THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING TORSION OF STRUCTURAL -SHAPES I. A. ElDarwish March, 196!4 IP-664

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TABLE OF CONTENTS Page LIST OF TABLES.........................................o......... i LIST OF FIGURES.................................................. iv INTRODUCTION.a... o.......... o................................ 1 THE RECTANGLE AND TRAPEZOIDAL SECTIONS........................... 6 SOLUTION OF THE TORSION PROBLEM BY USE OF DIFFERENCE EQUATIONS.o. 9 THE TORSION CONSTANT OF STRUCTURAL SECTIONS...................... 15 MAXIMUM SHEAR STRESS DUE TO TORSION................e..........O... 27 TORSION CONSTANTS FOR STRUCTURAL SHAPESo......................... 33 ACCURACY AND COMPARISON WITH PRIOR TEST RESULTS..................3 SUMMARY AND ACKNOWLEDGMENTS....... *......o...... O......... 000 40 REFERENCES.......o........................................ oo. 41 NOTATION............................................ 42 ii

LIST OF TABLES Table Page l....................... 7 2 Standard Slopes.................................. 8 3 Equations for D in Figure 7........................ 17 4 Comparison between Computed Torsion Constant J and Previous Test Results............................. 37

LIST OF FIGURES Figure Page ~'............................................ 8 2 a........................................ 12 34. ~.~e l.i ~ -.***e~~~ e ~ e. 12 5: Typical Example of Computer Output Giving Stress Function Ordinates Rounded Off as Shown................ 13 6 Basis for Tee or Angle Juncture Corrections for Torsion Constant of WF, I, Tee, Channel, Zee, and Angle Shapes..................................... 16 7 Values of (ac) for Parallel Flange Tee Shapes.,........ 19 Values of (a2) for I Section Tee Shapes Flange Slope 1/6.................................... 20 9 Values of (a3) for Parallel Flange Angle Section...... 21 10 Values of (a4) for Angle Section of Standard Channels Flange Slope 1/6.......................... 22 11 *s.......................................... 24 12 Angle Correction Coefficient (a) for Angle Intersection with Equal Thickness Legs....................... 26 13 Formulas for Stress Increase in Fillets Due to Torsion. 28 14 Coefficient of Stress Increase for Tee or WF Section with Parallel Sided Flange............................. 31 15 Coefficient for Maximum Stress Determination in Outside Face of Tee or I Section with 1/6 Flange Slope.. 32 16 Summary of Formulas for Torsion Constants for Structural Shapes................................. 34 iv

INTRODUCTION The torsion constant "J" and the shear stresses due to torsion, as covered herein for certain structural sections, relate to what is often referred to as "St. Venant Torsion,"(l) which involves certain conditions, as follows. The structural shape is uniform in cross-section and the torsional moment is applied by means of opposed couples at each end in such a way that there is no restraint against longitudinal displacement at any point on the end sections, The angle of twist per unit length "9" is thus uniform. If z is taken as the coordinate axis parallel with the bar, with x and y coordinates in the plane of any section, during twist there will be generated "warping" displacements "w" in the z direction, which are a function of x and y only. If one or more of the following violations of the foregoing conditions exist, the torsion problem is a superposition of Sto Venant Torsion and "warping" or "non-uniformrt torsion. (1) Cross-section is variable, (2) Torsional moment varies. (3) Longitudinal restraints are imposed at any location, In structural design applications, apart from torque-driven shafts, torsion is almost always non-uniform and is usually further complicated by beam bending due to transverse loads. Non-uniform torsion combined with bending is also a problem in lateral-torsional buckling problems. Non-uniform aspects of the torsion problem are covered in many books and references and this report is concerned only with providing basic data for St. Venant torsion as applied to structural shapes.

The torsion constant for St. Venant torsion has the same relation to unit twist of a shaft as the moment of inertia has to the unit curvature of a beam bent under uniform moment. Thus, for uniform torsion, Mz = J G o (1) The derivation of the general St. Venant torsion theory is available in standard texts on the mathematical theory of elasticity and will not be presented in this report, The general solution is obtained from the following equation, +2, 2- -a2G (2) Ir is the "torsion stress-function" and the torsion constant is given by -2 ff dx dy (3) at any location x, y, in the plane of the cross-section, the shearstress components are x y' Tzy = x and the magnitude of the resultant stress is therefore r = JT + T2 _= ()'2 + ( )2 zx zy 6x 6y Prandtl(2) in 1903 noted that if a thin membrane (devoid of shear strength, such as a soap film) is stretched across a hole having the shape of a cross section, with a slight differential in pressure on the two sides of the membrane, the equation of the surface displacement

