a Least Radius of Gyration Axis 3-3. Section Number. r X' I' S' r' r" Inches. Inches. Inches.* Inches.* Inches. Inches. .87 1. 17 4. II 1.76 1.06 .707 .62 A 97 .86 1. 19 4.41 I 91 1.05 .703 .62 A 97 .85 1. 21 4.70 2.05 1.04 .698 .62 A 97 .85 1.23 4.98 2.20 1.04 .694 .62 A 97 .89 1.26 3.38 1.23 1.27 .554 .65 A 99 .88 1.28 396 1.46 1.26 .551 .64 A 99 .87 1.30 4.52 1.68 1.25 .547 .64. A 99 .86 1.33 5.05 1.89 1.25 .543 .64 A 99 .86 1.35 5.55 2.09 1.24 .538 .64 A 99 .85 1.37 6.03 2.30 1.23 .534 .64 A 99 .84 1.39 6.49 2.49 1.22 .529 .64 A 99 .84 1.42 6.93 2.68 1.22 .524 M A 99 .83 1.44 7.35 2.87 1. 21 .518 .64 A 99 •55 1.68 6.26 1.89 1. 61 .368 .66 A loi .84 1.70 7,37 2.24 1. 61 .364 .65 A loi .84 1.73 8.43 2.58 1.60 .361 .65 A loz .83 1.75 9.45 2.91 1-59 .357 .65 A loi .82 1.77 10.43 3.23 1.58 •353 .65 A loi .82 1.80 11.37 3.55 1.57 .349 .64 A loi .81 1.82 12.28 3.86 1.56 .345 .64 A loi .80 1.84 13.15 4.16 1.55 .340 .64 A loz .80 1.86 13.98 4.46 1-55 .336 .64 A loz X.02 1. 61 7.78 2.29 1.60 .485 .76 A 103 i.ot 1.63 8.90 2.64 1.59 .482 .76 A 103 1. 01 1.66 9.99 2.99 1.58 .479 .75 A 103 1. 00 1.68 11.03 3.32 1.57 .476 .75 A 103 .99 1.70 12.03 3 65 1.56 .472 .75 A 103 .98 1.72 12.99 3 97 1.56 .468 .75 A Z03 .98 1.75 13.92 4.28 1.55 .464 .75 A 103 .97 1.77 14.81 4.58 1.54 .460 .75 A 103 .96 1.79 15.67 4.88 1.53 .455 .75 A 103 Column 9 contains the least radii of gyration for two angles with short legs back to back for all thicknesses of gusset-plates. i 132 TABLES, Table XII — Conttni^d. PROPERTIES OF STANDARD ANGLES. • I 2 3 4 5 6 7 8 Section Number. Dimen- sions. Thickness. Weight Foot. Area of Section. • Distance of Centre of Gravity from Back of Longer Flange. Moment of Inertia Axiz z-z. Section Modulus Axis i-i. bza t A X I S Inches. Inches. Pounds. Sq. In. Inches. Inches.* Inches.* A 105 A 105 A 105 A 105 A 105 A 105 A 105 A 105 A 105 A 107 A 107 A 107 A 107 A 107 A 107 A 107 A 107 A 107 6 X34 6 X34 6 X34 6 X3i 6 X34 6 X34 6 X34 6 X34 6 X34 6 X4 6 X4 6 X4 6 X4 6 X4 6 X4 6 X4 6 X4 6 X4 f ? II. 6 13.5 IS 3 17. 1 18.9 20.6 22.3 24.0 25.7 12.3 14.2 16.2 18. 1 19.9 21.8 23.6 25 4 27.2 3 42 3.96 4.50 5.03 5.55 6.06 6.56 7.06 7.55 3.61 4.18 4. 75 5.31 5.86 6.40 6.94 7.46 7.98 .79 .81 .83 .86 .88 .90 .93 .95 .97 .94 .96 .99 1. 01 1.03 1.06 1.08 1. 10 1. 12 3 34 3.81 4.25 4.67 5.08 5.47 5.84 6.20 6.55 4.90 5.60 6.27 6.91 7.52 8. II 8.68 9.23 9.75 1.23 1. 41 I 59 1.77 1 94 2. II 2.27 2.43 2.59 T.60 1.85 2.08 2.31 2.76 2 97 3.18 3 39 TABLES, 133 Table XII — Continued. PROPERTIES OF STANDARD ANGLES. 9 10 II 12 13 14 15 I Radius of Gyration Axis i-i. Distance of Centre of Gravitv from Back of Shorter Flange. Moment of Inertia Axisa-a. Section Modulus . Axis 2-2. Radir.8 of Gyration Axis 2-2. Tanirent of Angle a Least Radius of Gyration. Axis 3-3. Section Number, r X' I' S' r' r" Inches. Inches. Inches.^ Inches.' Inches. Inches. 99 .98 .97 .96 .96 .95 .94 .94 .93 X.17 1. 16 1. 15 1. 14 1. 13 1. 13 1. 12 I. II I. II 2.04 2.06 2.08 2. II 2.13 2.15 2.18 2.20 2.22 1.94 1.96 1.99 2.01 2.03 2.06 2.08 2.10 2.12 12.86 14.76 16.59 18.37 20.08 21.74 23.34 24.89 26.39 13.47 15.46 17.40 19.26 21.07 22.82 24.51 26.15 27.73 3.24 3.75 4 24 4.72 5 19 5 65 6.10 6.55 6.98 3.32 3 83 4 33 4 83 5 31 5.78 6.25 6.75 7.15 1.94 I 93 1.92 1. 91 1.90 1.89 1.89 1.88 1.87 1.93 1.92 1. 91 1.90 1.90 1.89 1.88 1.87 1.86 .350 .347 .344 .341 .338 .334 .331 .327 .323 .446 .443 .440 .438 .434 .431 .428 .425 .421 .77 .76 .76 .75 .75 .75 .75 .75 .75 .88 .87 .87 .87 .86 .86 .86 .86 .86 A 105 A 105 A 105 A 105 A 105 A 105 A 105 A 105 A 105 A 107 A 107 A 107 A 107 A 107 A 107 A 107 A 107 A 107 Column 9 contains the least radii of gyradon for two angles with short legs back to back for all thicknesses of gusset-plates. I 134 TABLES. Table XIII. LEAST RADII OF GYRATION FOR TWO ANGLES WITH UNEQUAL LEGS, LONG LEGS BACK TO BACK. Area of Least Radii of Gyration for Distances Least Radius Dimensions, Thickness, Two Angles, Back to Back. of Gyration Inches. Inches. Square Inches. for one Inch. 1 Inch. i Inch. Angle. 2}X2 "f^ 1.62 0.79 0.79 79 0.43 2iX2 t 3.09 0.77 0.77 0.77 0.42 2iX2 4 4.00 0.75 0.75 0.75 0.42 3 X2j i 2.63 0.95 0.95 0.95 0.53 3 X2i f 3.84 0.93 0.93 0.93 0.52 3 X2j ♦ 5.55 0.91 0.91 0.91 0.52 3iX2i i 2.88 0.96 1.09 1. 12 0.54 3iX2i i 5 SO 1. 00 1.09 1.09 0.53 3iX2j W 7.30 1.03 1.06 1.06 0.53 3iX3 A 3.87 1. 10 1. 10 1. 10 0.63 34X3 A 6.68 1.07 1.07 1.07 0.62 34X3 « 9.24 1.04 1.04 1.04 0.62 4 X3 A 4.18 1. 