ANALYSIS AND DESIGN FOR TORSIONALLY LOADED COMBINATION SECTIONS Submitted by: Raymond G. Makara in Partial Fulfillment for the uegree or: Master of Science in Engineering in the Civil Ehgineering Program -, /5,/9x3 Date YOUNGSTOWN STATE UNIVERSITY June 1983 . -7, ,-,.I "m. . ,",' , . .o. ,, ,, : ,# y., 8 8, '11 I' , > , I: ': ABSTRACT I 8, " ' I, , ' $3 ,, I r, ,#. ' .:: A, ., : ,:..,..,, ,<'.,,& : ' ':',,,{",' 'I " L, VL". ! !!, I! ra.>?n,, , , 8 8, ,/ . ,r.< 1, , .. s,':tJ:., {',!I ,LJ $4 In this study, a procedure for the design and analysis of unsym , ,. metrical ly 1 oaded combination sections was accompl i shed; combination sections being defined here as wide-flange shapes that have a channe section attached to the top flange. The torsional ,method of analysis took into account that the ent combination section resists the vertical and lateral loads while cur design practice usually considers the entire section to resist the v tical force and the top flange alone to resist the lateral force. In order to conduct the torsion analysis of such combination sections, certain warping and torsional factors were eval uated. Once these elastic section .properties were calculated, the torsional the0 was first verified and then applied to a beam loaded by a two wheel crane. The torsional method was compared to the so-called conventional method of analysis with the result that the conventional method of analysis is perhaps not as conservative in some cases as originally thought. Design aids for the torsional theory were developed in the format of design tables that list combination section properties for over 1 sections as well as-tables which list the maximum allowable len a crane beam for a given vertical load, lateral load, crane wheel b and steel strength. ACKNOWLEDGEMENTS During the course of working on this thesis, much information and time was required. It would have been impossible for the author of this manuscript to do all the thinking. E wo*ld therefore like to thank my advisor, Dr. Jack D. Bakos for all of the help, technical input, and criticisms which aided the completion of this thesis. The author would also like to thank V. E. Shogren for all of his help, input,. and use of technical literature. I would also like to thank my family for their understand- ing and encourwment in the pursuit of my education. ) * To all of these people I am indebted. , 2,, \ ,' $,. *' . 14"..,., , o This thesis is dedicated to my grandfather, John W. Susor Sr., who may not have understood a word of this thesis, but would have been proud of it. iif TABLE OF CONTENTS PAGE ABSTRACT.......... ....,... ................ .......... ii ................................... ACKNOWLEDGEMENTS iii TABU OF CONTENTS .................................. iv ..................................... LIST OF SYMBOLS . . v-vi .................................... LIST OF FIGURES vii ..................................... LIST OF TABLES viii CHAPTER 2. BEMDING AND TORSION THEORY OF CMBINATION SECTIONS .................... 6 - 19 3 APPLICATION OF THEE TORSIONAL THEORY OF .................... COMBINATTOM SECTTOVS 20 - 78 4. DISCUSBLONS AND COMCIXSLONS ............... 79 - 81 LIST OF SYMBOLS DEFINITION UNITS A Area of cross-section Cw Warping constant for a cross-section feet* e Distance from top of dambination section to shear center of top flange only in. ka i Modulus of elasticity Distanoe from shear center of a combination section.to bottom flange in. Distance from shear center of a combination section to top flange in. Vormal bending stress at a point ksi ks i Actual bending stress in compression Actual bending stress in tension ksi - ksi Normal warping stress at a point Allowable bending stress in compression ksi - ksi ksi in 4 Allowable bsnding stress in tension Shear modulus of elasticity Moment of inertia of section about X-axis - Product of inertia Moment of inertia of a section about Y-axis Moment of inertia of top flange of a combi- nation section about the Y-axis Torsional constant of a combination section - feet Length of a beam Length of an ^element between points i and j in. kip-in Bending moment about X-axis Bending moment about Y-axis kip-in Vertical wheel load kips kips Lateral wheel load Shear flow on a cross-section kip/in Radius of gyration comprising the compression flange plus one-third of the compression web area, taken about the Y-axis in. A constant involving sinh (see eq(3-6)) Crane rail height in. Crane wheelbase feet Applied torque on a span kip- in Thickness of an element between points i and j in. Shear stress on a beam web ksi Unit warping function in 2 Normalized warping function at a point A on the top flange of a combination section in 2 - Normalized warping function at a point B on the bottom flange of a combination section in 2 - A constant equal to (GK/EC,) * Ratio of I /I ycf Y Distance from shear center to an element in. - - Second derivative of the angle of twist with respect to the distance along the beam --- The angle of twist'of a torsionally loaded beam, at a point on the span --- LIST OF FIGURES FIGURE 11 Typical Combination Seotion... ............... 1-2 Conventional Design Loadiw Procedure. ........ 1-3 Unsymmetrical and Torsionel Loading of a Corabinertion Section. ..,..................., 2-1 Combination Section Subjeot to Unsymmetrical Be$ndiw**.*..*** ********a*** 1- ** ***.**8***+ 2-2 Beam Loading Diagram if S60.586L ............ 2-3 W-ing Ronaal Streee Distribution in a Wide- ............................. Flange Shape.. 2-4 Shear Flow in a Combination Section for the Determination of the Shear Cenr;er.....,.... 2-5 .Initially Assumed Shear Flow in a Combination .. Section for the Determination of C, rmd Wn, 2-6 Sup@rposition of Rro T6rpues on' a Cnvlo Beam. 3-1 Cmbinartion SeoZ;ion POP' the Determination of and C, by the Simpliiied Method......... PAGE LIST OF TABLES TABLE PAGE 2-1 Combination Section Warping Properties ........ 