-3is identical in form with Equation (2), as follows: "+a_ (6) ax2 ay2 T where = displacement of film in z direction. p = pressure differential causing film to deflect. T = tension force in the film per unit width (constant)O The analogy between Equations (2) and (6) illustrates the fact that the magnitude of the torsion stress function 4 may be thought of as a displacement, tracing out a continuous "'stress-function" surface, and that the integral in Equation (3) is the volume under the surfaceo The slope of the stress-function surface represents the magniftude of the shear stress component in the x-y plane in a direction 90~ to the direction for which the slope is evaluatedQ Contour lines, therefore, give the direction of maximum or resultant shear stress. The boundary of an open cross-section is equivalent to a contour line and on the boundary the value of r may be taken as zero. (It should be noted that the magnitude of maximum shear stress in general will vary along a contour line and will be inversely proportional to the spacing of the contour lineso) In the case of circular sections, plane right cross-sections prior to twist remain plane after twist, and the shear stress is proportional to the distance from the central axis of the shaft. In this case the torsion constant "JT" (Equations 1 and 3) is equal to the polar moment of inertia. The maximum shear stress at the boundary of the circular section is given by the simple expression Tm =~T- (7)

The polar moment of inertia would be the torsion constant for non-circular sections, also, if the resultant shear stress were everywhere proportional in magnitude to the distance from the center of twist and directed at 90O to radial lines, The torsion constant of a noncircular section is always less than the polar moment of inertia - increasingly so as a section becomes less and less compact. The present investigation is in part a sequel to earlier work whereby the membrane analogy was used to-.evaluate the torsion constant for the wide flange and I-beam section. (34) On the basis of this earlier work, the torsion constants of all rolled wide flange and I-beam shapes had been evaluated.(5) The advent of the electronic digital computer has provided a means whereby an accurate re-evaluation of the torsion constant could be made, avoiding the experimental errors involved in soap film tests. Thus, for the wide flange and I-beam sections the present study provides results that approximately confirm but provide greater precision than was possible in the earlier experimental work, For the channel and angle sections this report provides information that has not been fully available heretofore. The procedure for building up the formulas for the torsion constant of structural shapes follows the pattern suggested by Trayer and March(6) in their investigation of aircraft strut sections wherebythe added torsional rigidity due to the juncture of two rectangular component parts was denoted. as "a Dt4" D is the diameter of the largest circle that can be inscribed at the juncture of the two component parts

-5and a is either determined experimentally by means of soap film tests or numerically as in the present studies by means of the difference equations.

THE RECTANGLE AND TRAPEZOIDAL SECTIONS The rigorous solution of Equation (2) for the boundary conditions of rolled shapes such as the channel, WF beam, etco,- is not practicable. The relatively few "exact" solutions that are available include some shapes of only academic interest, such as the ellipse and equilateral triangle. The solution for the rectangular shape, however, as developed by St. Venant(l) is of considerable engineering importance, not only of itself, but by virtue of the fact that the usual structural shape is made up of component parts that are either rectangular or trapezoidal, The width-thickness ratio of any one of these parts is usually four or more, for which proportions the torsion constant can be accurately expressed by a simple formula. The "torsion constant" of a structural shape can then be approximated by summing the values of J for the component rectangular parts and adding the juncture effect connection as herein evaluatedo The evaluation of St. Venant's solution for the rectangular cross-section may be expressed as follows: bt3 4 (8) in which t = the thickness and b = the breadth of a rectangular section, and Tm MYt (9) b where V and y are factors depending upon the ratio ~ and are given by the following tableo -6

-7TABLE 1 Ratio b 2V y t 1.00 0.1928 0o 6753 lo.O O10o 1973 0 o7198 1.20 0 2006 0o 7578 lo 30 0.2031 0.7935 1,40 0 2050 o 8222 1.o 50 0.2064 0 o.8476 1.60 0,2074 0.8695 1.80 0.2086 0o9044 2.00 0.2093 0 o.9300 2.50 0,2099 0 o 9681 3.00 0.2101 o0.9855 4.00 0. o 2101 0.9970 ~~ O 0.2101 1.0000 The formula for the trapezoidal section, (Figure 1) as developed in Reference 4, can be written: 2 4 4 J =- (b/12)(t+t)(tl+t2)(t+t2) - - Vst1 (10) VL and VS are the end constants V, for the thick end and the thin end, respectively, of the trapezoid, VL = 0.10504 - oooo10000 S + 0.08480 s2 - 0.06746 S3 + 005153 s4 (11) and, VS = 0.10504 + o.ooo10000 s + 0.08480 s2 + o.o6746 S3 + 0.05153 S4 (12) in which S = the total slope of the section or tangent of the slope angle. For certain particular values VL and VS are calculated in Table 2.

