THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Civil Engineering Technical Report THE EFFECTS OF SHEAR DEFORMATION AND AXIAL FORCE IN BATTENED AND LACED STRUCTURAL MEMBERS Fung Jen Lin ORA Project 05154 under contract with: DEPARTMENT OF THE NAVY BUREAU OF YARDS AND DOCKS CONTRACT NO. NBy-45819 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR May 1968 Distribution of this document is unlimited

ACKNOWLEDGMENTS The author is deeply indebted to Professor Bruce G. Johnston, co-chairman of his doctoral committee, for suggesting the topic of this dissertation. His efforts of guidance during its preparation and reviewing the complete manuscript have been extremely valuable. The author wishes to thank all the other members of his doctoral committee for their valuable suggestions. He is also grateful for the initial research work that has been contributed by Dr. E. C. Glauser. The author wishes to express his appreciation to the Department of Civil Engineering, The University of Michigan, for granting a, tuition fellowship (paid by Support of Graduate Education NS 34570), and also to the Bureau of Yards and Docks (ORA Project 05154, administered by The University of Michigan's Office of Research Administration, and directed by Professor Bruce G. Johnston) for financial support under which much of the research was carried out, and also to the Computing Center of The University of Michigan for the use of the IBM 7090 computer and the Calcomp 763/780 digital plotter for the numerical computation and plotting the numerical results. The final typing and reproduction of the thesis was undertaken by the Office of Research Administration, The University of Michigan. For this valuable assistance the author is sincerely grateful. ii

TABLE OF CONTENTS Page LIST OF TABLES iv LIST OF FIGURES v LIST OF SYMBOLS vii ABSTRACT xi Chapter I. INTRODUCTION 1 II. THE SHEAR FLEXIBILITY OF STRUCTURAL MEMBERS 3 2.1. Definition of the Shear Flexibility 3 2.2. The Shear Flexibility of Laced Structural Members 5 2.3. The Shear Flexibility of Battened Structural Members 12 III. THE STIFFNESS PROPERTIES AND FIXED END MOMENTS OF STRUCTURAL MEMBERS WITH RIGID STAY PLATES AND CONSTANT SHEAR FLEXIBILITY 23 3.1. The Shear Flexibility of Solid-Web Structural Members 23 3.2. Fundamental Differential Equations 25 3.3. The Modified Slope-Deflection Equations 31 3.4. Fixed End Moments 55 IV. ELASTIC BUCKLING OF BATTENED OR LACED STRUCTURAL MEMBERS WITH RIGID STAY PLATES AND CONSTANT SHEAR FLEXIBILITY 74 4.1. Column with Hinged Ends 74 4.2. Column with One End Fixed and the Other Hinged 64 4.3. Column with Fixed Ends 75 V. NUMERICAL EXAMPLES OF BEAM AND FRAME ANALYSES 78 5.1. A Beam Analysis 78 5.2. A Frame Analysis 82 VI. SUMMARY AND CONCLUSION 88 6.1. Summary 88 6.2. Conclusions 90 APPENDIX 94 REFERENCES 112 111

LIST OF TABLES Table Page 5-2(1). The Geometric Arrangements and Elements' Properties of the Frame Structure 84 5-2(2). The End Moments (kip-ft) of the Frame Structure, and Its Correspondent Errors (%) which Occur in the Effects of Shear Deformation and Axial Force are Neglected 86 I. The Constant Shear Flexibility Parameter,I for Laced Structural Members 95 II. The Constant Shear Flexibility Parameter,u for Battened Structural Members. The Longitudinal and Batten Elements are WF Shapes and Channels Respectively 100 iv

LIST OF FIGURES Figure Page 2-1. Element of length "a," of a structural member under (a) bending, and (b) shear. 3 2-2. Five different lacing configurations for the laced structural members. 5 2-3. Typical panel of laced structural member. 7 2-4. The optimum arrangement of the diagonal elements. 11 2-5. Typical panel of battened structural member. 13 2-6. The limitation of the maximum local slenderness ratio a/r of the battened structural member with different connection factors i. 21 3-1. The structural member with rigid stay plates at the ends of the member. 24 3-2. The deformation of infinitesimal element dx, cut out from the structural member, due to shear force. 26 3-3. The structural member A-B subjected to an arbitrary external loading p(x) and joint displacement. 27 3-4. The slope-deflection constant C11 and carry-over factor rl2 of structural members with different rigid stay plates 6 and constant shear flexibility [i. 39 3-5. Two sets of force systems acting on the same structural member. 55 3-6. Fixed end moment Mfa for concentrated load W. 57 3-7. Fixed end moment Mfa for uniformly distributed loading w. 58 3-8. Fixed end moment Mfa for the moment loading M. 58 3-9. Fixed end moment Mf for concentrated load W with compression axial force. 61 v

LIST OF FIGURES (Concluded) Figure Page 3-10. Fixed end moment M for uniformly distributed load w with compression axial force. 61 3-11. Fixed end moment Mf for the moment loading M with compression axial force. 62 3-12. Fixed end moment M of structural members for a uniform fa, load w over full-length L, with different rigid stay plates 6, and constant shear flexibility A. 66 4-1. The critical loads of columns with different end conditions, constant shear flexibility pt, and rigid stay plates 6. 77 5-1(1). A structural member 0-1-2. 78 5-1(2). The battened structural member 1-2; (a) geometrical arrangement and (b) cross-section. 79 5-1(3). Moment diagrams for the Cases 1, 2, 3, and 4 for various numbers of panels. 81 5-2(1). A two-bay symmetrical frame structure. 83 5-2(2). Loading conditions. 85 5-2(3). Moment diagrams of the frame structure for the loadings 1, 2, and 3. 87 vi

LIST OF SYMBOLS A Area. Integration constant. Left end of structural member Ab Total cross-sectional area of batten or strut elements in one unit panel A Cross-sectional area of one of two longitudinal elements in a built-up structural member (half of total cross-sectional area of longitudinal elements in a symmetrical section) A Total cross-sectional area of diagonal elements acting within the unit-panel length "a" B Integration constant. Right end of structural member Cll...C22 Slope-deflection constants E Modulus of elasticity F Minimum yield point of the type of steel being used F.S. Factor of safety G Shear modulus I Moment of inertia of the entire longitudinal cross-section (excl-uding web material) Ib Moment of inertia of the battens of cross-sectional area A I Moment of inertia of the longitudinal element of cross-sectional area A c L Overall length of structural member with end stay plates M Bending moment M,Mab End moments of structural member M Fixed end moment at left end of the member fa M Moment loading vii

LIST OF SYMBOLS (Continued) M Bending moment due to external loading (excluding axial force) in a simple beam M Bending moment at a distance x from left end of the member x P Axial load in the structural member P Critical load of a column which is fixed at both ends cr, f P Critical load of a column which is fixed at one end and hinged at the other end P Critical load of a column which is hinged at both ends cr,h J2EI P Euler load for a column without stay plates, P L= P. P., for i = 1, 2, are the axial forces of the longitudinal sube1 ements as shown in Figure 2-5(b). These axial forces will vary from panel to panel along the length of the member, will be different on the two sides of longitudinal elements V,Vl1,V2 Constant shear forces V Shear force at right end of member V Shear force at left end of member, total stresses of strut elements as shown in Figure 2-3(b) Vd Total stresses of diagonal elements as shown in Figure 2-3(b) V Shear force due to external loading (excluding axial force) in a simple beam V Shear force at a distance x from left end of the member x W Concentrated load Z Semi-rigid connection constant a A unit-panel length in the lacing or batten arrangement of a builtup structural member b Distance between the axis of the two longitudinal elements of a built-up structural member viii

LIST OF SYMBOLS (Continued) k Nondimensional parameter, Equations (3-27) and (3-40) k* k* = (1 - 61 - 82)k k Nondimensional parameter, Equation (2-26) o 1 Length of a member, effective length of a structural member between end stay plates b m Height-length ratio between stay plates, m = - n Number of panels, n =p(x) Distributed load per unit length r12,r21 Carry-over factors rb,rc Radius of gyration of batten, longitudinal elements, respectively, where rb = I/A r2 = I /A Ib/Ab c c c s Slope of the lacing elements (diagonals) with respect to batten elements, s = a a b sopt. Optimum slope of s, for which [ is a minimum value opt. w Uniformly distributed load per unit length x Coordinate axis coinciding with the axis of the structural member in the undeflected rate y Deflection of the structural member Load parameter, a = P/P Shear angle 651,9 } 2 Length factors of stay plates 6 Length factor of stay plates, where 6 = 61 = 62 A1... 4 Shear displacements ix

LIST OF SYMBOLS (Concluded) %.Y\b'nc Shear shape factors o Joint rotation O 0'b End rotations of the structural member a b Nondimensional shear flexibility parameter Cl*f C1* = (1 - 6 - 62) 4 ra' b Connection factors of longitudinal, strut elements, respectively p Length factor oi. Actual stresses in the longitudinal sub-elements, where a. = P /Ac, for i = 1, 2 Rotation of cross-section per unit length due to bending moment alone Coefficient, Equations (3-66), (3-80), and (3-92) A12 Coefficient, Equations (3-67), (3-81), and (3-93) ~B12 md Amplification factor, Equations (2-24), (2-25), (2-28), and (2-29) Amplification factor, Equations (2-31) to (2-34) Member rotation x

ABSTRACT This dissertation presents a theoretical analysis of the elastic behavior of both the battened and laced structural member considering the effects of axial load, shear deformation, and connection rigidity of sub-elements, and the overall effects of axial load, moment gradient, and effective shear deformation of the complete member. The analytical solution is used to obtain modified slope-deflection equations to generalize a relation between applied forces and joint displacements. Types of the structural members may be catalogued here according to three different web configurations such as solid, battened, and laced. The arrangements of web elements are assumed to be the same throughout the effective length of the battened and laced structural members between end rigid stay plates. A nondimensional parameter, the shear flexibility, is defined so as to characterize the shear flexibility of the structural members and to take account of the effects of axial force, local joint connections, and local connection flexibility of the battened members. The fundamental linear second-order differential equation for the deflection curve of the structural member which includes the effect of shear deformation has been derived. The general solutions of this differential equation are of a fundamentally different nature for the cases of no axial force, compression, or tension axial force. By application of the natural boundary conditions to the general solution of deflected shape of the structural member, the solutions are set up in the forms of slope-deflection equations. In the xi

evaluation of the fixed end moments for a concentrated load, the reciprocal theorem is applied so as to make use of the deflection curves of the members which have been previously defined in the case of the homogeneous solution. From this basic expression one can derive fixed end moments for any combination of concentrated loads by simple summation or for continuously distributed loads by integration. Elastic buckling loads for structural members with rigid stay plates and constant shear flexibility have been evaluated for the cases of a column with hinged ends, a column with one end fixed and the other hinged, and a column with both ends fixed. Finally, numerical examples of beam and frame analyses are presented to provide a comparison with the ordinary beam and frame theories which neglect the effects of shear deformation and axial force. xii

CHAPTER I INTRODUCTION In addition to the deflection due to elongation and compression of fibers from bending moment, there is a further deformation due to shear and axial force and consequent strains in a beam. This is not usually considered in the analysis and design of frames made up of structural members of solid crosssect.on for which the influence of shear deformation is usually very small. This is due to the fact that the shear deformation is resisted in solid structural members by a continuous web which participates uniformly in the transmission of the shearing forces. The distortion caused by the shearing stresses in such a. case is relatively small except for very short members. However, the conditions are different in battened or laced built-up structural members, in whoich case the contribution of shear deformation to the total deflection may be appreciable. By neglecting the deformation due to shear, errors of considerable magnitude may be introduced in frame analysis. Many studies have been made by different investigators, such as Engesser (1,2) (3) (4) (1391',(' Muller-Breslau (191k.), Timoshenko (19356), Amstutz and %6) (qY4) (12) (13) Stuissi (1941),') Pippard (1948), Takekazu (1951), )Bleich (1952) 3) (1) Ppand (19o5), J,-nes 1952),(1) Koenigsberger and Mohsin (1956), (16,23)and Tomayo (1965)(22) to determine the critical loads as well as frame behaviors of built-up columns as affected by shear. More recently, Williamson and Margolin (1966),') have studied the effect of shear deformation on shears and moments in laced guyed 1

2 towers. Glauser (1967) has studied the shear effect here at The University of Michigan, but has not considered the effect of axial force or shear in the local sub-element. A nondimensional parameter of shear flexibility [i will be introduced to characterize the shear flexibility of battened and laced structural members. The parameter i will be evaluated so as to take account of the effects of axial force, local eccentric joint connections, and local connection flexibility' (11) of batten structural members. The effect of rigid stay plates at the ends of the structural member will be considered in the evaluation of the deflection curve for the member. The purpose of this thesis is to develop a. reasonably accurate yet comparatively simple evaluation of the shear flexibility 4 for a wide variety of cases, and to generalize the modified slope-deflection equations(5' ) and elastic stability for the battened and laced structural members. The shear flexibility parameter p., limitation of the maximum local slenderness ratio, slope-deflection constants and carry-over factors, fixed-end moments, and critical loads of the built-up structural members will be evaluated for structural design office use. (18) It is a further purpose to develop criteria to guide the designer to a decision whereby he might with reasonable accuracy neglect the effect of shear and axial force, either in a, battened sub-element, or in an entire member.

CHAPTER II THE SHEAR FLEXIBILITY OF STRUCTURAL MEMBERS 2.1. DEFINITION OF THE SHEAR FLEXIBILITY A nondimensional parameter p. is introduced to characterize the shear flexibility of structural members for a wide variety of particular battened or lacing arrangement. Consider an idealized element of the structural member whose length is defined by "a." An equilibrium state of the free element is shown in Figure 2-1. In general, this element is acted on by the axial force P, the shearing force V, and the bending moment M. The length of the uniform cross-section of a structural member between rigid stay plates at the ends of the member will be denoted as "Q." As Figure 2-1 shows, we separate the state of deformation into two parts: (a) bending, and (b) shear, and for each of these the axial force will be considered. simultaneously. The state of bending causes P P P.a a, (a) (b) Figure 2-1. Element of length "a" of a structural member under (a) bending, and (b) shear.

4 a rotation Oa of the cross-section while the state of shear causes the deformation 7 as shown. The shear flexibility is defined as the ratio of the changes in slope in length "a" due to shear deformation and due to the bending rotation, which is equal to y/Oa. Under bending alone the change in slope in length "a," is Ma EI While under shear alone, the change in slope is TV AG where EI is the flexural rigidity of the structural member, AG is the shear rigidity of the structural member, n is the shear shape factor. Thus the ratio y _ EI V (2.1) Oa aAG M It is desirable to define a parameter which will be equal to the foregoing ratio for a specific relationship between V and M and having the dimension 1/L. In this study V/M will be taken as a/X2 which is the same as that (5) (12) (8) adopted by Washio, Takekazu, Glauser. Other investigators Maugh, f4 (21) _ II (24) Gere, Williamson and Margolin have used ratios of V/M differing only by a numerical coefficient.

5 In evaluating the shear flexibility parameter [t, the effects of shear deformation as well as axial force on the behavior of the member sub-element will be considered. It should be noted that, with shear deformation considered, the deformed cross-section of the member is no longer a, plane perpendicular to the tangent of the deflected axis. 2.2. THE SHEAR FLEXIBILITY OF LACED STRUCTURAL MEMBERS Types of laced structural members may be catalogued according to five different lacing configurations as shown in Figure 2-2. They are parts of the laced structural members which consist of two main longitudinal elements, lacing b (a) (b) (c) (d) (e) Figure 2-2. Five different lacing configurations for the laced structural members. (diagonal) elements, with or without strut (transverse) elements. The two longitudinal elements are connected in one, two (or more) planes by the lacing bars and strut elements which serve as the web of the member. The assumption

6 of an equivalent solid member not requiring a consideration of shear deformation can be made if the properties and geometric configuration of the subelements are such as to make the shear parameter [ relatively small. The effect of shear also depends greatly on the moment gradient, which is a measure of the shear to moment ratio. The two main longitudinal elements are assumed to form a symmetrical section. A is the cross-sectional area of one of the two longitudinal elements. c The properties of both the diagonal and strut elements, are the same throughout the length ~ of the member. Ad is the total cross-sectional area, of all diagonal elements within one panel length "a." Ab is total cross-sectional area in one unit of strut elements. The force equilibrium condition of a typical panel is shown in Figure 2-3(b), assuming hinges at the ends of all strut and diagonal elements. Then we can obtain forces in the elements in terms of shearing force, where 1+ e V1 = v + V (2-2) 22a (2-3) a v2 = ~ v (2-3) V = - V1 - V2 = - (2-4) ab a, 1 X1V (2-5) vd l a be determined. Firstly, consider the shear displacement A1, a~s shown in

Note O indicates assumed hinges v 2~~~~~~~~~~~~~~~~~~~~~~~CV Vb 2~ 2 1 - - -1/.% //:/;I/ I / I / I I /2Cd configuration, (b) equilibrium relation, (c) equilibrium of internal i b (a) (b) (c) (d) Figure 2-5. Typical panel of laced structural member; (a) geometric configuration, (b) equilibrium relation, (c) equilibrium of internal forces, (d) shear deformation due to the lengthening and shortening of the diagonal elements, and (e) shear deformation due to the shortening of the strut elements.