17 1.27 1.27 0.65 4 X3 A 7.24 1. 21 1.24 1.24 0.64 4 X3 if 10.05 X.2I 1. 21 1. 21 0.64 5 X3 4.80 1.09 1.22 1.36 0.66 5 X3 ft 8.37 1. 13 1.26 1.41 0.65 5 X3 11.68 1. 17 1.32 1.47 0.64 5 X34 i 6.09 1.34 1.46 1.60 0.76 5 X34 1 9.84 1.37 I. 51 1.56 0.75 5 X34 |- 13.34 1.42 1.53 1.53 0.75 6 X34 f 6.84 X.26 1.39 1.53 0.77 6 X34 } 11.09 1.30 1.43 1.58 0.75 6 X34 15.09 1.34 I 49 1.64 0.75 6 X4 i 7.22 1.50 1.62 1.76 0.88 6 X4 11.72 1.53 1.67 1. 81 0.86 6 X4 i 15.97 1.58 1.68 1.86 0.86 TABLES. 135 Table XIV. PROPERTIES OF T BARS. E^pud Legs. z 2 3 4 5 6 7 8 DiMBN SIONS. Weight per Foot. Area of Section. Dist. Cent. Width of Depth of Thickness Thickness of Gravity from Out- Rirfe of Section Flange. Bar. of Flange. of Stem. Flange. Number. b d B to n' t to t.| A X Inches. Inches. Inches. 1 Inches. Pounds. Sq. Ins. Inches, T 5 I I i toA i toA .89 .26 .29 T 181 li Ij A" A ft::S 1.39 .41 .33 T 183 1: \P A" i 1.53 .45 .34 T 187 tV " i A" ^ 1. 61 .47 .36 T 189 1 I If A" i 1?' ^ 1.85 .54 .39 T 37 2 2- i "A 1 • « 6 3.7 I. OS .59 T 39 2 2 A" i rvr 1 4.3 1.26 .61 T 41 2\ 2i ■■ "A i "A 4.1 1. 19 .68 T 69 3 3 "A i " A 7.8 2.27 .88 T 97 3i 3i i "A f " A 9 3 1 2.74 .99 T 185 T 65 T loz li 3 3i It Unequal Legs, A " i TIT Tt "A IT 1.49 7.» 9-9 .44 2.07 2.91 .29 .71 1.20 136 TABLES, Tablb XrV — Contimted. PROPERTIES OF T BARS. Equal Legs — {ContiniAed), z 9 zo zz 12 13 14 Moment of Section Radius of Moment of Section Radius of Inertia Modulus Gyration Inertia Modulus Gyration Section Nufliber* Axis x-s. Axis x-i. Axis x-i. Axiss-9. Axis»-9. Axis s-s. I S r r S' F* Inchest Inches*. Inches. Inches^. Inches*. Inches. T 5 .03 .03 .30 .OZ • 02 .21 T x8z .04 .05 .32 .02 .04 .25 T 183 .05 .06 .33 .03 .05 .26 T 187 .06 .07 .35 .03 .05 .27 T Z89 .08 .08 .39 .05 •^z .29 T 37 .37 .36 .59 .18 .z8 .42 T 39 .43 .31 .59 .23 .23 .42 T 41 .51 .32 .65 .24 .22 .45 T 69 z.8a .86 .90 .92 .6z .64 T 97 3.x z.33 1.08 1.42 .81 .73 Unequal Legs — (CarUinued). T z8s .04 .05 .29 .03 .01 .28 T 6s Z.08 .60 .64 .90 .60 .28 T zoz 4.3 Z.54 Z.23 1.42 .8z .70 TABLES. m Table XV. STANDARD SIZES OF YELLOW PINE LtTMBER AND CORRESPONDING AREAS AND SECTION MODULI.* Nominal Size. JX 6 8 xo 12 x6 aiX 6 8 10 la \t 3X 6 8 10 la 8 10 la 4X Standard Site. ifX aix afX 3lX St 7* 9i "i I3i iSi 5i 7{ 91 ii< »3i Si iii 15} 3i Si 7\ 9\ "t 134 isi Area-ii, Sq. In. 9 1 xa.a 15-4 X8.7 ax. 9 35. a xa, x6 ax. as 30 34 4 9 4 9 4 9 X5.X ao.6 a6.x 3X.6 37 I 4a. 6 X4.X ax.x a8.x 35 6 50.6 58.x Section Modulus, 8.57 15 a3 a4-44 35 8a 49 36 65.03 XX. 34 ax. 10 33 84 40.60 68.34 90. xo X3.86 a5 78 41 36 60.60 83 53 xxo.xx 8.79 19 77 35.16 "3 91 X50.X6 Relative transverse strength of yellow pine Long-leaf xoo Cuban no Loblolly oa Short-leaf.. 84 Relative oompresrave strength. of yellow pine. With the grain Long-leaf xoo Cuban xis Loblolly 94 Short-leaf 86 Longitudinal shear at neatral axis IF » total safe uniformly distiib* uted load on b^im sup- ported at ends A »area of section of beam ft—BsIe intensity for longitudi^ nal shear W^iAfs, * Compiled from "A Manual of Standard Wood Construction/' published by The Yellow Pine Manufacturers' Association, St. Louis, Mo. \3^ TABLES. Table XV — Continued. Nominal Size. 6X 6 8 10 12 i6 i8 8X 8 10 la 14 i6 zoXio la \t xaXia 14 z6 z8 14X14 z6 zS z6Xi6 18 Standard Size. Six 7JX 5 7 9 II 13 15 17 7 9 II X3 15 9JX 9 II 13 15 iiJXii 13 15 17 i3iXi3 15 17 15JX15 17 Area A, Sq. In. 30 41 53 63 74 85 96 3 3 3 3 3 3 3 56.3 71.3 86.3 101.3 116. 3 90.3 109 3 128.3 147 3 133.3 155 3 178 3 201.3 182.3 209.3 236.3 240.3 271.3 Section modulus. 27.70 51.56 82.73 121.23 167.10 220.21 280.73 70.31 112. 81 165.31 227.81 300.31 142.89 209.39 288.56 380.39 253 48 349 31 460.48 586.98 410.06 540.56 689.06 620.67 791 14 Bending Moments. For a fiber stress of 1200 pounds per square inch the maximum bending moment in/oo(-pounds is looS, where ^S— the section modulus TABLES. 139 H Pm a /-s • Ok u s-* K »« e< -< H K (S 00 >»' o o 10 10 f« I |8 ua a > ki o gss^% s ssss SS^S f« M M M M > (A o 3 o m o into O>oo «> 8 ^ (8 ill 03 O (ft- H a o o Vi -So u o d t^doooo ooooo to C3 ^ « a P V e« lis V OM 5 %i •d a 0*2 S8SS 2 O H H « 8 -a o V too mo o « d « 10 10 100 1A ^ ^ rO 10 rON 1 888 8S as 80 O Q O O too O V)1A « M d n d M 80 O 00 t^t^ 00 00 t^ i^do 00 00 8 o o miA 8888 (1 M O M o in a V SSS8 8888S>8 )0 v5 o t<«oo t^ o2 H n o M V •a CO l4 o •I Q V a> « ^ 09 o ■»5'3'^«" ? ill 0) o Is If a> o P^CQ '4i^ •piH •vH gee no «4M «*H X40 TABLES. Table XVII. CAST-IRON WASHERS. I>iam.of boltd. Inches. D Inches. 