13 3-1 Section Properties of Combination Sections.... 21 - 25 3-2 Comparison Between the Exact Method and the Approximate Method for C, and Eb............ 26 3-3 Maximum Allowable Beam Lengths ................ 35 - 70 3-4 Typical Difference in Maximum Allowable Beam Lengths for Varying Rail Heights ............ 72 3-5 Comparison of Allowable Beam Lengths Obtained by Calculation and by Interpolation of Table 3-3e~emmme ............................ 73 viii Chapter 1 In many tndustrial buf 1 dings , overhead cranes are incorprorated into the lnnufacturing and handling of goods and equiulpmt. For example. cranes are usad extmsively in stml fabrication plants, in production or assably lines, in mi11 oparatfanr, in mining and cwntless othkr - - opemtiuns. Them i~anes serve these qwratfons by I ifting and trans- portJng leads frm me locati on to amtber. For this process to be iarglmW, a suitalrle. systa of cvanr beams and cglms, called a cram *my, mst be designed to hmdh all antlcipat@d leads the crane nay ecmunter,' The design of thsc crane beams is much mre compl IcaUd than ctasl~f ng an ordinary steel bar-, The design of these sophistf catad systems wit l b.e! di scussed in this paper. In the design of crane has, the structural engineer must take - - into account all of the lading condftions that could possibly be encoun- tered. Tha mJor Isad an a crme be- i s the di mcrt , ve~tlcal force or wheel load. This is the largest force on a crane baa and is usually listed l'n catalogs p~ovided by crane manufacturers. In addition, thds wheel load must be glran a percentage increase to accQunt far any inrpatt that wld k pmdubd when the crane load is raised and larrad'lnd due to any subsequent movement of the load. Thfs increase for impact is listed in section .1.3.3 of the design specifications of the American Institute of Steel Construction Manual of Steel Canstructian (I)* (herein referred 20 as "AISC Steel Code"). The next significant forces on a .- crane beam f s the lateral load that acts perpendicular to the beam - * Number in parentheses indicates reference cited. span and is produced by the rocking mjvement of the crane trol ley wheels. As out1 ined in section 1.3.4 of the AISC Steel Code (1) this force, which is assumed to act at the top of the crane rail, is specified as twenty percent of the lifted load plus the weight of the crane trolley. This force is divided equally between the two crane rails. The third force on the beam is the longitudinal force produced by the crane travel- ing along the crane rail. This force, which is resisted by the top flange of the beam, is usually not of much consequence and is often neglected. Lastly, the beam and rail weights must be considered. Any type of beam must be designed to resist lateral torsional buckling. In addition, the lateral force on a crane beam is not contained in the plane of the minor or major axes and does not pass through the sectiods shear center and thus, these considerations are critical in the design. The AISC Steel Code specifies the maximum unbraced length that may be used, while still utilizing the full, permissible compressive bending stress. If this maximum length is exceeded, the a1 1 owabl e compress i ve stress must be reduced. Thi s - allowable compressive stress is directly controlled by the size or lateral stiffness of the compression flange. So, a beam with a larger top flange has more lateral stiffness and is permitted to have a longer - - unbraced 1 ength. For most rolled beam shapes, the flanges are relatively narrow. Oftentimes, the size of the compression flange is increased to give the section a greater lateral stiffness. The most common procedure for reinforcing the top flange is by welding a channel section to that flange. This is called a built-up or a combination section and is shown in Figure (1-1). T CHANNEL SECXION Fig. 1-1: Typical Combination Section Such a profile i s often used for a crane runway beam. In the past, built-up crane beams, which are unsymmetrically loaded have been designed using a simp1 ified procedure. This procedure con- sf dcred the vertical wheel load to be carried by the entire bui 1 t-up section and the lateral force to be applied at the top of the channel and resisted by only the added channel and top flange. This simplifi- cation is shown in Figure (1-2). This procedure has been considered - Fig. 1-2: Conventional Design Loading Procedure conservative by most design engineers and thus not very precise since the lateral load is actually resisted by the entire cross section. In order to obtain a more exact stress analysis of the unsymmet- - rical ly loaded section, a more rigorous procedure uti 1 izing the equations for unsymmetrical bending of an elastic beam should be used. These equations are presented in many advanced strength of materials (3, 5) and steel design books (7, 8), but the use of these equations alone is insufficient since such an application assumes that the beam is loaded through its shear center. The shear center is defined as the point on a beam cross-section through which a force must act so that all twisting effects are eliminated. For a typical rolled wide-flange shape, and any other doubly-symnetric section, the shear center and the centroid are coincident. For a built-up section this is not the case. Obviously, the forces on a crane beam do not act through the shear center (See Figure 1-3). ENTER Fig. 1-3: Unsymmetrical and Torsional Loading - of a Combination Section Therefore, in addition to the unsymmetrical bending equations, a torsional analysis should be incorporated. The torsional analysis of wide-flanges, - - standard channels, and other individual shapg~; have been compiled by many other sources (2, 3, 4, 5). The Bethlehem Steel Company