Figure 2-3(d), due to the lengthening and shortening of the lacing elements in each panel. Then we obtain 3/2 A = 1 + (2-6) a, d Secondly, consider the shear displacement A2, as shown in Figure 2-3(e), due to the shortening of the strut elements in each panel. Then we have b Vb A2 = (2-7) a b where e and 5b are the connection eccentricity factors. They are dependent on the local joint geometry. Ab is the total cross-sectional area in one unit (panel) of strut elements. Therefore, the total angular rotation y caused by the shearing force V alone is A1 + A2 Y 1/2 (1+ + )a a 2b a 2 — 3/2 A =+ +. (2-8) 7 + EA a, + a, The state of bending causes a rotation 1a of the cross-section due to moment M, as shown in Figure 2-l(a), while the state of shear causes the shear deformation 7. VI2 Oa = EI (2-9) where I = moment of inertia of the member which is excluding web elements

9 I= 2 1 +' b2 A (2-10) r and A are the radius of gyration and the cross-sectional area of one of c C two longitudinal elements in a built-up structural member (half of total cross-sectional area of longitudinal elements in a, symmetrical section), respectively. I is the length of that portion of a member which has uniform panel arrangements with the same repeating properties of the web elements throughout. For laced structural members in general, 2 2 ( b << then I = b2 A (2-11) 2 c Then we get 4a = 2(L)2 V (2-12) b EA c From the definition (2-1), we can now obtain the following expression for the shear flexibility parameter Ii of laced structural members as follows: b 2C b 1 a, b d a da a, a b(2-13) If we change variables, s = slope of the diagonal elements with respect to strut elements.

10 a s = -T (2-14) b m = - (2-15) then the expression for [i becomes ____ Ac i(, 2)3/2 _ b 2 C _( _ d _ m + — + A (2-16) A..A s Ab s A Moreover, if we introduce a. new variable n, n = - (2-17) a, Then Equation (2-13) may be transformed to the expression: b2 = (I nb n + d+ m Ab (2-18) P-1 ( 1 + t a ) t a ~ m _~njl~ jC ~t \ + A A It is noted that the last term of Equations (2-13), (2-16), and (2-18) represent the contribution of the strut elements, Ab. This term should be omitted whenever strut elements, as shown in Figures 2-2(c) and 2-2(e), are missing, or whenever the strut elements, as shown in Figure 2-2(b) and 2-2(d), do not take part in the transmission of the shearing force of the structural members. We observe in Equation (2-16) that the value of the shear flexibility parameter [i becomes infinite as the slope of diagonal elements approaches either zero or infinity. Thus there are optimum values of s which will minimize the shear flexibility parameter, i. It is, therefore, important to know this geometric arrangement of diagonal elements.

11 Let s t. epresent the value for which the correspondent value of shear flexibility p becomes a minimum value. It implies that the first partial derivative of p. with respect to s is equal to zero. Therefore, the minimized condition is: A (l + opt )1/(2spt 1) - = 0 (2-19) The optimum slope s of the diagonal elements is plotted in Figure opt. 2-4 as a function of the area ratio Ad/Ab. It is noted that the minimum slope is NU/2, or 35.27 degrees for the case where strut elements are missing or stress-free and thereafter increases with the ratio Ad/Ab. 1.2....4 1.1. I. 5 15 2 25 3.9 H.8 0.7.' - AREA RATIOS Ad Figure 2-4. The optimum arrangement of the diagonal elements. If we substitute the value Ad/Ab from Equation (2-19) into (2-16), and restrict the value s to Sopt.' an expression for the minimum value of shear flexibility is obtained for the laced structural members:

12 2~b 1/2 A. m S opt.(1 s (2-20) min. 1 + Sa, opt. opt. A The values of the shear flexibility parameter jt, or min. of the laced structural members can be simply obtained from the Equations (2-13), (2-16), (2-18), or (2-20). The effect of rigid stay plates at the ends of the members will be introduced later. 2.5. THE SHEAR FLEXIBILITY OF BATTENED STRUCTURAL MEMBERS Figure 2-5(a) shows the basic elements of the battened structural member, which consist of two main longitudinal elements joined by batten elements. The two longitudinal elements are assumed to form a symmetrical section and connected, in one or two (or more) planes of batten elements, by means of rigid or semi-rigid joint connections. The batten elements serve as the web of the member, transmitting shear force in the member by virtue of their own shear and moment resistance, in combination with the local bending of the longitudinal elements. The geometric arrangement of the battened structural members is based on the typical unit of length a, and width b. The properties of the battened structural members are characterized by the moment of inertia, of the batten and longitudinal elements, and the semi-rigid connection constant which vanishes for a perfectly rigid joint connection. As we know, the battened structural members are highly redundant structures whose exact solution may be extremely laborious. We may assume, however, that the points of inflection in the batten elements of a symmetric structural system are at the midpoint, and that for the longitudinal elements with adequate

Note 0 indicates assumed hinces b b A _ _ - V 2 4 lb 2 metrical configuration, (b ) force equilibrium, and (c) shear deformation. metrical configuration, (b) force equilibrium, and (c) shear deformation.

14 shear resistance, this is also approximately true. When hinges at the midpoints of all elements are assumed the battened structural member becomes statically determinant. To determine the additional slope y of the deflection curve due to shear force V, we first consider the lateral displacement A3, as shown in Figure 2-5(c). This is due to the shear force V/2 with the axial force P. as follows: (a a)3 V A3 = 8 EI d (2-21) c Where M P x P1 + -- (2-22) 2 b M - X P2 - (2-23) d tan - 1)... Amplification factor (2-24) d k-~ k 0 0 or 2 2!7 4 62 6 + = k k1 k +... (2-25) d 5 o 105 o 94 o 2 2 _2 1 a, 0= EI 4E(r) (2-26) c c P. 1 = - (2-27) A c M is the bending moment at a distance x from end of the member. x P, for i = 1, 2, are the axial forces of the longitudinal sub-elements

15 as shown in Figure 2-5(b). These axial forces will vary from panel to panel along the length of the member and will be different on the two sides. I = the moment of inertia of one of the two longitudinal elements in a battened structural member. E = modulus of elasticity. If P is tension, then / taanh k\ 3 2 ~) (2-28) d k k / 0 0 or 4 k2+iL k4 62 6 d 5 o 105 o 945 o (2-29) Second, consider the lateral displacement Al, as shown in Figure 2-5(c), due to the angular rotation at the end of the batten elements as follows: The angular of rotation G, as shown in Figure 2-5(c), at each end of the batten elements is: abV 12 (2-30o) 12 EIb m where tan k =..Amplification factor (2-31) m k 0 or +1 - k2 +2 k4 +17 k6 + m 3 o 15 o 315 o

Ib = the total moment of inertia in one unit panel of the batten elements. In case of tension P, then ta.nh k = o0 (2-33) m k o or = 1- k2 + + k4- k6 +.. (2-34) m 3 o 15 o 315 o Then we can obtain a A1 = - 0 2 ab V = b (2-35) 2'I4 EI m 2 b Third, consider the lateral displacement A2, as shown in Figure 2-5(c), due to the shear deformation of the batten and longitudinal elements is aV (a b carc A2 = - + 2GAc) (2-36)'where fb and fc are shear shape factors of the individual batten and longitudinal elements, respectively, which depend on the shape of the individual element cross-sections. G - shear modulus a = connection factor'a,

17 Fourth, consider the lateral displacement A4, as shown in Figure 2-5(c), due to a semi-rigid connection. A4 a Z V (2-37) 4 m Equation (2-37) is restricted to small deformation due to connection nonlinearity and implies the condition: aZP i -aZP- (2-38) <K Z = semi-rigid connection constant, (78,10) i.e.: Z = 0 for rigid connection, Z = oo for hinged connection. From these results, the overall shear deformation due to shear force V is given by: Al + A2 + A3 + A4 7y = a/2 3 2 _a__ __ic + a aZ -Y = V I ab 0 + + a c + -a 0 + aZ ) (2-39) 12 EIb m bGAb 2GA 24 EI d 2( L.2E~~b b c c 2 m From Equations (2-9) and (2-10) we have a moment rotation 4a due to bending alone. This leads again, according to the definition (2-1), to the following expression for the shear flexibility parameter ~t of battened structural members: CJ- + 2;I 6- A A -3 ~~L92+ (k)]b~ s.2m % 2+2.6tal +-.(=-) +-2 b c cr ~ + 2a c a (a I-~~ C J~~~~~~~~~~

18 Again if we introduce new variables n = 1/a and m = b/~ we have a modified expression for.i: A 3.2% A A3 A + + (m)2m 2 C 1 + _ C+ 2.6+ + a2 2 + 2() + L 2 L6n rb b m mn Ab a c 12n r d n m (2-41) It is noted that the last term of Equations (2-40) and (2-41) represents the contribution of the semi-rigid connections. This term will vanish in case of perfectly rigid connection. Welded connections designed for full moment and joint shear may be assumed to provide full continuity, i.e., they are then termed "rigid." We observe by Equation (2-41) that the value of the shear flexibility parameter p. becomes infinite either as the number of panels n approaches zero, or as the semi-rigid constant Z approaches infinity, and the value of p. approaches zero as n approaches infinity which leads to the case of the solid structural member. Then from Equation (2-41) we have 1 m (2i = 2.6ro 2 + (_)2 (2-42) lim n+*o rc The values of the shear flexibility parameter np of battened structural members can be obtained from the Equations (2-40) and (2-41). The effect of rigid stay plates at the ends of the members will be introduced later.

19 In an actual situation, the axial forces P. (for i = 1, 2, as expressed in Equations (2-22) and (2-23)) of the longitudinal sub-elements, as shown in Figure 2-5(b), will vary from panel to panel along the length of the member, and will be different on the two sides of the longitudinal elements, due to the bending moment M. However, it is desired that the shear flexibility p. as x a constant value for the entire length 2 of the member in this analysis. To permit the assumption of constant shear flexibility A, we should determine a correct limit on the ratio of the local slenderness a/r, so that the influence of the axial forces P. will be small. P. (for i = 1, 2) are caused by the 1 1 axial force P, the bending moment M, or by these in combination, so as to develop the maximum allowable axial stress, in any one of the longitudinal subelements under any condition of external loading. The expressions for the amplification factors md and X, in Equations (2-25), (2-29), (2-32), and (2-34), respectively, are functions of the axial forces Pi or the axial stresses a., 1 1 and are approximately equal to 1 when the nondimensional parameter k is less than 1/3. Therefore, the analysis is applicable and of sufficient accuracy in the range of k < 1/3, so that the shear flexibility parameter t will be nearly O - a constant value through the entire length ~ of the battened structural member. Therefore, we have the condition: = 1, for k < 1/3 (2-43) d m o- / This approximation may cause maximum localized errors of the amplification factors ad' m n not more than 4.7%.

20 From Equation (2-26) and the condition (2-43), we obtain the limiting ratio of local slenderness to be: 2k a, 0 E (2-44) r a. c a \1i The maximum local slenderness ratio a/r will be obtained from Equation (2-44), by simply considering the factor of safety, F.S., which is defined by the applicable specification. Then the limiting condition, for the battened structural members, shall be such that the ratio of local slenderness is defined by: ar < (,2 F (2-45) rc M-a (F.S.). 1 where (F.S.)oi < F 1 y F is specified minimum yield point of the type of structural steel to be used. This limitation of the maximum local slenderness ratio a/r with different connection factors ~ is provided as shown in Figure 2-6. a It is noted that the upper bound value (for rigid connections) of the nondimensional parameter k is T/2, for which the amplification factors 1d and ~ become infinity. Then the battened structural member will collapse due to local (premature) failure. Therefore, we can conclude that if the local slenderness ratio a/r reaches n/a E/(F.S.)i, premature local failure will occur. The proposed limitation provided by Equation (2-45) should provide an adequate safety against this possibility.

21 70 60 50 H 40 10 5 10 15 20 25 30 35 40 45 50(ksi) (F. S. ) i Figure 2-6. The limitation of the maximum local slenderness ratio a/r of the battened structural member with different connection factors t. a

22 For the case of k in the range of 1/3 < k < T/2, the effect of axial load in the longitudinal panel elements should be considered. The shear flexibility parameter Mi will vary from panel to panel along the length of the member. A trial method may be used to obtain an average parameter [t for the entire length of member. Alternatively, each panel may be considered individually and a numerical method utilized, but this is outside the scope of this dissertation.

CHAPTER III THE STIFFNESS PROPERTIES AND FIXED END MOMENTS OF STRUCTURAL MEMBERS WITH RIGID STAY PLATES AND CONSTANT SHEAR FLEXIBILITY 3.1. THE SHEAR FLEXIBILITY OF SOLID-WEB STRUCTURAL MEMBERS The shear flexibility parameters, it, for battened and laced structural members have been established. Although the influence of shear deformations on the properties of structural member with solid webs is usually small, except for very short members, we need to relate the built-up structural member to an equivalent continuous solid member as a function of the parameter t. The relationship will be clear when the shear flexibility of a solid structural member (again characterized by the parameter [) is given by the following expression: The angular rotation y caused by shearing force V in a solid member is: = AG (3-1) where = shear shape factor of the solid structural member. A = the cross-sectional area of the solid structural member. The bending rotation ~a caused by the bending moment (again V22/a) alone is: VI2 a V2 a =- a = (3-2) 23

24 where I = the moment of inertia of the solid structural member. According to definition (2-1), the shear flexibility parameter p of the solid structural members is, therefore, obtained as follows: = ^GQ2 (3-3) where E and G are the modulus of elasticity and the shear modulus of the material, respectively, for structura.l steel, the value of the ratio E/G is equal to 2.6. It is noted that the parameter [i of the solid structural members is identical to the parameter 4 of battened structural members when the value n (number of panels) approaches infinity in Equation (2-42). A relation for the shear flexibility of a structural member with rigid stay plates will be developed. Let L be the total length of a, structural member with the length of rigid stay plates included. Then we ha-;e the geometric relation as shown in Figure 3-1. The structural member which has constant shear flexibility and bending stiffness is attached to supports A and B by means of rigid stay plates of length b1L and 62L at each end of the members. A B, ~ Figure 3-1. The structural member with rigid stay plates at the ends of the member.

25 From Figure 3-1, we have L = I + 61L + b2L = (1- 6 - 62)L (3-4) where 61 and 62 are nondimensional factors relating the length of the rigid stay plates to the total length L of the member. From Equations (3-3) and (3-4), we obtain (1 - 81 - 62)2 AGL2 It should be remembered that the application of [i by Equation (3-5) is only applicable in the region B. 3.2. FUNDAMENTAL DIFFERENTIAL EQUATIONS The contribution of the bending moment M and the shear force V to the x x deformation of an infinitesimal element dx, cut out from the structural member, is illustrated in Figure 3-2. The unit angle change ~ due to bending moment and the angular rotation 7 due to shear force, as shown in Figure 3-2, are given by the well-known formulas: M = _ -x (3-6) El

26 r i ra- -1x x _ a yt Figure 3-2. The deformation of infinitesimal element dx, cut out from the structural member, due to shear force. and Vx ~ = (3-7) AG The additional slope (the shear strain y) of the deflection curve due to the shear force V is seen from the detail of cut out section. Adding this shear x rotation 7 to the slope y', due to bending moment M only, gives the total m x slope: y' = y' + 7 (3-8) m Differentiating with respect to x produces y = yt + 7 (3-9) where y = = x m EI

27 Equations (5-6), (3-7), and (3-9) lead immediately to the fundamental linear second-order differential equation for the deflection curve of the structural member, i.e., M ~V y + - - ( )' = 0 (3-10) EI AG Consider now Figure 3-3 which presents the structural member A-B subjected to an arbitrary external loading p(x) in a general condition. p(x) -- P a, ( y -x _ Mb y! Figure 3-3. The structural member A-B subjected to an arbitrary external loading p(x) and joint displacement. Let M and V be the moment and shear force due to an arbitrary external loading S s p(x) on a simply supported beam of length L, when the axial load P is not acting. Then the total bending moment M, and the total shear force V, at a distance x x x from the support A in the member A-B, are simply given by: M = M + M +Vx+Py (-11) x s a a

28 V =V + V + Py (3-12) x S a m where M and V are the end bending moment and shear a a force, respectively. From Equation (3-8) is derived Ym = Y' - 7 m Substituting y' into Equation (3-12) which becomes m V 1.. (V + Va + PY') (3-13) GA Substituting the expressions for M and V from Equations (3-11) and (3-13), x x respectively, into Equation (3-10), we obtain 1 P 1 __a EI Y EI (Ms+ M + Vax) + GAsVa (3-14) EI EI s a a Qs+ ( GA where 31L < x < (1 - 32)L It is noted that Equation (3-14) illustrates a most important property of this kind of the second-order theory. As long as the axial load P is kept constant, then the effects of end moments, end-shear forces and the external loading p(x) can be superimposed. The ordinary beam theory as modified to include shear can still be applied since the effect of axial load P appears here only in a modification of the stiffness properties of the structural member.