3f 3 3i 3i 4 4i 6 6i 7i 8i 9i loi "i "i Inches. If li 3i ai 3 31- 3f 4i 4} Si Si 61 d' Inchet. T Inches. Weifht. Lbs. Bearing Area. Sq. In. \ 5.16 \ 6.69 li -7-78 »i 10.35 ai 11.68 3 16.61 Si 26.92 6 28.61 <>t 38.53 I7i 49-91 30 62.77 27i 77." 36 92.91 46 XZO.19 For sizes not given D TABLES. 141 Table XVIII. SAFE SHEARING AND TENSILE STRENGTH OF BOLTS. 1 Wrought Iron. Soft Stbku Diam. Gross Area. XT^A of Bolt. Net Afea. Single Tension Single Tension Shetu- Z3000 lbs. Shear x6ooolbs. 7 SOD lbs. xoooo lbs. Inch. Square Inch. Square Inch. perSq. In. perSq. In. perSq. In. perSq. In. 1 • 0.196 0.126 1470 X510 i960 3030 0.307 0.202 3300 3430 3070 3230 , , 0.442 0.303 3320 3630 4420 60x0 4830 6730 • 0.601 0.430 4510 5040 X 0.785 0.550 5890 6600 7850 8800 l\ 0.994 0.694 7460 8330 9940 xxxoo li 1.337 0.893 9300 10730 X3370 14290 If 1.485 X.O57 1 1 140 13680 X4850 X69IO ij 1.767 1.395 13250 15540 X7670 30730 li 3.405 1.744 18040 30930 34050 37900 3 3 14a 3.303 23560 37630 31420 36830 ^t 3 976 3.033 39830 36380 39760 48370 2\ 4 909 3.715 36820 44580 49090 59440 A 5 940 7.069 4.619 44550 55430 59400 73900 3 5 4^ 53030 65140 70690 86850 APPENDIX. I. Length of Keys, Spacing of Notches and Spacing of Bolts.— Let p = the end bearing intensity, q = the bear- ing intensity across the grain, and s = the intensity in i n 3 lihL. 1 .-^1 "j^m :9— 1 2 B Fig. I. longitudinal shear for the key. Then the length of the P key is / = -d, when end bearing and longitudinal shear are considered. As the key tends to rotate under the moment p(Py cross bearing stresses are produced and the maximum intensity must not exceed q. The length of the key based on g is / = d\l6-. This value of / is less than that found above for wooden keys, hence their length is controlled by end bearing and longitudinal shear inten- sities. Evidently square metal keys produce excessive 143 144 APPESDIX. cross bearing intensities and should only be used when p is taken as \q. For given values of p and q the proper length of metal keys is found from the second formula given above. Unless the pieces A and B are securely bolted together the rotation of the key will separate them. The rotating moment is \Ql = p=the stress per square inch in any member pro- duced by a full load ; the stress in any member produced by a load of one poimd acting at the left support and parallel to the plane of the support, usually horizontal; the length center to center of any member (inches) ; the modulus of elasticity of the material com- posing any member ; the total change in span produced by a full load. / = £ = Z) = Fig. 4. If S=the stress or horizontal force necessary to make D zero, the area of any member in square inches, D a = 5 = aE ♦Theory and Practice of Modern Framed Structures, Johnson, Bryan, Tur- neaure (John Wiley & Sons, N. Y.). Roofs and Bridges, Merriman and Jacoby (John Wiley & Sons, N. Y.). APPENDIX. 153 To illustrate the use of these formulas we will take a simple scissors-truss having a span of 20 feet and a rise of 10 feet. COMPUTATIONS FOR D AND S, Piece. Stress Produced by 1000- Ib. Loads. sq. in. 1&. lbs. inches. puX aE' Aa Eb ab ■ ah W + 3160 + 2100 + 800 — 2360 — iq8o 36 36 36 36 0.785 87.8 58.3 22.2 65 5 2522 +0.71 +0.71 0.00 -1.58 — 1. 00 84.8 84.8 63.2 126.5 80.0 .00528 .00351 . 00000 .013x6 .00336 .00000x18 .00000118 . 00000875 .00000170 0253X 2 .00001281 2 .05062 .00002562 .05062 .00002562 Let all members except bb' be made of long-leaf Southern pine 6"X6", and bb' consist of a i-inch roimd rod of steel upset at the ends. The value of E for the wood is i ,000,000 and for the steel 30,000,000. Computing D and 5, we find that the horizontal deflec- tion is very small, being only about ^-V inch, and the force necessary to prevent this is about 2000 potmds. In case the truss is arranged on the supports so that the span remains constant, the supports must be designed to resist a horizontal force of 2000 pounds. The actual stresses in the truss members will be the algebraic sum of the stresses produced by the vertical loads and the hori- zontal thrust. An inspection of the computations for D shows that the pieces aL and a'L contribute over one half the total value of D. If the area of these pieces is increased to 64 square inches, the value of D is reduced about 25 per cent. It is possible to design the truss so