publishes an excellent handbook (2) for torsion of common rolled shapes. The torsion analysis is also presented by other sources. But, no reference could be found that lists such torsional properties for the previously - .. mentioned combination sections. The intent of this paper is to compile the various torsion and warping properties of combination sections and and to apply them to the case of a torsionally, unsymmetrically loaded crane beam. Chapter 2 BENDING AND TORSION THEORY OF COMBINATION SECTIONS: As previously stated, the unsymmetrical bending equation (31 will be used for the crane beam analysis. The modified equation for bending stress in an unsymmetrically loaded beam is given as where, M = Bending moment about X-axis X IX = Moment of inertia about X-axis IY = Moment of inertia about Y-axis Ixy= Product of inertia - - = Coordinates of a point under consi- X'Y deration, using positive axes as ghown. = Angle between the plane of load, P, and the Xaxis. Fig. 2-1: Combination Sectlon Subject to Unsymmetrical Bending. . . In the case of a section that is symmetrical about one ' or both axes (see Fig. 2-1): so; and Substituting (2-3) into (2-4) yields: From Figure 2-1, it is seen that $ is related to the horizontal and vertical loads. More specifically, p,, cot f# = + The maximum live bending moment, Mx, may be easily calculated using a table of beam formulas. For a two wheel crane beam, the exact point of maximum moment varies with the span and crane wheelbase. There are two different loading conditions in which the maximum live moment might occur. Fig.. 2T2: . Beam Loading Diagram if S < 0.5863; First, if Ss0.586L (see Fig. 2-2), under load 1 at B = ~/2 - ~/4 . But, if S>O.586L, then with one direct wheel load at the point ~/2 on the beam span. Once the maximum 1 ive moment is calculated, EQ. (2-5) can be used - to calculate the maximum 1 ive unsymmetrical bending stress due to beam - action. This would, however, be an incomplete analysis since additional bending stress is caused by torsion on the section since the load does not pass through the shear center. This additional stress is called the warping normal stress. The warping normal stress is produced by the rotation of a beam about its shear center when a torque is applied. As this beam is twisted, cross-sections through the beam do not remain plane but warp out of plane. As the beam begins to warp, stresses normal to the cross-section develop (see Figure 2-3). L lot4 ST- PtSf81W71oN Fig. 2-3: Warping Normal Stress Distribution an a Wide-Flange Shape where - E = Modulus of.elasticity - - Wn = Normalized warping constant at a point on the cross-section 6" = Second derivative of the angle of rotation with respect to the distance along the length of the member. - It The value of 8 is a function of the torque on a beam, the beam - length, and torsion and warping constants of the cross-section. Many sources list equations used to evaluate dfor various loading condi- tions zinc! cross-sections (2,s) to evaluate 8". The values for the torsion and warping constants are a1 so needed. For wide f 1 ange shapes, these values are 1 isted in Part 1 of the AISC Steel Manual, For combi- nation sections, these constants are not, unfortunately, 1 i sted. The elastic properties for a 1 imited number of combinations are 1 isted. It will be ncce'ssary to develop expr@ssions for these required constants. The first parameter needed is the location of the shear center f~r conbination section. To locate it, the process outlined by Seely and Smith (3) will be used. Referring to Figure 2-4, a typical canbinatlon section is loaded through a point assumed to be the shear center. This force, Y,, will cause a shear flow on the cross-section to 'develop. This flow will produce forces F1, Fp, Fj, and F4; where, ql , the shear flow is substituting and solving; tb Likewise, F,, ad F, may be found to bet Now, taking the oummation of the moments about point A: ." 1 : Fig. 2-4; Shear Flow in a Combination Section for the Determination of the Shear Center - - . - Substituting for F1, F2, Fg, and solving yields3 12 Now, that the expression for the shear center has been found, the remaining torsional and warping properties must be determined. The general mathematical expressions for these torsional and warping pro- perties can be found in many references (2, 4, 8). C. P. Heins(4) has developed a numerical evaluation for standard steel sections, such as wide-flange shapes and channels. By expanding this numerical procedure, the warping and torsional properties for a combination section can be evaluated. The normalized warping function, Wni, at point i on the cross section is given as: where, w oi = unit warping function = foi L ij - tij = thickness of an element between i and j Lij = length of an element between i and j A = total area = ti jLi fo = distance from shear center to elemeiit The warping constant, C,, for the entire section is: with the terms the same as for eq.(2-11). The determination of both Wni and C,,, is best achieved utilizing A,. a tabular format. First, the combination section is considered to be a sequence of inter-connected rectangular plate elements with the ends and intersections of the plates numbered arbitrarily (See Figure 2- 5) A continuous flow is assumed across the section points 1-2-3-4- 5-6, with the flow on the elements 7-2 and 10-9-8-3 assumed to act from the free edges to the intersections. The first term to be calculated is wo= fbL . The values of ye are given in Table 2-1 and the sign of is determined by the rule that moving from point i to point j, if the shear center is located to the left with respect to the flow, the value of yeis positive. Thus, the values for wo at the edges and intersections can be determined since f. and Lij for each element can be easily tabulated. It is first assumed that