29 We now develop relations for the stiffness properties of the structural member. For convenience we let the external loading p(x) be equal to zero. Then from the equilibrium conditions we obtain: M =V - 0 (3-15) s s v =Vb (3-16) a b M + Mb V= - L (3-17) b L where * is the member rotation. Substitute these values of M, V, Va, Vb from the above expressions into Equations (3-11) and (3-13) for the bending moment and shear which becomes: M Ma Ptx - (M + Mb) + Py (3-18) x a aML M + X 1 a Vx = -- ( P + PY') (3-19) GA Then the fundamental differential Equation (3-14), for the flexible part, leads to: 1 +P 1 +M (3-20) 1+ y E, Y EI [P'x + (Ma + Mb) M (3-20) GA The general solutions of this differential equation are of a fundamentally different nature for the cases when the axial load P is greater than zero (compression axial force), equal to zero, and less than zero (tension axial force). The general solutions of the differential Equation (3-20), and their first derivatives are of interest and are given by the following expressions:

3o0 (1) Case P = 0 (No axial force) L x3 L x2 y=(M + b) EI - M -- + Ax + BL (3-21) Ma 6L a EI 2L L x 2L x' (M EI - M - — + A (3-22) (MMa EIa EIL (2) Case P > 0 (Compressive axial force) Ma+Mb x Ma y = AL cos k- + BL sin k- + xx + (3-25) L L P L P M + Mb x x a y' = - Ak sin k- + Bk cos k- +L + + (-24) L L PL (3) Case P < 0 (Tension axial force) M + M M x x a bx a y = AL cosh k- + BL sinh k- + Bx - (3-25) L L P L P M + M x x a b y = Ak sinh k- + Bk cosh k-+ - + PL(3-26) L L PL where k= L ( + ) (3-27) EI GA A and B are integration constants. Four boundary conditions are available for the evaluation of the integration constants A, B, and either the end-moments M, Mb, or the end-rotation @, a b a' Ob of the structural members. The four boundary conditions are: (i) x = L, y = 61L (3-28) M + M (ii) x = 51L, Y' = (1 + GA) 0 - A (Pt + ) (3-29) GA a ~'A T

31 Similarly, (iii) x = (1 - 62)L, y = (t-0b62)L (3-30) M + Mb (iv) x = (1 - 62)L, y' = (1 + A (p* + (3-31) GA b GA L 3.3. THE MODIFIED SLOPE-DEFLECTION EQUATIONS By application of the four boundary conditions (3-28), (3-29), (3-30), (3-31) to the general solution of deflected shape of the structural member, the coefficients for the modified slope-deflection equations may be determined. The equations are: EI M = L [C11+a + C129b - (C11 + C12),] (3-32) EI b L a b N = L [C,,Qa + C22Qb - (C21 + C22) tI (3s33) Or, to show the relation to the moment-distribution procedure, they may be written: EI M = C11 L [9 + r129 - (1 + r12) ] (3-34) a L a b EI Mb = C22 L [r21@ + @b - (1 + r21) 8] (3-35) 0 L a b where C11, C12, C21, and C22 are the well known slope-deflection constants which depend upon the properties of the structural member and, in addition, are functions of the axial load P. rl2 and r21 are carry-over factors in the direction from point A to B and from point B to A, respectively, as shown in Figure 3-3, and therefore are defined by:

32 C12 r12 C2 (3-36) C11l r21 = (3-37) C22 For convenience, the dimensionless axial load parameter ao introduced as: _ = p- (3-38) P e where P is the Euler load for a column without stay plates, i.e., e T2EI P = 2 (3-39) e L The parameter k which has been defined in Equation (3-27), now can be transformed in terms of nondimensional parameters 1i and a in the Equations (3-5) and (3-38), respectively. Thus we have k2 = c2(1 + t2at*) (3-40o) where -* = ( - ~_ - ~&2 )24 (3-4 ) For convenience, let k* = (1 - - 2)k (3-42) Then, the four boundary conditions (3-28), (3-29), (3-30), and (3-31) lead to a system of four simultaneous linear equations for the joint rotations O and 0; the member rotation n; the integration constants A, B; and the member a b

33 end-moments M and Mb. Now it will also be transformed in terms of nondimensional parameters i, 61, 52, a, and k as follows:

(1) Case a = 0 (No Axial Force) 3 2..3 ML _ 1 - a t 0 0 -O 6 2 6 EI 2)3 (I-2 - 52) 2 -2 l 2 1 0 2 - MbL * +- - 81 * +2 11 0 -1 0 0 A 2 2 = o (3-43) | * + s (1 - (2)_2 + ~)2 - 52+ - 01 0 0 -1 0 B 2 2 "b

(2) Case a > 0 (Compressive Axial Force) 12 -IT rC2a> Cos kl1 sin k1 - 0 52 - 2 MbL I 2 IT2 Cos k(l - t2) sin k(l - t2) 0 52 -52 1 1 1 2 2 ~L+ xf (2 + - --- - k sin k1 k cos k6S -1- r2Ct* o 1 + iaCz* A = 0 + + - k sin k( - 2) k cos k(1 - 62) - -1 02QP 1+ 2aL * l344 (3-44 )

(3) Case a < 0 (Tension Axial F 15 cosh kC1 sinh k~1 -ti 0 E It5a ITI l52-21 cosh k(l - t2) sinh k(l - 0 2) 012 -~~~~~~~~ 7~2) LA-'- E2 1 12A C*+ 1 i*+-J —- k sinh kr1 k coshkM1 -1 - aT2 0 1 + a TI2 1 + -4- k sinh k(l - F2) k cosh k(l - )2) 0 -l - 1 + m2C24 B (5-45)

37 To formulate the modified slope-deflection equations, the sets of four linear nonhomogeneous equations (3-43), (3-44), and (3-45), in four unknowns M Mb, A, and B will be solved by Cramer's Rule in terms of @ 0, b' and V. Then, the solutions will be set up in the forms of Equations (3-32), (3-33), (3-34), and (3-35). Where (1) Case a = 0 C = 4[53* + 61 + 51(1 - 2) +( (1- 2)-46) C: =C( - -2)2(1 + 262) - F1(3 - 261) - 6(1 - - 62)3t1 12 = 21 l )(1 - 61 - 62)4(12p + 1) (3-47) 022 =+~ 4 *. (1 - +1)2 + (1- 61)62 + 2( -6 C2(1 - E1 - 62)3(121 + 1) (3-48) rm = — l -52)2(1 + 262) - 2(3 - 261) - 6(1 - 61 - 62)3) 2 F -(1 62 (3 6-1 +- 62)( + +1(l - 62) + (1 - 32) ) (3-49) l - 221 + 22) 1(3 - 25-) - 6(1 -, - (3-50) (2) Case c > O 1 - k* cot k* + [P* + 81(1 - 62)] -l) (1 - - 2) [*(1 + ~y~*) tan —1 k csc k* - 1 - k(61 + 62) cot k* + (8162 - M*) a2O C2= ( - 61 - 62)[ (l + i2c*) tan k-_ 1 (3-32)

38 1 - k* cot k* + [1* + (1 - 61)62] 2aC C22 = (3-53) (1- l - 2) 2 (1 + J2l*) tan - k csc k* - 1 - k(6 + 62) cot k* + (6162 - I2 ) J ( r12 = 1 - k* cot k* + [4* + ~6~(1 - 62) ] a2. k csc k* - 1 - k(6l + 62) cot k* + (6162 - 2*) 2aO ( r21 = 1 - k* cot k* + [ * + (1 - 1).] 2 (3-55) a (3) Case C < 0 k* coth k* - 1 - [1* + 61(1 - 62)] 2 (-56 (1 - 6 - 2)[1 - (1 + t2au*) tanh 2 C 1 - k csch k* + k(61 + 62) coth k* + ( *,- 6162) Tr 2C ( - -) - k — (1 -+ C2*) tanh (3-57) k* coth k* - 1 - [ + (1 - 61)62] (aC22 2 k (3-58) (1 6 - 5 2) - k- (1 + IfT2a*) tanh -k 1 - k csch k* + k(61 + 62) coth k* + (~I* - 6162) i2a 12- = k* coth k* - 1 - [4* + 61(1 - 62)] a,,, 1 - k csch k* + k(61 + 62) coth k* + ( * - 6162) (-6 k* coth k* - 1 - [I* + (1 - 61)652] a The numerical values of the stiffness constant C1L and its corresponding carry-over factor rl2 are presented in Figures 3-4 for the special (symmetrical) case of 61 = 52 = 8.

co 0 CARRY-OVER FACTOR r SLOPE-DEFLECTION CONSTANT C ~ ~ ~. ~~~~~~~~12 1 CD ckA Fj o ~3 I. I~~ —- ~ ~~~ ~~~ P.. H H0 H- CtH H, / H I 7 H0 O -/.... CDO I 0 I I I I I- // CD. t *~ -i~x-~3 - i —' - 0~ ~~~~~~~~~~~~~~~~~~~~~~~~i.0 0~~0 P 0 H /'///// o~~~~~~~~~~~~~~~~~~~~~~~~~~~~ H L -.. F j I)

CARRY-OVER FACTOR r SLOPE-DEFLECTION CONSTANT C ""-I......... I 4~~~ N) ~~ C) N)~ ~o 0 N H C H N CD L CD Fxj C) C CD /!~~~~~~~~~~~~~~~~ H H1C)/ ~~~~~~~~~~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~~~~~~~.. F-J~~~~~~~~~~~~~~~~~~~- ~~D 1. I I I __ -1~~ ~ ~ ~ ~~/

CARRY-OVER FACTOR r SLOPE-DEFLECTION CONSTANT C!. * 1* 15 *, 0~ CS0 ~ - 0I- 03 *~~~~~~~2j 0~~~~~~~~~~~~~~~~~~~~~0 70; CD I~~~~~~~~~~~~~~~~~~~~~ t d.. ~~~~~~~~~~~~~~~~~ - A I - * I I I I I I' ~D CD~~~~~!!/1 ~~~~~~~~~/ 0 I

8.- =. 0 6 r-I u 6 s~C 5 -I ~ 06 - U 4 04........... 2 E- 1I _ _ _ _ _ __-1 1. 4 1.2 rq..6....... i.- 6 I 0. = [o 4.4 -1.0 -.8.6 -.4 -.2 0.2.4.6.8 1.0 LOAD PARAMETER x = P/Ps Figure.-4 (Continued).

CARRY-OVER FACTOR r1 SLOPE-DEFLECTION CONSTANTC * 0 0~~~~~~~~~~~~~CN0 Fj_ I I I I I I I~ _ _ _ CD o

CARRY-OVER FACTOR rl2 SLOPE-DEFLECTION CONSTANT C I 1 H- H- HI * *~ ~ I Ft i, I C)0 ~ 0'~' ~' ~ H y OO0 (D,

CARRY-OVER FACTOR SLOPE-DEFLECTION CONSTANT Cll I 0~0 t~ ~e 0 9 ~ ~ l O 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 ~~~~~~~~~r2j C) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~)C! 1 H'.' Z I. ~~ -' 0I -,,,1 0~-.:: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~cox X,,\ CD ~~~~~~~~~~~~~~~~~~~~~~'~II H. 00 C)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r)

CARRY-OVER FACTOR r SLOPE-DEFLECTION CONSTANT Cil 121 I~ I H H (D ~ ~ o D n o n I — ~~~~~ —-----------— 7~ ~ ~ ~ ~ ~~II -F cr CD I ~~l 07 F-J~~~~~~~~~~~~~~~~~~~4: C)~~~~~~~~~~~~~~~~~~~~~~~~I CD r. r-3 cL.- I I I II CD C)

CARRY-OVER FACROR rl2 SLOPE-DEFLECTION CONSTANT Cll I N F 0 Io I Io 00 o 0 l FJ OH 0. H n m l 0o cc n ciHJ. II 00 0.... CD~~~~~~ / /

CARRY-OVER FACTOR r1 SLOPE-DEFLECTION CONSTANT C11 I H...H H... H- 4 O 0L ) 0 0 O C Y,- O H-) [~ U Un 0% -J O 0; H tj* 0 07 0:r CI II II I oti~ H if. ~ ~, I: CD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C C~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ I II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I o

CARRY-OVER FACTOR r SLOPE-DEFLECTION CONSTANT C11 12 11 I..... H H H I 0 O 0 6 0 0% O0 O6 ~ ~~ O ~ ~O ~o 0 0 0 H it N 0 N C4 taj 0 C N)t 4 0 H N)C r LJ oLnj C ~ ~E, b,,~...... h E3~P vl o ~I, \ \ II ti i:q -~':~ o I.- o c — -7 0 P II r~. CD ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A 00 I- I t I I 1 I -1 t -1 I I CD - ~S 2h a~~~~~ cP~ ~~~~ oD i I i I I

CARRY-OVER FACTOR r12 SLOPE-DEFLECTION CONSTANT C11 I H..... HI ~- ~ 1' 0 t~ C % O0 0 t) 0 H- t~ O t) U 0%U., J o0 00 tQ C,,-J, F....... ~ 0Y~~~ H * o o II di- c)o (D oTJ 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ H 0D ON~~~ o ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 I- I I 00~~~~~~.,_., h

CARRY-OVER FACTOR rl2 SLOPE-DEFLECTION CONSTANT Cll 12 11 I I H H H H C o ) C:n O oo 00 o o A oH o0 H. Ul w1 o. 00 o1 tl;l\ E 0 lltd I I'o Id O III I 1 I I I f I I I 00 O L izz z H>

CARRY-OVER FACTOR r1 SLOPE-DEFLECTION CONSTANT C1 H 4 1 N C> K) ~ On 0) C) N) 4~. H C) H ),J 4 Ul YnJ O 0~CD 1-307 t~~~~~~~~~~~j~~~IC CD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C dzzo ~ ~ ~ ~ co IIJ

CARRY-OVER FACTOR r12 SLOPE-DEFLECTION CONSTANT C11 I I H- H- HI......... I H- I 0 [S) O O 0:0 0 P H 0 H [ W I-U1 O I-J 00 $~~~~~~~~~~ - I - \31 -~ II o i~~~~~~~ IIMO I~IC hO CD ~~d ~I I i I I..... I o~~aLw ~s

CARRY-OVER FACTOR r12 SLOPE-DEFLECTION CONSTANT Cll I I H- H- H — I.*....... ~ H v CM O O~ 0 0 0 [O ~ O H M LO v n1 O - O 00 PI M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CO bTp m I~~~~~~~~~ I,\~ L- L_ I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I *~- - -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (C.. 0 0 ~~~~ ~/ co~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 / ~ ~ ~ I ~ C.. oI

3.4. FIXED END MOMENTS In the evaluation of fixed end moments we can apply the reciprocal theorem and use the deflection curves of the previous presentation. Consider the two sets of force systems which are acting on the same structural member A-B as shown in Figure 3-5. Mfa W P P A B B L L (a) (b) Figure 3-5. Two sets of force systems acting on the same structural member. According to the reciprocal theorem, we have the relation: Mf a + Wy + MfbO = M0 + MO, then M = - Y (3-61) a where Mf = the fixed end moment at left end, as shown in Figure 3-5(a). W = an arbitrary concentrated force acting on a, structural member, a distance pL from its left end. y = deflection of the structural member, a distance pL from its left end, caused by end moment M. a

p = nondimensional length factor which prescribes the location of the concentrated load W. 0 = end rotation caused by end moment M a. a, To determine the deflection curves of the structural members in terms of the member properties and end moments Ma and Mb we again consider the sets of four linear nonhomogeneous equations (3-43), (3-44), (3-45) in four unknowns, two integration constants A and B; two end rotations @, ~b. a b We apply Cramer's Rule again to solve for A, B in terms of M, Mb, and ~, thus obtain: (1) Case = O (1 - 62)3 + 22(1 81 a, A = b L ( 62 + + 62() - 22) + 8l)] + (1 + 262) (1 - 62) 2- 13] I (3-62) l1 SL2_~ M L B =5% Ij 1 a + 6 1 (3-63) L'2 3 EI 3 EI Mb = r1l2M and -0 then M L A = a,_ (35-64) A12 EI M L B = ~ a (3-65) B12 EI where =A12 - (1 + rl2)(61 + 62) ~* + 62(1 - 62) + (1 - 62)2 +A12 521 g~~6 - rl(1 + 2 ( 262)] + 621L - B —(l + r12)] (3-66)

57 = l (1 + r12) * (+ r2]3 (3-67) B12 (3 67) The deflection curve y and the rotation @ caused by end moment M are: a a M L2 Y = (1 + rl2 3 _ p + a(3-8) -y + P + r a (3-68) 6 2 A12 B12 EI M L 1 a (3-69) a, C11 EI Then, from Equation (3-61) we can obtain the fixed end moment Mfa, as shown in Figure 3-6 caused by concentrated load W: M -WLC11 + rl2 (2-70) Mfa 6 2 A+ 12 B12 1L W 2 fa IpL Figure 3-6. Fixed end moment M for concentrated load W. From this basic equation (3-70), we can derive fixed end moment for any combination of concentrated loads by simple summation and for continuously distributed loads by integration. As examples, the fixed-end moments Mfa for uniformly distributed load w and for moment loading M are determined as follows: 0