that the change of span is very small by simply adjusting the sizes of the truss members, increasing considerably those members whose distortion contributes much to the value of D. The application of the above method to either wood or steel trusses of the scissors type enables the designer to avoid the quite common defect of leaning walls and sagging roofs. W^ APPENDIX. , 155 5, Tests of Joints in Wooden Trusses, — In 1897 a series of tests was made at the Massachusetts Institute of Tech- nology on full-sized joints. The results were published in the Technology Quarterly of September, 1897, and re- viewed by Mr. F. E. Kidder in the Engineering Record of November 17, 1900. The method of failure for three types of joints is shown in Fig. 5. 6. Examples of Details Employed in Practice. — The fol- lowing illustrations have been selected from recent issues of the Engineering News, the Engineering Record, and The Railroad Gazette. Fig. 6 . A rotindhouse roof -truss, showing the connection at the support with arrangement of brickwork, gutter, down-spouts, etc. The purlins are carried by metal stirrups hanging over the top chord of the truss. Fig. 6a. Details of a Howe truss, showing angle-blocks and top- and bottom-chord splices. Fig. 66. A common form of roof-truss, showing detail at support. The diagonals are let into the chords. The purlins stand vertical and rest on top of the truss top chord. Fig. 6c, A comparatively large roof -truss of the Pratt type of bracing, showing details of many joints. A large number of special castings appear in this truss. Fig. 6 J. .Howe truss details, showing connection to wooden column, knee-brace bolster, cast-iron angle-block, and brace-connection details. Fig. 6e. Scissors-trusses, showing five forms in use» and also three details which have been used by Mr. F. E. Kidder. 156 APPENDIX. Fig. 6. — ^Roundhouse Roc^, Urbana Shops, Peoria and Eastern R.R. Fig. 6/. A steel roof-truss, showing details. The pur- lins are supported by shelf-angles on the gusset-plates ex- tended. The principal members of the web system have both legs of the angles attached to the gusset-plates. ^ \ M^ K i s - issssis*?;! - ^ S * S4 _ . t'V---^ ^ .7'8-— — 4 e.-_=..T'fl'- +. jj'a-- 0^ pMlbi("'"" "" Oik SiiJln - .CHORD DETAILS, PUNING MiU. fiOOF TRUSS, ■' IiG. 60.— Canadian Pacific R.R., MontreaL Fig. 66.— Boston and Maine R.R., Concord, N. H. Outline of Main Truss of Forestry Building. Fig. 6g. A steel roof-truss with a heavy bottom chord. The exceptional feature in this truss is the use of flats for web tension members. APPENDIX, 159 li'^lte'f^ l*Bolt« Fig. td. — ^Howe Truss, Horticultural Building, Pan- American Exposition. 4-1)^ Bolta Fig. 6e. — Scissors-trusses and Details Used by Mr. F. E. Kidder. FlG. 6/,— Roof-truss' of Power-house, Boston and Maine R.R., Concord, N. H. FlQ. 6f .— Roof-tniss, P«oria and Eastern R.R.. Urbana. APPENDIX. l6l Fig. 6A. A light Steel roof-truss, showing arrangement Fig. 6A. — Power-house, New Orleans Naval Station. ■ SECTION THROUGH EAVES . Fig. 6i.— Pennsylvania Steel Company's New Bridge Plant. of masonry, gutters, down-spout, etc. In this roof the purUns rest on the top chord of the truss, and any tipping l6a APPENDIX. or sliding is prevented by angle-clips and f-inch rods, as shown. Fig. 6i, Detail of connection of a steel roof-truss to a steel column. The illustration also shows gutter, down- spout, cornice, etc. J Fig. 6A. — General Eleclric Machine-shop, Lynn, Mass. Figs. 6; and 6fe. Details similar to those shown in F^. 6t, but for lighter trusses. APPENDIX. 163 7. Abstracts from General Specifications for Steel Roofs and Buildings. ^y Charles Evan Fowler, M. Am. Soc. C. E. GENERAL DESCRIPTION. 1. The stnicttire shall be of the general out- Diagram, line &nd dimensions shown on the attached dia- gram, which gives the principal dimensions and all the principal data. (2, 72.) 2. The sizes and sections of all members, together with the strains which come upon them, shall be marked in their proper places upon a strain sheet , and submitted with proposal. (1,72.) 3. The height of the building shall mean the Qearances. distance from top of masonry to tmder side of bottom chord of truss. The width and length of building shall mean the extreme distance out to out of framing or sheeting. 4. The pitch of roof shall generally be one fourth. (6.) LOADS. The trusses shall be figured to carry the fol- lowing loads : 5. Snow Loads. SnowLoad. Pitch of Roof. Location. 