Efficient modeling and order reduction of new 3D beam elements with warping via absolute nodal coordinate formulation

Abstract

To describe the particular mechanical behaviors of beams with both uniform and non-uniform cross sections, such as the bidirectional bending, torsion-bending coupling, the torsion-related warping, the cross-sectional stretch, and Wagner effects, a series of efficient higher-order beam elements (HOBEs) is proposed in the frame of the absolute nodal coordinate formulation (ANCF). In the proposed HOBEs, a new mixed kinematic description of beam elements is introduced via the warping functions and slope vectors. Compared with the existing HOBEs using Lagrange polynomials, the additional degrees of freedom per element proposed to accurately describe the warping deformation are dramatically reduced. Moreover, the tremendous Von-Mises stress on the cross sections in the existing HOBEs does not occur in the proposed new HOBEs. Compared with the classical nonlinear finite elements formulations, the complete 3D strain state with the higher-order terms allows the cross-sectional stretch and avoids the expensive calculations of the extra warping and Wagner strain measures and their derivatives. Moreover, the transverse integration allows an arbitrary section shape to vary along the beam axial direction. Thus, these new HOBEs benefit from the efficient warping description in the classical FE and inherit the advantages of 3D-continuum theory in the ANCF. In addition, the shear locking is alleviated due to the ability to capture the non-uniform distribution of shear stress, and the Poisson locking is addressed via the enhanced continuum mechanics approach. Finally, the proposed HOBEs are validated and compared using statics and dynamics undergoing complex significant deformations on various benchmarks, FEs, commercial codes, and experimental data.

Appendices

Appendix

The existing HOBEs

The existing HOBEs (e.g., B42 [26, 27, 29], 3363 [28], 34X3 [28], and B4 [26]) introduced here are the complete polynomial based on Pascal's triangle. And some existing HOBEs with incomplete polynomial can be found in the works by Matikainen et al. [24] and Ebel et al. [28], just to name a few.

The quadratic beam element BE42

To test the performance of locking alleviation, the higher-order BE42 was studied by the standard CMA [26, 27] and subsequently studied by the strain split method [29]. Compared with the BE24, the main improvement in the BE42 is using the displacement field given by Eq. (5) with quadratic interpolation in the transverse directions. As shown in Table 1, the BE42 possess three additional directional derivatives for each node of the two-node to describe the cross-sectional warping. As displayed in Table 2, the vector of element nodal coordinates is defined as ${\mathbf{e}} = \left[ {{\mathbf{e}}_{1}^{{\text{T}}} \;\;{\mathbf{e}}_{2}^{{\text{T}}} \;} \right]^{{\text{T}}} \in {\mathbb{R}}^{42}$. Thus

$$ {\mathbf{S}}^{{{\text{BE}}42}} = {\mathbf{s}}^{{{\text{BE}}42}} \otimes {\mathbf{I}} = [\underbrace {{{\mathbf{s}}_{1}^{{{\text{BE24}}}} \;\;S_{1,22} \;\;S_{1,23} \;\;S_{1,33} }}_{{{\mathbf{s}}_{1}^{{{\text{BE42}}}} }}\;\;\underbrace {{ \cdots_{{}} }}_{{{\mathbf{s}}_{2}^{{{\text{BE42}}}} }}\;] \otimes {\mathbf{I}} \in {\mathbb{R}}^{3 \times 42} . $$

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What is more, the beam element BE42 studied in [26, 27, 29] is also the beam element 3273 studied in [28].

The cubic beam element 34X3

As shown in Table 2, the beam element 34X3 is proposed using the displacement field given by Eq. (5) considering cubic interpolation in the transverse directions. The vector of nodal coordinates is defined as $\;{\mathbf{e}} = \left[ {{\mathbf{e}}_{1}^{{\text{T}}} \;\;{\mathbf{e}}_{2}^{{\text{T}}} } \right.\;\;\;\left. {{\mathbf{e}}_{3}^{{\text{T}}} \;\;{\mathbf{e}}_{4}^{{\text{T}}} } \right]^{{\text{T}}} \in {\mathbb{R}}^{120}$, where ${\mathbf{e}}_{I}^{{\text{T}}} = \left[ {{\mathbf{r}}_{I}^{{\text{T}}} \;{\mathbf{r}}_{I,y}^{{\text{T}}} \;{\mathbf{r}}_{I,z}^{{\text{T}}} \;{\mathbf{r}}_{I,yy}^{{\text{T}}} \;{\mathbf{r}}_{I,yz}^{{\text{T}}} \;{\mathbf{r}}_{I,zz}^{{\text{T}}} \;{\mathbf{r}}_{I,yyy}^{{\text{T}}} \;{\mathbf{r}}_{I,yyz}^{{\text{T}}} \;{\mathbf{r}}_{I,yzz}^{{\text{T}}} \;{\mathbf{r}}_{I,zzz}^{{\text{T}}} } \right]\;$, $i = 1,\;2,\;3,\;4$. Thus,

