370 ___wIC-ENGINEERING THEORY OF PRISMATIC MEMBERS CHAP. 12 12-7. Refer to the sketch for Prob. 12-3. Determine the reaction R andcentroidal displacements at x = L/2 due to a concentrated force Pi2 appliedto the web at x = L/2. Employ the force method.12-8. Refer to Example 12-7. Assuming Equation (h) is solved for Z1,discuss how you would determine the translation u2 at x = L/2. 1312-9. Consider the four-span beam shown. Assume linearly elastic be-havior, the shear center coincides with the centroid, and planar loading. I(a) Compare the following choices for the force redundants with respectto computational effort: 1I Restrained1. reactions at the interior supports2. bending moments at the interior supports(b) Discuss how you would employ Maxwell's law of reciprocal deflections Torsion-Flexure ofto generate influence lines for the redundants due to a concentrated force moving from left to right. a Prismatic Member Dn,,. 4 a ."� n /77 77* /7 13-1. INTRODUCTION12-10. Consider a linearly elastic member fixed at both ends and subjectedto a temperature increase The engineering theory of prismatic members developed in Chapter 12 isbased on the assumption that the effect of variable warping of the cross sectionT = a + a2x2 + a3x3 on the normal and shearing stresses is negligible, i.e., the stress distributionsDetermine the end actions and displacements (translations and rotations) at predicted by the St. Venant theory, which is valid only for constant warpingmid-span.12-11. Consider a linearly elastic member fixed at the left end (A) and and no warping restraint at the ends, are used. We also assume the crosssection is rigid with respect to in-plane deformation. This leads to the resultsubjected to forces acting at the right end (B) and support movement at A. that the cross section twists about the shear center, a fixed point in the cross i Determine the expressions for the displacements at B in terms of the supportmovement at A and end forces at B with the force method. Compare this section. Torsion and flexure are uncoupled when one works with the torsional approach with that followed in Example 12-2. moment about the shear center rather than the centroid. The complete set ofgoverning equations for the engineering theory are summarized in Sec. 12-4. Variable warping or warping restraint at the ends of the member leads toadditional normal and shearing stresses. Since the St. Venant normal stressdistribution satisfies the definition equations for F, 3I2, M3 identically, theadditional normal stress, a, must be statically equivalent to zero, i.e., it mustsatisfy ffa' , dA f x2(,rrl I A = ' 3U',;, A = O (I i 18 The St. Venant flexural shear flow distribution is obtained by applying the i! engineering theory developed in Sec. 11-7. This distribution is statically equiva-lent to F2, F3 acting at the shear center. It follows that the additional shear stresses, (o2 and a13, due to warping restraint must be statically equivalentto only a torsional moment: R SfCr1 2dA 0 (13-2) To account for warping restraint, one must modify the torsion relations. Wewill still assume the cross section is rigid with respect to in-plane deformation. 371-- C 372 373 :· i_ -CI-· sI·II·iWI-i~ RESTRAINED TORSION-FLEXURE OF PRISMATIC MEMBER CHAP. 13 In what follows, we develop the governing equations for restrained torsion. We start by introducing displacement expansions and apply the principle of virtual displacements to establish the force parameters and force-equilibrium equations for the geometrically linear case. We discuss next two procedures for establishing the force-displacement relations. The first method is a pure-displacement approach, i.e., it takes the stresses as determined from the strain (displacement) expansions. The second method is similar to what we employed for the engineering theory. We introduce expansions for the stresses in terms of the force parameters and apply the principle of virtual forces. This cor-responds to a mixed formulation, since we are actually working with expansions for both displacements and stresses. Solutions of the governing equations for the linear mixed formulation are obtained and applied to thin-walled open and closed cross sections. Finally, we derive the governing equations for geomet-trically nonlinear restrained torsion. 13-2. DISPLACEMENT EXPANSIONS; EQUILIBRIUM EQUATIONS The principle of virtual displacementst states that JffaT 3& d(vol.) = JffbT Au d(vol.) + 'fpT Au d(surface area) (a) is identically satisfied for arbitrarydisplacement, Au, when the stresses (a) are in equilibrium with the applied body (b) and surface (p) forces. We obtain a system of one-dimensional force-equilibrium equations by introducing expan-sions for the displacements over the cross section in terms of one-dimensional displacement parameters. This leads to force quantities consistent with the dis-placement parameters chosen. We use the same notation as in Chapters 11, 12. The X1 axis coincides with the centroid; X2, X3 are principal inertia axes; and x2, x3 are the coordinates of the shear center. We assume the cross section is rigid with respect to in-plane deformation, work with the translations of the shear center, and take the dis-placement expansions (see Fig. 13-1) as l = ul + 2x3 -C) 3X2 + f a12 = Us2 -t)1(X3 - X3) (13-3) f13 = U3 + (O1(x2 - X2) where ¢ is a prescribed function of x2, X3, and-1. u1, us2, Us3 are the rigid body translations of the cross section. 2. 091, c02, (03 are the rigid body rotations of the cross section about the shear center and the X2, X 3 axes. 3. f is a parameter definining the warping of the cross section. The variation over the cross section is defined by 4). Note that all seven parameters are functions only of x. For pure torsion t See Sec. 10-6. SEC. 13-2. DISPLACEMENT EXPANSIONS; EQUILIBRIUM EQUATIONS (i.e., the St. Venant theory developed in Chapter 11), one sets f = o, = const and 0 = ,. For unrestrained variable torsion (i.e., the engineering theory developed in Chapter 12), one sets f = 0. Since there are seven displacement parameters, application of the principle of virtual displacements will result in seven equilibrium equations. x3 1s3 ---!. e9 / Shear center tIs2 X2 1 1I I -4(3 I X3 l X27 2c Centroid Fig. 13-1. Notation for displacement measures. The strain expansionst corresponding to (13-3) are el = U1, + 0)2. 1X3 -C03, I2 + ft 1 E2 = 3 = 23 = 0 Y12 = Us2, 1 -03 -0)1. (X3 --3) + f 2 713 = U3,1 + 092 + )1, (X2 -2) + f, 3