A mixed finite element method for beam and frame problems

Abstract

In this work we consider solutions for the EulerBernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation.A finite element solution method is presented from a three-field variational form based on an extension of the Hu-Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects -identical to the behavior of flexibility based elements.The advantages of the approach are illustrated with a few numerical examples. IntroductionThe development of computational models for beam bending problems dates from the earliest days of structural analysis and the literature is too extensive to fully cite here. Mike Crisfield considered solution of beam problems from many perspectives as indicated in approaches contained in his books [1-3] and papers with co-workers [4][5][6][7][8]. Most of his work was for finite displacement applications -many using co-rotational formulations for which he was well known. Here we would like to remember him for his pioneering work in this field of endeavor.In a displacement formulation, the nonlinear straindisplacement relations are postulated and polynomial interpolation functions are used for the displacement approximation [9][10][11]. Because the postulated displacement interpolation functions are approximate in nonlinear material and geometric behavior, each structural member needs to be discretized into several elements in order to capture the actual variation of deformations along its axis. This fine discretization results in a large number of degrees of freedom in the final numerical model of the structure, thus, reducing the computational efficiency of this approach. Alternatively, higher order displacement interpolation functions can be used. This approach results in several internal degrees of freedom that need to be condensed out during the element state determination. Even so, mesh discretization is often required for accuracy. With both of these approaches numerical instabilities are not uncommon, particularly under cyclic loading conditions. Problems with the displacement formulation of beam elements encou...

Mixed Tetrahedral Elements for the Analysis of Structures with Material and Geometric Nonlinearities

“…Distinctive feature of the present formulation is a Gauss-point-discontinuous strain interpolation. Denoting by e e g ∈ R 6 the strain at Gauss point g of the element e, say x e g , the following choice is made (e.g., see [4]):…”

“…Distinctive feature of the present formulation is a Gauss-point-discontinuous strain interpolation. Denoting by e e g ∈ R 6 the strain at Gauss point g of the element e, say x e g , the following choice is made (e.g., see [4]):where δ x − x e g is a Dirac delta measure centered at x e g . The substitution of (3) and (4) into the functional (2) yields its finite element approximation.…”

Quasi-hinge beam element implemented within the hybrid force-based method

“…In extreme cases, nodal forces cannot be determined accurately by direct integration over the whole element length and cross-section [10,11] because the element deformation is too complex to model using cubic Hermitian shape functions, especially when the inelastic deformation gradient is very steep. In such cases the equilibrium equations will not be fully satisfied [12]. Typical techniques to improve the accuracy of displacement-based solution methods when handling steep deformation gradients in inelastic zones are (i) through mesh refinement, though this necessarily leads to longer computation times and (ii) through the use of higher order elements which again leads to longer computation times.…”

“…Typical techniques to improve the accuracy of displacement-based solution methods when handling steep deformation gradients in inelastic zones are (i) through mesh refinement, though this necessarily leads to longer computation times and (ii) through the use of higher order elements which again leads to longer computation times. Thus, the computational efficiency of displacement-based methods can be reduced and numerical instabilities are possible, particularly under cyclic loading [1,12]. In this case, an alternative approach can be to use a force, rather than a displacement-based solution strategy.…”

Finite element response sensitivity analysis using three‐field mixed formulation: general theory and application to frame structures

“…Mixed frame elements are more accurate in nonlinear analysis than displacement-based elements and are a possible alternative to the recently established force-based elements [19]. Examples are available in the recent literature for monolithic beams [19][20][21] and for composite beams with deformable shear connection [22,23]. This paper focuses on the formulation of finite element response sensitivity analysis in the case of a nonlinear three-field mixed approach derived from the Hu-Washizu variational principle, considering both quasistatic and dynamic loadings.…”