On shear and extensional locking in nonlinear composite beams

Laulusa, A.; Reddy, J. N.

doi:10.1016/s0141-0296(03)00175-5

Cited by 8 publications

(11 citation statements)

References 14 publications

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“…Following the same derivations as that are carried out in Laulusa and Reddy [2004], e.g., making use of the one-dimensional constitutive equations and of the straindisplacement relations, δU can be expressed in terms of the cross sectional constants, and of the quantities: δu i , δθ i , δθ i , u i , θ i , c i = cos θ i and s i = sin θ i , where u i and θ i are the three displacements and three rotations, while ( ) designates derivative with respect to x 3 . These derivations are omitted here in the interest of saving space and not repeating previously published results.…”

Section: Reduced Integration Beam Element (Rie)mentioning

confidence: 99%

“…As in Laulusa and Reddy [2004], the six generalized displacements (three displacements u i and three rotations θ i ) are interpolated by identical shape functions of the 2-noded, 3-noded, and 4-noded element. That is, the six kinematic unknowns are interpolated by linear, quadratic and cubic Lagrange polynomials, respectively.…”

Section: Reduced Integration Beam Element (Rie)mentioning

confidence: 99%

“…The approach adopted in Laulusa and Reddy [2004] to overcome shear and extensional locking was to use the reduced/selective integration technique [Hughes (1980)]. It was found in Laulusa and Reddy [2004] that the uniform reduced integration rule with one-, two-, and three-point quadrature rules for the linear, quadratic, and cubic interpolations, respectively, gave the best results.…”

Section: Reduced Integration Beam Element (Rie)mentioning

confidence: 99%

“…The locking is of two kinds, shear and extensional (or membrane), and it can be very pronounced for elements with low order interpolations, e.g., linear and quadratic polynomials. It is well known that the 2-noded (linear) element that simulates a thin beam exhibits an extremely severe shear locking while it was shown in a recent study [Laulusa and Reddy (2004)] that the 3-noded (quadratic) element presents a very pronounced extensional locking when large deflections occurred without extensional deformation. In the same study, it was also shown that for the 4-noded (cubic) element, both the shear and extensional locking could be significant.…”

mentioning

confidence: 99%

“…It was shown in Laulusa and Reddy [2004] that the uniform reduced integration rule provides the best results though several linear elements must be used. Yet, for all the cases investigated, no artificial mechanisms, such as spurious energy modes were noted, unlike plate and shell elements in which these mechanisms can arise.…”

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confidence: 99%

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A Critical Evaluation of Various Nonlinear Beam Finite Elements

Laulusa¹,

Reddy

2012

Int. J. Comput. Methods

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The characteristics of interdependent interpolation and mixed interpolation nonlinear beam finite elements are investigated in comparison with the equal-order interpolation element with uniform reduced integration. The stiffness matrix of the 3-noded and 4-noded equal order interpolation elements is identical to that of the 2-noded interdependent interpolation element if the internal nodal degrees-of-freedom are eliminated. The extension of the latter to include nonlinear kinematics by approximating the extensional displacement and the twist rotation with quadratic and cubic Lagrange polynomials yields unsatisfactory results. The 2-noded, 3-noded, and 4-noded mixed interpolation elements using one-, two-, and three-point quadrature rules, respectively, are shown to be equivalent to the corresponding uniform interpolation elements employing the same quadrature rules. The equivalence is established in the framework of nonlinear kinematics and anisotropic couplings. Int. J. Comput. Methods 2012.09. Downloaded from www.worldscientific.com by FUDAN UNIVERSITY on 05/16/15. For personal use only. A. Laulusa & J. N. Reddy bending-twist and bending-extension couplings. Inclusion of all these features makes the development highly complex.A relatively simple beam finite element is the displacement based uniform interpolation element. But, this element is known to suffer from locking, i.e., the beam element behaves stiffener than it actually is [Reddy (2006)]. The locking is of two kinds, shear and extensional (or membrane), and it can be very pronounced for elements with low order interpolations, e.g., linear and quadratic polynomials. It is well known that the 2-noded (linear) element that simulates a thin beam exhibits an extremely severe shear locking while it was shown in a recent study [Laulusa and Reddy (2004)] that the 3-noded (quadratic) element presents a very pronounced extensional locking when large deflections occurred without extensional deformation. In the same study, it was also shown that for the 4-noded (cubic) element, both the shear and extensional locking could be significant.The locking is often overcome by using the reduced/selective integration technique. It was shown in Laulusa and Reddy [2004] that the uniform reduced integration rule provides the best results though several linear elements must be used. Yet, for all the cases investigated, no artificial mechanisms, such as spurious energy modes were noted, unlike plate and shell elements in which these mechanisms can arise. As there is no mathematical proof that demonstrates the viability of the reduced/selective integration technique, some researchers view it more as an ad hoc technique that happen to work. In fact, the equivalence established in Malkus and Hughes [1978] between some mixed models and the reduced/selective integration displacement model gave a certain foundation to the latter approach, though linear kinematics was considered.Within the framework of linear theory, Bathe [1996] has introduced the mixed interpolation beam element ...

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Section: Reduced Integration Beam Element (Rie)mentioning

confidence: 99%

Section: Reduced Integration Beam Element (Rie)mentioning

confidence: 99%