point 1 has wo = 0 and, the summation of the results of f&L around the loop 1-2-3-4-5-6 yields the wo at the respective points. NOW, in order to calculate the wo at point 7 and the wo around the loop 10- 9-8-3, the values of wo at points 2 and 3 are used. These values are known since they were calculated in the previous loop. Since the flow - is known to act from 7-2 and from 10 to 9 to 8 to 3, the wo at the points 7, 8, 9, and 10 can be found directly as shown in Table 2-1. Now, the equation for Wn can now be evaluated since the wo are known. In Table 2-1, the areas, tij Lij , and the sum of the ( w6<+wo j)- tij Lij are listed. The expression for wni can now be evaluated as 'ni = Ebb3 - w oi (2-13) Therefore, by using equation (2-13), the values of Wn at the points on the section can be determined and are 1 isted in Table 2-1. Now, with the values of Wn at the points on the cross-section known, the warping constant, Cw, can be determined. Using EQ (2-12), the expression for Cw is found to be: 1'4 ~ig. 2-58 Initially Assumed Shear Flow in a Combination Section for the Determination of Cw and Wn Another torsional property required is the termr where r G = shear modulus of elasticity E = modulus of elasticity Cw = warping constant K = i<(b. t?) 3 11 and ti is always the smallest dimension and bxt 1 i ' Refering to figure(2-5), K may be expressed as follows: Now, with the values of K and C, found, they may be substituted into EQ (2-15) and a value for can be evaluated. The value for 9 usually is much less than zero and does not lend itself suitable for cornp;l ing - into a tabular form. Therefore, the reciprocal of is often tabulated a as is the case for wide-flange shapes 1 isted in the AISC Steel Manual. It is given as: NOW, with the expression for the torsion and warping properties I4 evaluated, the value of 8 may be determined as described before. , For the live load case of a two-ntrrsel crane which was considered previously in an unsymmetrical banding mode, the expression for &for the two load cases (S 5 0.5861. and S > 0.5861) will be evaluated. Case 1: S 5 0.586L For this case, two wheel loads are applied on the beam span. The two loads must be considered separately and the ~rinciple of superposi - tion is used by taking the sum of the two vglue~ for @"at a point. I Ses Figure 2-6. Fig. 2-6r.Superposition of Two Torques on a Crane Beam - ~ro* Roark and Young (5) , the vilue ' of ~"f o& a concentrated intermediate torque on any beam is given asr *m-,;..7[ ;, 'L L,b 7. > 'A b'? , ' 1 lid_ I'r where, TA = reaction torque at end, A Q: = first derivative of eat left end @it= second derivative of 0-at left end To = applied torque on span. I - - - -- -- * - F& a- crane beam, the applied torque, To, is defined as8 where, P~ '= lateral force at top of crane rail -*t = distance from shear center to the top of the section RH = crane rail height. It must be mentioned that the boundary conditions to be used in n evaluating 8 are for a beam with ends that are resisted from rotating a, - irA r , ,. .I,>. 1 about the shear center but not resisted from warping out of pl'ane. This is consistent with actual design practice because the crane beam ends are bolted on the bottom flange to form a seat while the top flange is connected to a stationary object such as a building column to prevent rotati on. Now, referring again to Figure (2 - 6) , two torques must be con- - If sidered. First, e must be evaluated at X = B for the torque applied at X = B. The boundary conditions for this case are given as: . - ., . By applying these boundary conditions to EQ (2 - 19) and evaluating at the point X = B, yields: (I Next, the value for @"at the point X = B with To applied at the point X = B + S must be evaluated. The boundary conditions for this loading are given as: Again, substituting these boundary conditions into . eq. (2-19) and evaluating at x =r B yields.: Now, using the principle of supeqosition, it is I@ possible to evaluate 8 at the point x = B, which is the , point of maximum moment. Therefore, Case 2: ~-0.586L - - .. . . . - ..-- . .- For this case, the maximum moment due to the wheel loads occurs with one wheel located at midspan. For a concentrated torque at the midspan of a beam with the same end condition as in Case 1, the formula for B" has been evaliated by many sources. From Roark and ~oung'(5), It the equation for (9 is given as: _ _. -.re . . .'.. Now, with the expressions for B" evaluated, and -the expressions - for the,torsion and warping constants for any combination section, a crane beam can now be analyzed or designed easily and accurately. Chapter 3 APPLICATIONS OF THE TORSIONAL THEORY OF COMBINATION SECTIONS In order to analyze or design a torsionally loaded combination section as a crane beam, elastic section properties for the sections are required. As noted before, although the AISC Steel Manual lists some elastic section properties for some thirty different combination sections, no torsion or warping properties are given. Utilizing the equations developed for these properties as in Chapter 2, a computer program was developed to conveniently compute the elastic torsional, and warping properties for a 1 arge quantity of possible combination sections (See Appendix "A" for a listing). The output from this pro- gram was neatly arranged into a tabular form and is presented by Table 3-1. In calculating the elastic section properties of the combination sections in Table 3-1, a check can be made for the values of Eb and . In a paper by Kitipornchai and Trahair (6) dealing with monosym- metric I-Beams, an approximate solution for these two properties was outlined. In this approach, the ratio of the moment of inertia for the top flange versus the moment of inertia of the entire sectiot- both- about the Y-Y axis, is calculated (See Figure 3-1). -C .. - -:Y -. .