SEC 12 2 12 Engineering Theory of Prismatic Members INTRODUCTION St Venant s theory of flexure torsion is restricted to the case where 1 2 There are no surface forces applied to the cylindrical surface The end cross sections can warp freely The warping function consists of a term due to flexure Os and a term due to pure torsion t Since k is independent of x the linear expansion F1 t A ll M12 2 1 13 Y2 331 satisfies the definition equations for FI M 2 M 3 identically the normal stress correction is self equilibrating i e it is statically equivalent to zero Also the shear flow correction is statically equivalent to only a torsional moment since 12 3 satisfies the definition equations for F2 F 3 identically In the engineering theory of members we neglect the effect of variable warping on the normal and shearing stress i e we use the stress distribution predicted by the St Venant theory which is based on constant warping and no warping restraint at the ends In what follows we develop the governing equations for the engineering theory and illustrate the two general solution procedures This formulation is restricted to the linear geometric case In the next chapter we present a more refined theory which accounts for warping restraint and in vestigate the error involved in the engineering theory 12 2 12 1 FORCE EQUILIBRIUM EQUATIONS FORCE EQUILIBRIUM EQUATIONS In the engineering theory we take the stress resultants and couples referred to the centroid as force quantities and determine the stresses using 12 1 12 3 and the pure torsional distribution due to MT To establish the force equilibrium equations we consider the differential element shown in Fig 12 1 The statically equivalent external force and moment vectors per unit dxI 2 12 1 clxl 2 1 F d tr 2 l is the exact solutiont for alt The total shearing stress is given by 12 2 6 1S C 1 f where at is the pure torsion distribution due to 4 and af represents the flexural distribution due to Of We generally determine caf by applying the engineering theory of shear stress distribution which assumes that the cross section is rigid with respect to in plane deformation Using 12 1 leads to the following expression for the flexural shear flow see 11 106 qB qA Q3 2 Q2 3 12 3 The warping function will depend on xl if forces are applied to the cylindrical surface or the ends are restrained with respect to warping A term due to variable warping must be added to the linear expansion for al This leads to an additional term in the expression for the flexural shear flow Since 12 1 t A linear variation of normal stress is exact for a homogeneous beam Composite beams e g a sandwich beam are treated by assuming a linear variation in extensional strain and obtaining the distributions of ao from the stress strain relation See Probs I I 14 and 12 1 xt dil v 2 Fig 12 1 Differential element for equilibrium analysis length along X are denoted by b nii Summing forces and moments about 0 leads to the following vector equilibrium equations note that F F M M dF b dx1 dM dxT dxl 330 dx 71 x F a THFORY OF PRISMATIC MEMBERS l cl rlNUILL i 332 CHAP 12 We obtain the scalar equilibrium equations by introducing the component expansions and equating the coefficients of the unit vectors to zero The re sulting system uncouples into four sets of equations that are associated with stretching flexure in the XI X 2 plane flexure in the X X 3 plane and twist Note that the shearing force is known once the bending moment variation is determined The statically equivalent external force and moment components acting on the end cross sections are called endforces We generally use a bar superscript to indicate an end action in this text Also we use A B to denote the negative and positive end points see Fig 12 2 and take the positive sense of an end Stretching X2 dF b 0 xY 1A 2 Flexure in X 1 X2 Plane FA2 dF2 b2 0 dxl dM 3 m3 f F2 dx Flexure in X1 X 3 333 FORCE DISPLACEMENT RELATIONS SEC 12 3 12 4 X1 00 B3 Plane X3 dF3 b3 0 dM 2 rn2 dx1 Fig 12 2 Notation and positive direction for end forces F3 0 force to coincide with the corresponding coordinate axis The end forces are related to the stress resultants and couples by Twist Mt 0 ml dxt FBj This uncoupling is characteristic only of prismatic members the equilibrium equations for an arbitrary curved member are generally coupled as we shall show in Chapter 15 The flexure equilibrium equations can be reduced by solving for the shear force in terms of the bending moment and then substituting in the remaining equations We list the results below for future reference dM 3 d x d2 M3 M dir dxl n 113 1 din b2 0 dxl 12 5 Flexure in X 1 X 3 Plane dMz F3 d2 M22t dn2 d2M dx2 M2 d dx J b3 o Fj x L MBj Mj L FAj rEjr ICj3r 0 MAJ MIAj Mj x o j 1 2 3 12 6 Aminus sign is required at A since it is a negative face 12 3 Flexure in X 1 X2 Plane F2 I U r i 3 dx FORCE DISPLACEMENT RELATIONS PRINCIPLE OF VIRTUAL FORCES We started by selecting the stress resultants and stress couples as force parameters Applying the equilibrium conditions to a differential element results in a set of six differential equations relating the six force parameters To complete the formulation we must select a set of displacement parameters and relate the force and displacement parameters These equations are generally called force displacement relations Since we have six equilibrium equations we must introduce six displacement parameters in order for the formulation to be consistent Now the force parameters are actually the statically equivalent forces and moments acting at the centroid This suggests that we take as displacement 334 CHAP 12 ENGINEERING THEORY OF PRISMATIC MEMBERS parameters the equivalent rigid body translations and rotations of the cross section at the centroid We define Ctand cas u lj equivalent rigid body translation vector at the centroid C ji equivalent rigid body rotation vector SEC 12 3 FORCE DSPLACEENT RELATIONS 335 Evaluating the right hand side of b we have cli 3di AP 12 7 d d AF AI dx t AM C M dx c Using the second equation in a c takes the form By equivalent displacements we mean fJ force intensity displacement dA F A 12 8 M c Note that 12 7 corresponds to a linear distribution of displacements over the cross section whereas the actual distribution is nonlinear owing to shear de formation In this approach we are allowing for an average shear deforma tion determined such that the energy is invariant We establish the force displacement relations by applying the