-8TABLE 2 STANDARD SLOPES VL VS 1/6 0.090o45 0.12441 1/20 0.10026 0.11026 1/50 0.10307 0.10707 /C o.010504 0.1050o4 til~ _ _ _ _ _ _ _ _ _ _ t2 L _1 FiF re 1 Figure 1

SOLUTION OF THE TORSION PROBLEM BY USE OF DIFFERENCE EQUATIONS To provide solutions for complex boundary conditions not readily evaluated by analytical procedures, Equation (2) may be replaced by a "difference equation" and the solution obtained for any cross-section to any desired degree of approximation dictated by need and available computation time. For manual or desk calculator use, "relaxation" procedures of solution were initiated by Christopherson and Southwell.(7) Shaw(8) has applied these procedures with special emphasis to the torsion problem and has extended the application to include non-linear stress-strain relationships and closed sections. The foregoing references should be consulted for a more complete discussion of difference equations for the torsion problem and for a review of relaxation methods. In one form of the difference equation procedure the value of the torsion stress-function, 4, is determined approximately at the points of intersection (nodal points) of a network of squares formed by equally spaced lines (Figure 2), parallel to the x and y axis in the plane of the cross-section. For any point "O" (Figure 2), Equation (2) may be written in terms of neighboring nodal points as follows: *1 +2 + 3 + Ji4 - 4 *O) 2 HX2 G G (13) in which H is the mesh size~. may be replaced by ~, where ~ is a dimensionless quantity, taken so that 22Gs (14) -9

-10Ro = any convenient constant introduced for the purpose of making the 0's large numbers for ease of numerical manipulation. Equation (13) with this substitution becomes (01 + 02 + 03 + 0 4- 4 00) R, (15) Points on the boundary will not usually coincide with the points of the square net. For this case, a special finite difference equation may be derived (1) Referring to Figure 3a, the nondimensional finite difference equation at point "0" for Equation (2) will be: 01 + 2 + 3 + 4 0 (1 + 1 +1 + 1 R (M+N+R+K) (16) M N R K M N R K 4 when M = K = 1, and 02 = 03 = 0 (Figure 3b), Equation (16) reduces to 01 04 0 (2 + +) R (+2 1 (17) R N 14 when M = K - N = R = 1, Equation (16) reduces to Equation (15) as derived for a square net. Special'block" relaxation techniques developed to aid convergence in manual solution are not particularly helpful in computer analysis, The computer is programmed to make a simple repetitive scanning of the node points with. at each point "0" calculated by Equation (15), rewritten as follows: It is helpful if a guess 3 04 + Rvalues o f n the) It is helpful if a guess can be made for initial values of 0. In the present investigation there was a sequential variation of dimensional parameters in the analysis of structural shape intersection regions. In the first analysis of a given set of parameters, initial 0fs were set

-11at zero, but for succeeding analyses, for which minor variations were made in a particular dimensional parameter, the initially assumed values were those of the previous computer run. Thus convergence subsequent to the first run of a sequence was rapidQ In order to calculate the volume under the stress function surface, as approximated by nodal values of 9, the computer met conditions illustrated in Figure 40 Whenever the volume could conveniently be calculated under a 2H by 2H square, as in regions A of Figure 4a, the double application of Simpson's Rule gave a good approximation. Referring to Figure 4b, ff dxdy - (2H (05 + 06 + 07 + 08).36 + 4(01 + 02s + 03 + 04) + 6 o00 (19) When a single whole square was involved, as in regions B of Figure 4a, the volume was computed by the following equation, as used in the same situation by Shaw,(8) which incorporates a correction for the usual convexity of the stress-function surface. (The numbers are for one quadrant of the 2H by 2H surface in Figure 4b). Jff Odxdy _ (05 X 1 - Ro) H2 (20) B For the various triangular areas along the curved boundary of Figure 4, as represented by Figure 4c, the volume was calculated by NBIH2 + V- = —(i0 + 02 + 3(21) The results of a typical computer calculation are shown in Figure 5 for the intersection region of a web and sloping flange such

-12Iy () 4H 4KH 3 RH H'H ~MH NH - 3 H H H H M H Figure 2 Figure 3 (a) H Figure 3 (b) Figure 4 (a) 2H 5 I 6 - 2 2H I X 4 XI Figure 4 (b) Figure 4 (c)

w=4H ASSUMED PARABOLIC VARIATION T| 75 1 00| 75 l OF THE STRESS FUNCTION 75 10oo 75 75 100 75 75 100oo 75 R= 50 75 100 75 D = 9.186 H 75101 76 76 101 76 77 103 77 r= 8H 79 106 80 82113 88 5 88 124 105 26 96140130 62 106 161 163 109 118 185 2021161 7 130 210 2 33 229 168 62.'~~~~~~~~~~~ ~ ~~~ASSUMED PARABOLIC VARIATION _ 141;34 281 287'2-5 82 -- _85 5. OF THE STRESS FUNCTION 145 252 312 334__802 217 7 6 6,i54 263 332 366 370 350 312 263 213 168 13; 105 79 53 27 154 264 33 7 378 393 387 366 336 302 t69 241 216 192 170 148 126 106 86 65 44 22 " l 47252323 366 387 389 379 361 339316294 7 57 23721902 8 168152135 118101 85 69 52 36 18 1\32 226 288 327 348 355 351 341 32j 312 296 281 267 253 240 227 214 201 189 176 164 151 139 126114 101 88 75 107 t79 227 257 274 o 21 80 27 67 57 2 247 238 219 210 201 193 4 176167159 1511 14 134 126 117 109 100 66o 107133 150 159 163 164462 1 L9 1)43- 1 3 349 130 126 121 117113 109o 105100 96 9 2788 81 79 75 b= 28 H Figure 5. Typical Example of Computer Output Giving Stress Function Ordinates Rounded Off as Shown.