- Also, the shear center location of the too flange; e, is: 22 ' tlbldc - e = 7 (3-2) X The expressions for 'i kd b are defined as :: . . . . . . . . ;a = (1 -1)h , ... 'b = Ah . -- - -. -- -- - - - - - - - - - - --- - - - - - A - - - -- - - - . - - - - -- -- - - Y . . TABIJ3 3-1 (cont.) --. , - . - ;.. - .P -!! Section properties of I - _ ,*- . .,T7 &.. LvT7Ei combination sections L I -- - t 'I - .: ELASTIC PROPERTIES TORS I ON PROPEWIES 'IOTAL - SECTION AREA I, IY Y1 r~ 5, K cw A W~~ w~~ , - IU In L ln4 In In In I& 1n6 In I n 2 1n2 u4 I -8.48 61-20 -1.12 et.54 -15.42 54-83 -9.14 6C.86 -1.99 66-6l I -16.49 54-46 -1C.80 60.68 I ! W 18X 65-ClSh33 494.9 . 83.26 . -2.75 6c.l7 tr82.7 15-45 -17.66 56-05 -12.01 6C-50 3 66-91 -1s-2C 86-51 -11.54 d5.74 -IS.S~ d4-8d -21-56 84-45 -24.24 63-62 -&a24 LC-31 I -3-03 64-34 2-34 tl-IS. -10.06 kc-06 UilX SO-C12120 -4.50 64-55 1.4s 6d-UI -12.35 5%-16 -6.42 64.15 C-33 6d-28 W2LX 62-C12EZO -13.55 34-8s bdi?lx (62-C15h33 -4-95 82-41 W21X 68-C12>20 528.2 111,lF -15.63 I4028 -6-22 82*)@ UUUUUU I11111 .r.?Y*UY DaOP+a' XXXXXX *.?.S)*wIZ NCYNNNN z11115 'Y Fig. 3-1 r Combination section for the Determination of Eb and C, by the Simplified Method But, b = Eb, $he distance from the shear center to the bottom flange. Therefore; The expression for the warping constant is given asl! . . By incorporating these solutions into the computer program for - elastic section properties, a comparison of the different values for Eb and Cw canbe made. Although the values for the approximate solution are not included in the elastic section properties of Table 3-1, they were calculated and compared. The values for C, and Eb for the-two - methods agreed qui ti? well as is shown in TABLE 3-2. - TAaLE.3-21 Comparison Between the E&ct Method and the - -..- Approximate Method for C, and %. I , I SECTION 1 ~14~30-~i0~lj. 3 I ~18X5O-C12X20.7 I W24X84-C12X20.7 W3OXl~b-C15X33,9 W36X:5O=CiBxM. 7 6 C, (in.) E+, (in.) $ diff. 0.52 0.49 -' 2.64 1 *90 2.06 Eq(2-14) 1857 1 6175. 29809, 62722. 148221. Eq(3-4) 5847. 6205. 30594. 63912, 151281. $ diff. 0 983 0 -62 0.85 L.00 1.22 Eq(2-11) 12.81 16.39 19.22 2.5.14 30 48 Eq(3-3) 12 92 16.49 19.39 25 39 30 55 In the determination of the normal warping stress, the value of $is needed. The value of B" for a given loading condition and be* span may be evaluated using formulas or design charts. Roark, and Young (5) give fonnular for obtaining el' for a mu1 titude of loading conditions. The Bethlehem Steel Corporatf on pub1 ishes a torsional design handbook tt (2) that contains design charts for a rapid determination of 0 for . a I fmitd quantity of load eases. For the loading condition consisting of two crane wheels on a simply supported beam span, the equations for @"ware given as EQ. (2-21) or EQ. (2-22). EQ (2-21) tan be rewrftten as: EQ. (2-21) is valid only if. S 5 0.586L. The expression for R is cumber- . some to handle and evaluate, but a very efficient design chart can be - developed ~lating k, el' . Calculating a large quantity of values for e", a plot can then be made as shown as Figure 3-2. In order to use the chart, the value of f% for a given condition .is - - located on the abscissa. Then, movfng vertically until the correct curve for the value of 851 s found. It is necessary to only move horizontal 1y to the left and read the value for R on the ordinate. If the given value for Pa fa1 1 s between two curves on the chart, 1 inear interpolation - may be used to yield a satisfactory valuemfor R. If the wheel base - S > 0.586L is encountered, .- . - the chart cannot be used. '1n this. instance, however, the value of e"as given by EQ. (2-22) can be easily-calculated. The following example showing the analysis of a crane beam will demonstrate the use of these tables and the torsion theory. Example 3.11 Given the beam section and load combination shown above, find the maximum live bending stresses using both the more exact torsion theory and the usually accepted conservative method. Since ~e0,586L, eq(2-7) is used to evalute M.,. Thus, My = 137.8 kip-in. The allowable bending stresses may be evaluated - - using the AISC Steel Manual (1) Specifications, Section 1.5.1.4.5. or this combination section,, the allowable tensile bending stress is: FbF = 0.60Fy = 22.0 ksi The allowable compress~on.~tress can be evaluated using either Code eq(1.5-6a) or eq(1.5-6b), whichever applies. Since or 53.2 6 ~/r, 6 119.2 Therefore, Code eq(1.5-6a) is used to calculate the alLowable compressive bending stress. Hence, Fb~ =20.4ksi With the allowable stresses calculated, the actual bending stresses can now be evaluated. The stresses at points A and B will first be evaluated using the conserv- ative method. For this method, the bending stress 9or.4be -.a twp (comprersion) flange is given by the equation1 n M c ;$;$$$&[;.n- lxcx 4 i,,, -4 2 b6 = 3- + \dl !id# X ycf Iycf = moment of inertia of the top flange -= distance to pdint under consideration Ox ' Cy from the X-axis and Y-axis, respectively where 4 For the combination section in this example, Iycf=149 in. Now, evaluating the maximum compressive stress at point Aa b~ = 13.9 ksi FbC .