Construction

# Shear deformable beams and plates : relationships with by C M Wang; J N Reddy; K H Lee

By C M Wang; J N Reddy; K H Lee

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Extra info for Shear deformable beams and plates : relationships with classical solutions

Example text

12) into Eq. \ dM:?. dw^ ^" - i ? 2 D,,w§{x) = ( 5 ? 14) Prom Eq. 15) dx^ dx M•'•XX £ = MXX i l - aPxx = -D^x EUminating dcp^/dx from Eqs. 17) and using Eq. 18) Finally, substituting Eq. 18) into Eq. 19) 44 S H E A R D E F O R M A B L E B E A M S A N D PLATES Now, we wish to simplify the Reddy-Bickford beam theory by neglecting the second-order derivative term in Eq. 19). This amounts to reducing the order of the theory from six to four. We obtain D. DxxW^ix) = DXXWQ{X) + -C2 X'- D, i?. 20) - C3X - C4 In summary, we have the following relations from Eqs.

The relationships SHEAR-FLEXURAL STIFFNESS MATRIX 41 allow interdependent interpolation of t^o ^^^ 0 ^^^ the rank deficiency is removed, resulting in an efficient and accurate locking-free finite element for the analysis of beams according to classical as well as refined beam theories. 1 Relationships B e t w e e n T B T and E B T As discussed in Chapter 2, the shear force, bending moment, slope and deflection of Timoshenko beam theory can be expressed in terms of the corresponding quantities of the Euler-BernouUi beam theory.

1a. 3 The shear correction coefficient for the Timoshenko beam element is taken to be Kg — 5/6. SHEAR-FLEXURAL STIFFNESS MATRIX 49 The structure is analyzed using the aforementioned stiffness method according to the Euler-BernouUi theory and the Timoshenko beam theory. The simpUfied Reddy beam element essentially gives the same results as the Timoshenko beam element, and hence is not included. The exact Timoshenko beam element [A = Dxx/{KsAxz) = EI/{GAKs) and iB = 0] is denoted by UBE. The results are also compared with those predicted by two other commonly used Timoshenko beam finite elements, namely the linear equal-interpolation reduced-integration element (RIE) and the consistent interpolation element (CIE) [see Reddy 1993, (19976,19996)].