$$ {\mathbf{S}}^{{{\text{34X3}}}} = {\mathbf{s}}^{{{\text{34X3}}}} \otimes {\mathbf{I}}{ = }[\underbrace {{{\mathbf{s}}_{1}^{{{3433}}} \;S_{1,22} \;S_{1,23} \;S_{1,33} \;S_{1,222} \;S_{1,223} \;S_{1,233} \;S_{1,333} }}_{{{\mathbf{s}}_{1}^{{{\text{34X3}}}} }}\;\;\underbrace {{ \cdots_{{}} }}_{{{\mathbf{s}}_{2}^{{{\text{34X3}}}} }}\;\;\underbrace {{ \cdots {}_{{}}}}_{{{\mathbf{s}}_{3}^{{{\text{34X3}}}} }}\;\;\underbrace {{ \cdots_{{}} }}_{{{\mathbf{s}}_{4}^{{{\text{34X3}}}} }}] \otimes {\mathbf{I}}. $$

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Warping

5.1 Rectangular beam

Assuming that the warping function of the rectangular cross section takes the form of the product of hyperbolic and sine functions as follows [11],

$$ \omega \left( {y,z} \right) = yz + \sum\limits_{n = 1,3,5 \cdots }^{\infty } {c_{n} \sin \left( {\frac{{n{\uppi }}}{H}z} \right)} \sinh \left( {\frac{{n{\uppi }}}{H}y} \right), $$

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where $c_{n} = ( - 1)^{{{{\left( {n + 1} \right)} \mathord{\left/ {\vphantom {{\left( {n + 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}}} \frac{{8H^{2} }}{{n^{3} {\uppi }^{3} \cosh \left( {{{n{\uppi }W} \mathord{\left/ {\vphantom {{n{\uppi }W} {\left( {2H} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2H} \right)}}} \right)}}$, H and W are the height and width. For a square beam, H = W holds in Eq. (33).

I-section beam

The approximate warping function for the I-section is assumed as follows [10,11,12, 17],

$$ \omega_{{{\text{top}}\;{\text{flange}}}} \left( {y,z} \right) = y\left( {z - \overline{h}} \right),\;\omega_{{{\text{web}}}} \left( {y,z} \right) = - yz,\;\omega_{{{\text{bottom}}\;{\text{flange}}}} \left( {y,z} \right) = y\left( {z + \overline{h}} \right), $$

where $\overline{h} = H - T_{{{\text{flange}}}}$ is the distance between the mid-lines of the flanges and displayed in Fig. 7.

In the I-section beam, the top flange, web, and bottom flange have their own warping functions. Thus, the warping-related function $\overline{S}$ can be written as the following compact form,

$$ \overline{S}_{I} = \omega \left( {y,z} \right)f_{I} \left( x \right) = y\overline{z}f_{I} \left( x \right), $$

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where $\overline{z}_{{{\text{top}}\;{\text{flange}}}} = z - \overline{h}$, $\overline{z}_{{{\text{web}}}} = - z$ and $\overline{z}_{{{\text{botton}}\;{\text{flange}}}} = z + \overline{h}$.