-14as occurs at a corner of a standard channel. At each extremity, the boundary value of ~ has been assumed parabolic, as for the interior of a narrow rectangular or trapezoidal section. The figure written in each square represents the computer calculation for ~ at the upper right hand corner rounded off without digits to the right of the decig mal point. The values actually used by the computer to calculate J were accurate to several additional significant figures.

THE TORSION CONSTANT OF STRUCTURAL SECTIONS The essential or missing element required for the accurate calculation of the torsion constant of a structural shape is illustrated by Figure 6 for the four cases herein considered. On the left side, labeled (a), are the arbitrary assignments of rectangular or trapezoidal areas and end conditions which account for the bulk of the torsion constant contributed by these regions. When a rectangular or trapezoidal element is bounded at either end by a dashed line (Figure 6), the variation in stress-function along the dashed line has been assumed parabolic and equivalent to the variation across the thickness in the interior of a rectangle of indefinite extent, rather than zero as it would be at the end of a rectangular cross section. The length of the extremities were several times the thickness of the part beyond the fillet tangent point, as illustrated in Figure 5, thus assuring complete evaluation of the juncture effect to any reasonable degree of needed accuracy. At the juncture, various choices could be made as to end conditions or arrangement of trapezoidal and rectangular components - it is important only that the same conditions be reflected in the way the torsion constant is synthesized in the formulas to be presented later for the various structural shapes. The total correction represents the difference between the computer calculated constant for the tee or angle in Figure 6b and the summation of constants for the rectangles or trapezoids in Figure 6a by Equations (8) or (10), with appropriate deletion of the negative -15

-16COOBRECOION x, l x (a) (b) K X2tWtX2-2 1 24 (a) (b) F —-- X3-~J ~ -XF 73 3 W H RHX Case 4. Angle 2. alSection of xi (a) (b) Constant of WF, I, Tee, Channel, Zee, and Angle Shapes.

-17terms for ends of cross-sections which have dashed lines. The correction coefficients ca are made dimensionless by the introduction of D4 where D is the diameter of the largest circle that can be inscribed at the juncture point. Thus, (Computed J of (J of arbitrarily chose 1tee or angle) segments shown in (22).~~D4 t Figure6) J Corresponding to the four cases of Figure 6, formulas for'"D" are given in Table 3. TABLE 3 EQUATIONS FOR D IN FIGURE 7 (S is the flange slope, i,e., tangent of the slope angle) Case 1 (t+r)2 + w(r + Parallel Flange (t+r) + w(r +23) Tee 2 r + t Segment Case 2 (F+t) + wr + w) I-Section (F+t3 + w(r + (24) Tee - =F + r + t3 Segment where F =rS( L +1 -l -) 2r Case 3 Angle D3 2 {(3r + w + t) Segment - /2(2r + w)(2r + t)} (25) Case 4 ofAngle Segmeta ndard D = 2 {(3r + w + H) of Standard 4 Channel - 2(2r + w)(2r + H)} (26) where H =t2 - r (S + 1 - 1+ S2)

-18Computer analyses were made for approximately thirty different proportions for each of the four cases shown in Figure 6. For each case there was a systematic variation between zero and one of the Villet radius)/(flange thickness) ratio (r/t) or (r/t2) and the (web,)/ (flange) thickness ratio, (w/t) or (w/t2). The resulting values of a for the four cases are plotted in Figures 7, 8, 9, and 10, respectively, by solid lines. For structural sections having flanges with slopes other than 1 in 6, a good approximation for a may be made by interpolating linearly between S = 0 and S = 0.1667. Also shown on Figures 7, 8, 9, and 10 are dashed lines plotted from the following empirical formulas and applicable in the range for which they are shown as a reasonable approximation for ao 1 = -.0420 +.2204 W +.1355 r -.0865 wr -.0725(H) (27) t t t2 t w r wr ~2 = -.0836 +.2536 t +.1268 r o6-.080 -- 858(t) (28) a2t2 +0t2 8 (28) 2 =O90 + t2621 +.1231 -.0752 - o945(w) (29) O3 =..0908+ 262lW. t t2 t a -,1325 +.3015 +.1400 r - 1070 - "4 ~t t -.107070956(W_)2 (30) In each of the foregoing cases (Equations 27 - 30) the empirical formula gives a good approximation for o in the range 0,2 < (r/t) < 1.0 0.5 < (w/t) < 1.0 Generally, the corrective term a D4 is positive, but for Case 2 (the sloping flange tee intersection) it may become negative when w/t2 is less than 0'4 for varying amounts of r/t2 ranging from

W I I I I I I I IVALUES OF r/t 0.16 rllr 1 ~~~~~~~~~~~~~~0.8 0.14 0.6 — Nw 0.12 0.14 0.10 0.2 0.08 U-O - 0.06 - U) Q> 0.04' --- DOTTED LINES CORRESPOND TO 0.0 2 _ EQUATION 27 0 0. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9. VALUES OF (w /t ) Figure 7.Values of (al for Parallel Flange Tee Shapes.

