Microstructural topology optimization of periodic beam structures based on the relaxed Saint-Venant solution

Abstract

Design optimization of beam structures is a significant topic as beams are an efficient load-carrying component in engineering applications. Most of the earlier researches concentrate on the structural optimization of beams with invariant cross-section topology, and design optimization of periodic beam structures, which cross-section topology varies along the axial direction, has not been extensively investigated. This work presents a novel approach based on the relaxed Saint-Venant solution to conduct microstructural topology optimization of periodic beam structures for minimum structural compliance. One benefit of adopting Saint-Venant solution based compliance formulation is that the strain energy induced by transverse shear loading is incorporated in the structural compliance, which is not reflected in that based on first-order homogenization of the asymptotic homogenization (AH) theory, so that a more reasonable objective function is adopted for the optimization problem. In addition, a material connectivity constraint, which is constructed by restraining the ratio of the strain energy calculated from relaxed Saint-Venant solution to that obtained from first-order homogenization of the AH theory, is further proposed to prevent material separation or to strengthen material connection through the thickness direction. The detailed sensitivity analysis of the objective function and the constraints are carried out with the adjoint method, and several numerical examples are given to show the validity of the relaxed Saint-Venant solution based compliance formulation and the effectiveness of the proposed material connectivity constraint.

Appendices

Appendix 1 Sensitivity analysis of the strain energy e ^SV

In Appendix 1, we conduct sensitivity analysis of the strain energy e^SV in (33) with the adjoint method. First, it is easy to deduce that

$$ {e}^{\mathrm{SV}}\left(\mathbf{x}\right)=e\left({{}^{\mathrm{n}}\mathbf{V}}^{\alpha },{{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}}^p,\mathbf{x}\right),{{}^{\mathrm{n}}\mathbf{V}}^{\alpha }={{}^{\mathbf{n}}\mathbf{V}}^{\boldsymbol{\upalpha}}\left({{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}}^p,\mathbf{x}\right),{{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}}^p={{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}}^p\left(\mathbf{x}\right) $$

(63)

where x is the vector of design variables. Thus the sensitivity of e with respect to ith design variable x_i can be written as:

$$ \frac{\partial {e}^{\mathrm{SV}}}{\partial {x}_i}={\left(\frac{\partial e}{\partial^{\mathrm{n}}{\mathbf{V}}^1}\right)}^{\mathrm{T}}\frac{\partial^{\mathrm{n}}{\mathbf{V}}^1}{\partial {x}_i}+{\left(\frac{\partial e}{\partial^{\mathrm{n}}{\mathbf{V}}^2}\right)}^{\mathrm{T}}\frac{\partial^{\mathrm{n}}{\mathbf{V}}^2}{\partial {x}_i}+\sum \limits_{p=1}^4{\left(\frac{\partial e}{\partial^{\mathrm{n}}{\overset{\sim }{\mathbf{u}}}^p}\right)}^{\mathrm{T}}\frac{\partial^{\mathrm{n}}{\overset{\sim }{\mathbf{u}}}^p}{\partial {x}_i}+\frac{\partial e}{\partial {x}_i} $$

(64)

We start with the first term at the right hand side of (64) by introducing the following Lagrangian ${e}_{L_1}$ as:

$$ {\displaystyle \begin{array}{c}{e}_{L_1}=e+{\left({{}^{\mathrm{n}}\boldsymbol{\lambda}}_{\mathrm{m}}^1\right)}^{\mathrm{T}}\left[{\mathbf{T}}^{\mathrm{T}}\mathbf{k}\mathbf{T}{{}^{\mathrm{n}}\mathbf{V}}_{\mathrm{m}}^1+{\mathbf{T}}^{\mathrm{T}}\mathbf{k}{\mathbf{T}}_2\left({}^{\mathrm{n}}\mathbf{U}+l{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}\right){C}^1-l{\mathbf{T}}^{\mathrm{T}}{\mathbf{T}}_2{}^{\mathrm{n}}\overline{\mathbf{f}}{\mathbf{C}}^1\right]\\ {}\kern1.75em +\sum \limits_{p=1}^4{\left({{}^{\mathrm{n}}\boldsymbol{\upmu}}_{\mathrm{m}}^p\right)}^{\mathrm{T}}\left[{\mathbf{T}}^{\mathrm{T}}\mathbf{k}\mathbf{T}{{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}}_{\mathrm{m}}^p+{\mathbf{T}}^{\mathrm{T}}\mathbf{k}{{}^{\mathrm{n}}\mathbf{u}}^p\right]\end{array}} $$

