wIC 370 ENGINEERING THEORY OF PRISMATIC MEMBERS CHAP 12 12 7 Refer to the sketch for Prob 12 3 Determine the reaction R and centroidal displacements at x L 2 due to a concentrated force Pi2 applied to the web at x L 2 Employ the force method 12 8 Refer to Example 12 7 Assuming Equation h is solved for Z 1 discuss how you would determine the translation u at x L 2 2 12 9 Consider the four span beam shown Assume linearly elastic be havior the shear center coincides with the centroid and planar a Compare the following choices for the force redundants loading with respect to computational effort 1 reactions at the interior supports 2 bending moments at the interior supports b Discuss how you would employ Maxwell s law of reciprocal deflections to generate influence lines for the redundants due to a concentrated force moving from left to right Dn 7 7 77 4 a n I I 1 a 2x 2 Restrained Torsion Flexure of a Prismatic Member 7 13 1 12 10 Consider a linearly elastic member fixed at both ends and subjected to a temperature increase T a 13 a3 x 3 Determine the end actions and displacements translations and rotations at mid span 12 11 Consider a linearly elastic member fixed at the subjected to forces acting at the right end B and support left end A and movement at A Determine the expressions for the displacements at B in terms of the support movement at A and end forces at B with the force method Compare this approach with that followed in Example 12 2 i INTRODUCTION The engineering theory of prismatic members developed in Chapter 12 is based on the assumption that the effect of variable warping of the cross section on the normal and shearing stresses is negligible i e the stress distributions predicted by the St Venant theory which is valid only for and no warping restraint at the ends are used We also constant warping assume the cross section is rigid with respect to in plane deformation This leads to the result that the cross section twists about the shear center a fixed point in the cross section Torsion and flexure are uncoupled when one works with the torsional moment about the shear center rather than the centroid The complete set of governing equations for the engineering theory are summarized in Sec 12 4 Variable warping or warping restraint at the ends of the member leads to additional normal and shearing stresses Since the St Venant normal stress distribution satisfies the definition equations for F 3I M identically the 3 additional normal stress a must be statically equivalent2 to zero i e it must satisfy ffa dA i R f x2 rrl I A 3U A O I i 18 The St Venant flexural shear flow distribution is obtained engineering theory developed in Sec 11 7 This distribution by applying the 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 equivalent to only a torsional moment SfCr 1 2 dA 0 13 2 To account for warping restraint one must modify the torsion relations We will still assume the cross section is rigid with respect to in plane deformation 371 372 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 i C CI SEC 13 2 DISPLACEMENT EXPANSIONS EQUILIBRIUM EQUATIONS 373 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 1 s3 X2 e9 center Shear tIs2 1 1I 13 2 DISPLACEMENT EXPANSIONS EQUILIBRIUM EQUATIONS I 4 3 I X3 The principle of virtual displacementst states that JffaT 3 d vol JffbT Au d vol l 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 X 1 axis coincides with the centroid X2 X 3 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 t 1 X3 Us2 f13 U3 O1 x2 X3 13 3 Fig 13 1 Notation for displacement measures The strain expansionst corresponding to 13 3 are el U1 E2 C03 I2 ft 1 23 0 03 0 1 713 U 3 1 092 X3 1 X2 f 2 f 3 3 2 13 4 JfffrT ie d vol F 1 Aul1 F2 Auls2 1 A 3 F3 Aus3 1 A 2 M2 A 2 1 M 3 A03 I MT Al 1 M Af 1 MR Af dx 1 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 X 2 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 0 2 1X3 3 Y12 Us2 1 X2 where is a prescribed function of x2 X3 and t See Sec 10 6 Centroid Using 13 4 the left hand side of a expands to 2x3 C 3 X2 f a12 X2 7 2c For pure torsion b where the two additional force parameters are defined by M Sll dA MR o12 2 13 0 3 dA 13 5 Note that M has units of force …