1/2 1/3 1/4 i/S 1/6 Pounds per Horizontal Square Foot. Southern States and Pa- cific Slope. 7 10 10 12 15 20 20 25 22 27 35 37 Central States 30 35 45 50 Rocky Mountain States. . . New England States Northwestern States l64 APPENDIX, Wind Load. ^* '^^ wirid prcssuTc Oil trusscs in pounds pef square foot shall be taken from the following table : Pitch. Vertical. Horizontal. Nom 1/2=45000' 19 19 27 1/3 =33" 41' 17 12 22 i/4 = 26»34' IS 8 18 1/5 =21" 48' 13 6 IS 1/6 = 18° 26' II 4 13 (7.) 7. The sides and ends of buildings shall be figured for a uniformly distributed wind load of 20 pounds per square foot of exposed surface when 20 feet or less to the eaves, 30 pounds per square foot of exposed surface when 60 feet to the eaves^ and proportionately for intermediate heights. (6 . ) c^vo^ 8. The weight of covering may be taken as follows : Corrugated iron laid, black and painted, per square foot : No. 27 26 24 22 20 18 16 .90 i.oo 1.30 1.60 1.90 2.60 3.30 poimds For galvanized iron add 0.2 poimds per square foot to above figures. Slate shall be taken at a weight of 7 pounds per square foot for 3/16" slate 6"Xi2", and 8.25 pounds per square foot for 3/16" slate i2"X24", and proportionately for other sizes. Sheeting of dry pine-boards at 3 pounds per foot, board measure. Plastered ceiling hung below, at not less than 10. pounds per square foot. APPENDIX, 165 The exact weight of purlins shall be calcu- lated. 9. The weight of Fink roof -trusses up to 200 '^Jf^^*'' feet span may be calculated by the following for- mulae for preliminary value : w = .065 + .6, for heavy loads ; w = .045 + .4, for light loads. (40, 45.) s = span in feet ; t£;= weight per horizontal square foot in pounds. 10. Mill buildings, or any that are subject to ^g^®^ corrosive action of gases, shall have all the above loads increased 25 per cent. 1 1 . Buildings or parts of buildings, subject to strains from machinery or other loads not men- tioned, shall have the proper allowance made. 12. No roof shall, however, be calculated for Minim am Load. a less load than 30 potmds per horizontal square foot. UNIT STRAINS. Soft-medium Iron. Steel. 13. Shapes, net section. 15000 (5 y.) Tension only. Bars 14000 17000 Bottom flanges of rolled beams. ... 1 5000 Laterals of angles, net section 20000 (57.) Laterals of bar. . . . 18000 (41.) 14. Flat ends and fixed Compression 7 only. ends 12500 — 500—- 7 Z= length in feet center to center of connections; r =least radius of gyration in inches. (59.) i66 APPENDIX. Flanges. Combined. Shearing. Bearing. Bending. Laterals. Bolts. 1$. Top flanges of built girders shall have the same gross area as tension flanges. 1 6. Members subject to transverse loading in addition to direct strain, such as rafters and posts having knee-braces connected to them, shall be considered as fixed at the ends in riveted work, and shall be proportioned by the following formula, and the tmit strain in extreme fiber shall not exceed, for soft-medium steel, 15000. Mn P (52, 62.) 5 = strain per square inch in extreme fiber ; M = moment of transverse force in inch-pounds; n = distance center of gravity to top or bottom of final section in inches ; / = final moment of inertia ; P = direct load ; A = final area. Soft-medium Soft Steel. Steel. 17. Pins and rivets loooo Web-plates 18. On diameter of pins and rivet-holes .... 20000 19. Extreme fiber of pins. Extreme fiber of pur- lins (57.) 7000 20000 (57.) 25000 15000 (49.) 20. Lateral connections will have 25 per cent, greater unit strains than above. 21. Bolts may be used for field connections at two thirds of rivet values. (17, 18.) r APPENDIX. TIMBER PURLINS. 22. In ptirlins of yellow pine, Southern pine, or white oak, the extreme fiber strain shall not exceed 1200 potinds per square inch. .(50.) CORRUGATED-IRON COVERING. 26. Corrugated iron shall generally be of 2^- inch corrugations, and the gauge in U. S. standard shall be shown on strain sheet. 27. The span or distance center to center of roof-purlins shall not exceed that given in the following table : i$7 27 gauge 2' o" 26 gauge 2' 6" 24 gauge 3' o" 22 gauge 4' o" 20 gauge 4' 6'- 18 gauge 5' o" 16 gauge 5' 6" (48.) 28. All corrugated iron shall be laid with one corrugation side lap, and not less than 4 inches end lap, generally with 6 inches end lap. (32.) 