(65)

where ${{}^{\mathrm{n}}\boldsymbol{\uplambda}}_{\mathrm{m}}^1,{{}^{\mathrm{n}}\boldsymbol{\upmu}}_{\mathrm{m}}^p\left(p=1,2,3,4\right)$ are the master degrees of freedom of adjoint vectors ⁿλ¹, ⁿμ^p(p = 1, 2, 3, 4), which satisfy the periodic boundary condition: ${{}^{\mathrm{n}}\boldsymbol{\uplambda}}^1=\mathbf{T}{{}^{\mathrm{n}}\boldsymbol{\uplambda}}_{\mathrm{m}}^1,{{}^{\mathrm{n}}\boldsymbol{\upmu}}^p=\mathbf{T}{{}^{\mathrm{n}}\boldsymbol{\upmu}}_{\mathrm{m}}^p$. Conduct sensitivity analysis in (65), collate terms containing $\frac{\partial {{}^{\mathrm{n}}\mathbf{V}}_{\mathrm{m}}^1}{\partial {x}_i},\frac{\partial {{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}}_{\mathrm{m}}^p}{\partial {x}_i}\left(p=1,2,3,4\right)$ and set them to zero. Then the term $\frac{\partial e}{\partial {{}^{\mathrm{n}}\mathbf{V}}^1}\frac{\partial {{}^{\mathrm{n}}\boldsymbol{V}}^1}{\partial {x}_i}$ in (64) can be readily obtained as:

(66)

where the vectors ⁿλ¹, ⁿμ^p(p = 1, 2, 3, 4) are determined by the following adjoint equations:

$$ {\displaystyle \begin{array}{c}{\mathbf{T}}^{\mathrm{T}}\mathbf{kT}{{}^{\mathrm{n}}\boldsymbol{\uplambda}}_{\mathrm{m}}^1+{\mathbf{T}}^{\mathrm{T}}\frac{\mathrm{\partial e}}{\partial {{}^{\mathrm{n}}\mathbf{V}}^1}=\mathbf{0},{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^1=\mathbf{T}{{}^{\mathrm{n}}\boldsymbol{\uplambda}}_{\mathrm{m}}^1\\ {}{\mathbf{T}}^{\mathrm{T}}\mathbf{kT}{{}^{\mathrm{n}}\boldsymbol{\upmu}}_{\mathrm{m}}^{\mathrm{p}}+\mathrm{l}{C}^{1p}{\mathbf{T}}^{\mathrm{T}}\left({\mathbf{T}}_2\frac{\mathrm{\partial e}}{\partial {{}^{\mathrm{n}}\mathbf{V}}^1}+{\mathbf{T}}_2\mathbf{k}{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^1-\mathbf{k}{\mathbf{T}}_2{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^1\right)=\mathbf{0},{{}^{\mathrm{n}}\boldsymbol{\upmu}}^p=\mathbf{T}{{}^{\mathrm{n}}\boldsymbol{\upmu}}_{\mathrm{m}}^p\end{array}} $$

(67)

Similarly, the second term at the right hand side in (64) can be calculated as:

(68)

where the adjoint vectors ⁿλ², ⁿν^p(p = 1, 2, 3, 4) are determined by the following adjoint equations:

$$ {\displaystyle \begin{array}{c}{\mathbf{T}}^{\mathrm{T}}\mathbf{kT}{{}^{\mathrm{n}}\boldsymbol{\uplambda}}_{\mathrm{m}}^2+{\mathbf{T}}^{\mathrm{T}}\frac{\mathrm{\partial e}}{\partial {{}^{\mathrm{n}}\mathbf{V}}^2}=\mathbf{0},{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^2=\mathbf{T}{{}^{\mathrm{n}}\boldsymbol{\uplambda}}_{\mathrm{m}}^2\\ {}{\mathbf{T}}^{\mathrm{T}}\mathbf{kT}{{}^{\mathrm{n}}\boldsymbol{\upnu}}_{\mathrm{m}}^p+l{C}^{2p}{\mathbf{T}}^{\mathrm{T}}\left({\mathbf{T}}_2\frac{\partial e}{\partial {{}^{\mathrm{n}}\mathbf{V}}^2}+{\mathbf{T}}_2\mathbf{k}{{}^{\mathrm{n}}\boldsymbol{\lambda}}^2-\boldsymbol{k}{\mathbf{T}}_2{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^2\right)=\mathbf{0},{{}^{\mathrm{n}}\boldsymbol{\upnu}}^p=\mathbf{T}{{}^{\mathrm{n}}\boldsymbol{\upnu}}_{\mathrm{m}}^p\end{array}} $$