58 (i) Uniformly distributed load w. P_ 2L piL W=wd(pL) Mfa / w L _ll10 d(pL) Figure 3-7. Fixed end moment M for fa uniformly distributed loading w. M12 c + r12 3 )dp (-71) Mfa wL C I 6 P 2 A12 p 12 For fully loading, P1 = 0, P2 = 1 Mfa 2 w L - wL2 C 1-521 + rl2 3 )d + M =-(51L)- WL Cj11f 6 Al2 + )dp~+ 0 fa 2 16 2 B12 ~1 Mf L2C 11 - + 1 r12 1 - - ) 1 fa Cll2Cll 6 A12 2 ~)2 +2 )2 + ~B12(1- 1- 23 (3-72) (ii) Moment loading M: 0W (a) (b) Figure 3-8. Fixed end moment Mf for the moment loading M 0

59 Consider the condition as shown in Figure 3-8(a) and (b), they are identical if A(pL) is approaching to zero. Then M = lim W A(pL) 0 (pL)+ o then a = - WL A(pL) C11 f( + p) - f(p) Ap-* 0 Mfa M Cll f'(p) (3-73) where f(p) 1+ r12 3 p2 6 2 A12 B1(374) d(1 r (A12 2B12( f1(p) df(p) =1 + r 2p2. dx 2 - p+A12

(2) Case a > 0 Similarly we can obtain M L [62k cos k(l - 62) + (1 + 2 i) in k(l - 62)] - + [ ~ ) sin k61 - 61k cos k61] EI E A = Trece(k(61 + 62) cos k* ~ [1 + -152)> l] sin k*) ML MbL [62k sin k(l - 62) - (i + c2o*) s k(l - 62)1 - - [(1 + xea4*) cos k61 + 61k sin k61] EI EI B = - (I-7 2 a(k(6j + 62) cos k* + [1 + ( 12- 6i62> a] sin k*) it2EI Mbr=12rl a 9 P=a L2 then M L A=~~ (5-78) A12 EI M L a (39) B12 El where [1 + JT2ai*][sin k(l - 62) + r12 sin k61] + k[62 cos k(l 62) - r1261 cos k](8 A12 2af~k(b1 + 62) cos k* + [1 + (4* - 5152)eCf sin k*J = [+ Iitrc*][cos k(l - 62) ~ r12 cos k61] + k[r1261 sin k61 - 62 sin k(l - 62)] B12 T2afk(b, + b2) cos k* + [11 (+ * - 516 ) Taa sin k*J Then the deflection curve y and end rotation 0a caused by end moment Ma, are: M L2 Y cos kp sin kp + r12 P a A12 B12 I EI

M L G 1 a (3-83) a C11 EI Then, the fixed end moment Mfa for concentrated load W with compression axial force P is expressed in the following equation (3-84), as shown in Figure 3-9. Mfa - WL C( 2 cos kp - B12 sin kp + r2 p — 84 Mfa 111((~ A12 B12 sn2a P TI2a(3 Mfa pL W P P |a- L - L F.igure 3-9. Fixed end moment Mf for concentrated load W with compression axial force. Similarly, the fixed-end moments Mf for uniformly distributed load w and for moment loading M are determined as follows: 0 (i) Uniformly distributed load w (as shown in Figure 3-10): p2L Mp I pL L Figure 3-10. Fixed end moment M for uniformly distributed load w with compression axial force.

62 P2 1+ r22 1 M - wL C11 f cos kp - ~ sin kp + 2 P 2 )dp fa, Al2 BL2 C a( a P1 (3-85) For fully loading, P1 = 0, P2 = 1, then 32 i A12 M = - C+ - [sin k(l - 52) - sin kS1] fa w 2C1' k + k [cos k(l - F2) - cos kF1] + 2 12 [(1- )2 _ 1] - (1- - 2, k 2el-c (5-86) (ii) Moment loading M (as shown in Figure 3-11): o P P g L L Figure 3-11. Fixed end moment Ma for the moment loading M1 with compression axiaf force. Mf = - M C11 k(- A sin kp - cos kp + r12) (3-87) fa o A12 B 12

(3) Case ca < 0: [k52 cash k(l - 62) + (1 + ir2Qai) sinh k( - 62)1 -- + [(1 + E2I*) sinh k61 - k61 cosh k61] [k6~~ cosh k~~~l - s2 ~EI E A IT2c((k(6i + 62) cosh k* + [1 + - 6162>x2a] sinh k*3 (~-88) M L ML [k62 sinh k(l - 62) + (1 + jt2at*) cosh k(l - 62)1 - + [(1 + ir2ci*) cosh k61 - k61 sinh k61] bl B T2a(k(3 +632) cash k* + [1 + (* 1- 12) 62> sinh k*) (3-89) i2EI Mb=rl12Ma L2' then M L A ~~a __ (5-90) A12 El M L B=-~~ a (5-91) B = - B12 EI where (1i + LaC ) [sinh k(l - 62) + r12 sinh k611 + k[62 cash k(l - 62) - r1263 cash k611 A12 =c2a(k(6 + 62) cash k* + [1 ~ ( 1* -162) da] sinh k*) (1 + it2aJg) [cash k(l - 62) + r12 cash kl]_ + k[62 sinh k(l - 62)- r1261 sinh k61 (3-J) B12 = iT2(k(6 + 62) cash k* + [1 ~ ([j* - 5612)er al sinh k*)

64 Then the deflection curve y and end rotation @ caused by end moment M a a are: ML2 (A12 cosh kp + B12 sinh kp + + r (394) A12 Bl2 jT2a i t ar El (I94) M L 1 a C11 EI Then M = - WL C11(A2 cosh kp - B12 sinh kp + 2 fa A12 B12 2 P c (3-96) Similarly, the fixed-end moments Mfa for uniformly distributed load w and for moment loading M are determined as follows: (i) Uniformly distributed load w: P2 Mf= wL2 Cll f (A12 cosh kp - B12 sinh kp + p - )dp P1 (3-97) For full loading, P1 = 0, P2 = 1, then - WL2 65 A12 Mf= - wL? C11l + [sinh k(l - 52) - sinh k61] fa 2C11 k B12 - k [cosh k(l - 62) - cosh k61] 1 + r l2 21 (3-82 + 2 —a i[(- 6 2)2 - - 6,' - l ) (3-98) (ii) Moment loading M: M_ = -M C11 k(~1 snk B2o1+ r )2 Mfa - - M C11 k(4 sinh kp - 4 cosh kp +..2ok ) (3-99) fa o A12 B12

65 Fixed end moment Mfa of structural members for a uniform load w over fulllength L with different rigid plates 6 (where 6 = 51 = 62) and constant shear flexibility t is presented in terms of wL2 as shown in Figure 3-12.

-Ma -Mfa FIXED-END MOMENTS fa FIXED-END MOMENTS wL2 ce o HmC........ P~i.. 0>'D C) CD MMWW CD Hnm M P 0 o~ ~ ~ ~~~~I 0 p~ (DCO 0 c't-fr H i H- c~ 0t~ c+ P, H~1 P II CO0 ~1CO 01) 00 o jH I I I I I 1I

-Mfa -Mfa FIXED-END MOMENT 2 FIXED-END MOMENT L wL I 0 0 0 0 H H- H- H- H- 0 0 0Z 0 H- H- H- H- H- ~ 1-~- - O~ O0 0 [~ ~, O~ 0 0 t) 0, C) 00 Q 4 co 00 0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~0 0~~~~~~~~~~~~~~1 -~C ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~II c~~ 0 _ Co I~~~~~~~~~~~~~~~~ ON,' T-~ Ct- t~l ) P. w~ "~ (D N'L. H ~ I- ~ 0 -~~~~~~~~~~~~~

~Mf -Mfa -Mfa FIXED-END MOMENT 2 FIXED-END MOMENT w2 L2 wL * * 0 0 S S~ ~~ ~~ ~~~~~~ S 4). ON 00 0 _Q _ h 00 CD toJ 4S m 00 00 C) 00 0 O ~ ~ ~ ~ ~~~0C (D tz) t-' ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a VV ~ CL~~~~~~~~~~~~~~~~~~~~C (D ~ ~ ~ ~ ~ ~ I1I ct II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Z r. M II~~~~~~~~~~~~~N =1 13~~~~~~~~~~~~~~~~~~~~~~C)

-Mfa -Mfa FIXED-END MOMENT 2 FIXED-END MOMENT f2 wL wL Io o oP ~ o r o o o o ~ ~ ~ o o oo -0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0% > C) ~ m~ O CD I I I *~ II n H I -~

f -tfa FIXED-END MOMENT fa FIXED-END MOMENT 2 wL wL *..% *% *C** ** 00 * * * * 10 0 0 O O 0- - 1- O O H H H-H H 0 0- 0 F0J HMH 0 1-30Mtl~~~~~~~~~~~~~~0 o \ PID 5 oo 0~~~~~~~~~~~~~~~~~~.

-Mf -Mf a FIE-NaMMN FIXED-END MOMENT FIXED-END MOMENT 2 2 wL wL 0 0 S 0 0 0 0 S 0 00 0 0 0 0 0 00 0 cI 0 0 0 0 H- H- H- H - 0 0 0 0 H- H- H- H- H- [ HO A_ 0. 0L 00 0> to 0% 00 0X t: 00 C) o I I C) ~~~~~~~~~~~~~~~~~~~~~~~~~~C (D ~ _3 0 (0 t~i OD PI t) dl D' ICI II a~~~~~~~~~~ 0 id~~ ~~~~~~~~~~~ol \,, IIIIII'o

-mfa -M.fa FIXED-END MOMENT 2 FIXED-END MOMENT a 2 2 wL wL 1 ~ ~ ~ ~ b n 0 0 0 S) 0P 6~ 6 0 0 0 0 03 0 V 00~~~~~~~~~~~~K 0 C )0 H''Y C) 0 (D ~~~~~~~~~~d ~ ~ ~ ~ ~ ~ ~ ~ d I L ~~~ ~r I 1- I I I I I I I t -AI

-Mfa -Mfa FIXED-END MOMENT 2 FIXED-END MOMENT M 2 wL wL * * 0 * - 0 - - 6 0 * 0 0 * * 0 * 0 * O 0 0 0 H H H H H 0 o o o o H H H H ) 0I 0 N ~ 0 0 0 o 000 0 P) oP Co 0 D) cn 00 0 0 00 o I I I i I I I ____ __ ___ _TT f o ~~ - ~- -~ 6, 0 - 0 Hro

CHAPTER IV ELASTIC BUCKLING OF BATTENED OR LACED STRUCTURAL MEMBERS WITH RIGID STAY PLATES AND CONSTANT SHEAR FLEXIBILITY The lowest elastic buckling loads will be evaluated for different endconditions of built up columns, as determined by the eigenvalues of certain submatrices of the system of Equation (3-44). 4.1. COLUMN WITH HINGED ENDS The critical load P of the column which is hinged at both ends causes cr,h the matrix (3-44) with the unknowns 0 0 Eb, A, B to be singular, since the a natural boundary conditions give M = 0, Mb = 0, and = O. Then cos k6l sin k61 -81 0 cos k(l - 82) sin k(l - 62) 0 52 - k sin k61 k cos k61 -1- _ r24* 0 - k sin k(l - 62) k cos k(l - 52) 0 1 I 2a* Then we have the buckling condition: tan k* + 1 + (2)* - 6i2) = (4-1) The lowest root of Equation (4-1) yields the critical load Pcr,h' 4.2. COLUMN WITH ONE END FIXED AND THE OTHER HINGED The critical load P of a column which is fixed at one end and cr, f-h hinged at the other causes the matrix (3-44) with the unknowns Ma, A, B, 0b 74

75 to be singular, since the natural boundary conditions give Mb = 0, Q = 0 a and = 0. 1- -1 2 cos k61 sin k61 0 62 -27~ cos k(l - 62) sin k(l - 62) 62 =0 1 - * + - k sink61 k cos k61 0 1 ~* +2 - k sin k(l - 62) k cos k(l - 62) -1 a* Then we have the buckling condition: k*n tan k* + [* - 1)6 = 0 (4-2) 1 + ea1t* + (1 - 31 F)2 The lowest root of Equation (4-2) yields the critical load Pcr,f-h 4.3. COLUMN WITH FIXED ENDS The critical load P of the column, which is fixed at both ends, again crf causes the matrix (3-44) with the unknowns M,,Mb' A, B to be singular, since the natural boundary conditions give 0 = 0, 0 = 0 and 0 = Then a b ~=. Te 1- 61 6 I 12a cos k6l sin k61 6C2 1-62a 52 - F)2 cos k(l- 62) sin k(l- 652) TI2 JO2a = 0 1 1. * + j- k sink61 k cos k61 1 1 ~* + 2 * 2 k sin k(l - 82) k cos k(l - 2) 2 - - - Then we have the buckling condition: sin k* [ -k* cos k — sin = 0 2 2(1+ ~2~")

76 There are two solutions in Equation (4-3), one of them is the symmetrical buckling condition, and the other corresponds to the antisymmetric buckling pattern. We are interested here only in the symmetrical buckling shape, since the critical value is always smaller than for the case of antisymmetry. Therefore, we consider only the symmetrical buckling condition which is: k* sin - = 0 (4-4) 2 For the lowest root of this equation gives: k(l - 51 - 62) - 2 = O (4-5) Equation (4-5) yields the critical load P cr, f The critical loads of columns with different end conditions, constant shear flexibility Ci, and rigid stay plates 6, are provided in Figure 4-1.

UI~~PP Pcr, f-h k P e a) U 0 P P crf-h P4 1:14 Pcr,h O P e 0~~~~~~~~~~~~~~~~~~ H 3 1.08,.06,.04, 02, 0 I,0 H U __ (=0 o.__ ~~__ _ 0.05.10.15.20.25.30.35.40.45.50 SHEAR FLEXIBILITY PARAMETER i Figure 4-1. The critical loads of columns with different end conditions, constant shear flexibility pi, and rigid stay plates 5.

CHAPTER V NUMERICAL EXAMPLES OF BEAM AND FRAME ANALYSES 5.1. A BEAM ANALYSIS A structural member 0-1-2, as shown in Figure 5-1(1), is composed of two parts 0-1 and 1-2 with fixed ends at 0 and 2. Part 0-1 is solid-web member, with assumed shear flexibility equal to zero. Part 1-2 is a battened structural member with rigid joint connections (Z = 0), and with appreciable shear flexibility. However, parts 0-1 and 1-2 are assumed to have the same nominal flexural rigidity EI, and length L.,I l EIl ~0 2~12 L L L=na Figure 5-1(1). A structural member 0-1-2. As shown in Figure 5-1(2), the battened structural member 1-2 is composed of the following elements: longitudinal elements: 14WF43 batten elements: 2-9[15 The geometrical arrangement of the battened structural member is: a, = 30 in. b = 20 in. 78

79 a N 14WF43 (a) (b) Figure 5-1(2). The battened structural member 1-2; (a) geometrical arrangement and (b) cross-section. Four cases will be compared here for n = 2, 6, and 10 as follows: Case 1. Results are obtained considering the effect of shear deformation. Case 2. Results are obtained considering the effect of shear deformation but neglecting the effect of the local connection factor (i.e., a = 1). Case 3. Results are compiled from STRESS computer program by MIT. Case 4. Results are obtained from ordinary beam theory (neglecting the effect of shear deformation). The shear flexibility i, stiffness C11, carry-over factor r12, of the structural member 1-2, and bending moments M0, M1, and M2 due to concentrated load W are tabulated as follows:

80 Case n Cll r12 WL WL WL 2 1.030 1.22 -.63 -.553.250.053 1 6.114 2.26.11 -.323.250 -. 176 10.041 3.00.334 -.279.250 -.221 2 1.457 1.16 -.72 -.593.250 ~093 2 6.162 2.02.01 -.347.250 -.152 10.058 2.76.27 -.290.250 -.209 2 1.46 -.56 -.503.258.021 3 6 2.54.15 -.308.251 -.189 10 3.22.35 -.272.250 -.227 4 4.00. 5 -. 250.250 -. 250 The bending moment diagrams are presented in Figure 5-1(3).

81.50 n= 2.25 H O z CASE 2 CASE 1 z -.25.CASE CASE 4 -.50/ MEMBER 0-1-2 (a).50 n= 6.* 25 0 z 2 o -— CASE 1 -.25 -CASE -CASE 4 -.50 MEMBER 0-1-2 (b).50 n = 10.25 0 H/2 -25 IEMBER 0-1-2 (c) Figure 3-1(3). Moment diagrams for the Cases 1, 2, 3, and 4 for various numbers of panels.

82 5.2. A FRAME ANALYSIS A two-bay symmetrical frame structure, as shown in Figure 5-2(1), is composed of: Two-type 1, laced structural member DE and EF to be used as roof truss beams. The two longitudinal elements (10WF45) are connected, in two planes by the diagonal and strut elements (9[15). Two-type 2, battened structural members AD and CF to be used as exterior built-up columns. One-type 3, battened structural member BE to be used as an interior builtup column. The type 2 and type 3 battened structural members which consist of two main longitudinal elements (10WF45) with batten elements (9[20). The two longitudinal elements are connected in two planes of batten elements, by means of perfectly rigid joint connection (Z = 0). The geometrical arrangements and elements' properties of the frame structure are given in Table 5-2(1).

83 Rigid Stay Plates 1| 2 2 2 1 -1 l op (a) Elevation of Frame Structure 10" 1'-8" Rigid Stay Plates 98-6" 98A -6" C 98j-6" 98'-6" (b) Idealization of the Frame Structure 10WF45 Ln 9[20 <' oh1 H 9[20 1OWF45 Cross-Section 1-1 Cross-Section 2-2 Figure 5-2(1)., A two-bay symmetrical frame structure.