0.14 _ VALUES OF r 0.14 t 0.12 0.08 ~ ~ ~~0.8 0.10 0.(/ 0.4 -l0.20. 0.04 0.1 06 0 Figr 8. Vaue ofJ/ (i 2 fo I Secio LTe Shape Flng Slp 1/6.OREPND o' ~ -0.2~.' ~~; U)0.04 I 00..j~ 1 0.024 CORRESPONDTO -0.42~ ~' VALUES OF (w/tO Figure 8. Values of (U2) for I Section Tee Shapes Flange Slope 1/6.

0.16 0.14 VALUES OF r/t~ 0.12 — W- I 0.8. 0.10 ", I I r i I " —-~~~~~~~~~~7t- I:r I I I i I r I o~~~~~~~~ —- - U- 0.08 WII 0. 0 t,,-~~~'- ~. - 0 0.0..7 > ~ ~ ~ ~ ~ 0.04 DOTTED LINES CORRRES PON D TO 0.0 2 EQUATION 29! 03 0 0.1 0.2 0.3 0.4. 0,5 06 0.7 B 0.9. VALUES OF (w/t) Figure 9. Values of (a3) for Parallel Flange Angle Section.

0.16 0.14 0.2 r L 6 V, 0.10; — IL -.. 0.10 0~~~~~~~~~~~~~~~~~~~~~~~~06 w a~~~~~~~~~~~~~~~~~~~~~~~~o 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 VALUES OF (w/to) Figure 10. Values of 00) for Angle Section of Standard Channels Flange Slope 1/6.

-23 - O up to about 0.4. This can be explained by means of Figure 11, which is Case 2 for both w/t2 and r/t2 equal to zero. In this case J of the trapezoidal segments of Case 2(a) (Figure 6) would be calculated by Equation (10) without any end loss (VS or VL) terms. Obviously, the stress-fvinction surface must have a zero slope in the long direction at the central discontinuity, as shown for actual computed values in Figure 11. The assumption of parabolic variation across the thickness at the ~ (adjoining large ends of the two trapezoidal segments) gives a greater stress function value and an incorrect discontinuity of surface. In Figure 11 the maximum ordinates of a parabolic variation of the stress-function, on which Equation (10) is based, is plotted (without end loss) along the longitudinal sloping centerline of adjoining trapezoidal sections of dimensions t2 = 8 H, S = 1/6, and b _ 24 H. In the same figure the computed centerline stress function (0) is also plotted. The value of J obtained from the computer is 1871.6 H4 as compared with 1920 H4 by Equation (10) (without end loss). Twice the area of the hatched area in Figure 11 times two thirds of the width of the trapezoid at its center of gravity is found to be 45,.0 H4, which very nearly accounts for the foregoing difference. For the angle section with legs of equal thickness, the value of a can be interpolated at the extreme right boundary of Figure 9. Figure 12 gives a more accurate evaluation of a for angles with legs of equal thickness, particularly when the fillet radius is large. In this particular case, the values of a were computed for fillet radii up to six times the flange thickness. This is far greater than the range of rolled structural angles but was computed to afford a comparison

-24400 ~ COMPUTED ~ STRESS FUNCTION (J=3743.26 H4) STRESS FUNCTION ORDINATES CORRESPONDING TO EQ. 10- NO END LOSS (J= 3840 H4) 300 200 Ro= 50 100 24 H Fibre 11.

-25with the work of other investigators in these ranges. Figure 12 also plots the empirical formulas suggested by Trayer and March,(6) which were based on their tests. In addition, the results of both experimental tests and numerical analyses by the relaxation technique, as carried out by Cullimore and Pugsley,(9) are plotted together with the two straight line formulas that they propose. The present computer analyses plot midway within the scatterband of results of Cullimore and Pugsley for small fillet radii and agree very well with their results as the radius becomes large, Also shown on Figure 12 is the plot of a proposed empirical formula that agrees well with the computer analyses made for this report. a~r =87~(2 = 0.0728 + 0.o571 () o.oo0049 (~) (31) t

-260.30 a =0.0728a0.0571(r/t)- 0.00 49 (r/t )2 ~0.~25 HI~TRAYER B4MAR"PC"H;'EQUATION (31) 0.25... 07,, 0.074 6(,/t) 0.20. l l X'/' 7:-EXPERIMENTAL:_CULLIMORE ~! I I.~ "' /.1 RELAXATION J aPUGSLE 3 0.15 a Oi 15 0.017 (r/t) / a o +0.42 (r/t) g~~~t~~'..' CULLIMORE E PUGSLEY 0.05 2 3 4 6 7 0 I 2 S3 4 5 6 7 RATIO OF FILLET RADIUS/THICKNESS ( r/t) Figure 12. Angle Correction Coefficient (ca) for Angle Intersection with Equal Thickness Legs.