(69)

Examining the equations for ⁿμ^p and ⁿν^p in (67) and (69), we define vectors ⁿξ^α(α = 1, 2), whose FE formulations are as follows:

$$ {\mathbf{T}}^{\mathrm{T}}\mathbf{kT}{{}^{\mathrm{n}}\xi}_{\mathrm{m}}^{\alpha }+l{\mathbf{T}}^{\mathrm{T}}\left({\mathbf{T}}_2\frac{\partial e}{\partial {{}^{\mathrm{n}}\mathbf{V}}^{\alpha }}+{\mathbf{T}}_2{\mathbf{k}}^{\mathrm{n}}{\boldsymbol{\uplambda}}^{\alpha }-\mathbf{k}{\mathbf{T}}_2{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^{\alpha}\right)=\mathbf{0},{{}^{\mathrm{n}}\upxi}^{\alpha }=T{{}^{\mathrm{n}}\upxi}_{\mathrm{m}}^{\alpha } $$

(70)

and vectors ⁿμ^p and ⁿν^p can be accordingly expressed as:

$$ {{}^{\mathrm{n}}\boldsymbol{\upmu}}^p={C}^{1p}{{}^{\mathrm{n}}\boldsymbol{\upxi}}^1,{{}^{\mathrm{n}}\boldsymbol{\upnu}}^p={C}^{2p}{{}^{\mathrm{n}}\boldsymbol{\upxi}}^2 $$

(71)

Substitute (71) into (66) and (68), and collate relevant terms. The first two terms in (64) can be rearranged as:

$$ {\displaystyle \begin{array}{c}\sum \limits_{\alpha =1}^2{\left(\frac{\partial e}{\partial {{}^{\mathrm{n}}\mathbf{V}}^{\alpha }}\right)}^{\mathrm{T}}\frac{\partial {{}^{\mathrm{n}}\mathbf{V}}^{\alpha }}{\partial {x}_i}=\sum \limits_{\alpha =1}^2{\left(\frac{\partial {\mathbf{C}}^{\alpha }}{\partial {x}_i}\right)}^{\mathrm{T}}\left[{\left({}^{\mathrm{n}}\overset{\sim }{\mathbf{V}}\right)}^{\mathrm{T}}\frac{\partial e}{\partial {{}^{\mathrm{n}}\mathbf{V}}^{\alpha }}+{\left(\mathbf{k}{}^{\mathrm{n}}\overset{\sim }{\mathbf{V}}\right)}^{\mathrm{T}}{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^{\alpha }-l{\left({\mathbf{T}}_2{}^{\mathrm{n}}\overline{\mathbf{f}}\right)}^{\mathrm{T}}{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^{\alpha}\right]\\ {}+\sum \limits_{\alpha =1}^2\left[{\left({{}^{\mathrm{n}}\boldsymbol{\uplambda}}^{\alpha}\right)}^{\mathrm{T}}\frac{\partial \mathbf{k}}{\partial {x}_i}{{}^{\mathrm{n}}\mathbf{V}}^{\alpha }+{\left({\upxi}^{\alpha}\right)}^{\mathrm{T}}\frac{\partial \mathbf{k}}{\partial {x}_i}\left({}^{\mathrm{n}}\overline{\mathbf{u}}{\mathbf{C}}^{\alpha}\right)-l{\left({\mathbf{T}}_2{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^{\alpha}\right)}^T\frac{\partial \mathbf{k}}{\partial {x}_i}{}^{\mathrm{n}}\overline{\mathbf{u}}{\mathbf{C}}^{\alpha}\right]\end{array}} $$

(72)

where ${}^{\mathrm{n}}\overset{\sim }{\mathbf{V}}={\mathbf{T}}_2\left({}^{\mathrm{n}}\mathbf{U}+l{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}\right)$.