TABLE 5-2(1) THE GEOMETRIC ARRANGEMENTS AND ELEMENTS' PROPERTIES OF THE FRAME STRUCTURE Long. Element Batten Element Dia. Element Member Type* n a b L I Size c c c Size b rb Size d in. in. in. 4 in.2 in. in.2 in. in.2 DE, EF 1 16 72 72 98'-6" 54,580 10W45 15.24 9[15 8.78 9115 8.78.0113 AD, CF 2 6 40 20 20 -0" 2,766 loW45 13.24 2.0 1.32 9[20 11.72 3.22 1.45.0898 BE 3 6 40 40 235'-0' 10,746 10W45 13.24 2.0 1.32 9[20 11.72 3.22 1.45.3910 *Type 1 is laced structural member (where =.875 =.944). Types 2 and 3 are battened structural members (where a =.775). a

Consider three loading conditions as shown in Figure 5-2(2): w W w 4: v A B C A B C A B C (a) (b) (c) Figure 5-2(2). Loading conditions. (a) Loading 1, dead load (w ) only. (b) Loading 2, dead load (w ) + wind load (w ). (c) Loading 3, dead load (w ) + appreciable footing rotation a, where w = 1.25 Kips per ft, wh =.50 Kips per ft, and =.001 radians(. The end moments will be determined simply by using the slope-deflection equations, and the equilibrium conditions of each joint and the whole frame structure in the following two cases. Case 1. Considering the effects of shear deformation and axial force. Case 2. Neglecting the effects of shear deformation and axial force (ordinary frame analysis). The solution of the end bending moments, and the moment diagrams of the framed structure are presented in Table 5-2(2) and Figure 5-2(3), respectively. The corresponding errors which result if the effects of shear deformation and axial force are neglected, are also indicated in percentages.

TABLE 5-2(2) THE END MOMENTS (KIP-FT) OF THE FRAME STRUCTURE, AND ITS CORRESPONDENT ERRORS (%) WHICH OCCUR IN THE EFFECTS OF SHEAR DEFORMATION AND AXIAL FORCE ARE NEGLECTED AD DE EB EF FC Load- Case M M M M M M M M AD DA | DE ED MEB ME EF FE FC CF ing 1 67.78 263.75 -263.75 1,418.19 0 0 -1,418.19 263.75 -263.75 -67.78 1 2 128.95 257.90 -257.90 1,387.o03 o o -1,387.03 257.90 -257.90 -128.95 Error 90.2 -2.2 -2.2 0.0 0.0 -2.2 -2.2 90.2 00 1 11.22 247.47 -247.47 1,444.76 -57.42 -60.52 -1,387.34 280.71 -280.71 -124.54 2 |2 177.78 249.30 -249.30 1, 417.47 -60.88 -84.08 -1,356.59 266.50 -266.50 -180.12 Error 59.2.7 -1.9 6.0 38.9 -2.2 -5.1 -44.0 1 668.76 310.44 -310.44 1,496.31 -258.20 -258.79 -1,238.11 435.83 -435.83 -263.88 3 2 995.73 517.63 -517.63 1,478.65 -399.74 -596.26 -1,078.91 431.17 -431.17 -350.68 Error, 48.9 66.7 -1.2 54.8 130.4 -12.9 -1.1 2.9

87 1,418.19 1,387.03 (-2.2%) NOTE: 263.75 CASE 1 257.90 }; ------ CASE 2 (-2.2%) h /111| I I Unit: Kip-Ft. D ^ _ _,. I F. 67.78 128.95 (90.2%) A'r B C (a) Loading 1, Dead Load Only 1,444.76 1,387.34 1,417.47 1,356.59 (-1.9%) (-2.2%) 247.47 280.71 249.30 266.50 (.7%) (-5.1%)2 310.44, ] [ 259460.52 360 8 77.78 124.54 84.08 (6%) (5932 89i80.12 A B C (-44%1 (b) Loading 2, Dead Load + Wind Load 1,496.312, and 1,478.65' 2' (-12.9%) 310.44 435.83 258.20 431.17 516.7.63 ~\399.74 n (-1.1%, (54.8%) D E - F ~.~73 596,.2 350.68 (c) Loading 3, Dead Load + Bad Footing Rotation 1, 2, and 3.

CHAPTER VI SUMMARY AND CONCLUSION 6.1. SUMMARY This dissertation presents a theoretical analysis of the behavior of battened and laced structural members for an ideal perfectly elastic material and for small deformations. The analytical solution is made by application of classical procedures and modified slope-deflection equations are developed to generalize a relation between applied forces and joint displacements. If the displacements of the joints in a structure are known, it is a comparatively easy matter to obtain the bending moments and shear forces at any location of the structure. A nondimensional parameter p for the shear flexibility is defined as the ratio of the change in slope in a unit panel length due to shear deformation to the change in slope due to the bending rotation for a relationship between shear and moment. The parameter p is calculated with consideration of the effects of axial force, local shear deformation, local joint eccentricity, and local connection flexibility of battened structural members. Types of the structural members may be catalogued here according to three different web configurations such as solid, battened, and laced structural members. The battened and laced structural members are built up from two (or more) main longitudinal elements which are assumed to form a symmetrical section. The two longitudinal elements are connected in one, two (or more) planes by the diagonal and strut elements or batten elements. The arrangements of web 88

89 elements are assumed the same throughout the effective length of the flexible portion of the structural member between end rigid stay plates. The shear flexibility parameter is evaluated in terms of the properties of the composed elements, the geometric arrangements, the local joint connection factors, and the axial force. The optimum slope of the diagonal element is evaluated for the laced structural member to minimize the influence of shear deformation. In an actual situation, the axial forces of the longitudinal sub-elements will vary from panel to panel along the length of the members, and also will be different on the two sides of the longitudinal elements due to the lateral external loadings. However, ultimately the analysis assumes the shear flexibility p. as a constant value for the entire length of the structural member. To permit the assumption of constant shear flexibility, a limit on the ratio of the local slenderness has been determined for the battened structural members, so that the influence of the actual axial forces will be relatively negligible. The upper bound of the premature local failure is also investigated for the battened structural members. The fundamental linear second-order differential equation for the deflection curve of the structural member has been derived for which shear deformation is considered. As long as the axial load is kept constant, then the effects of end moments, end-shear forces and external loading can be superimposed. The ordinary beam theory can still be applied since the effect of axial force appears here only in a modification of the stiffness properties of the structural members. The general solutions of this differential equation

9o are of a fundamentally different nature for the cases when the axial force is equal to zero, greater than zero (compression axial force), and less than zero (tension axial force). By application of the natural boundary conditions to the general solution of deflected shape of the structural member, the solutions are set up in the forms of slope-deflection equations. Therefore the slopedeflection constants are obtained in terms of the shear flexibility parameter, the length factors of stay plates, and effect of axial force. In the evaluation of the fixed end moments for a concentrated load, the reciprocal theorem is applied so as to make use of the deflection curves of structural members which have been previously defined for the homogeneous solution (no external lateral loading). From this basic expression one can derive fixed end moments for any combination of concentrated loads by simple summation and for continuously distributed loads by integration. Elastic buckling of the structural members with rigid stay plates and constant shear flexibility have been evaluated for the cases of a, column with hinged ends, a column with one end fixed and the other hinged, and a column with fixed ends. Finally, numerical examples of beam and frame analyses are presented to provide a comparison with analyses using the ordinary beam and frame theories which neglect the effects of shear deformation and axial force. 6.2. CONCLUSIONS 1. The property of the shear flexibility parameter t (a) For the laced structural members:

91 (i) The parameter p. becomes infinite as the slope of diagonal elements approaches either zero or infinity. (ii) The optimum slope of the diagonal elements is J-/2 whenever strut elements are missing, or whenever the strut elements do not take part in the transmission of the shearing force of the structural members. When stressed strut elements are part of the system, the optimum slope increases with the ratio of cross-sectional area, of the diagonal elements to the strut elements. (iii) The parameter [i increases with increase of the heightlength ratio of the member, and with the ratio of cross section of the longitudinal elements to the diagonal elements. (b) For the battened structural members: (i) The parameter p becomes infinite either as the number of panels approaches zero, or as the semi-rigid constant approaches infinity (i.e., hinged connection). The parameter p. approaches zero as the number of panels approaches infinity. (ii) The parameter p. is inversely proportional to the ratio of slenderness of the member, and proportional to the heightlength ratio. 2. If the local slenderness ratio a/r reaches n/~ ~E/(F. S.)a, premature local failure will occur. The proposed limitation a/r < 2/35 E/(F.S.)a. should provide adequate safety against this possibility.

92 3. For tension axial force, when 2 (1 - - 8 2 then the equivalent flexural rigidity of the structural member becomes infinite, and therefore leads to a trivial solution of the deflection curve of a structural member. Beyond this condition, the general solution of the deflection curve is not applicable. 4. The slope-deflection constant C11 and carry-over factor are almost linearly proportional to the axial load parameter a in the range of Jaj <.15 and p. <.10. Beyond this range, however, (i) The constant C11 will decrease as the compression axial force increases, and may either decrease or, for very small values of t, increase as the tension force increases. (ii) The carry-over factor will increase rapidly with increasing compression axial force. The increase is approximately proportional to the parameter a for small values of a. 5.For a structural member of normal design proportions with a certain constant shear flexibility p. there exists an axial load parameter a, for which the carry-over factor is very nearly independent of the effect of the end rigid stay plates. 6. Effect of shear deformation varies with the magnitude of the axial force, properties of the bracing systems, and the height-length ratio of the member.

93 7. For a symmetrical structural member with uniformly distributed loading, the fixed-end moments are independent of the effect of the shear flexibility when there is no axial force. 8. The influence of the end rigid stay plates is of considerable significance since large increases in the carrying capacity can be achieved by very short end rigid stay plates. 9. The influence of the shear deformation may be neglected in the range of ~L <.004, and lal <.50, for which the errors of the stiffness constant C11, carry-over factor, and fixed-end moments will be not more than 5%. However, beyond this range large errors will be introduced in the analysis if account is not taken for the influence of shear deformation.

APPENDIX 94

95 TABLE 1 THE CONSTANT SHEAR FLEXIeBILITY P ARAMETER. FOR LACEO SIRUCTLRAL MEtBERS (1) VALUES OF (1+ia)/b FOR T.E..CASE CFNO. STRUT ELEMENTS aa/b =.4.6.8 1.0 1.2 1.4 1.6 A /Ad 9 /b3. 4 o 3904.3304.3282.3536.3170.4547.5248 6. 1 735.-14619..1.458..'1571. 1765.202 1.2332 8.0976.0826.082C.0 84.CS93.1137.1312 A...O'........,0'65......;0529.~ 0-52'-5...-C:566.O63 5. 012 8.0'84-W2 12.0434.0367.0365.C393.0441.CSC5.0583 14..0319.,0270 a.028.C289 -;0324..37.04 — ~8-. 16.C244.0207.02C5.C221.0248.C284.0328 18....0193.0163.0162.C175..0196.C225.0259 20.0156.0132.0131.0141.015.C182 0210 4.7808.660E.6563.7071.7940.90S4 1.0495 6.3470.2937.2917.3143.3529.*4042.4665 8.1952.1652.1641.1768.1985.2273.2624 10.1249.1057.105C.1131.1270.1455.i67 4 12.C868.0734.0729.0786.0882.1010.1166 14.0637.053.0536.....0577.0648.0'7?2-.0857 16.0488.0413.04O10.C442.0496.0568.0656 18.0386.0326.0324.C349.0392.C449.0518 20.0312.0264.0263.C283.0318.0364.0420 4 11713.9913 o9E45 1.C607 1.1911 1.3641 1.5743......6....520'6...440.43'75..474..524.603. 6997 8.2928.2478.2461.2652.2978.3410.3936 10.1874.1586.15575.1.6 1i906...2183..,2'5- 9. 6 12.1301.1o01.10s4.1179.1323.1516.1749 14.0956.0809.0804.C866.0972.1114.1285 16.c732.062C.0615.c663.0744.0853.0984 18.0578....0490...-04 8. C524.0588.064.0777 20.0469.0397.03S4.0424.0476.0546.0630 4 1.5617 1.3217 1.3126 1.4142 1.5881 1.81EE 2.0991 6.6941.5874.5834.6285.7058.80E3.9329 8.3904.3304.3282.3536.3970.4547.5248.10...'"...2-49C9.211.'''5.2-CC....2263.2541.2910.3358 8 12.1735.1469.1458.1571.1765.2021.2332 14.1275.1079.1072.1154.12$6.1485.1714 16.0976. 0826.082C.0884.0993.1137.1312 18..077 1.0653.0648.0698...0784- l.C8-8.1 037 20 0625.052S.0525.C566.0635.C728.0840 4 1.9521 1.6521 1.64CE 1.7678 1.9851 2.2735 2.6238 6. 86b76. 7343.71,29292.7~~857. -88 23 I 1.014 ~1.1661i 8.4880.4130.4102.4419.4963.5684.6560 10-.3-23-123.2643.22.2-. —3176-6-2,.2-.3638.....,-'.419.8 10 12.2169.1836.1823.1964.2206.2526.2915 1...1594. 1349.1339.1443. 162C. 185-6-.2 42 16.1220.1033.1026.10 1241.1421.1640 18.0964.8o16.081C.C873,0980.1123.1296 20.0781.0661.o065.o0707.0794.9c9.o105

96 TABLE I THE CONSTANT SHEAR FLEI.eILITY PARAMETER 1P FOR LACED STRUCTURAL MEP:EERS (2) VALUES OF 11+ a )/b FC. d Ab - aC b P.5 aa/b =.4.6.8 1.0 1.2 1.4 1.6 Ac/Ad k/b 4 * 5467.4346.4063.4161.4491.4953.5638 6.2430.1931.18C6.1849.1996. 2219.*2506 8.1367.1086.1016.1040.1123.1248.1410 10.C875.0695.C65C.C666.071S..C7S.0902 2 12.0607.0483.0451.0462.0499.C555.0626 14.C446.0355 0332..0340.0367.C4C8.0460 16.0342.0272.0254.0260.0281.0312.0352 18.0270.0215.02C1.C205.0222.0247.0278 20.0219.0174.0163.C166.018C.02CC.0226 4 1.C933.8692.8126.8321.8982.99E7 1.1277 6.4859.3863.3611.3698.3992.4439.5012 8.2733.2173.2031.2C80.2246.2497.2819 10.1749.1391 ~13CC.1331.1437.1598.1804 4 12.1215.0966.09C3.C925.0l98.1110.1253 14.C893.071C.0663.6C79.0733.C815.0921 16.0683.0543.050e.C520.0561.0624.0705 18.0540.0425 ~C4CI ~C411.0444.043.055720.0437.0348.0325.C333.0359.C399.0451 4 1.6400 1.3038 1.2189 1.2482 1.3473 1.49E0 1.6915 6.7289..57-94..541.... 5 547.598" 8..6-8... 7518 8.4100.325S.3047.3120.3368.3745.4229 10.2624.2086.195C'.197.2156.23s7.2706 6 12.1822.1449.1354.1387.1497.1664.1879 14.1339.1064.09c5.1019.1100.12-23.* 1381 16.1025.0815.C762.C780.0842.0936.1057 18.0810.0644.0602.0616.0665.C74C.0835 20.0656.0522.04EE.0499.0539.05C9.0677 4 2.1867 1.7383 1.6251 1.:642 1.7964 1.9973 2.2553 6.9719.7726.7223.7397...7984.8877 1.0024 8.5467.4346.40o3.4161.4491.49s3.5638 10.3499.2781.26CC.2663.2874.31S6.3608 8 12.2430.1931.18c6.1849.196.2219s.2506 14 41785.1419.1327.1359.1466.16.0.1841 16.1367.1086.1016 1040. 1123.1248.1410 18.1080.085E.08C3.0822.0887.09E6.1114 20.0875.0695.065C.0666.0719.C7SS.0902 4 2.7334 92.1729 2.0314 2.C 803 2.2455 2.4967 2.8191 6 1.2148.9657.9029..246...8o.1096 1...2530 8.6833.5432.5075.5201.5614.6242.7048 10.4373.3477.~3250.3328.35i93.3955.4511 10 12.3037.2414.2257.2311.2495.2714.3132 14.2231.1774.1658.1698'.1i833.2038. 2301l-... 16.1708.1358.127C.1300.1403.1560C.1762 18.1350.1073.1(3.1027.1109.1233.1392 20.1093.0869.0813-.C832.O8!98.C9~9.1128