29. AH valleys or junctions shall have flashing extending at lea^t 12 inches under the corrugated iron, or 12 inches above line where water will stand. 30. All ridges shall have roll cap securely fastened over the corrugated iron. 3 1 . Corrugated iron shall preferably be secured to the purlin by galvanized straps of not less than five eighths of an inch wide by No. 18 gauge.; these shall pass completely arotmd the purlin and have each end riveted to the sheet. There Timber. Covering'. Valleys. Rid^res. Fastenings. l68 APPENDIX. shall be at least two fastenings on each purlin for each sheet. 32. The side laps shall be riveted with six- potind rivets not more than six inches apart. (28.) psnish Angle. 33. At the gable ends the corrugated iron shall be securely fastened down on the roof, to a finish angle or channel, connected to the end of the roof purlins. DETAILS OF CONSTRUCTION. Tension Mem- 37- All tcnsion members shall preferably be bers. composed of angles or shapes with the object of stiffness. 38. All joints shall have full splices and not rely on gussets. (65.) 39. All main members shall preferably be made of two angles, back to back, two angles and one plate, or four angles laced. (67.) 40. Secondary members shall preferably be made of symmetrical sections. 41. Long laterals or sway rods may be made of bar, with sleeve-nut adjustment, to facilitate erection. 42. Members having such a length as to cause them to sag shall be held up by sag-ties of angles, properly spaced. Compression 43. RaftcTS shall preferably be made of two Members. angles, two angles and one plate, or of such form as to allow of easy connection for web mem- bers. (65.) 44. All other compression members, except APPENDIX. 169 substruts, shall be composed of sections sjmimet- ricallv disposed. (65.) 45. Substruts shall preferably be made of symmetrical sections. 46. The trusses shall be spaced, if possible, at PutMus. such distances apart as to allow of single pieces of shaped iron being used for purlins, trussed pur- lins being avoided, if possible. Purlins shall pref- erably be composed of single angles, with the long leg vertical and the back toward the peak of the roof. 47. Purlins shall be attached to the rafters or columns by clips, with at least two rivets in rafter and two holes for each end of each purlin. 48. Roof purlins shall be spaced at distances apart not to exceed the span given tinder the head of Corrugated Iron. (27.) 49. Purlins extending in one piece over two or more panels, laid to break joint and riveted at ends, may be figured as continuous. 50. Timber purlins, if used, shall be attached in the same manner as iron purlins. 51. Sway-bracing shall be introduced at such sway-brac- points as is necessary to insure ease of erec- tion and sufficient transverse and longitudinal strength. (41.) 52. All such strains shall preferably be car- ried to the fotmdation direct, but may be ac- counted for by bending in the columns. (62.) 53. Bed -plates shall never be less than one- Bed-piates. half inch in thickness, and shall be of sufficient I7P^ APPENDIX. thickness and size so that the pressure on masonry will not exceed 300 potinds per square inch. Trussies over 75 feet span on .walls or masonry shall have expansion rollers if neces- ■ sary. (54.) Anchor-bolts. 54. Each bcaring-platc shall b^ provided with twQ anchor-bolts of not less than three fourths of an inch in diameter, either built into the masonry or extending far enough into the masonry to make them effective . (53.) . Punching. 55. The diameter of the punch shall not exceed the diameter of the rivet, nor the dianieter of the die exceed" the diameter of the ptmch by more than one sixteenth of an inch. (56.) • ; Punching and 56. AH rivct-holes in steel may be punched, Reaming. and in case holes do not match m assembled members they shall be reamed out with power reamers. {71.) Effective 57. The effective diameter of the driven rivet Diameter of Rivets, shall be assumed the same as before driving, and, in making deductions for rivet-holes in tension members, the hole will be assumed one eighth of an inch larger than the imdriven rivet. (13, 17.) ' Pitch of 58. The pitch of rivets shall not exceed twenty times the thickness of the plate in the line of strain,; nor forty times the thickness at right angles to the line of strain. It shall never be less than three diameters of the rivet. At the ends of compression members it shall not exceed APPENDIX. 