CAD-integrated analysis of 3-D beams: a surface-integration approach

Abstract

Most engineering artifacts are designed and analyzed today within a 3-D computer aided design (CAD) environment. However, slender objects such as beams are designed in a 3-D environment, but analyzed using a 1-D beam-element, since their 3-D analysis exhibits locking and/or is computationally inefficient. This process is tedious and error-prone. Here, we propose a dual-representation strategy for designing and analyzing 3-D beams, directly within a 3-D CAD environment. The proposed method exploits classic 1-D beam physics, but is implemented within a 3-D CAD environment by appealing to the divergence theorem. Consequently, the proposed method is numerically and computationally equivalent to classic 1-D beam analysis for uniform cross-section beams. But, more importantly, it closely matches the accuracy of a full-blown 3-D finite element analysis for non-uniform beams.

Appendix: boundary integration for the timoshenko beam

In Timoshenko beam theory, the displacements are approximated via (32), where u ₀(x), w ₀(x) and θ ₀(x) are the axial, bending and rotation of a transverse normal about the y-axis. To avoid locking, we use the shape functions associated with the T2CL6 Timoshenko beam element described in [18]. This beam element (see Fig. 16) has a total of ten degrees of freedom with one degree of freedom w _α that is eliminated.

Fig. 16

The T2CL6 beam element

The unknown functions u ₀(x), w ₀(x), and θ ₀(x) are approximated in this beam element via:

$$ \begin{aligned} u_{0} (x) & = N^{u} \hat{d}_{0} \\ w_{0} (x) & = N^{w} \hat{d}_{0} \\ \theta_{0} (x) & = N^{\theta } \hat{d}_{0} \\ \end{aligned} $$

where:

$$ N^{u} = \left\{ {\begin{array}{*{20}c} {(\xi^{2} - \xi )/2} \\ 0 \\ 0 \\ {( - \xi^{2} + 1)/2} \\ 0 \\ 0 \\ {(\xi^{2} + \xi )/2} \\ 0 \\ 0 \\ \end{array} } \right\};\quad N^{\theta } = \left\{ {\begin{array}{*{20}c} 0 \\ 0 \\ {(\xi^{2} - \xi )/2} \\ 0 \\ 0 \\ {( - \xi^{2} + 1)/2} \\ 0 \\ 0 \\ {(\xi^{2} + \xi )/2} \\ \end{array} } \right\}; $$

are the usual quadratic shape-functions while the shape function for the w ₀(x):

$$ N^{w} = \left\{ {\begin{array}{*{20}c} 0 \\ {(\xi^{2} - \xi )/2} \\ {L(\xi^{3} - \xi )/12} \\ 0 \\ {( - \xi^{2} + 1)/2} \\ {L( - \xi^{3} + \xi )/6} \\ 0 \\ {(\xi^{2} + \xi )/2} \\ {L(\xi^{3} - \xi )/12} \\ \end{array} } \right\}^{\text{T}} $$

is obtained by applying the constraint discussed in [18] to avoid locking. Notice that w ₀(x) is approximated via a polynomial that is one order higher than θ ₀(x).

Further, the beam stresses are given by:

$$ \begin{aligned} \sigma_{xx} & = E\left( {u_{0,x} + z\theta_{0,x} } \right) \\ \sigma_{xz} & = kG\left( {\theta_{0} + w_{0,x} } \right) \\ \end{aligned} $$

where k and G are the shear correction factor and shear modulus, respectively. Accounting for the axial and shear strain energies, it is easy to show that the stiffness matrix is now given by:

$$ K_{ij} = \int\limits_{\Upomega } {\left[ \begin{gathered} E\left( {N_{i,x}^{u} + zN_{i,x}^{\theta } } \right)\left( {N_{j,x}^{u} + zN_{j,x}^{\theta } } \right) \hfill \\ + kG\left( {N_{i}^{\theta } + N_{i,x}^{w} } \right)\left( {N_{j}^{\theta } + N_{j,x}^{w} } \right) \hfill \\ \end{gathered} \right]{\text{d}}\Upomega } $$

i.e.,

$$K_{{ij}} = \int\limits_{\Upomega } {\left\{ {\begin{array}{*{20}l} {E\left[ {\begin{array}{*{20}l} {N_{{i,x}}^{u} N_{{j,x}}^{u} + z\left( {N_{{i,x}}^{u} N_{{j,x}}^{\theta } + N_{{i,x}}^{\theta } N_{{j,x}}^{u} } \right)} \\ { + z^{2} N_{{i,x}}^{\theta } N_{{j,x}}^{\theta } } \\ \end{array} } \right]} \\ { + kG\left[ {N_{i}^{\theta } N_{j}^{\theta } + N_{i}^{\theta } N_{{j,x}}^{w} + N_{{i,x}}^{w} N_{j}^{\theta } + N_{{i,x}}^{w} N_{{j,x}}^{w} } \right]} \\ \end{array} } \right\}} {\text{d}}\Upgamma$$

Reducing the integration to the boundary results in the boundary form of the Timoshenko T2CL6 stiffness matrix:

$$K_{{ij}} = \int\limits_{{\partial \Upomega }} {\left\{ {\begin{array}{*{20}l} {E\left[ {\begin{array}{*{20}l} {zN_{{i,x}}^{u} N_{{j,x}}^{u} + {\frac{{z^{2} }}{2}}\left( {N_{{i,x}}^{u} N_{{j,x}}^{\theta } + N_{{i,x}}^{\theta } N_{{j,x}}^{u} } \right)} \\ { + {\frac{{z^{3} }}{3}}N_{{i,x}}^{\theta } N_{{j,x}}^{\theta } } \\ \end{array} } \right]} \\ { + kG\left[ {\begin{array}{*{20}l} {zN_{i}^{\theta } N_{j}^{\theta } + zN_{i}^{\theta } N_{{j,x}}^{w} } \\ { + zN_{{i,x}}^{w} N_{j}^{\theta } + zN_{{i,x}}^{w} N_{{j,x}}^{w} } \\ \end{array} } \right]} \\ \end{array} } \right\}} {\text{d}}\Upgamma$$