MAXIMUM SHEAR STRESS DUE TO TORSION A number of approximate formulas have been developed to give an indication of the stress concentration factor in a fillet for a structural member in uniform torsion. For the rectangular cross-section of large b/t ratio, the maximum shear stress at the center of the wide face is Mt T= _ (32) When the breadth-thickness ratio of a rectangle is less than 4 the maximum torsional stress is less than as given by Equation (32), dropping down to 0.6753 times this value for the square section. St. Venant(l) gives a complete tabulation. (See Table 1.) For an angle section with equal legs, it is obvious because of symmetry about a line from the corner at 45 degrees to the angle face that, for angles having large width-thickness ratios, the maximum stress due to torsion is at the intersection of the 45 degrees, line from the corner and the inside angle fillet. As the fillet radius becomes large, the stress concentration effect must decrease but the actual stress due to torsion increases because of the greater thickness of material in this juncture region. If the total stress increase is given as a percentage increase above that in the body of the angle away from the fillet, as given by Equation (32), results of various investigators are plotted in Figure 13. The curves in Figure 13 that rise at the right side of the figure reflect the increasing effect of thickness that overbalances the -27

-28240.. (I) I w -J U. 200,. z U) FOPPL 160 C:1 TIMOSHENKO W TREFFTZ La. WESTERGAARD AND MINDLIN 0 w 0 ~__.SOAP FILM TESTS LL 120 W 120 ~~BY CUSHMAN ~u ~~, SOAP FILM TESTS W BY TAYLOR ",,~ ~ COMPUTER LCI 40 -- 89.. Figure 13. Formulas for Stress Increase jin Filets Due to Torsion 0 0.4 0.8 1.2 1.6 2.0.ran RATIOT Figure lj5. Formulas for' Stress Increase in Fillets Due to Torsion.

diminishing effect of stress concentration as the fillet radius increases. This is true, of course, for the curve labeled "computer solution" which is based on the results of the present investigation. These results agree very well with those of Huth(lO) who wrote a rather complete discussion of the earlier results shown on Figure 13. Huth suggests that "the stress concentration in angle sections with generous fillets may be lowered considerably by rounding off the outside corner in such a way as to keep the thickness of the section everywhere approximately constant." Such a suggestion overlooks the fact that more is gained due to increased torsion constant as this juncture increases in thickness than is lost by the so-called "stress concentration factor for generous fillets" which is not a stress concentration but merely an indication of the effect of increased thickness on shear stress. Thus, for very "generous fillets," the actual stress for a given torsional moment will always continue to decrease as the fillet radius increases, all other dimensions remaining the same. It may also be noted that the curve resulting from Timoshenko's approximation solution(ll) does diminish as the fillet radius increases and thus is probably a very good approximation of the actual stress increase since it agrees very well in the lower ranges with that obtained in the present investigation as well as the similar results obtained by Huth. It is believed that the results of the present investigation may be considered to be quite accurate since the mesh size used was relatively much smaller than would be feasible for desk calculator operation using relaxat ion techniques. For the T juncture section there is a maximum stress on the outer face of the flange at the intersection with the midplane of the web.

-30This is an important stress location since at this point shear stress is combined with maximum shear stress due to flange bending that occurs in problems of nonuniform torsion. Here, also, the stress reflects only the increased thickness of the juncture since there can be no stress concentration effect on the flat surface. Figures 14 and 15 give values of b to be used in the following Equation (33) as a measure of maximum shear stress as a function both of fillet radius and webbed flange thickness. T =-x - (33) max J For the I beam with flange slope of 1/6, "t2" (see Figure 6) is to be used in Equation (33) in the place of "t". Stresses will be greater than as given by Equation (33) in the fillets on the inner side of the flange but the location can no longer be predetermined as in the case of the angle with equal thickness legs. A similar observation could be made regarding the stresses on either face of the angle channel type segment or the angle with different thickness legs. The results of the present program are available to permit a more complete study of these stresses but these will be reported upon in a later paper,

-311.5 MAXIMUM STRESS EXCEPT IN FILLET 1.4.. " 1.3 60 C') VALUES OF w/t 1 Figure 1. Coeffi.cient of Stress Increase for Tee or WF Section With Parallel Sided Flange. 000,1V L UE OF0 w / t 001 Figure l~~~~~~... Coeffcien ofSrs nrasefr0Te or WF Section With Parallel Sided Flange.~~~0001

-321.4 -.. MAXIMUM STRESS t EXCEPT IN FILLET ww~ VALUES OF rt / t/ 1 1.2 ~ " a' 0 00fr 0.0.. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 VALUES OF w/t~ with 1 - Flange Slope. with 1 -6 Flange Slope.