97 TABLE I THE CONSTANT SHEAR FLEXIBILITY PARAMETER FOR LACED STRUCTURAL MEIPBERS (3) VALUES OF ap(l+ a)/b FCF Ad/b = 1.0 5 a/b =.4.6.8 1.0 1.2 1.4 1.6 A/A d -/b 4 *7029.5388 *4844.4786.5012.5440.6029 6 *3124.2394.2153.2127.2228.2418 *2680 8 *1757.1347.1211.1156.1253.1360.1507 10.125..0862.0775.C766.0802.C87C.0965 2 12.C781.0599.0538.C532.0557.C6C4.0670 14.0574.044GC. 3..0-C391 0409.0 444.0492 16.0439.033 7 030C.0299.0313 *034C 0.037 7 18.0347.0266.023S.C236.024E.C265.0298 20.0281.0216.01S4.C191.020C.0218 *0241 4 1.4058 1.0775.9688.9571 1.0024 1.C88C 1.2058 6.6248.4789.43C6..4254.4455.4835. 5359 8.3515.2694.2422.2393.25C6.2720.3014 10.2249.1724.155C. 1531.1604.1741.1929 4 12.1562.1197.10C6.1063.1114.12CS.1340 14.1148.088C.0751.C781.0818.C8E8.0984 16.C879.0673 *06C6.0598.0626 *068C.0754 18.0694.0532.0478.C473.0495.0C57.0595 20.0562.0431.03EE.C383.04C1.G435.0482 4 2.1088 1.6163 1.4532 1.4357 1*5036 1.6319 1*8087 6.9372.7183.645.6.381.6683.7253.8039 8.5272.4041.3633.3589 *3759.40E0.4522 10.3374 *2586.2325.2297 *2406.2611.2894 6 12.2343.1796.1615.1595.1671.1813.2010 14.1721.131S.11lE.1172.1227.1332.1476 16.1318.101c.09CE.C897.0C40.1020.1130 18.1041.0798. 071E., 709.0743.8ec6.-0893 20.0844 O0647.05E1.0574.0601.0653.0723 4 2.8117 2.155C 1,9376 1.9142 2,0048 2.1759 2,4116 6 1. 2496.957.8612.*E508.8910.9671 1 —0718 8.7029 *5388.4844.4786.5012.5440.6029 10.4499.344.8.31"C.3063. 32C8.34E 1.3858 8 12.3124.2394.2153.2127.2228.2418 *2680 14.2295.175S.1582.1563.1637.1776 *1969 16.1757.1347.1211.1196 *1253.1360 *1507 18. 1388 *1064,095;.. C945. 0990.1075.1191 20.1125.0862.0775 1 C766.0802.087C.0965 4 3.5146 2.6938 2.4221 2.3928 2.5059 2.7199 3.0145 6 1.5621 i.1972 1.i07tE 1.C635 1.1138 1.2CE 1i.3398 8.8e787.6734.6055.5982.6265.6800.7536 10.5623.431C.3875.3828.4010.4352.4823 10 12.3905.2993.2691.2659..2784.3022.3349 14,2869.2199..1971.193.2046,2220.2461 16.2197.1684.1514.1495.1566.17CC.1884 18. 1736. 33C. 116. 1 182. 1238.1343.1489 20.1406,10C78.096S.0957.1002.1088.1206

98 TABLE I THE CONSTANT SHEAR FLEXIBILITY PARAMETER 1i FOR LACED STRUCTURAL MEMBERS (4) VALUES OF p(l+Ea)/Eb FCR Ad/Ab = 1.5 Saa/b =.4.6.8 1.0 1.2 1.4 1.6 AC/Aa P/b 4.8592.6429.5625.5411.5. 533.5 e8.6 420 6.3819.2857.25CC.2405.2459.2616.2853 8.2148.1607.14C6.1353.1383.1412.1605 10.1375.1029.O9CC.0866.0885.C942.1027 2 12.0955.0714. 0625.C601.0615.0C654.0713 14.0701.0525 -.04 5. 0442.0452. 0481.0524 16.0537.0402.03.52.0338.0346.0368 o0401 18.0424.0317.-027E.0267.0273.02S1.0317 20.0344.0257.0225.0216.0221.0235.0257 4 1.7183 1.2858 1.1251 1.0821 1.1065 1.1772 1.2839 6 7637..5 71-5..50CC.4809.4918-i8- 5232.5706 8.4296.3215 o2813.2705.2766.2943.3210 10.2749.2057.18CC 1731.177C.1884.2054 4 12.1909.1429.1250.1202.1229.13C8.1427 14.1403.105C.0<18.C883.O003.0961.1048 16.1074.0804.0703.0676.0692.0736.0802 18.0849.0635.0556.0534 o0..0 46.0581.0634 20.0687.0514.045C.0433.0443.0471.0514 4 2. 5 7 7 5 9288 1 687 1.6232.1 6598 1 7659 1. 9259 6 1.1456.8572.750C.7214.7377.7848.8559 8.6444.4822.421S.4058.4.150.4415.4815 10 *4124.3086.*27CC.2597.2656.2825.3081 6 12.2864.2143.1875.1804.1844.1962.2140 14. 2.104.1574.378.3..* 132325. 1355.1442.1572 16.1611i.1205.1055.1014.lC37.1104.1204 18. 1273.0952 0833.C802.0820.0812.0951 20.1031.0772.0675.0649.0664.0706.0770 4 3.4367 2.5717 2.2501 2.1642 2.2131 2.3545 2. 56. 6 1.5274 1.143C 1.00CI.9619.9836 1.0464 1.1412 _ 8.8592.6429.56625.5411.553 33.5865.6420 10.5499.4115.36CC.3463.3541.3767.4108 8 12.3819.2857.25CC.2405.2459.2616.2853 14.2805.2099.1837.1767.1807.1922.2096 16.2148.1607.14C6.1353.1383.1472.1605 18.1697.12'70. 1111.1069.1093.1163.1268 20,1375.1029.0osCC,866.0885 C942. 1027 4 4.2959 3.2146 2.8127 2.7053 2.7664 2.94c31 3.2098 6 1.9093 1.4287 1.2501 1.2023 1.2295 1.3081 1.4266 8 1. 0740.8037 ~ 7032. 6.763. 65916 ~ 7.358.8024 10.6873.5143.45CC.4328.4426.4709.5136 10 12.4773.3572.3125.3006'.3074.3270.3566 14.3507.2624.2296.'22C8..2 2 58.24C3.2620 16.2685.2009.1758.1691.1729.1839.2006 18.2121.1587.1389.1336.1366.145-.1585 20.1718.1286.1125.1082.1107.1177.1284

99 TABL E I THE CONSTANT SHEAR FLEXIEILITY PARAMETEP p FOR LACED STRUCTURAL MEPEERS (5) VALUES OF p(l+.)/b FCFA A 2.0:-+ a )b d b E a/b = *4.6.8 1.0 1.2 1.4 1.6 A /A _-/b 4 1.0154.7471.64C7.6036.6054.6 33 3.6810 6. 4513.3320.2 8 41.2684- 2 6. 2 --- 0 28 15 3027 8.2539.1868.1602.1509.1513.1583.1703 1i0.1625.119. ~1025.-C966. 0969 ~1013.1090 2 12.1128.083C.0712.C671.0673.07C4.0757 14. C829.061C. 0523.C493. 04-4.0517.0556 16.0635.0467. 04CC.0377. 0378 C3S6.0426 18. 0501.0369.0316. 0298. 025-5 -.0313 -.0-3-36 20.0406.029S.0256.C241.0242.0253.0272 4 2.0308 1.4942 1.2813 1.2071 1.2107 1.2665 1.3620 6.9026.6641'.5695., 365.5381.5629.6053 8.5077.3735.32C3.3018.3027.3166.3405 10.3249.2391.205C.1S31. 137.2026.2179 4 12.2256.1660.1424.1341.1345.14C7.1513 1..-.4'.. —. 1 122C -.1-0-.46.. —----—. —0.-5.,8.5 —-0c8-8.. 8....-i-0-.-4. 11i16.1269.0934. 0801.0754.0757.07S2.0851 18.1003.0738.063 3. C596. 0598. —.0625 -.-067 3 20.0812.0598.0513.0483.0484.05C7.0545 4 3.0463 2.2413 1.922C 1.8107 1.8161 1.89S8 2.0430 6. 1.3539'.9961...-8542.e047.....-8071-...8444.9080 8.7616.5603.480C5.4527.4540.474.5108 10.4874.3586.3075.2897.2906.3040.3269 6 12.3385.2490.213E.2C12.2018.2111 *2270 14.2487.1830.156S.1478.1483.155 1.1668 16.1904.1401.1201.1132.1135 * 11e7.1277 1.8... -- --.150 1 4 -.-. 07.-s —....., — -897. 9- 8.oo1009 20.1219.0897.076S.0724 *0726.076C.0817 4 4.0617 2.9883 2*5626 2.4142 2.4214 2.5331 2.7241 6 1..8052 1.-3282 1.3 i C 1.0730 1.0762 1.1258 1.2107 8 1*0154.7471.64C7.6036.6054.6333.6810 10- -. —-,-6499.4781.41C. 863.3-387 4 I.-4-, c 03-.4358 8 12.4513.3320.2847.2682.2690.2815.3027 14.3316 * 2439-.2-02-.1971.*-1S77.2068.2224 16.2539 1868.1602.1509.1513.1583.1703 18.2006.1476.1266.1192.1156 12 1. 1345 20.1625.1195 *1025.S966.0969.1013.1090 4 5.077.1 3.7354 3.2033 3.0178 3.0268 3.1663 3.4051 6 2.2565 1.6602 1.4237 1..3412 1.3452 1.4073 1.5134 8 1.2693. 9339.80C8.7544.7567 7916 _.8513 10.8123.5977.5125.4828.4843.5066.5448 10 12. 5641.4150.3559.3353.3363.3518.3783 14.4145.3049.2615 2. 463. 2471.25E5.2780 16.3173.2335.20C2.1886.1892.1979.2128 18.2507.1845.1582.1490..1495.1564.1682 20.2031.1494. 1281.1207.1211.1267.1362

100 TABLE II THE CONSTANT SHEAR FLEXIBILITY PARAMETER p FOR BATTENED STRUCTURAL MEMBERS. THE LONGITUDINAL AND BATTEN ELEMENTS AKt WE SHAPES AND CHANNELS RESPECTIVELY. THE VALUES OF 1 FOR q =2.0, r1~=1.5, E =.85, Z=0........... (.1..).... I.Ab -= -] —_.- __- _- _.A_ - ~~Z/a = 6 _ 8 10 ~~~~~~~12 114 _ 1 6 1 8_ _._.....................a._...-..........._.......R.a__= etlO _...........1_2_ 1_.__._...........~..............!6._..4._.___~h~.__.~~.r......_!__... 9C 6.2289.I674.1344.1140.1002.0903.0829 8.1212.0884.0710.0603.0931.0480.0442 10..0768 05 60.0.0450.0382.0337.0305.o0281 12-,.00544.0396.0318.0271.0239.0216.0199 80 14.0415.0302.0243.0206.0182.0165.0151 16.I0334.0244.0196.0166.0146.0132.0122 l8.0280.0204.0164.C139.0123.0111.0102 20.0242.0177.0142.0120.0106.0096.0088 __6 _.3218.2304.1815.1514.1312.1167. 1,058.1650.1175.0924.0771.0669.0596.0542 10.r1015.0720.0566.0472 0410.0366 0.333 12.0698.0495 ~ 0389.0324.0282.0251.0229 ~00 14.0518.0367.0288.0240.0209.0186.0170 16.0406.0288.0226.0189.0164.0146.0133 18.0332.0235.0185.0154.0134.0119.0109 20.0280.0198.0156.0130.011;3.0101.0092 6.4357.3076.2394.1975.1693.1491.1340;3.2190.1534.1189.0980.0840.0740.0666 10 1320.0920C.0711.0586.050 2.0443.0399 12.0891. 0619.0478.0393.0337.0297.0268 I?0 14.0649.0450.0347.0286 0245 0216.0195 16.0500.0346.0267.0220.0188.0166.0150 18.0401.0278.0214.0176.0151.0133.0120 20.03 3.0230.0178.0146.0125.0111.0100 6 _.5705.3991.3079.2520.2144.1876.1675 F~i. 2830.1959.1503.1227.1043.0912.0815 10. 1684. 1!58 ~ 0385. 0721.,0612.0535.0478!12. "1122.0768. 0585.0476. 0404.0353.0316 140 14.0807.0551.0419.0340.0289.0253.0226 16.0613.0418.0318.0258. 0219.0191.0171 18.0486.0331.0251.020 )4.01!73.0151.0135 20.0398.0271.02C6.0167.0142.0124.0111 6.7261.5046.3870.3149,2666.2320.2061 8.3569.2451_.1866.1513. 1278.1111.0987 I0.2104.1433.1086.0877. 0740.0643.0571 12.1389.094C.0.70.0572.0482.0418.0372....'0 14. 0O90.0667.0502.0404. 0340. 0295.0262 16.0746.0501.0377.030'3..0255.0221.0196 18.0586.0393.0295.C237.0199.0173.0154 2.0...0475.... 3.9..: 0239.0.192.....0 6.1.0140....0_124

101 TABLE II THE CONSTANT SHEAR FLEXIBILITY PARAMETER P FOR BATTENED STi. UCTURAL MEMBERS. THE LCNGITUDINAL.AND BATTEN ELEMENTS ARE WF SHAPES AND CHANNELS RESPECTIVELY. THE VALUES OF P FOR r,.=2.0, qb=1.5, E =.85, Z=O (2.). 1, 0 /A 1.0 _____._..~._..~_. (... a..__~. -......1.__0, AJ~~b.......-.....! —?-0-.....................~....................6a.... 10 _ 12 14 16 18 k/r 9Z/b 6.3618.2671,2141.1804.1572.1401.1272 8,1874.1381,1108.0935,0815,0729.0662. 10.1177.0866.0695.0587.0512.0458.0417 12,.0831.0612.0491.C414,.0362.0324.0295 80 14. 0636.0468. 0376,0317, 0277.0247,0225 16.0515.0379, 0304.0257.0224.0200.0182 1 8.0434.0 320.0257,0216.0189.0169.0153 20. 0378.0279.0224 C189.0164.0147 0133 6.5124.3733.2958.2467.2128.1881.1693 3.2554.1853.1467.1223.1056.0935.0843 10. 1545.8.118.0e4.0737.0637.0564.0509 1 2.1054.0762.0602.0502.0434.0385.0347 100 14,0780.0563,0445.0371.0321.0285,0257 16.0,611.0442.0349.0291.02,52.0223,0202 18.0501.0362.0286.*C239.o0206.0183.0165 20.0424.0307.0243.0202.0175.0155.0140 6.6969.5035.3961.3280.2812,2471.2211 3.3 391 2435. 1910 ~ 1580. 1354. 1191 1067 10.2001.!43.1120.0926,0794.0698.0626 12.1333 09 5 C.0743.0614.0526.0463.0415 20 14. 0964.0686.-0536.0443. 0380.0334.0300 16.0739.0526.0411.0339.0291,0256,0230 18,0593.0422.0329.0272.0233.0205.0184 20.0492.0350.0274,C226.0194.0170.0153 6. 9152.6576.5147.4243.3622.3168.2824 8.4382.3124.2435.2003. 1708 1494,1332 10,2544. 1803, 1401. 1151. 0981.0858.0765 12.1667.1176, 0912.0748.0637.0558,0498 1_40 1.. 118 _ -6.0 8.3 5 0646.0 530. 0451. 0395. 0352 16,0895.0629.0487.C399.0339.0297.0265 18. 0707.0496.0384 ~ C314. 0268. 0234.0209 20.0578.0406,0314.0257. 0219.0191.0171 6 1. 1672.8354.6516.5355.4556.3974.3532 8.5527.3920.3042,2492.2117.1845.1640 10.3172.2234.1'726.141 1.1197.1043.0927 12..2.053..1_439 t 1 C18.0904.0767.066 8.0593 160 14.1443.1007,0775.0631. 0-535.0465.0414 16,1077.0750.0576,0469.0397,0345.0307 18. 0841. 058.4, 0448. 0365.0309.0269, 02 39 20.0680.0472.0362.0294.0249.0217.0192

102 TABLE II THE CONSTANT SHEAR FLEXIBILITY PARAMETER p F'R BATTENED STRUCTURAL MEMBERS.'THE LC NGITUDINAL AND BATTEN ELEMENTS ARE WF SHAPES.AND CHANNELS RESPECTIVELY. THE VALUES OF p FOR n_=2.0, T.=1.5, E =.85, Z=0 (...)_.=- /x 1.0,__ A I __ b c -W ——. —_= 6 8 _ ~~~ ~~10 12 14 16 1.8 "%L'c.4947.3667.2938.2469.2141. 1900.1715 8.2537.1878.1505.1266.1099.0977.0883 10,1585.1173.0940.0791.0687,0611.0553 12. 1119.0828 0664 o.0558. 0485.0432.0391 90 14.0857.0634.0508.0428.0372.0330.0299 16.0695.0515.0413.0347.0301.0268.0242 1.8. 0589. 0436.0349.0294.0255.0227.0205 20.0514.0381.0305.0257.0223.0198.0179.7029.5162.4102.3420.2945.2596.2328 8.3458 o.2531.2009.1675.1444.1274.1144 10. 2075.1516.1202.1002.0864.0763.0686 12.1409. 1028. 0815.0680. 0586. 051.8.0466 100'14.1041.0759.0602.0502.0433.0383.0344 16.0817.0596.0472.0394. 0340.0300.0270 18.06.70.0489.0388.0323_.0279.0246.0222 20.0569.0415.0?29.0275.0237.0209.0188 6.9580.6994.5528.4586.3931.3450.3081 8 _.452.3335._2630.7180.1869.1641.1467 10.2682.1941. 1 529.1266. 1086.0954.0853 i 2.1775.1282.1C8.0835., 0716.0629.0563 120)!1. 1278.0922.0725.0600. 0514.0452.0405 16.0979.0706 o.0554.~0459.0394.0346 0310 18.0785.0566.0444.o 0368.0315.0277.0248 220.0652.0470.0369.0306.0262.02 3-0.0206 6. 2598..9161..72._1-.5967.5099.446 1.3972 8.5935.4288.3366.2779.2373.2076.1850 10.3404.2448.1917.1581. 1349.1181.1052 12.2211.1585.1239.1021.0871.0762.0679 140 1.4. 1564.!1119.0873.0719.0613.0537.0478 16 I. 177.0840.0656.0540.0460.0403.0359 2... 8..0928.0662.-0516..0425.0362.o 031'7.0282 20.0758.0541.0422.0347.0296.0259.0231 6 1.6082 1.1662.9163.7560.6446.5628.5002 8.74_86..5389..._421 7.3471. 2956.2580.2.2 93 10.4240.3035.2367.1945.1655.1443.12 83 12. 271.8 ~ 1 93.7. 1507.1237.1051.0917.0815 160 14.1897.1348.1047.C858.0729.0636.0565 16.1408.0998.0774.0634.0539.0470.0417 18.1096.0775.0601.0492.0418.0364.0324 20. 0884. 062_5. 04.84..0_396...0337....029_3.0261!