~*20.4 ksi Q.K. The tensile stress at point B is calculated as8 Next, the bending stresses will be calculated using the more exact torsion theory.. First, the unsymmetrical bending stresses will be calculated using eq(2-$1. Thys, Calculating the compressive bending stress, The tensile bending stress is now calculated as, . fbTD 17.21 ksi FbT = 22 .'0 ksi . Now, the warping no.rmal stresses have to be calculated and added to the unsymmetrical bending stresses -L,T,ptg P ,?,* -8 84 : !bl; li .calculated above. The warping normal stress can be eval- ,$ uated using eq(2-9); - I1 where @ is found using eq(3-5), with To = P (E + m) Yt To = 1.5(1.88 + 4.25) To = 9.20 kip-in. So, the value of 8. is calculated as; 0" = ,-1(0.687) e"= 0.10468/~ . So, the normal warping stress (compressive) at point A is: bw~ = E W~ g'' bw~ = 7.30(0.10468) bw~ = 0.8 ksi and, the normal warping stress (tension) at point B is: bwB = 8 WnB 6" bwB = 61.42(0.10468) bw~ = 6.4 ksi . Now, adding the warping normal stresses to the unsymmetrical bending stresses at points A and B will give the total bending stresses for the torsion method. So, bA = 13.2 + 0.8 and k: bA = 14.0 ksi 5 FbC = 20.4 O.K. ; h ', - fbB = 23.6 ksiS FbT = 22.0 ksi No Good AS shown, -the allowable tension stress in the bo'ttom - flange is exceeded, so the torsion method of analysis indicates the beam is overstressed while the wconservativew method indicates it is not overstressed. Thus, it appears the wconservativew method may-not always be conservative. In comparing the stresses calculated by the two methods, a very interesting item is observed. The torsion method of analysis and the conventional.method yielded almost equal values for the compressive bending stress; 13.9 ksi versus 14.0 ksi. But, the conventional method underestimated the tensile bending stress , grossly. The convention& method yielded a tensile bending stress of 14.2 ksi while the torsion analysis yielded a tensile bending stress of 23.6 ksi. So, it seems the conventional method is conservative only with respect to the compressive stress and is unconservative with respect to the tensile stress. In the design of a crane beam, the process is not straight-forward due to the many unknown quantities encoun- tered. Most often,the designer will know the required beam span and the capacity of the crane (along with all corre- sponding manufacturers' dimensions and wheel loads) that are to be employed. Therefore, a suitable combination section must be chosen. This usually requires a trial and error - procedure, but the design can be considerably shortened if, for a given wheel loading condition and combination section, the maximum allowable span for the beam was known. By expanding the computer program used to calculate section - properties for combination sections (see Appendix A), a set of tables has been constructed in which the maximum allow- able lengths have been listed for a variety of wheel loads, lateral loads, and wheelbases. Also, tables are given for - either 36.ksi or 50 ksi grade steel. These appear as 'sable 3-3. The combination sections listed in Table 3-3 ... c m m N m r? XXX n' m m -.Id ... h' F h' um.7 XXX a u. a d-4 0 :, if? ooo~oo~mmrm~~~m~~m~~m~~~m~mm~~ .............. m d~~~~o~~~cu~~~rmmmmmu~~w~~~mm~v(~ g 't: drddduddddrddddNNNNNNNm a 7 .. 0' P m X LP. 4 U I C w d Y .- P z - Y9': cmo RFN XXX %'LPN d - d UUU I I I 990 CI 9. m XXX 3uE: 1.1 1 333 - *..* mmmm nmnm xxxx ..** F.NmN -emu xxxx mP3VIED Lddd IRNWtn "-I(- UUUU IIII CUe6 =Dee UUUVU IIIII emam.* eeee= XXXXX -. 0 .- IT .- ntl'n.K-1 3PLr33 UUUU IIII d9Ca dddL .I-.-- XYXX e - P. P. mmm~ PI 4 a Y OOOOQ NI-MCnmOhNmOmNNHQ00m~mmmc a Y ? ooooo~oscusc~.nos~~m-.rw~smm~(~~~oD~ ~*.eme*****e*.e...e ~~emee***e k .p ae $ ~oo~oocoIo-ow-NN~N~~~~c~-~--~~~ r-r--r-----d-~d&-mwm~ a TABLE 3-3 (oont.)r Maximum allowable beam lengths 0 1 P, = 36 ksi S~CTI~K , Wl2ILb-ClOX15.3 W17X 26-C12X20.7 W14X JQ-C10X15.3 W14X30-C12XZ0.7 WlbY3h-ClZX20-7 HIOX36-C15X33.9 W18X 50-Cl2X20.7 UllX50-C15X33.9 H21X62-Cl2X20.7 W21X62-C15X33.9 W21X6R-CL2X20.7 Y21X 68-Cl5X33.9 W24X6O-C12X20.7 W74X68-C15X33.9 h24X R4-C12X20.7 W24X 84-C15X33.9 W27X 84-C12X20-7 H27X Y4-C15X33.9 U27X94-Cl2X30.7 W27X94-C15X33-9 W?OX 99-C15X33.9 H30X '39-Cl9X42.7 W30Xl16-ClSX31.9 WlOXllb-ClRx/12.7 W13XI18-C15X33.9 W33X11fl-CI8X42.7 h31X141-C15X33.9 W13X141-Cl3X42.7 H36X150-C15X33.9 W36X150-ClRX42.7 Noter A value of 0.00: indicates allowable web shear striss exceeded. .WEEL 'MAD pX (kips) I 55-0 0.00 0.00 0.00 0.00 0-00 0-00 0-00 0-00 10.25 12.17 11-17 12.75 11-75 13-25 13.33 15-08 13.50 15-42 14.50 16-58 16-50 18.00 19p42 21.42 19-92 22.00 23.00 25.50 24.25 27-00 50.0 0.00 0.00 0.00 0-00 0.00 0.00 0.00 0.00 11-33 12.83 12-17 13-58 12-50 14.08 14.17 16.08 14.33 16-50 15.42 17-75 17-58 19.25 20.92 23.08 21.42 23-15 21-83 27-61 26.17 29.33 60.0 0.00 0.00 0.00 0.00 0-00 0-00 0.00 0.00 9-33 11.25 10.17 12-13 10.75 12.58 12.67 14.25 12-83 14-58 13-75 15-67 15-58 16.92 18-25 20000 18-61 20.67 21.50 23-75 22.58 25.17 ..- 30.0 0.00 0.00 0.00 0.00 9.83 11-42 13.42 14.83 15.92 18.08 17.00 19.33 17.33 20.00 20.42 23.75 20.50 24.17 22.50 26.50 26-00 28-83 32.17 35.92 32.75 36.75 36.83 42.75 37.83 43.92 5.0 26.25 20.75 30.83 33.67 33.42 47-58 41.00 54-00 43.25 56.58 43.50 56.75 44.92 59.54 45.92 59.17 47.58 61-08 68-42 61.50 61.50 73.17 63.75 75.42 65.75 77.5R 67-25 18-50 68-83 80.08 20.0 8.50 9-62 10.42 11.67 13.67 15.00 1767 19-13 21.58 24.92 23.25 26.92 23-67 27.32 2R.58 33.92 28-61 34.33 30.42 38.25 37.33 41.50 40.58 47.R3 41.83 49.25 42.83 50.00 43.75 51.00 65.0 0.00 0100 0.0C 0-OC 0.00 0.00 0-00 0.00 0-00 0-OC 9.33 11.33 9.83 12-00 12-00 13-51 12.25 13-92 13-00 14-03 14-75 16.00 17-25 18-03 17-67 19.42 20.11 22.33 21.25 23-58 . 35.0 0.00 0.00 0-00 0.00 0.00 0.00 12.25 13-62 16-33 16-11 15.25 17.25 15.58 17-92 1@.C8 20.92 18.33 21.33 19.92 23.33 23-00 25.33 28-08 31.17 28.58 32.00 34.00 38.08 35.42 40.58 25.0 0.00 0.00 e.17 9-11 11.92 12-92 15.08 16.83 18.17 20.75 19.42 22.33 15-83 23.17 23.67 27.67 23.75 2e.08 26.25 31.08 30.50 33-03 37.08 42.67 38.33 43.67 39.67 46.17 40.58 41.25 10.0 15-17 16.25 17.31 18.75 22.67 25-50 31.58 36.CO 34.67 45-33 14.92 45.50 36.00 46.92 36.83 47.31 18.08 48-83 38-67 49.17 49.17 58-50 59.92 60.17 52.50 61.92 53.5H 62-58 54.81 63.81 5.0 11.67 12-42 13.17 l+.CO lC.5R 18-42 22.17 35-08 27.59 37.C9 30.00 34.63 30.42 36.17 32.17 41.42 23-33 42.75 31.92 43-17 W.CO 51.25 54-61 52-83 66-00 54.25 47.CO 54-92 66.08 56.00 70.0 0.00 0.00 0.00 0-00 0.00 0-00 0-00 0-00 0-00 0-00 0.00 0.00 9-17 11-17 11.25 12-92 11.58 13.25 12-50 14-11 14-08 15.25 16-33 17.92 16-15 18-42 19-08 21-08 20-00 22-25 40.0 0.00 0.00 0.00 0.00 0.00 0.00 10.83 12.33 13-17 14.75 13-92 15.67 14.33 16.25 16.42 18.92 16.67 19.33 18-08 21.00 20-75 22.75 25.00 27.75 25.50 28.50 30.08 33-61 31.83 35.75 75.Q 0-00 0.00 0.00 O-CO 0-CO 0-CO 0.00 0.00 0-00 0-00 0.00 0-00 0.00 0.00 10-50 12-42 10.83 12-15 12-00 13-58 13-50 14-58 15-50 11-00 16-00 11-58 18-17 20.00 19-08 ZLoC0 45.0 0.00 0.00 0-00 0.00 0.00 0.00 9.50 11.25 12.33 13-61 12.92 14.50 13.25 15.00 15-17 17.33 15.33 11.75 16.58 19.17 19.00 20.75 22.75 25.08 23.25 25.83 27.17 30-25 28.67 32.17 ".".T7'1S': u'croomo -N-.NN m N XXXXXXL 9NSNMLFN ....... UUUVUUU IIIIIII ' + csecseca ' g. ~hlrnc*rnrny\ PteP * F o m m~m xxx mN&n I*..,.. *.