TORSION CONSTANTS FOR STRUCTURAL SHAPES The value of "'a" applicable to a particular structural shape is determined from Figures 7, 8, 9, or 10, or, less approximately from Equations (27), (28), (29), or (30). The correction for the juncture effect then can be computed and added to the known contributions of the various component parts to the torsion constant. For the special case of the angle with legs of equal thickness, a may be obtained from Figure 12 or by Equation (31). After a is determined, the torsion constant J for any one of the nine different structural shapes shown in Figure 16 is readily calculated by the appropriate equation (Equations 34 through 41). -33 -

-34t b 2 bt3 + 1 (d-2t) w3 + 2alD14 3 3 -.420 t4 (34) L, tet 1 | C J - (b-w) (tl + t2)(t12 + t22) d + 2 wt + 1 (d-2t2) w3 + 2a2D24 w 3 3 - 4 vstl4 (35) b b r r _ J= bt3 + (d-2t) w3 3 3 + 2a3D34 -.420 t4 (36) d w w d'I b b-w (tl + t2)(t12 + t22) + 2 wt23 6 3 2 + (da2t2) w3 + 2ca4D4 - 2Vstl -.210 t24 (3' Figure 2uy F l r Figure 16. Summary of Formulas for Torsion Constants for Structural Shapes.

cd i J 1 bt3 + (dt) w3 + clD14-.210 t4 3 3 -.105 w4 (38) tI[ J= (t1 + t2)(t2 + t22) + 2 + (d-t2) w3 + a2D24 - 2VStl4 1+ t,51 + (') 5F ~~. o: (39 -.105 w4 (40.) It -b - w (tl + t2)(t12 + t22) + w t23 12 3 + i (d-t2) w3 + a4D44 - Vstl4 I It,.t.105 t24 -.105 w4 (41) Figure 16. (Continued)

ACCURACY AND COMPARISON WITH PRIOR TEST RESULTS The accuracy of the computer program results were tested by comparison with the known torsion constant for the rectangular section. Using a section of thickness 8 H and breadth 48 H, the computer analysis gave a value of J = 7320,58 H4 while the St. Venant solution gave a value of J = 7331.42 H4, a difference of about 0.15 per cent. The effect of mesh size "Ht" was also studied. For the channel segment, as shown in Figure 5, with r/t2 1 and w/t2 = 0.5, as shown, CX for the juncture correction factor was found to be 0.080927. For a mesh size 2/3 of that shown on Figure 5, an a of 0.081334 was obtained and an H2 extrapolation of the two results yields a value of 0.08166, thus indicating a probable error in the values of o of less than 1 per cent for this particular case. Since the juncture correction factor for the channel is usually less than 10 per cent, this means that the overall error in the torsion constant would usually be less than 0.10 per cent and the final values of the torsion constant can be considered correct to three significant figures. Although no current experimental tests were made, a comparison with earlier tests made by one of the authors is given in the first seven lines of Table 4. The present investigation did not disclose any great amount of error in the evaluation of a by the earlier soap film tests.(4) The average of the absolute values of the per cent variation tabulated in Column 5 of Table 4 for the seven earlier tests was 1.76 per cent for the currently proposed formulas for c as compared with 2.26 per cent for a values calculated by the results of the soap film tests. Thus -36

-37TABLE 4 COMPARISON BETWEEN COMPUTED TORSION CONSTANT J AND PREVIOUS TEST RESULTS (1) (2) (3) (4) (5) (6) (7) Torsion Constant J By Test Nominal From from Percentage 2 a D4 Test No. Size Dimensions Reference 3 Variation 2 c D4 T-140 6WF20.2408.243.9.0277.115 T-220 8WF31.4582.451 - 1.6 oo645.1407 T-250 x 10x12 1.989 1.960 - 1.5.0839 o.0421 (62) T-260 12I31.8.817.804 -.6.0920.1126 T-300 12I55 3.290 3.380 2.7.484.1471 T-310 12WF190 47.290 48.870 3.3 4.293.0908 T-330 12WF65 2.206 2.222 0.7.246.1115 Ship chan- 6 In., nel$* 15.3 lb..211.206 - 2.4.0124.0588 Heavy chan- 6 In., nel* 15.5 lb..4249.400 - 5.9 o 468.1101 Standard chan- 10 In., nel* 15.3 lb..216.256 18.5.01151.0533 Standard chan- 15 In., nel* 33.9 lb. 1.0229 o 08 5.o 6 o 0606 0592 ~Test numbers used in Reference 3, which gives complete information on measured dimensions. *Nominal dimensions are used, tests from Reference 12. $Flange slope S =.03480 XSection no longer manufactured.