Next we tackle the third term in (64) and introduce the Lagrangian ${e}_{L_2}$ as:

$$ {e}_{L_2}=e+\sum \limits_{p=1}^4{\left({{}^{\mathrm{n}}\boldsymbol{\upeta}}_{\mathrm{m}}^p\right)}^{\mathrm{T}}\left[{\mathbf{T}}^{\mathrm{T}}\mathbf{k}\mathbf{T}{{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}}_{\mathrm{m}}^p+{\mathbf{T}}^{\mathrm{T}}\mathbf{k}{{}^{\mathrm{n}}\mathbf{u}}^p\right] $$

(73)

Following the same procedure, the sensitivity can be obtained as:

$$ \sum \limits_{p=1}^4{\left(\frac{\partial e}{\partial^{\mathrm{n}}{\overset{\sim }{\mathbf{u}}}^p}\right)}^{\mathrm{T}}\frac{\partial^{\mathrm{n}}{\overset{\sim }{\mathbf{u}}}^p}{\partial {x}_i}=\sum \limits_{p=1}^4{\left({{}^{\mathrm{n}}\boldsymbol{\eta}}^p\right)}^{\mathrm{T}}\frac{\partial \mathbf{k}}{\partial {x}_i}{{}^{\mathrm{n}}\overline{\mathbf{u}}}^p $$

(74)

where vectors ⁿη^p(p = 1, 2, 3, 4) are determined from

$$ {\mathbf{T}}^{\mathrm{T}}\mathbf{kT}{{}^{\mathrm{n}}\boldsymbol{\upeta}}_{\mathrm{m}}^p+{\mathbf{T}}^{\mathrm{T}}\frac{\partial e}{\partial^n{\overset{\sim }{\mathbf{u}}}_{\mathrm{m}}^p}=\mathbf{0},{{}^{\mathrm{n}}\boldsymbol{\upeta}}^p=\mathbf{T}{{}^{\mathrm{n}}\boldsymbol{\upeta}}_{\mathrm{m}}^p $$

(75)

Thus substitute (72) and (74), the sensitivity of strain energy e^SV can be arranged as:

$$ {\displaystyle \begin{array}{c}\frac{\partial {e}^{\mathrm{SV}}}{\partial {x}_i}=\sum \limits_{\alpha =1}^2{\left(\frac{\partial {\mathbf{C}}^{\alpha }}{\partial {x}_i}\right)}^{\mathrm{T}}\left[{\left({}^{\mathrm{n}}\overset{\sim }{\mathbf{V}}\right)}^{\mathrm{T}}\frac{\partial e}{\partial^{\mathrm{n}}{V}^{\alpha }}+{\left(\mathrm{k}{}^{\mathrm{n}}\overset{\sim }{\mathbf{V}}\right)}^{\mathrm{T}}{{}^{\mathrm{n}}\boldsymbol{\lambda}}^{\alpha }-l{\left({\mathbf{T}}_2{}^{\mathrm{n}}\overline{\mathbf{f}}\right)}^{\mathrm{T}}{{}^{\mathrm{n}}\boldsymbol{\lambda}}^{\alpha}\right]+\sum \limits_{p=1}^4{\left({{}^{\mathrm{n}}\boldsymbol{\upeta}}^p\right)}^{\mathrm{T}}\frac{\partial \mathbf{k}}{\partial {x}_i}{{}^{\mathrm{n}}\overline{\boldsymbol{u}}}^p\\ {}+\sum \limits_{\alpha =1}^2\left[{\left({{}^{\mathrm{n}}\boldsymbol{\lambda}}^{\alpha}\right)}^{\mathrm{T}}\frac{\partial \mathbf{k}}{\partial {x}_i}{{}^{\mathrm{n}}\mathbf{V}}^{\alpha }+{\left({\upxi}^{\alpha}\right)}^{\mathrm{T}}\frac{\partial \mathbf{k}}{\partial {x}_i}\left({}^{\mathrm{n}}\overline{\mathbf{u}}{\mathbf{C}}^{\alpha}\right)-l{\left({\mathbf{T}}_2{{}^{\mathrm{n}}\boldsymbol{\uplambda}}^{\alpha}\right)}^T\frac{\partial \mathbf{k}}{\partial {x}_i}{}^{\mathrm{n}}\overline{\mathbf{u}}{\mathbf{C}}^{\alpha}\right]+\frac{\partial e}{\partial {x}_i}\end{array}} $$