**.*****.*......**..*. ~~~em~~emmmm~~~~~~cv~a-ohi~~~~~u --~~~~N~~Fc~cFu~u~Y~uuu.~~YY~L~~vIs 7 * oooooooo~mwmmmwmmwww~~~~m~~~~m ~oooooww-~m~m~~~~~~w~~~m.cc~an~m~ .............................. o~o~oooo~e~~~~~-~~~mm.cm~rmw~-~~ r--rrr-r-rrrrrN-N ~~~VOOOOOmOmmmm~mm~~rmm~~m~wmre 35990000m~9~7y~~y7~~";(r;4",33~":93 ***** OOOOOOOOWPO~OOONON -mmmerneee~om m-ea-drr-mrrdrNwN oowooow~~m~~mm~~m~mr~m~n~m~~~m oovooooo~~eerm-rmmmem~n~mmo~orn ***********e**************m*** $ p0000000rn~0~~ --rnrm~ee+a~er-~-rnwm r-e-derrd-ddNNNNN ~ooooooo~~n~wm~m~~~~~~~mm~m~mmm ~'~o~~~~~~~e~~~~~qooeqe~~y~-wm~~m a********W* *we w****~ o~oo~mwwe~~-~~-e~~mm\n~~w~~-~mmc --r-rr--r--rd-~d~~NNN = oooooo~~~w~m~m~wm~~m~mm~e~~~mm , ~~~~~~.te~mmme~~~mw~~~r~~enm~mw .............................. OOOOOOW~O-~~~~~m~meeew~m~mewme rrrrr4drrdd&rrNNNNNNN~ ooooooowm~~m-rn-mm~wrnmmmeommmmm oooooooo~~e~~w~mme-memo-m-mom- ' *e.*.*.* ..................... C OOOOOOOOON~~~~~~~~~WWON~N~~O~N A * -~--~-----~---NNNNNN~N~ m. P 4 GOOOOF~NWOFICPWNNQ~~W~.)~NF~O~~WP y 7 ~~~~mmsommmm~oeeo~mo~mew~m~~mm Y ............................. d~~~cw~~.~-m~e~mec~~eoowe~~weeum W * ~--~-@---~W-N-NNNNNNF~~ PI. p'~ooo~oo~~m~~~w~~~nmm~~mm~~~mmm ~ooo~vu~~emmv~mm~m~ma~~mme~wm~~ *.*.***.***********.*****.*mm. c oOOOW~O~N~m~m~~~~OaJNNU~L~4~~NC ---e-------~-~m~~m~mmrnma oo~~~w~mw~e~~~~~wm~~~mm~w~c~~m 0009WQ)rCINO9-Cm9mm(h~9+@NCWmmrCO~ ' .***.**.**.**.**..***...****.* 5 c~c-m~-rnwemmm~~~~~oneo-m-~mee~ rre---r--~-~~~~~rnarnmmmmm s = 00mtu~000wwmc~~~m0~mwm~~ 5 OOOe~OOOOCme~e~~muwm~e~m~~~ap- .............................. C o~-we-r.meecocr-s.-ecr.~m~meuom-m~ V mdrrrd~d~~~~~~~~mnm(~,.tm~m.t - *N~C~M-C~WN~~OOO*FO~-(DWN-~NC~L~ - ~.~-mmk-*wmr~rmovr~ew~-mmeemuo-c .............................. ~r r~w~-~m~w~owonm-m-~eu~m~em~uwe -.I -&~~-NNNNNN~N~N~~~OUCIJ~~~U _ ~mm~~m~~m+~m-a~nm~nw~omwommmmc ~~~~emuew-ema~umme~--(~m~~~~m~~ . - ........................... eo~-mmw-mm~om-~mm~m~~u~weeoc-~ ~-r--~~w~~n~m~rn~~.~mmucwmueueu w~-m~~wmmonme~~m~~~mmp.r~m~~omo~~, 5 OOPO.~~CP~~~~WPO~~NUW~~~~~~~O~~~NO~O ' .............................. - ~NN.c~cP~~~~~o)~(F~P-c-o~~NO~-~N~~ - rodrd-~~~m~m~mmmmqmeee.~me~n4~.4~h ~~~emw~~~m~rn~~eoenwm~m~rn~~comw m~~-~m-me~wam~m~em~cnmw-mmeem-m e *.**.*~**.**.*..**..****.. *** m~mom~--NmNmawemm~ee~mcee~der)c - ~~-NNN~cIucI.I~w~u~~~~~v~~~IP~~I~~~w~ - O~O~~OCNNC*~OOFON~~OW~W~N~~N~~* m"y"'CCey~y~yy;~y~3",":y":y~~y",~": * ** ***** ' WF~~~O~-O~-~N~~~~Q~QWOCNN~~UW~ ~mwmmmmmem~m.-m~w~m~mr-a~ar~-a- m*ncc~-e-ewece-p-ece~-~-e-ec'p.e ............................ w~wee~tmon~c~mcm~no~nn~~n~m~m~n~~ -~-~~m~mnm~m~~wn~~~~~umeme~~me ; XXXXxXxxXXXXXxXxxXXxXXXXXxXXXx ONONNaN(PNmNVhmNmNmNmW5In6L4eDWcCMCn & d~-dd~~---~~-dd~.-~---.r----d~ - UUUUUUUUUuvUUUUUUVUUUUUvuUU~Uv 1111111111~ 1111111111l11111111 = ~~OO~~QONNQQQ~U~~UU~J~~~WQ~-CC . - ~N~mmmmw~9e~e~m6mceva~====ff~~~ xxxxXXxxxXXxIxxxXXx~XXxxxxxxxx ~~~eee~o~~~~~~ue~ewc~o~~mmm~nrte --------NNNN~N~NN~NN~P~~P~F~~(CI 123213~3111XXZ23313t3333f133:XL Ld-m UUUU - -- - - cmwn mmmocw.m~rn~~~mw~cc~mc~ommys c ~rnmwW~mmmacc~mm~a-eaP~ae~~-~~~*~~ I I I ?Ye"; *Om mwm xxx PNm I* - UUU LA& xxx *em - - rll ZSZ d$d$dc; NmhlmNm xxxxxx 2!22z:2 xxxxxx rrrrreu NNhlNwcU sIZ=13 TABLE 3-3 (oont.)r Maximum allowable beam lengths kq. 2 Sz'7e-O 4 Py = 0. lop, F = 50 ksi Y ' Note: A value of 0.00 indicgites allowable web shear strese exceeded. sf(: 7 IT?, Wl2X 2b-CIOX15-3 WlZX26-CI2X20-7 U14X~0-C10X15-3 Wl4X 30-Cl2X20.7 W16X 36-C12X20-7 W16X3h-C15X33-9 Hl8150-CIZXZO-7 Wl8X50-C14X33-9 WZlX 62-C12XZO-7 W2lX 62-C15X33.9 WZLX6fl-ClZX20-7 W21X 68-CI5X33-9 Y24X60-Cl2X20.7 HZ4X09-C15X33-9 U24X 04-Cl2X29-7 W2OXB4-C15X13-9 H21XB4-CI2X2O-1 W27X 84-C15X33-9 U27X 94-C12X20-7 U27X 94-C15X33.9 W39X 99-ClSX33-9 WlOX99-ClnX42-7 k3OX116-Cl5X33-9 U30Xl16-Cl8X42-1 W33XlL8-C15X33-9 W31N118-CldX42-7 W33XI41-C15X33-9 64-32 51-75 45-33 41.33 38-50 36-29 34-42 32i75 31-25 29-83 27-58 25-58 23-92 ,22050 21-33 W33XI4l-Cl8X42-7 75-92 60-50 5)-OR 48-42 45-00 42-50 40.25 38.25 36-42 34.08 31-17 28-92 26.92 25-33 23-92 Hl6Xls0-C15133.9 66-33 52-83 66-25 42-17 39-25 37-00 35-17 33-58 32-00 30-67 29-00 26.83 25-08 23-58 22-33 W36X150-Cltl1422-7 77-42 61-67 540C8 49.25 45.83 43.25 41-08 39-08 31-25 35-58 33.00 30-50 28-42 26-67 25-17 WHEEL LOAD P, (kips) ' 29-58 15-92 30-58 37-25 38.58 51-00 40.08 52-92 42-00 55.25 42-25 55-33 43-50 57-08 44-42 57-50 46-00 59-25 46-75 59-61 59-67 71-25 61-67 13.11 63-50 75.17 59-92 52-58 41-92 44.58 41-92 39-50 35.17 31.58 28-83 26-58 24-15 23-17 21-92 2011S . 1 11-17 12-25 12.75 13-75 16-11 18-25 21-50 24.83 26-42 31-50 27.00 34-25 27-83 35-17 28-58 36-92 29-50 38-00 30.00 38-25 38.17 45-58 39-42 46.75 40-50 11. 1 18-33 20-00 21-08 13-25 28-58 32-92 32.25 42-50 33.67 44-25 33-92 44-42 34-92 45-75 35-59 46.08 36-15 47-42 37-33 47-67 47-67 56-92 49-25 58-42 50.67 75.1 8-75 9-83 10-58 12-00 13-92 15-50 18-00 20-58 21-83 25-75 23-58 27.92 23.15 28-67 2633 34-OB 27-25 35-00 27-83 35-50 35.42 42-17 36-67 43.50 31-75 15.Q 13-75 14.75 15-42, 16-83 20-17 23.00 27-5R 32-08 29.67 39-00 29-83 39-00 30-75 40-17 31-33 40-50 32-33 41-61 32-83 41-83 41-83 49-92 43-17 51-25 44-42 30.0 0-00 0-00 8-67 9-92 12-42 13-67 15-75 17-8? 