171 Length of Compression Members. four diameters of the rivet for a length equal to the width of the members. 59. No compression member shall have a length exceeding fifty times its least width, unless its unit strain is reduced accordingly. (14.) 60. Laced compression members shall be xie-piates. staved at the ends by batten-plates having a length not less than the depth of the member. 61. The sizes of lacing-bars shall not be less ^acm bars, than that given in the following table, when the distance between the gauge-lines is 6 8 10 12 16 20 24 or less than 8" if'Xi" 10" ii"xr 9 12" if'X"' 16 ff Jf X // // 2rx •"' 20 :^x '^T^ 24" 2\"y.\ above of angles. (62.) They shall generally be inclined at 45 degrees to the axis of the member, but shall not be spaced so as to reduce the strength of the mem- ber as a whole. 62. Where laced members are subjected to Bending, bending, the size of lacing-bars or -angles shall be calculated or a solid web-plate used. (13, 14, 61.) 63. All rods having screw ends shall be upset Upset Rods, to standard size, or have due allowance made. 64. No metal of less thickness than i inch shall Variation in weight. be used, except as fillers, and no angles of less 1J2 APPENDIX, than 2-inch leg. A variation of 3 per cent, shall be allowable in the weight or cross-section of material. WORKMANSHIP. PWgedSur. 65. All workmanship shall be first class in every particular. All abutting surfaces of com- pression members, except where the joints are fully spliced, must be planed to even bearing, so as to give close contact throughout. (38.) 66. All planed or turned sxirfaces left exposed must be protected by white lead and tallow. Riveu. 67. Rivet-holes for splices must be so accu- rately spaced that the holes will come exactly opposite when the members are brought into position for driving-rivets, or else reamed out. (38, 70, 71.) 68. Rivets must completely fill the holes and have full heads concentric with the rivet-holes. They shall have full contact with the surface, or be cotmtersunk when so required, and shall be machine driven when possible. Rivets must not be used in direct tension. 69. Built members when finished must be free from twists, open joints, or other defects. (65.) Drilling. 70. Drift-pins must only be used for bringing the pieces together, and they must not be driven so hard as to distort the metal. (71.) s^^^^joms. 7^- When holes need enlarging, it must be done by reaming and not by drifting. (70.) DrawiMsand 72. The dccision of the engineer or architect Specinca- tiooa. shall control as to the interpretation of the draw- yfPPENDIX. 173 ings and specifications during the progress of the work. But this shall not deprive the contractor of right of redress aSter work is completed, if the decision shall be proven wrong, (i.) STEEL COLUMN UNIT STRAINS. QD 12500 -500^- l+r. DD l+r. uD l+r. DD l+r. DD 3.0 1 1000 7.6 8700 12.2 6400 16.8 4100 .2 10900 .8 8600 .4 6300 17.0 4000 .4 10800 8.0 8500 .6 6200 .2 3900 .6 10700 .2 8400 .8 6100 .4 3800 .8 10600 4 8300 13.0 6000 .6 3700 40 10500 .6 8200 .2 5900 .8 3600 .2 10400 .8 8100 -4 5800 18.0 3500 .4 10300 9.0 8000 .6 5700 .2 3400 .6 10200 .2 7900 .8 5600 .4 3300 .8 lOIO) 4 7800 14.0 5500 .6 3200 5.0 1 0000 .6 7700 .2 5400 .8 3100 .2 9900 .8 7600 .4 5300 19.0 3000 .4 9800 10. 7500 .6 5200 .2 2900 ,6 9700 .2 7400 .8 5100 .4 2800 .8 9600 .4 7300 ^50 5000 .6 2700 6.0 9500 .6 7200 .2 490D .8 2600 .2 9400 .8 7100 .4 4800 20.0 2500 •4 9300 II. 7000 .6 4700 .2 2400 .6 9200 .2 6900 .8 4600 .4 2300 .8 9100 •4 6800 16.0 4500 .6 2200 7.0 9000 .6 6700 .2 4400 .8 2100 .2 8900 .8 6€o3 .4 4300 •4 8800 12.0 6500 .6 4200 SHEARING AND BEARING VALUE OF RIVETS. Diameter of Rivet in Inches. Area of Rivet. Single Shear at 1 0000 Lbs. per Sq. In. Frac- tion. Deci- mal. 1" A" I" •5 .5625 .625 .6875 •75 .8125 875 9375 .1963 2485 .3068 .3712 .4418 •5185 .6013 .6903 i960 2480 3070 3710 4420 5180 6010 6900 r Bearing Value of DifFerent Thicknesses of Plate at aoooo Lbs. per Sq. In. ( *» Diam. of Rivet X Thickness oi Plate X 20000 Lbs.). I'f 2500 2810 i 344 375 4070 4380 4690 -// 313 3520 3910 4290 4600 iff SoSo 5470 585c 375» 4210 4690 5160 5630 6090 6570 7030 A" 4920 5470 6010 6560 7110 7660 8200 Iff 6880 7500 8120 8750 9370 // 8440 9150 9840 10550 \.ff 10160 10940 1 1 720 H ft 12890 INDEX. I PAGB Agriculture, Dept. of 22 American Ry. Eng. and M. Assn 25 Angles, connections 104 end cuts 107 Angle-blocks 83, 84, 87 Bearing, across