103 [TABLE'I I THE CONSTANT SHEAR FLEXIBI L ITY PARAMETER p FCR BATTENED JSTRUCTURAL MEMBERS. THE LONGITUDINAL AND BATTEN ELEMENTS ARE WF SHAPES AND CHANNELS RESPE CT IVELY. THE VALUFS OF P FOR rl =2 O r =1 5 =.85, Z=O (4) rb/r = 1.0, A /Ab = 2.0 =/a = 8 12 4 16 18 Q/r Q/b 6.6275. 4664 3 7 36 3133 2711.2398 *2157 8. 3200 2. 37. 9 03. 597. 1383 122 5 *1104 ].. 1994. 1480 o 1185.00995.0863 o.0765.0689 1?.1407.1 044.0836.0702 * 0(609.0540 *0486 80 ]4.1078.(800 r)641.0538.0466.043.0373 l'6 ~.0876, n650.521.0437. 0379 0336.0303 I..8 0.743.0552. 9),-142.0 371.0 321. 0284. 0256 20 0.61. 83.0387.0325. 0281.0249.0224 6. 893.T591. 5245.4373.3762.3310.2963 8.4362. 3209.2551.2127. 1831 1613.1445 i0, 2605.1 91.13. 5 20. 1 28. 1092.0962.0863 12.17 65,1295 1 029 0858.0739.0651.0584 100 14.1303.0956.0759.0633.0545.0481.0431 1.6.1022.0 75 C. 0596.0497. 0428. 0377. 0338 18.0839.0616.0489.0408.0351.0310.0278 20.0713.0524. O416.0347. 0299.0263.0236 6 1.2192 2.7952 7095.5892. 5051.4429 *3952 8.3.5702.4235.3350.2781.2383.2091.1867 10. 363.2452.1937.1607.1377.1209.1080 12.2216.1613.1273.1056.005.0794.0710 I 20.9 14.1593.1158.0 913.0757.0649.0570.0509 1 6. 1 219.0885.0698.0579.0496.0436 *0389 8.0977.0710.0560.0464.0398.0349.0312 20.0812.0590.0465.C386.0331.0290.0259 6 1, 6045 1.1 746. 9283. 7690. 6576. 5753. 51 21 8. 74-87 *5452.4297.3555. 30439.2367 10.426;5.3093.2433.201.1 718.1503.1339 12.2756.1 994.1566.*293.1104.0966.0861 140!14. 1943.1402.1100.0908. 0776.0679 7.0605 16. 459.1052.0825 *0680 0582 81.0508.0453 8. 1148 *082 7,0648.0535.0457.0400,0356 20.0938.0676.0529.0437.0373.0326.0291 6 2 Q.0493 1.4970 1.1809 9766.8337.7282.6472 8.9445 6858.5392.4451. 3796.3314.2945 10.5308,3835.3)308.2479.2112.1844.1639 12.338..2_435 96..190..569.1336. 1 166.1036 j160 14.2350.1688.1319.1]085.0923.0806.0716 16.1740.1247.0973.0800;0681.0594.0528 18. 135].0967.07 54.C620. 0527. Q460.0409 2.O 1089 _.0778. 0607 0499. 0424 037C. 0329

104 TABLE [1 THE CONSTANT SHFAR FLEXIBILITY PARAMETER P FOR BATTENED STRUCTURAL M'EM'BERS. THE LCNGITUDINAL AND BATTEN ELEMENTS ARE WF SHAPES AND CHANNELS RESPECTIVELY. THE VALUES OF P FOR rl =2 b l =1 =.85, Z=0 (5) rb/_rc = IO5LAY =..5 Z/ = -6 8 1__ 10 _ 12 _ 14.16 18 -~/r Il/b 6.1704.1236.a993.C848.0752.0684.0634 8. 0961.06965. )560. C478. 0424.0386.0358 10.0637.0462. 0371.0317.0281.0255.0237 12.0466.0338.0272.0232.0205.0187.0173 q80 14.0364.0 265.021. 3.C181.0160.0146.0135 16.0299.0217.0175.0149.01i31.0!19.0110 18.0254.0185.0149.0127.0112.0101.0093 20 0?23',0162.0130.0111.0098.0088 0081 2.312 I.12 4.1272.1061.0924. 0827.0756..1264.0886.0693.0578.0503.0451.0413 10.0814.0570.0446.0372.0324.0290.0266 12.0580.0406.0318.0265.0231.0207.0189 14.0442.0310.0243.0203.01 76.0158.0144 16.0354.0249.0195.0163.0141.0127.0116 1]8.0294.0207.0162.0135.0118.0105.0096 20. C025.0177. 0139.0116.0101.0090.0082 6.3058.2102.1614.1325.1136.1004.0907 8.1638.1120.0858.0703.0603.0533.0482 10. 1035.0706.0540.0443.0.'580.0336.0304 12. 0724.0494 0378.0309 0265.0235.0212?.......14.0543.)37C.02 83.0232.0199.0176.0159 16.04?7.0292.0223.0183.01 5 7.0'139.0126 18.0349.02 3 9.0183.0150.0!29.0114.0103 20.0294,020,00155.C127. 0!09.0096.0087 6.3942.2668.2021.1638.1389 _.1215.1087 8 2082.1398.10.54.0853.0722.0632.0566 10.1298.0868.0553.0528.0447.0391.0350 12.0897.0599.0450.0363.0307.0269.0241!40 14.0664.0443 0333.0269.0227.0199.0178 16.05!.6.0345.0259.0209.01t77.0155.0139 113...04.17..0279.-.0210._0169. 01" 43. 2 5 01 9_12 20.0347.0232.0175.0141. 012 0.0105.0094 t6.4962.3322.2491.2000.1680.1458'.1295 8.2595..1720.1282.1026.086080.0745.0662 10.1603.1057.0785.0626.0525.0455.0404 12. 1097.0721.0534.0426.0357.0309.0274 160 14.0805.0528.0391.0312.0261.0226.0201 16.0620.0407.0302.0240.0201. 0174.0155!8.0497.0326.0242.0193.0161.0140.0124 20.0410.0269.020C.0159.0133.0115.0102

105 TAB R LE I.I THE CONSTANT SHEAR FLEXIBILITY PARAMETER p FOR BATTENED STRUCTURAL MEMBERS. THE LC NGITUD'[NAL AND BATTEN ELEMENTS'ARE WF SHAPFS &NO CHANNELS RESPECTIVELY. THE VALUES OF 1 FOR 2.0, =0.5,.85 ZO 0 x r I 5 __AA$Ab I's 0 _(6) r./r..5)~ A/ =_ 1.0 __1._ ____.. _-_.. 175,_..._...__0. /a =. 8. 1 12 16 1 8 _LZrc R/b s.2449.1794.1440.1220.1071.0963.0882 8.1373.1005.0807.o0684.0600.0540.0495 1 -. 0914.0670..o33.o3 0455.o0400.0360.0329 32.0676.04 95.0398.C337.0295.0265.0243 ~n 14.0535.0393.0315.C267.0234.0210.0192!6.04t45.0327.0262.0222.0194.0)174.01 59 13.0381.0282.0226.0191.0167.0150.0136 20 *.0340. 0250.0201.0 169.0148.0132.0120 6'.3312.2374.1871. 1.561.1352.1202.1089 3. 1781 1274.1003.0837.0725 0645 0585 10.1144.0817.0643.0537.0465.o 0414.0376 1.2318.0584. 0460.0384. 0333.0296 0269 1 0 1.4. 0628. 0450. 0354. C296. 0256.0228.0206.6.0.08.0364.0287.0239.0207.C184... 0167 18.0426.0306.0241.C201.0174.0155 _.0140 20.3. 0.8.0265.02C9.0174.0151.0134.0121 6.4371.3087.2402.1982.1699.1496.1345 8. 2287 1606. 1?47. 1 028.0881.0777.0698 10. 1430.1003 o.0777.0640.049.0484.0436 12. 0999. 0700. 0542.0447. 0'38.03 3 8.0304. 20 14.0750.0526.0408.0336.0288.0254.0229 16. 0594.0/+4 17. 0324.0267. 0229.0202.0181 18.0489.0344.0267.0220.0189.0166.0150 20 0415.0292. 0227. C187. 0161 0.142.0127 6.5625.3931.3031.2480.2110.1846.1648 8.2886.2002.1537.1255.1067.0933.0834 10.] 7 7 2 o.1224 o.0938.0765.0650.0569.0508 12.1216 a.0838.0642.0523.0444.0389.0347!.4.) 14.0899.0620.o 0474.C386.0328.0287.0257 16.0701.0483.037C.0302.0256.0224.0200 18.0569.0393.0301.0245.0208.0182.0163 20.0476.0329.0252.0206.0175.0153.0137 6.7073.4906.3758.3056.2%85.2250.1999 3.3579.2459.1873.1518. 1282. 1115.0990 10.2169.1481.1124.0C10.0767.0667.0592 1'2.1469.1000.0758.0612.0516.0448.0398!60 14. 10'7"3.0729.0552.0446.0376.0'326.0290 1 6. 0826.0562.0425.0343. 0289.0 251.0223 18.0663.0 451.0341. 02'76.0232.0202.0179 20.0949.0373.0283.0229.0193.0167.0149

lo6 TABLE 11 THE CONSTANT SHEAR FLEXIBILITY PARAMETER 1P FOR BATTENED STrRUCTURAL MEMBERS'THE LCNGITUDINAL AND BATTEN ELEMENTS ARE WF SHAPES AND CHANNELS RESPECTIVELY. THE VALUES OF p FOR n =2.0, l,_=1.5, E =.85, Z=0......fi...... (_.___ _..~.8 10.........14..........2.__a-...._..... 8o._..... 16 18.A/r 2,/b 6.3193.2352.1:86.1592.1390.1242.1130 8. 78 131 4 1054 0890. 0777. 0695 0632 O.130?. 0878.0704.059.4.0519.0464,0422 12.0886.0653,0523.0442.0385.0344,0313 s80 1.4.0706.05 _-.0417,0352_.0307,0274,0248 16, 059 0.0436.0350.0295.0256.0229.0207 18.0512.0379.0303.0256.0222.0198.0179 20.0457.0338.0271.02?8.0198,0176.0159 6.4311.3123,2471.2061.1780.1577.1422 8.2299,1 662.1314 1 096.0947.0839.0758 10.1473.1064.0841.0702.0606.0538.0485 12.1055.0763.0603.0503.0435.0385,0348 100 14.0814.0589 _ 0466.0388. 0336.0297.0268 16.0661.0479.037q.0316. 0273.0242.0218 18. 0558.0405.0 320 0267. 0231.0204.0184 20.0485.0352. 0279.0233. 0201.0178.0160 6.5684.4071.3190.2638.2261.1989.1782 8.2935 2?093 ~ 1636.1352 1159.1020.0915 10.~1826.1299.1015.C838.0719.0632,0567 1 2. 1273..0906...0707 C 584.0501.0441.0395 120 14.0958.0682.0533.0440.0377.0332.0298 1,6 0761.0542.0424.0350.0300.0264.0237 18 06 2 9.0449 0351. 0290 0249.021.9 0196 20.0536.0383.0300.0248.0213.0187 e.0168 6.7309.5193.4041.3322.2832.2477.2209 8,3690.2605.2020.1657.141,1235.1102 10.2247.1580.1223.1002.0853.0746.0666 12.1536. 1 078 0833.0683. 0581.0509, 0454 140 _ 14.1134.0796.0615.0504.0429,0375.0335 16.0885.0622.0481.0394.0335,0293.0262 18.0720...._0506.0392,0321.0273,.0239.0213 20,0605.0426.0329,0270.0230.0201.0180 6.9185.6489.5025.4111.3490.3042.2703 8.4564.3197.2463.2010.1704.1484.1318 10.2735.1906.1464.1193 ~.01,0879.0781.1_2.1841._.1279.0981.0798.0676.0588.0522 160 14.1341.0930.0713.0580.0491.0427.0379 16.1032.0716,0549,0446, 03'78.0329.0292 18. 0829.0575 ~ 0441. 0359, 0303.0264.0235 2_0__' 068 7.0478..0366._0298..02_.5.2.0 29.....0195

107 TABLE II THE CONSTANT SHEAR FLEXIBILITY PARAMETER p FOR BATTENED STRUCTURAL MEMBERS. THE.'fPNGITUDINAL AND BATTEN ELEMENTS ARE WF SHAPES AND CHANNELS RESPECTIVELY. THE VALUES OF p FOR r =2.0,,,1.5i =85, Z=/ L8)rtbc=.b./r'. _.._AA 2.0... _.. __ 8 10 12 14 16 1 8 k/r 9/b 6.3938.29 1.2.333.1964.1709.1521.1378 8. 2197.1623.1301.1096.0954 o.0849.0770 10.1469.i-1086.0871.0 733.0638.0568.0514 1 2.1095.081C.0649 o.0546.0475.0423.0383 80 14.0876.0649.0520.0437.0380.0338.0305 16. 0736. 0545 ~ 0'37 0367. 0319. 0283.0256 18.0641.0475. 0381.0320.0278.0246.0222 20.0574 0425. 0341. 0286. 0248. 0220. 0198 6 ~ 5311.3873.3071.2561.2209.1951.1755 8.2816.2050.1624.1354.1169.1033.0930 10.1803.1311.1039.0866.0748.0661.0595 12.1293.0941.0746.0622.0537.0474.0427 100 14.1000.0728.0577.0481.0415.0367.0330 16.0815.0594.0471.0393.0339.0299.0269 18.0690.0504 o040C.0333.0287.0254.0228 20.0602.0440.0349.0291.0251.0221.0199 6.6996.5056.3977.3294.2824.2481.2220 8.3584.2579.2025.1676.1437.1263.1131 10.2221.1596.1252.1036.0888.0781.0699 12.1548.1111.0872.0721.0618.0544.0487 120 14. 1166.0838.0657.0544.0466.0410.0367 16.0928.0667.0524.0434.0372.0327.0293 18.0769.0554.0435. C360. 0309.0271.0243 20.0657.0474.0373.0309.0264.0232.0208 6.8992.6456.5051.4163 *3553.3108.2770 8.4495. 3208.2502.2059.1756.1536.1370 10.2721.1936.1507.1239.1057.0924.0824 12.1855.1318.1025.0843.0718.0628.0560 140 14.1370.0973.0757.0622,.0530.0464.0414 16.1070.0760.0591.0486.0414.0363.0323 18.0872.0620.0483.0397.0338.0296.0264 20.0733.0522.0407.0335.0285.0250.0223 6 1.1297.8073.6292.5167.4395.3834.3407 8.5548.3935. 3054.2502.2126.1853.1647 10.3301.2331.1804.1476.1253.1092.0970 12.2213.1559.1204.0984.0835 _.0728.0647 160 14.1609.1131.0874.0714.0605.0527.0469 16. 1238.0871.0672.0549..0466.0406.0361 18.0995.0700.0540.0442.0375.0326.0290 20.0826.0582.0449.0367. 0312.0272.0241

lo8 TABLE II THE CONSTANT SHEAR FLEXIBILITY PARAMETER p FnR BATTENED STRUCTURAL MEMBERS~ THE LCNGITUDINAL AND BATTEN ELEMENTS ARE WF SHAPES 4ND CHANNELS RESPECTIVELY. THE VALUES OF p FOR n =2.0, q,=1.5, =.85, Z=0 (27) rb/rC =2.O2 A /A.5....= 8 10 12 14 16 18 p1/b.~,/rc R/b 6 Z.1500.1082.0870.07.45.0664.0607.0566 8 *.0873.0631.0507.0434.0386.0353.0329 10.0591.0427. 0343.0294 ~0261.0238.0221 12 ~ 043:8.0317?0255 o0218 01 94,0176.0164 80 14 034-7.0251.0202.0172.0153.0139.0129 16.0287.0208.0167.C143.0126.0115,0106 18.0245.0179.0143.0122.0108.0098.0090 20.0216.0157.0126.0107.0095.0086.0079 6.1995.1386 o1081.0903.0788.0708.0650 8.1129,0784.0611.0511.0445.0400.0368 10,0744.0517.0404.0337.0294 0264.0242 12.0539.0375.0293.0245.0213.0191.0176 100 14.0416.0290.0227.0189.0165.0148.0136 1. 6.0336.0235.0184.0154.0134.0120.0110 *1. 020281, 0197.0154,0129.0112.0100.0092 20.0242.0170.0133.0111.0097.0087.0079 6.2604.1761.1342.1098.0941.0834.0756 A.1445.0975 0742.0607.0520 0461 0418 10. 093 5.0631.. 0480. 0393.0337. 0298 ~ 0271 12.0665. o450. 0343.0280 0240.0213.O1 93 120 14.0505,0342.0261.0213. 0183,062.0147 16.0402.0273.0208.0171.0146.0129.0117 18.0331.0225 ~0172.0141.0121.0107.0097 20,0280.0191. 0146.0120.0103.0091.0082 6.3325.2205.1651.1330.1124'.0983.0881 8 ~1820.1202.0897 0722.0610,0533.0478 10.1163.0767.0572.0460,0389.0340.0305 12.0818.0540.0403.0324.0274.0239.0215 140 14.0613, 0405.0303.0244.0206.0180.0161 16.0482.0319.0239.0192,0163.0142.0127 18.a0393.0261..01095. 0157 01 33.0116.0104 20.0329.0219.0164.0132.0112.0098.0088 6.4157.2718 *.2008.1598.13336.1156.1027 8 _ 2254.1 46 _.1077. 0 855 _ 071 4.0618 A.0548 10. 1427.0925. 0679. 0539. 0449.0389.0345 12,.0994.0644, 047.3, 0375. 03'13,0271.0240 "60 14.0740.0'+80.0352.0279.0233.0202.0179 16.0577.0374.0275.0218.0182.0158.0140 18.0466.0303.0223.0177.0148.0128.0114 2_0...038 7.025?...!6,.......01..48.0123.01..7.~.00_95...