-38there is improvement, most of which accrues from Test T-25, which was 6.7 per cent in error by the old formula. In a discussion of Reference 3, Seeley and Putnam(12) reported on tests that they made of four channel sections and these are compared with the proposed channel shape formula in the last four lines of Table 4. The agreement is not particularly good, especially in the case of the 10 in., 15.3 lb. channel and the fact that the investigators used a shear modulus of 12,000 ksi in calculating their torsioniconstants indicates that the reported test values are lower than they should be. Seeley and Putnam point out that there was considerable deviation between the measured dimensions of the channel and the nominal or handbook dimensions but they used.the nominal values and do not report the measured values. It is possible that the high value reported for the 10 inch channel is due to the fact that this channel might not have maintained its shape of cross-section during tests, The investigators measured only the angle of twist of the flange. Since this section has the thinnest web of any channels tested the actual twist of the web may have been greater than that of the individual flanges. Seeley and Putnam correctly point out that the variation in their test results is greater than the variation between a more accurate evaluation of the torsion constant and the simple approximation bt3 J bt3 (42) 3 They also correctly suggest that in using this equation neither the end loss correction nor juncture correction should be included since these two corrections tend to compensate. Including either correction without the other gives a poorer approximation than neglecting both.

-3 9The justification for the present evaluation of accurate formulas for channel and angles together with improvement in the earlier formulas for I beams and wide flange shapes is for the purpose of permitting more accurate evaluation for handbook listings. It is to be hoped that torsion constants will be listed in future steel handbooks as they provide a basis for the simplest and most accurate solution of problems involving combined torsion and bending as well as problems involving the lateral buckling of beams and columns made of I beam or WF shapes. As indicated previously, the agreement between the formulas for angles with legs of equal thickness and experimental results obtained by Cullimore and Pugsley(9) is good, (Figure 12).

SUMMARY AND ACKNOWLEDGMENTS This paper has provided a basis for the handbook tabulation of torsion constants for all currently manufactured hot rolled structural shapes with the exception of bulb angles and similar special shapes. Whereas such previous evaluation was necessarily made by means of s6ap film tests or by tedious desk calculator relaxation procedures, the advent of the digital computer has permitted the development of formulas of greater accuracy than hitherto was possible. Special acknowledgment is due the University of Michigan Computing Center facilities which permitted use of the many hours of computer time necessary for the many runs needed. -40

REFERENCES 1. Saint Venant, "Memoire sur la Torsion des Prisimes,.. Memoires des Savants etrangers, 1855, T. XIV, pp. 233-560. 2. Prandtl, L. "ZiUr Torsion von.Prismatischen Staben" Physikalische Zeitschrift, IV, 1903, p. 758. 3. Lyse, Inge, and Johnston, B. G. "Structural Beams in Torsion," Transactions of the Amer. Soc. of Civil Engineer, Vol. 101, 1936, p. 857. 4. Johnston, B. G. "Torsional Rigidity of Structural Sections," Civil Engineering Magazine, Nov. 1935, p. 698. 5. "Torsional Stresses in Structural Beams," Bethlehem Steel Co. Booklet, 1803. 6. Trayer, George W. and March, H. W. "The Torsion of Members Having Sections Common in Aircraft Construction," Technical Report of the Advisory Committee for Aeronautics, No. 334, 1930. 7. Christopherson, Do G. and Southwell, R. V. "Relaxation Methods Applied to Engineering Problems," Proc. Royal Soc., London, Series A, No. 934, Vol. 168, p. 317, Nov. 1938. 8. Shaw, F. S. "The Torsion of Solid and Hollow Prisms in the Elastic and Plastic Range by Relaxation Methods," Australian Council for Aeronautics. Report ACA-11, Nov. 1944. 9. Cullimore, M.S.G. and Pugsley, A.G. "The Torsion of Aluminum Alloy Structural Members," Research Report No, 9, The Aluminum Development Association, London, July, 1952. 10. Huth, J. H. "Torsional Stress Concentration in Angle and Square Tube Fillets," ASME Journal of Applied Mechanics, Vol. 17, No, 4, December, 1950. 11. Timoshenko, S. and Goodier, J. N. "Theory of Elasticity," McGraw-Hill, 1951. 12. Seely, F.B. and Putman, WV J. Discussion of Reference 3. Loc. Cit., PP, 907-912. -41

NOTATION D1, D2, D3, D4 diameter of circle inscribed at juncture F a constant (a length) G shear modulus of elasticity H a constant (a length), network space J torsion constant Mz torsional moment N ratio of triangle base to mesh space R ratio of triangle altitude to mesh space Ro an arbitrary number S slope of flange T tension force per unit width in a membrane V volume under stress function surface V, VS, VL coefficient of end loss b overall breadth of flange d overall depth of section p pressure under a membrane r fillet radius t, tl, t2, t3 thickness of flange (See Figure 6) x, y, z coordinates G angle of twist per unit length TZXTZyTzr.Tm shear stress coefficients of juncture correction to torsion constant 7 coefficient for maximum shear stress in a rectangular cross-section 6 coefficient of stress increase at tee juncture

-43 - z displacement of membrane dimensionless form of stress-function, 4 torsion stress function w web thickness

UNIVERSITY OF MICHIGAN II3 9015 02845 337IIlllIl ll 3 9015 02845 3317