In the previous works [10,11,12], the shear-related energy part is modified by the approximate warping function of Eq. (34) and the shear correction factors. However, a so-called “geometrically exact cross section deformable thin-walled beam” in [17] was proposed via more cross-sectional modes than [10,11,12]. As a result, it can get a more detailed description of the cross-sectional deformation and thus lead to a more flexible torsion response. Similarly, in this study, BE24c-ω and MGD30c-ω are examined with k₂ = k₃ = 0.3868 [2]. Moreover, to get more cross-sectional modes, the quadratic elements BE24c-ω-II and MGD30c-ω-II are also examined without shear energy correction. The dimensionless shape functions related with the additional nodal coordinates ${\mathbf{r}}_{{i,y^{2} }}$, ${\mathbf{r}}_{i,yz}$, ${\mathbf{r}}_{{i,z^{2} }}$, ${\mathbf{r}}_{{i,y^{3} }}$, ${\mathbf{r}}_{{i,y^{2} z}}$, ${\mathbf{r}}_{{i,yz^{2} }}$, and ${\mathbf{r}}_{{i,z^{3} }}$ are depicted in Table

Table 11 Computation time of the unbalanced shaft (T₁: the pre-processing time; T₂: the solving time; T₃(= T₁ + T₂): the total time; $T = T_{3}^{{}} /T_{3}^{s} \times 100\%$;$T_{3}^{s}$: the total time of the standard one)

12, same rules with the BE24c-ω-II.

Hollow rectangular beam

According to the work of Chandra et al. [84], the warping function for the hollow rectangular beam in Fig. 7c is assumed to be

$$ \omega \left( {y,z} \right) = yz\tfrac{{\left( {W - H} \right)}}{{\left( {W + H} \right)}}. $$

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Tapered beam

In the tapered beam of a linearly varying cross section height, the upper and lower limits of integral terms for the initial configuration are linear functions of the axial generalized coordinates [47]. That is, $z_{u} = {H \mathord{\left/ {\vphantom {H 2}} \right. \kern-\nulldelimiterspace} 2} + k_{s} x$ and $z_{l} = - {H \mathord{\left/ {\vphantom {H 2}} \right. \kern-\nulldelimiterspace} 2} - k_{s} x$, where l, H, and ks are the length, height, and slope of beam element, respectively. Thus,

$$ z\left( x \right) = z_{u} - z_{l} = H + 2k_{s} x. $$

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It is worth noting that the nonlinear functions of x for the upper and lower integral limits in Eq. (37) can account for the beam configuration of a nonlinearly varying cross section height.

Substituting Eq. (37) into Eq. (33) to get the warping function of the tapered beam when n = 1 as follows.

$$\omega \left( {y,z\left( x \right)} \right) = y\left( {H + 2k_{s} x} \right) + \frac{{8H^{2} }}{{{\uppi }^{3} \cosh \left( {{{{\uppi }W} \mathord{\left/ {\vphantom {{{\uppi }W} {\left( {2H} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2H} \right)}}} \right)}}\sin \left( {\frac{{n{\uppi }}}{H}\left( {H + 2k_{s} x} \right)} \right)\sinh \left( {\frac{{n{\uppi }}}{H}y} \right).$$

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The integral of an irregular section

The integral of an irregular section is usually divided into n independent integrals of regular sections. Thus, the element elastic energy $U^{\rm E}$ in Eq. (26) can be rewritten as.

$$ U^{\rm E} = \sum\limits_{i = 1}^{n} {U_{i}^{\rm E} } = \sum\limits_{i = 1}^{n} {\frac{1}{2}\left( {\int_{{V_{i} }} {{{\varvec{\upvarepsilon}}}:{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{E} }}^{0} } :{{\varvec{\upvarepsilon}}}dV_{i} + T_{i} W_{i} \int_{{L_{i} }} {{{\varvec{\upvarepsilon}}}:{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{E} }}^{\nu } } :{{\varvec{\upvarepsilon}}}dL_{i} } \right)} . $$

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I-section beam

The I-section is divided into n = 3 rectangular regions to integrate independently, and the subscript i = 1, i = 2, and i = 3 in Eq. (39) represents the web, up flange, and bottom flange, respectively. $T_{1} = H - 2T_{flange}$,$\;W_{1} = W_{web}$, $T_{2} = T_{3} = T_{flange}$, and $\;W_{2} = \;W_{3} = W_{flange}$

Hollow rectangular beam

The hollow rectangular is divided into n = 4 rectangular regions to integrate independently, and the subscript i = 1, i = 2, i = 3, and i = 4 in Eq. (39) represents the top, bottom, left, and right rectangular regions. $T_{1} = T_{2} = T_{3} = T_{4} = T$,$\;W_{1} = W_{2} = W$, and $\;W_{3} = \;W_{4} = H - 2T$.