(76)

where the terms $\frac{\partial e}{\partial {x}_i},\frac{\partial e}{\partial^{\mathrm{n}}{V}^{\alpha }}\left(\alpha =1,2\right),\frac{\partial e}{\partial {{}^{\mathrm{n}}\overset{\sim }{u}}^p}\left(p=1,2,3,4\right)$ can be readily obtained as:

(77)

$$ {\displaystyle \begin{array}{c}\frac{\partial e}{\partial {{}^{\mathrm{n}}\mathbf{V}}^{\alpha }}=\left[-{F}_{\alpha }+\frac{F_{\alpha }}{l}\mathbf{k}{}^{\mathrm{n}}\mathbf{u}\mathbf{C}{\left({}^{\mathrm{n}}\overline{\mathbf{u}}\right)}^{\mathrm{T}}\right]\left(n\mathbf{k}{{}^{\mathrm{n}}\mathbf{u}}^{\mathrm{SV}}-\frac{\left(n-1\right) nl}{2}\mathbf{k}{}^{\mathrm{n}}\overline{\mathbf{u}}\mathbf{C}\mathbf{Q}\right)\\ {}=\left[-{F}_{\alpha }+\frac{F_{\alpha }}{l}\mathbf{k}{}^{\mathrm{n}}\mathbf{u}\mathbf{C}{\left({}^{\mathrm{n}}\overline{\mathbf{u}}\right)}^{\mathrm{T}}\right]n\mathbf{k}{{}^{\mathrm{n}}\mathbf{u}}^{\mathrm{SV}}+\frac{\left(n-1\right) nl}{2}{F}_{\alpha}\mathbf{k}{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}\mathbf{C}\mathbf{Q}\end{array}} $$

(78)

$$ \frac{\partial e}{\partial {{}^{\mathrm{n}}\overset{\sim }{\mathbf{u}}}^p}=\left\{n\left[-\frac{1}{l}{\left({}^{\mathrm{n}}\mathbf{u}{\mathbf{C}}^p\right)}^{\mathrm{T}}\mathbf{k}{{}^{\mathrm{n}}\mathbf{V}}^{\mathrm{s}}+{\left({\mathbf{C}}^p\right)}^{\mathrm{T}}{\mathbf{F}}^{\mathrm{ext}}\right]+\frac{\left(n-1\right) nl}{2}{\left({\mathbf{C}}^p\right)}^{\mathrm{T}}\mathbf{Q}\right\}\mathbf{k}{{}^{\mathrm{n}}\mathbf{V}}^{\mathrm{s}} $$

(79)

where ${\mathbf{F}}^{\mathrm{ext}}={\mathbf{F}}^{\mathrm{EBT}}+\frac{l}{2}\mathbf{Q}$, and the sensitivity of the effective compliance matrix C can be readily obtained from that of the effective stiffness matrix D as:

(80)

The sensitivity of the strain energy e^AH in (39) can be then obtained as

$$ \frac{\partial {e}^{\mathrm{AH}}}{\partial {x}_i}=\frac{nl}{2}{\left({\mathbf{F}}^{\mathrm{EBT}}\right)}^{\mathrm{T}}\frac{\partial \mathbf{C}}{\partial {x}_i}{\mathbf{F}}^{\mathrm{EBT}}+\frac{\left(n-1\right)n{l}^2}{2}{\left({\mathbf{F}}^{\mathrm{EBT}}\right)}^{\mathrm{T}}\frac{\partial \mathbf{C}}{\partial {x}_i}\mathbf{Q}+\frac{\left(n-1\right)n\left(2n-1\right){l}^3}{12}{\mathbf{Q}}^{\mathrm{T}}\frac{\partial \mathbf{C}}{\partial {x}_i}\mathbf{Q} $$

(81)

Appendix 2 Analytical strain energy formulation for the I-section base cell

We first derive the closed-form strain energy formulation ${e}_{\mathrm{analytical}}^{\mathrm{SV}2}$ in (44) of the I-section base cell subject to the transverse shear force and the bending moment in Fig. 4(a). As the thickness of the flange and the web is assumed much smaller than the dimensions of the cross-section, where the mechanics of thin walled structures can be applied, the normal stress σ₃₃, which admits linear distribution along x₂ direction, and the shear flow q in Fig. 15(b), which admits linear and quadratic distribution on the flange and the web respectively, can be readily calculated through mechanics of thin-walled structures. The stress components on different parts of the I-section in Fig. 15(c) are given by:

$$ {\displaystyle \begin{array}{c}{\sigma}_{33}=\frac{1}{I_1}{x}_2\left({x}_3-\frac{l}{2}\right)\\ {}{\tau}_{32}=\frac{1}{2{I}_1}\left(\frac{h^2}{4}+\frac{t_1}{t_2} hb-{x}_2^2\right)\kern0.75em \mathrm{on}\ {\Omega}_3\\ {}{\tau}_{31}=\frac{h}{2{I}_1}\left(\frac{b}{2}-{x}_1\right)\kern1em \mathrm{on}\ {\Omega}_2\\ {}{\tau}_{31}=\frac{h}{2{I}_1}\left(\frac{b}{2}+{x}_1\right)\kern1em \mathrm{on}\ {\Omega}_4\\ {}{\tau}_{31}=-\frac{h}{2{I}_1}\left(\frac{b}{2}+{x}_1\right)\kern0.5em \mathrm{on}\ {\Omega}_1\\ {}{\tau}_{31}=-\frac{h}{2{I}_1}\left(\frac{b}{2}-{x}_1\right)\kern0.5em \mathrm{on}\ {\Omega}_5\end{array}} $$

(82)

where ${I}_1=\frac{t_2{h}^3}{12}+\frac{h^2{bt}_1}{2}$ is the moment of inertia around x₁ axis. Then the strain energy can be analytically calculated as:

$$ {\displaystyle \begin{array}{l}{e}_{\mathrm{analytical}}^{\mathrm{SV}2}={t}_2{\int}_{-h/2}^{h/2}\left(\frac{\sigma_{33}^2}{2{E}_s}+\frac{\tau_{32}^2}{2{G}_s}\right)d{x}_2+{t}_1{\int}_{-b/2}^0\left(\frac{\sigma_{33}^2}{2{E}_s}+\frac{{\left.{\tau}_{31}^2\right|}_{\varOmega_1}}{2{G}_s}\right)d{x}_1+{t}_1{\int}_0^{b/2}\left(\frac{\sigma_{33}^2}{2{E}_s}+\frac{{\left.{\tau}_{31}^2\right|}_{\varOmega_2}}{2{G}_s}\right)d{x}_1\\ {}+{t}_1{\int}_{-b/2}^0\left(\frac{\sigma_{33}^2}{2{E}_s}+\frac{{\left.{\tau}_{31}^2\right|}_{\varOmega_4}}{2{G}_s}\right)d{x}_1+{t}_1{\int}_0^{b/2}\left(\frac{\sigma_{33}^2}{2{E}_s}+\frac{{\left.{\tau}_{31}^2\right|}_{\varOmega_5}}{2{G}_s}\right)d{x}_1\\ {}=\frac{l^3}{24{E}_s{I}_1}+\frac{h^2{b}^3l{t}_1}{48{G}_s{I}_1^2}+\frac{h^3l{t}_2}{G_s{I}_1^2}\left[\frac{h^2}{240}+\frac{hb}{24}\left(\frac{t_1}{t_2}\right)+\frac{b^2}{8}{\left(\frac{t_1}{t_2}\right)}^2\right]\end{array}} $$

(83)

Fig. 15

Schematic illustration of the I-section. a Cross-section of the I-beam; b Shear flow on the cross section; c Domains of the I-section

For the I-section subject to bending moment ${\left.{M}_1\right|}_{\omega^{-}}=\frac{l}{2},{\left.{M}_1\right|}_{\omega^{+}}=-\frac{l}{2}$, which is adopted to calculate analytical e^AH, the normal stress σ₃₃ is readily calculated as:

$$ {\sigma}_{33}=\frac{1}{I_1}{x}_2 $$

(84)

and the strain energy can be derived as:

$$ {e}_{\mathrm{analytical}}^{\mathrm{AH}2}={t}_2{\int}_{-h/2}^{h/2}\frac{\sigma_{33}^2}{2{E}_s}d{x}_2+{t}_2{\int}_{-b/2}^{b/2}\frac{{\left.{\sigma}_{33}^2\right|}_{x_2=\frac{h}{2}}}{2{E}_s}d{x}_1+{t}_2{\int}_{-b/2}^{b/2}\frac{{\left.{\sigma}_{33}^2\right|}_{x_2=-\frac{h}{2}}}{2{E}_s}d{x}_1=\frac{l^3}{8{E}_s{I}_1} $$

(85)