APPENDIX F A DISPACEMENT-BASED BEAM EEMENT WITH SHEAR DEFORMATIONS Never use a Cubc Functon Approxmaton for a Non-Prsmatc Beam F. INTRODUCTION { XE "Shearng Deformatons" }In ths appendx a unque development of a dsplacement-based beam element th transverse shearng deformatons s presented. The purpose of ths formulaton s to develop constrant equatons that can be used n the development of a plate bendng element th shearng deformatons. The equatons developed, hch are based on a cubc dsplacement, apply to a beam th constant cross-secton subected to end loadng only. For ths problem both the force and dsplacement methods yeld dentcal results. To nclude shearng deformaton n plate bendng elements, t s necessary to constran the shearng deformatons to be constant along each edge of the element. A smple approach to explan ths fundamental assumpton s to consder a typcal edge of a plate element as a deep beam, as shon n Fgure F..
APPENDIX F- STATIC AND DYNAMIC ANAYSIS z, DEFORMED POSITION INITIA POSITION x,u -s +s - + Fgure F. Typcal Beam Element th Shear Deformatons F. BASIC ASSUMPTIONS In reference to Fgure F., the follong assumptons on the dsplacement felds are made: Frst, the horzontal dsplacement caused by bendng can be expressed n terms of the average rotaton,, of the secton of the beam usng the follong equaton: u - z (F.) here z s the dstance from the neutral axs. Second, the consstent assumpton for cubc normal dsplacement s that the average rotaton of the secton s gven by: N +N +N (F.) 3
SHEAR DEFORMATIONS IN BEAMS APPENDIX F-3 The cubc equaton for the vertcal dsplacement s gven by: N (F.3a) + N + N 3β + N 4β here: - s N, +s N, N 3 - s and N 4 s(- s ) (F.3b) Note that the term ( s ) s the relatve rotaton th respect to a lnear functon; therefore, t s a herarchcal rotaton th respect to the dsplacement at the center of the element. One notes the smple form of the equatons hen the natural coordnate system s used. It s apparent that the global varable x s related to the natural coordnate s by the equaton x s. Therefore: s (F.4) Thrd, the elastcty defnton of the effectve shear stran s: u + ; z hence, - (F.5) Because, the evaluaton of the shear stran, Equaton (F.5), s produces an expresson n terms of constants, a lnear equaton n terms of s and a parabolc equaton n terms of s. Or: 4 ( - ) - s β - (- 3 s ) β - s +s - - - (- s ) α (F.)
APPENDIX F-4 STATIC AND DYNAMIC ANAYSIS If the lnear and parabolc expressons are equated to zero, the follong constrant equatons are determned: β ( - ) (F.7a) 8 β (F.7b) The normal dsplacements, Equaton (F.), can no be rtten as: N +N +N 3 ( - ) +N α (F.8) 8 4 Also, the effectve shear stran s constant along the length of the beam and s gven by: ( - ) - ( + ) - (F.9) 3 No, the normal bendng strans for a beam element can be calculated drectly from Equaton (.) from the follong equaton: ε u z - z [ - +4 s ] s x (F.) In addton, the bendng stran ε x can be rtten n terms of the beam curvature term ψ, hch s assocated th the secton moment M. Or: ε zψ x (F.) The deformaton-dsplacement relatonshp for the bendng element, ncludng shear deformatons, can be rtten n the follong matrx form:
SHEAR DEFORMATIONS IN BEAMS APPENDIX F-5 ψ s 3 / / / 4 or, u B d (F.) The force-deformaton relatonshp for a bendng element s gven by: G da E da z V M ψ α or, σ d C f (F.3) here E s Young' s modulus, G α s the effectve shear modulus and V s the total shear actng on the secton. The applcaton of the theory of mnmum potental energy produces a 5 by 5 element stffness matrx of the follong form: ds CB B K T (F.4) Statc condensaton s used to elmnate to produce the 4 by 4 element stffness matrx. F.3 EFFECTIVE SHEAR AREA { XE "Effectve Shear Area" }For a homogeneous rectangular beam of dth "b" and depth "d," the shear dstrbuton over the cross secton from elementary strength of materals s gven by: ] [ τ τ d z (F.5)
APPENDIX F- STATIC AND DYNAMIC ANAYSIS here, τ s the maxmum shear stress at the neutral axs of the beam. The ntegraton of the shear stress over the cross secton results n the follong equlbrum equaton: 3 τ V (F.) bd The shear stran s gven by: z [ G d ] τ (F.7) The nternal stran energy per unt length of the beam s: 3 E I da V τ 5bdG (F.8) The external ork per unt length of beam s: E E V (F.9) Equatng external to nternal energy e obtan: V 5 Gbd (F.) Therefore, the area reducton factor for a rectangular beam s: 5 α (F.) For non-homogeneous beams and plates, the same general method can be used to calculate the shear area factor.