Eccentricity effects in the finite element modelling of composite beams
Highlights
► A correction technique is introduced to avoid eccentricity related numerical errors in the modelling of composite beams. ► It is shown that the error can be significant in practical applications. ► Effect of loading type on the eccentricity related numerical errors is illustrated. ► Effect of relative layer stiffness on the numerical error is illustrated.
Introduction
Composite beams and laminated plates that consist of components juxtaposed with a shear connection find widespread applications in structural engineering. A mathematical model for composite beams with flexible shear connectors was introduced by Newmark et al. [1], in which two Euler–Bernoulli beams are connected by assuming that vertical separation does not occur between the components. Subsequently, several displacement-based finite element formulations were developed based on Newmark’s model, which include the works of Arizumi et al. [2], Daniels and Crisinel [3], Ranzi et al. [4], [5] and Dall’Asta and Zona [6]. However, in many practical cases the interlayer connections are very stiff between the two components such that the interlayer slip is negligible. In such cases, displacement-based finite element formulations based on flexible shear connectors may suffer from the so-called slip-locking phenomenon because of the coupling between the displacement fields in the discretized form of Newmark’s model [7]. Erkmen and Bradford [8] used the kinematic interpolatory strategy and assumed strain-mixed formulations to alleviate locking behaviour for stiff connections. An exact finite element formulation is also presented in [8]. A mesh free approach that eliminates slip locking was recently developed by Erkmen and Bradford [9].
Special solutions, however, are not always available for a design engineer in which case the modelling options are limited to standard conventional beam type finite elements. Therefore, a convenient practice in the modelling of composite beams especially for the cases with stiff connections is to connect the two conventional beam type finite element components by using a rigid bar to connect the end nodes of the two components or use master–slave type kinematic constraints to express the nodal degrees-of-freedom of one of the members in terms of the other. However, this type of modelling leads to eccentricity related numerical errors as initially shown by Gupta and Ma [10] in a composite cantilever beam problem. A similar type of error in multiple-point constraint applications for built-up plates and shells was pointed out by Crisfield [11]. Recently, Erkmen et al. [12] adopted variational multiscale approach to correct numerical errors in the applications of master–slave type constraints when forming composite beams.
In this study, a simple correction technique is introduced for the applications of master–slave type constraints to form composite beams and it is shown that the eccentricity related numerical errors in the stiffness matrix can be completely corrected by using extra fictitious elements and springs. The correction terms are obtained by selecting interpolation functions that do not violate the zero-slip constraint between the components in the point-wise sense. Since numerical errors are avoided the developed solution provides identical results to those that can be obtained using the elementary solution for composite sections, e.g. transformed area method in classical mechanics of solids books [13].
Section snippets
Displacements and strains
The composite member is made up of a top and a bottom Euler–Bernoulli beam elements, which are referred to as members 1 and 2, respectively. Deformations of each component can be expressed in terms of the axial displacements of the centroids and the vertical displacements of the members, where throughout the document. Axial strain in each component of the composite beam can be determined by using the Euler–Bernoulli beam kinematics, hence in terms of the axial displacement gradients
Finite element formulation
A displacement based finite element formulation can be developed by employing the total potential energy functional, i.e.where the first integral is the elastic bending energies of the two components, in which is the cross-sectional area of the component and L is the span of the composite member, and is the work done by the external forces. From Eqs. (1), (2), the total potential energy functional in Eq. (2) can be written as
Numerical errors due to eccentricity
In order to illustrate the numerical behaviour, a 2 m span composite cantilever beam is analysed herein. As shown in Fig. 1, the top component of the composite beam has the modulus of elasticity of E1 = 200 × 103 MPa, cross-sectional area of A1 = 6 × 103 mm2 and the moment of inertia of I1 = 112.2 × 103 mm4. The bottom component has the modulus of elasticity of E2 = 26 × 103 MPa, cross-sectional area of A2 = 7.1 × 103 mm2 and the moment of inertia of I2 = 124 × 106 mm4. The kinematic constraint conditions of no vertical
Conclusion
In this study, a simple correction technique is introduced to avoid eccentricity related numerical errors in the modelling of composite beams with the conventional beam type finite elements. From the exact homogenous solution of the problem the interpolation functions that satisfy the kinematic constraints in the point-wise sense are obtained and the correct stiffness matrix is developed. The resulting stiffness matrix is contrasted with that of the conventional finite element and the missing
References (13)
- et al.
Non-linear analysis of composite beams by a displacement approach
Comput Struct
(2002) - et al.
Slip locking in finite elements for composite beams with deformable shear connection
Finite Elem Anal Des
(2004) - et al.
Tests and analysis of composite beams with incomplete interaction
Proc Soc Exp Stress Anal
(1951) - et al.
Elastic–plastic analysis of composite beams with incomplete interaction by finite element method
Comput Struct
(1981) - et al.
Composite slab behaviour and strength analysis. Part I: Calculation procedure
J Struct Eng (ASCE)
(1993) - et al.
A direct stiffness analysis of a composite beam with partial interaction
Int J Numer Methods Eng
(2004)