109 TABLE I I THE CONSTANT SHEAR FLEXIBILITY PARAMETER W FOR BATTENED STRUCTURAL MEMBERS, THE LCNGITUD-INAL AND BATTEN ELEMENTS ARE WF SHAPES AND CHANNELS RFSPECTIVELY. THE VALUES OF P FOR rc =2.0, r =1 5 a = 85 Z=O (1.). rb/rc 2-.0, A/ = 1. AC-/ Q/a = 6 8 10 12 14 16 18 Q/r Q/b 6. 2040.1487.1194.o1S.07895. 0810.0746 8.1197.0874.0702.0596 *0525.0475.0437.10.0822.0601.0482.0410 0360.0325.0299 12.00621.0454.065.0309.0272.0245 *0225 8') 14.03500.0366.0294.0249.021 8 *0196 01 80 16.042'0.308.0247.0209.0183.0165.0151 1.8.365 0269 0215. 269 215.C18 0159 *0143.0130 20.0326.024C.0192.0163.0142.0127.0116 6.2678.1898.1491 1244 1080 *0964.0878 ~.1 ~ 5!t. 071. 0 84. 0702 0609. 0544.0495 10.! 003.0712.0559.0467. 04.05. 0.361.0329!?.0735.0522.0411.0343.0297.0265.0241. 0..)' 14.0575.0410.0322,0269.0233.0208.0189 16.0471.0 33 7. 026 5. C21 0 -i9.0171.0155 18 0400.0286.0226.0188. 063 45.0131 20.0349.0250.0197.0165.0143.0127.0115 6.3462.2405.1857.1527.1309.1155.1042 8.,1900. 316.1015.0834.0715.0632.0570 1t.12"51.0 853. 065.5 C5 1.0463.0409.0369 12.082.8 0612.04-72.0388 0333.0294 *0265 20 14. 0676.0470.0363,0299.0256.0226.0204 1.6.0543 0 379. 02 93.C241. 0207.0182.0164 18.0453.0317.0 245.020).03t73 *.0153.0137 29.0388 *0272.0211.0174 _.0149.0131.0118 6.4391.3005 2290 1 863.1581_, 1383. 1237 8. 2.3,,2. 609. 1223.0993.0842.0737.0659 1.0. 1502. 102?2 0776.0630. 0534.0467.0418 12. 1058.0720 0547.0444.0377. 0330.0295 40 14.0799.0544.0414.0336.0285.0250 *0223 16.0633.0432. 0329.0268.0227.0199.0178 1 8.0520 03 56.*0272.0221.0188.0164.0147 20.0440.0302.0231.0188.0160.0140 40125 6. 546.4.3699 2792.2251 1896.1646.1462 8.28' 7. 1. 94 7. 1464 *. 177. 0990 0859.0763 10f. 1818. 218.0914.0734. 0617. 0535.0475 - 2. 1264 0847. 0635.0510.O0429 0372.0330 1 60 14.0943.0632.0474.0381.0320. 0278.0247 16.0739.0496. 0372.0299. 0252. 0218. 0194 18.0601. 0404.030(4. 0244.0206 () 7.0178. 015 9 20.0503.0339.0255. 206. 01773.0150.03O 3

110 TABLE II'THE CONSTANT SHEAR FLEXIBILITY PARAMETER P FOR BATTENED STRUCTURAL MEMBERS. THE LCNGITUDINAL AND BATTEN ELEMENTS ARE WF SHAPES AND CHANNELS RESPECTIVELY. THE VALUES OF P FOR Ti =2.0, rb=1.5, =85, Z=0 (11) rb/rc =.__2 —-A/-.. —_ 1.5 *5 _. Z/a = 6 8 10 12 14 16 18 Q/r Q/b 6.2580.1892.1518.1285.1127 *1012.0926 8. 1521 111 7.0896.0758.0664 0.596 *0545 1O.1054 *0775. 0621,.0525.0460 *0412.0376 _ _12 1.0804.0591 0474.0401.0350. 031 3 *0285 E80 14 *0653.0481. 0386.0325 ~ 0284.0254.0231 16 0554. 0409. 0328. 0276.0241 0215.0195 18.0485.0359 0287 *0242.0211.0188 *0170 20.0436.0323.0259.0218.0189 o0169 *0153 6.3360.2410 ~ 1900.1585.1373.1220.1105 8.1893.1 357. 1070. 0893. 0773.0687.0623 10 o1262.0906.0715.0596 0516 *0459.0415 12.0931 *0670. 0529.0441.0382 *0339.0307 14.0734.0529.0418.0349.0302.0267.0242 16.0607.0438. 0346.0289.0250 ~ 0221 *0200 18.051 9.0376. 0297.0248.0214.0190.0171 20 *0456.0330. 0261.0218. 0188.0167 *0150 6.4320.3048.2371.1956.1677.1477.1328 8.2355. 1658. 1288. 1 062 0910.0802.0721 10. 1526.1074.0835 0688.0590.0520.0467 12.1098.0774.0602.0496.0425.0375.0337 2 0 14. -0846. 0598. 0465 o C384 * 0329 02 90 *0260 16.0685.0485. 0378.0312 o 0267 *0236.0211 18. 0575.0408 * 0318.C263. 0225 * 0198.0178 20.0496.0353.0276.0228 * 01 95.0172 *0154 6. 5457 *3805. 2930 *2396.2038 * 178 3 1592 8 * 2905.2016.1548. 1264 ~ 1075. 0940 *0840 10. 1842.1276.0980.0800.0680 *0595 *0531 12.1299.0901 o0692 *0565.0480 *0420,0375 140 1 4 _ 0984. 0683 0525 C 4 29 0365.0319 *0285 16. 0783. 0545. 0420.0343. 0292 *0255.0228 18. 0648..0452. 0348._ O285. 0242~ *0212. 01 89 20.0551.0385.0297 *0243.0207. 0181.0162 6.6771.4679.3576 2904 * 2456.2136 *1898 8 _.3541. 2430. 1850.1499. 1265.1100.0977 10.2208. 511 1148.09C29.0784.0682.0606 12. 1534. 1049.0797.0 645.0 544. 0473.0420 160 14.1146.0784. 0596.0482.0407.0354.0314 16.0901.0617.047C.0380.0321.0279.0248 18.0735.0505. 0385. 312. 0263. 0229.0203 20.0619.0426.0325.0264.0223.0194.0172

111 TABLE II THE CONSTANT SHEAR FLEXIBILITY PARAMETER P FOR BATTENED STRUCTURAL MEM8ERS. THE LONGITUDINAL AND BATTEN ELEMENTS ARE WF SHAPES AND CHANNELS RESPECTIVELY. THE VALUES OF p FOR qc =20 OB=l., =.85, Z=O (1.2) r /r =2.0 A/A 2.0....._ —b_..c........ —..~. _..e/_.A....b_........../...._........... 6__/.......-...._.8.10.... _ 12'14 16 18 k/r k/b 6.3120.2297.1842.1555.1358.1215 1106 8.1846.1360.1090.0920.0803.01't8.0653 10.1-286.0948.0760.O641.0559.0499.0453 12.0986.0728.0584.0492.0428.0382.0346 680!~ -.0806.0596.0477.0402.0350.0311.0282 16.0687.0509.0408.0343.0298.0265.0240 18.0605.0449.0359.C302. 0262.0233.0210 20. 0547.0405.0325.0273.0237.0210.0189 6.4042.2922.2310.1927.1665.1476.1332 8 0.2277' 1 644.1299 1084 0937.0830.0750 i0.1522.101. 0870.0726.0627.0556.0502 12. 1128.0817. 0646.0539. 0466.0412.03'7 2 100 1'4.0893. 06.49.0513.0428.0370. 0327.0295 1.6.07 42. 540 0427. 0356..0308.0272,0245 18.0638.0465.0368.C307.0265.0234.0211 20.0562.0411.0326.0271.0234.0207.0186 6.5178.3692.2386.2385.2045.1799.1614.3.2810.1999.1561.1290.1106.0973.0873 10.1822.!.296.1'012.0836.0717.0631.0566 12.1314.0936.0731.0604.0518.0456.0409 14.1016.0726.0568.0469. 0402.0354.0317 16. 0826.0591.0463.0383.0328 0289.0259 18.0696.0499. 0391.01324.0278.0244.0219 20.0603.0434.0340.C282.0241.0212.0190 6.6524.4604.3570.2929.2495.2183.1947 8.3447.2422.1874.1535 1307.1144.1021 10.2181..15.3.1183.G969.0825.0722.0644 12.1540.1081.08.36.0685.0583.0510.0455 140 14.116Q.0822.0636.0521.0444.0388.0347 16.0934.0658. 0510.041.8. 0356.0312..027a.18.0775..05.7 _547.2.025 C348..0297 0.0260 _.02 32 20.0662.0468.0364.0299.0255.0223.0199 6. 8078.5659.4360..3558.3016.2626.2334 8.4184.2913.2236.1820.1541.1342 1192 10.2599.1804.1383.1125.0952.0828.0736 12.1804.1252.0959.0780.0660.0574.0510 160 14.1349.0936.0718.0584.0494.0430.0382 16.1063.0739.0567.0461.0391.0340.0302 18.0870.0606.0466.0379'.0321.0280.0248 20.0734.0513.0394.0321.0272.0237.0211

REFERENCES 1. Engesser, F., "Die Knickfestigkeit Gerader Stabe," Zentralblatt der Bauverwaltung, Vol. 11, 1891. 2. Engesser, F., "Uber die Knickfestigkeit von Rahmenstaben," Zentralblatt der Bauverwaltung, Vol. 29, 1909. 3. "Mller-Breslau, H., "Uber excentrisch gedruickte gegliederte Stabe," Sitzungsberichte d. k. Preuss., Abad. d. Wissenschaften, February 1910. 4. Timoshenko, S., "Theory of Elastic Stability," First Edition, 1936; Second Edition, 1961. 5. Washio, K., "A Study of the Slope-Deflection Method Considering Effects of Axial and Shear Forces," Proceedings of Japan Institute of Architecture, No. 17, March 1940. 6. Amstutz, E. and Stissi, F., "Die Knicklast gegliederter Stabe," Schweizerische Bauzeitung, August 1941. 7. Johnston, B. G. and Mount, E. H., "Analysis of Building Frames with SemiRigid Connections," Proc. Amer. Soc. Civ. Engrs., March 1941. 8. Maugh, L. C., "Statically Indeterminate Structures," First Edition, 1946; Second Edition, 1964. 9. Pippard, A. J. S., "The Critical Load of a Battened Column," Phil. Mag., January 1948. 10. Lothers, J. E., "Elastic Restraint Equation for Semi-Rigid Connections," Proc. Amer. Soc. Civ. Engrs., September 1950. 11. Hoff, N. J., Boley, B. A., and Nardo, S. V., "Buckling of Rigid-Jointed Plane Trusses," Proc. Amer. Soc. Civ. Engrs., 116, 1951. 12. Takekazu, T., "A Solution of Frames with Trussed Members," The Reports of Yokohama National University, Engineering Dept., Vol. 1, 1951. 13. Bleich, F., "Buckling Strength of Metal Structures," McGraw-Hill Book Co., 1952. 14. Pippard, A. J. S., "Battened Columns," Studies in Elastic Structures, 1952. 112

113 REFERENCES (Concluded) 15. Jones, B. D, "A Theory for Struts with Lattice or Batten Bracing," Structural Engineer, May 1952. 16. Koenigsberger, F. and Mohsin, M. E., "Design and Load Carrying Capacity of Welded and Battened Struts," Structural Engineer, June 1956. 17. Winter, G., "Lateral Bracing of Columns and Beams," Proc. Amer. Soc. Civ. Engrs. 84, ST2, March 1958. 18. Johnston, B. G., "Design Criteria for Metal Compression Members," First Edition, 1960; Second Edition, (John Wiley), 1966. 19. Japan Long Column Research Council, "Elastic Stability Survey," CoronaSha, 1960. 20. Hetenyi, M., "An Analytical Study of Vierendeel Girders," Proc. of the 4th U. S. National Congress of Applied Mechanics, Vol. 1, 1962. 21. Gere, J. M., "Moment Distribution," Van Nostrand Co., Inc., 1962. 22. Tamayo, J. Y. and Ojalvo, M., "Buckling of Three-Legged Columns with Batten Bracing," Proc. Amer. Soc. Civil Engrs. 91, ST1, February 1965. 23. Mohsin, M. E., "The Critical Load of Unsymmetrical Welded Battened Struts," Structural Engineer, November 1965. 24. Williamson, R. A. and Margolin, M. N., "Shear Effects in Design of Guyed Towers," Jour. of the Struct. Div., ASCE, Vol. 92, October 1966. 25. Fenves, S. J., Logcher, R. D., Mauch, S. P., and Reinschmidt, K. F., "Stress," A User's Manual, Third Printing, December 1966, The M.I.T. Press.

Unclassified Security Classification DOCUMENT CONTROL DATA - R&D (Security claaeification ol title, body ol abatrect and Indexing annotation muet be entered when the overell report ia claeassied) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The Regents of The University of Michigan Unclassified The University of Michigan 2b. GROUP Ann Arbor, Michigan 3. REPORT TITLE The Effects of Shear Deformation and Axial Force in Battened and Laced Structural Members 4. DESCRIPTIVE NOTES (Typs of report and Incuelsve datee) Technical Report S. AUTHOR(S) (Let nanme. fir t name, initial) Lin, Fung Jen 6. REPORT DATE 7-. TOTAL NO. OF PACES 7b. NO. OF REFS May, 1968 113 25 Be. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) NBy-45819 b. PROJECT NO. o05154-4-T C. Sb. OTHER RE PORT NO(S) (Any othernumber, that may be asslgned thls report) d. 10. A V A IL ABILITY/LIMITAtION NOTICES Distribution of this document is unlimited. 1. SUPPL EMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Department of the Navy Bureau of Yards and Docks Washington, D.C. 13. ABSTRACT This dissertation presents a theoretical analysis of the elastic behavior of both the battened and laced structural member considering the effects of axial load, shear deformation, and connection rigidity of sub-elements, and the overall effects of axial load, and effective shear deformation of the complete member. The analytical solution is used to obtain modified slope-deflection equations to generalize a relation between applied forces and joint displacements. A nondimensional parameter, the shear flexibility, is defined so as to characterize the shear flexibility of the members and to take account of the effects of axial force, local joint connections, and local connection flexibility of the battened members. The fundamental linear second-order differential equation for the deflection curve of the member which includes the effect of shear deformation has been derived. By application of the boundary conditions to the general solutionof deflected shape of the member, the solutions are set up in the forms of slope-deflection equations. In the evaluation of the fixed end moments for a concentrated load, the reciprocal theorem is applied so as to make use of the deflection curves which have been previously defined in the case of the homogeneous solution. Elastic buckling loads for the structural members have been evaluated for the cases of a column with hinged ends, a column with one end fixed and the other hinged, and a column with both ends fixed. Finally, numerical examples of beam and frame analyses are presented to provide a comparison with the ordinary beam and frame theories which neglect the effects of shear deformation and axial force. D 1 JAN 64 1473 Unclassified Security Classification

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UNIVERSITY OF MICHIGAN 111111113 90 15 03466 5771 3 9015 03466 5771