18.83 22-08 20.25 23-75 20-50 24-42 24.50 29-50 24-33 29-58 25-92 32-83 32-0@ 36-11 34-50 40-83 35-50 35..) 0.00 0-00 7-42 8-42 10-83 12-42 14-17 16.00 16.83 19-50 17-92 20.92 18.25 21-58 21-58 25-67 21-50 25-92 23-61 28-50 27-92 31-42 32-50 38-42 33-50 411.r) 0-00 0-00 0-00 0-00 9-42 11-08 13-00 14-58 15.33 17-61 16-25 18-83 16-58 19-42 19-33 22-92 19-42 23-17 21-25 25-42 24-92 27-92 30-58 34.58 30-83 45.r) 0-00 0-00 0-00 0.00 8-33 9-75 12-08 13-50 14-17 16-17 15-00 17-25 15-25 17-83 17-75 20.83 17.75 21-08 19-33 23-00 22-67 25-25 27.50 31-00 27-83 50.0 0-00 0-00 0.00 0-00 0-00 0.00 11.00 12-58 13-25 15-08 14-00 16-00 14-25 16-50 16-42 19-17 16.50 19-50 17-92 21-17 20-92 23-25 25-08 28-25 25.42 55.0 0-00 0-00 0-00 0-00 0-00 0-00 9-92 11-92 12-96 14-17 13-17 15-00 13-42 15-50 15-33 17-83 15-50 18-17 16-75 19-61 19-42 21-58 23-17 26-00 23-50 60.0 0-00 0-00 0.00 0-00 0-00 0-00 9-00 10.83 11.83 13-42 12-50 14-17 12-75 14-58 14-50 16-75 14-58 17-08 15-75 18-42 18-25 .2O-17 21-58 24-17 22-00 70.0 0-00 0-CO 0-00 0-00 .0-00 0-00 0-00 0-OC 10-08 12-17 11-OC 12-83 11-42 13-25 13-08 15-08 13-25 15-33 14-25 16-50 16-42 18-00 19-11 21-33 19-50 65.0 0.00 0.00 0-00 0-00 0.00 0-00 0-00 0-00 10-92 12-75 11-92 13-42 12-17 13-83 13-75 15-83 13-92 16-11 11-92 17-42 17-25 19-00 20-33 22-67 20-61 75.0 0100 0-00 0-CO 0-CO 0-00 0-00 0-00 0-CO 9-33 11-58 10-17 12-33 10-67 12-15 12-58 14-42 12-75 14.61 13-5@ 15-75 15-61 11-17 18-25 20-25 18-58 m \ Q). -P ca. -- 0 4 a C -rl I a-l I I listed in part one of the AISC Steel Manual, ,. To use the table as a design aid, it is first necessary to select the correct table. There are listings for wheelbases of 4'4, 5'-0, 6'-0, 7 Also, the listings are repeated for varying lateral loads, P~ , which are expressed as functions or Px, i.e. P is ' Y either O'.@ Px, 0. lop,, or 0. 12Px. Thus, the designer must _ . use the table with the wheelbase. lateral load, and grade I. .'* . . -,-: .,,,. *: .A :,tl of steel that corresponds to the design situation. Once e, ..'i-si -* ,,, , < , -4 >iJ' I .> the proper table is selected, the column for P,, the direct . i *. wheel load, must 'be selected so that the design situation, The values of P minimum of 5 kips to a maxim- of 75 kips, with inter- mediate values of Px in multip'les of 5 kips. with' the correct value tor Pi selected, the de le'ngth in this column that is closest to tha actual length used. Obviously, the length selected cannot be less than the design length needed. The combination sectionathat corresponds to this length should be used. 'I, In constructing these design tabaes, some simpl,$ TO FIND: using Table 3-3, find the lightest suitable $b,rnn!$ ,', .4~~,l,, ,.:\ combination section for both 36 ksi and 50 ksi J 5.. grade steels. ., servative. This is dramatically shown in Example 3.1. In the conserva- tive method, the top flange of the combination section is assumed to carry the entire lateral force while in the torsional theory, the entire cross-section resists the lateral load. Since the first method over- estimates the compressive stress in the top flange, the method is con- sidered to be conservative. When a channel is mounted to the top flange of a wide-flange shape, . c which, in turn, is to be unsymmetrically loaded, three things are accom- . - k,.: '';':,'& plished. First, the stresses in the top flange due to the vertical f!, .. ,L-- 2. ,h8. 2 -, load are reduced since the neutral axis is shifted closer to the top. - Second, the bending stress in the top flange due to the lateral load is also reduced since there is a greater section modulus in that direc- tion. And third, the radius of gyration of the built-up top flange is increased and thus, the member is less 1 ikely to fail by latemT torsional buck1 ing. A1 so, the AISC specifications address this by permitting a larger allowable compressive stress. The resulting bui 1 t- up wide-flange shape is also more effective because the shear center is shifted toward the top flange, thereby making the torque on the sec- - 1. tion smaller so the warping stresses are, .in turn, lower. While the addition of the channel is beneficial for the top flange, it is detrimental to the tension flange. Since the neutral axis is shifted toward the top, the reduced, often by a very significant amount. This will, in turn, yield a higher bending stress in the bottom flange. This fact was highly evident when Table 3-2 was initially developed. Obviously, the per- &&! .' SECT ICh*.8X~'&PEb*r5X~'Yl'~7X~8tX'v5)I~*lY'#~3t'Rl' lexletet) .112--~'.'300- KR~TC tb* 1~3irl~~k~~l\rrrC~C,hTC~~~Ebr~ I,IX vf~ rPTrEt? -. - /13 1~3 ~C~~D~(~*,~~,*~*,T~~*-C~,IZ~*X*~F~~~~F~*Z~F? 02.FlC*L*f7~lvF& iP7.2 L lh h3lTE(hr 105) L 17 FC.:"JT(4>,' jiCTIC%',~OX,'K',iXfi'tS>,'Ci',7X,'L4b1,5X~'hh5') L LR hRITE(hr IOC!lHCTb,kThrkCTCrbTCtXU qAtCb,ihd,IrhE 119 LC6 FC3NbT(' 'd'tlir'Y' ,I?,~-C'rIZ,~X~tF4.1~F~~3~FE.Z,FlC-l~ZFS~2) 1 20 kRITE(h,l3C)PXtPY~S,ZPrFY 121 134 FCR!'41( lkO/2Xr' PX*' ,F5. 1,4Xt'PY=' rF5*294X, *bC;EELS&SE*'cF5.1~4X., L'PJIL HElGkT~'rFh.?,4X~'FY~'tF5*1/1PCI 122 SC4 IF (J J-1) S0Q rFCS FC2 L23 SCS ~PITE (6. 1311k~~k~h?k~kc'~~*n~~,~~ 1hb~ 124 121 FcPPPT(' ~'r[Z,'~'r[3r'-C*r12t*X*,F40L~5X**P431PL SPPhsqrF7-21 1' FT.'q$Xt'TENSICh $TRESS CChlRCLS'/lP1I 1 25 GC TI: 907 126 SC8 URITE~~~~~~)HCT~~~T~~~GTC~~TC~LFI~~L 127 122 FCfSP41(* 'A',I~*'X*, [Zr '-C'*(2,'X'tF4r1*5Xra'*3FI,YLP SFItha',F7.2, L* Ff .',:x,*CC~TRESS~~~ :TRESS CCAT~CCS'/lkLl 11.' cc TC ';:7 L Lv SCS hfl1TE(~~,13?lk~lk~~\ff'r~k~ili,~~TC 1 :a 1?3 FCFrb1 (' ~'~I?~'A'~~~~'-C'~I~,'X'~F~.~IS~~'J~LC~~~LE SPEAR 1 STGESS EXCEECEC1/ lk 1) 131 SC7 CChTIhCE 1 32 GC TC ZCC L 23 LCCC CChT IhLE STCP EhC SLeRCLTIPE ST'IESStft ,Rlr FYvF bLL1 R=EL/ET rLloSCPT (LCZOCO/FY I MZ=SCRT (SLCDCO/FY I - lF(E-GE-ELIIGC TC SLC FALL=C.CC*F" PvLarc .. --- -- IF(R-GT->LZ)GC TC SZC F8LLafY*(2./3*-FY*R**2/15300CC) PEltPh --.- ---. 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