fibers of steel 33, 166, 173 across wood fibers 33, 34, 139 on end fibers of wood 29, 32, 139 on inclined wood surfaces 30 on round metal pins in wood 31 Bolsters, see Corbel. Bolts, anchor 78, 103, 170 bearing on wood 31 bearing values for steel 33, 166, 173 shearing values for 43, 44, 141 size of 49, 141 spacing of 145 Center of gravity 8 Columns, metal 27 steel 28, 173 wood 24, 25, 26 Compression, see Bearing and column. Corbel, use of 62, 63, 64, 65, 78 Covermg for roofs 46, 111, 112, 113, 114 ! Details, examples, from practice 155 Dimension, least, defined for struts 22 Drawings 103 Equilibrium, conditions of 1 forces to produce 2 internal , 18 175 176 INDEX. PAGB Equilibrium of forces in plane 1 polygon 7, 9, 12, 14, 15 Expansion of trusses 103 Fiber-etiress, see Stress. Forces, direction of 20 inside treated as outside 20 moments of parallel 9 more than two imknown 20 not in equilibrium 2 parallel 7, 9 Forestry, division of 22 Fowler, C. E 27, 163 Frame-lines 103 Gusset plates 96, 107 Gyration, least radius of 27, 134 Hooks, metal 67, 73, 84, 147 Iron, wrought, in tension 45 Johnson, A. L 22 Joints, designs in wood 61-95 designs in steel 101-107 tests of 155 Keys 143 Kidder, F. E 155, 159 Knee-brace 148 • Loads, apex 48, 54 Local conditions, effect upon design 50 Metal, columns of 27 Moisture, classification for wood 23 Moments, parallel forces 9 pins and bolts 44 vertical loads 14 Multiplication, graphical 12 Notches 143 INDEX. 177 PAOB Parallel, forces, see Forces. Pins, bearing against wood 32 bending strength of 44 splitting effect in wood 32 Pipe, in angle blocks 89 Pitch, defined for roof trusses 47 used in practice 48, 163 Polygon, equilibrium 3 force 1 through three points 12 Pm-lins, angle 104, 169 attachment of 91 wood 63, 167 Badius of gyration ....'. 27, 134 Kafters 62 Keactions, application of equilibrium polygon in finding 6 due to inclined loads 16 inclined 7-16 vertical 16 Hesultant, defined 3 Rivets, bearing values for 44, 173 diameter of 44 field 101, 107, 166 shearing values for 44, 173 tie 98 weight of 118 Rods, round 120 upset 61, 120, 171 Boilers, expansion 103 Boof , covering 46 pitch of 47 weight of Ill, 164 Boof-truss, design in steel 96-107 design in wood 61-96 function of 46 loads on ♦ , 48 span of 46 transmission of loads to 48 wind loads for 47 weight of 56, 166 Safety, factor of 26 Scissors truss 161 178 INDEX. PAOS Shear, bolts and pins 44, 173 longitudinal, for steel 35 longitudinal, for wood 34, 139 transverse ior steel 44, 173 transverse for wood 45, 139 Shapes, steel 49, 122-135 Specifications, for steel trusses 163 Sleeve-nuts 121 Snow 51, 163 SpUces in wood 85-90 in steel 102 Square, defined 40 Strength of materials 22-45 Stresses, character of 18, 19 fiber 39-44 String, in equilibrium polygon, defined 5 Strut, see Column. Supports, at end of trusses 77, 102 Timber, sizes of 48, 137 Transverse strength, see Stresses. Tumbuckles 49 Upset ends on rods 49, 61, 120 Vose, R. L. . : 90 Washers, cast iron 140 Wind, assumed action of 15 loads 47, 164 Wrought iron 43, 45 TABLES Areas to be deducted for rivet-holes 117 Bearing, across fibers of wood 34, 139 end, for wood 30, 139 end, bolts in wood 31 for pins and rivets 33, 166, 173 on inclined surfaces of wood 31 Columns of wood 24, 25 of steel 28, 173 INDEX. 179 PAGK Dimeneions of bolt-heads 119 right and left nuts 121 timber 137 upset screw ends 120 washers 140 Least radii of gyration 134 Lumber, commercial sizes 137 Pitch of roofs 48 Properties of steel angles, equal legs 126 of steel angles, unequal legs 128 of steel channels 124 of steel I beams 122 of steel T bars 135 • Bight and left nuts. 121 Safety factors 25 Shear, longitudinal for wood 35, 139 transverse for pins and rivets 44 transverse for wood 45, 139 Sizes of rivets in beams, channels, etc 116 Spacing of rivets 116 Strength of bolts 141 of timber 139 Transverse strength of timber 139 Upset screw ends 120 Washers, cast iron 140 Weights of bolt-heads 119 of brick and stone 110 of corrugated iron Ill of glass 112 of masonry 109 of metals 110 miscellaneous 114 of rivets 118 of shingles 112 of slate 113 of terra cotta 114 of tiles 114 of tin 114 of washers 140 of wood 109 Short- TITLE Catalogue OF THE PUBLICATIONS OF JOHN WILEY & SONS New York London: CHAPMAN & HALL, Limited ARRANGED UNDER SUBJECTS Descriptive circulars sent on application. 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