Isogeometric analysis-based design of post-tensioned concrete beam towards construction-oriented topology optimization

Abstract

Topology optimization as a computational approach is used to find an optimal structure with specific objectives such as an ultimately lightweight design. The optimization is normally performed under a series of constraint functions to ensure expected structural performance for safety and resilience. To promote the use of topology optimization in structural engineering, it is crucially important to identify and apply suitable constraints for realistic construction. In this paper, a conceptual attempt aiming for a construction-oriented topology optimization framework is presented. An isogeometric analysis module using NURBS curves for geometric description and discretization is employed in stress analyses, which incorporates the density-based SIMP optimization approach to generate the optimized topology. A Drucker–Prager criterion is adopted to constrain the stresses within the unequal limits of tension and compression for the use of concrete type material. To prevent slim components of potentially high complexity in construction, a minimum-width control is applied as an additional geometric control. In this paper, the conceptual framework is demonstrated using a prestressed concrete beam, while the optimization is performed to find suitable density distribution of concrete and a NURBS-described tendon profile after imposing the mapped prestressed action. The iterative optimization processes are presented and demonstrated to investigate the effect of tensile-compressive strength ratio and the minimum width, which reflects the construction-oriented vision pursuing a more regularized topology after applying the constraints in optimization.

Appendices

Appendix 1: The curvature sensitivity analysis of the tendon curve

The formulation of curvature can be written as:

$$\kappa \left( \xi \right) = \frac{{\left| {{\mathbf{C}}^{^{\prime}} \left( \xi \right) \times {\mathbf{C}}^{^{\prime\prime}} \left( \xi \right)} \right|}}{{\left| {{\mathbf{C}}^{^{\prime}} \left( \xi \right)} \right|^{3} }} = \frac{{\left| {\mathbf{A}} \right|}}{{\left| {\mathbf{B}} \right|^{3} }}$$

(40)

where ${\mathbf{C}}^{\prime}\left( \xi \right),{\mathbf{C}}^{\prime\prime}\left( \xi \right)$ are the first and second derivative values of the tendon curve, respectively. For the sake of formulation simplicity, we define terms ${\mathbf{A}} = {\mathbf{C}}^{^{\prime}} \left( \xi \right) \times {\mathbf{C}}^{^{\prime\prime}} \left( \xi \right)$ and ${\mathbf{B}} = {\mathbf{C}}^{^{\prime}} \left( \xi \right)$. Therefore, the expression of Eq. (40) could be rewritten as:

$$\frac{\partial \kappa \left( \xi \right)}{{\partial y_{i} }} = \frac{{\mathbf{A}}}{{\left| {\mathbf{A}} \right|\left| {\mathbf{B}} \right|^{3} }}\frac{{\partial {\mathbf{A}}}}{{\partial y_{i} }} - \frac{{3{\mathbf{B}}\left| {\mathbf{A}} \right|}}{{\left| {\mathbf{B}} \right|^{5} }}\frac{{\partial {\mathbf{B}}}}{{\partial y_{i} }}$$

(41)

where the term ${{\partial {\mathbf{A}}}}/{{\partial y_{i} }}$ is resulted in $({{\partial {\mathbf{A}}}}/{{\partial y_{i} }}) = ({{\partial {\mathbf{C}}^{^{\prime}} \left( \xi \right)}}/{{\partial y_{i} }}) \times {\mathbf{C}}^{^{\prime\prime}} \left( \xi \right) + {\mathbf{C}}^{^{\prime}} \left( \xi \right) \times ({{\partial {\mathbf{C}}^{^{\prime\prime}} \left( \xi \right)}}/{{\partial y_{i} }})$; and the term ${{\partial {\mathbf{B}}}}/{{\partial y_{i} }}$ is straightforward: $({{\partial {\mathbf{B}}}}/{{\partial y_{i} }}) = ({{\partial {\mathbf{C}}^{^{\prime}} \left( \xi \right)}}/{{\partial y_{i} }})$.

According to the formulation of the first and second derivative values of the tendon curve, the corresponding derivatives are given as follows:

$$\begin{gathered} \frac{{\partial {\mathbf{C}}^{^{\prime}} \left( \xi \right)}}{{\partial y_{i} }} = \sum\limits_{j = 1}^{{N_{{{\text{refine}}}} }} {\left( {\frac{{\partial R_{j,p}^{{{\text{tendon}}}} \left( \xi \right)}}{\partial \xi }\frac{{\partial {\mathbf{P}}_{j}^{{{\text{tendon}}}} }}{{\partial y_{i} }}} \right)} \hfill \\ \frac{{\partial {\mathbf{C}}^{^{\prime\prime}} \left( \xi \right)}}{{\partial y_{i} }} = \sum\limits_{j = 1}^{{N_{{{\text{refine}}}} }} {\left( {\frac{{\partial^{2} R_{j,p}^{{{\text{tendon}}}} \left( \xi \right)}}{{\partial \xi^{2} }}\frac{{\partial {\mathbf{P}}_{j}^{{{\text{tendon}}}} }}{{\partial y_{i} }}} \right)} \hfill \\ \end{gathered}$$

(42)

Appendix 2: Normal direction sensitivity analysis of the tendon curve

The formulation of normal direction can be written as:

$${\mathbf{n}}\left( \xi \right) = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right] \times \left[ {\begin{array}{*{20}c} {\cos \theta_{\xi } } \\ {\sin \theta_{\xi } } \\ \end{array} } \right]$$

(43)

where $\theta_{\xi }$ represents the angle at the location $\xi$, and the expression could further take the form as $\left[ {\begin{array}{*{20}c} {\cos \theta_{\xi } } \\ {\sin \theta_{\xi } } \\ \end{array} } \right] = ({{{\text{d}}{\mathbf{x}}\left( \xi \right)}}/{{{\text{d}}s}}) = ({{{\mathbf{J}}_{1}^{{{\text{tendon}}}} }}/{{\left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|}})$. Therefore, the derivative of normal ${{\partial {\mathbf{n}}\left( \xi \right)}}/{{\partial y_{i} }}$ and is as below:

$$\frac{{\partial {\mathbf{n}}\left( \xi \right)}}{{\partial y_{i} }}{ = }\left[ {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right] \times \left( {\frac{1}{{\left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|}} - \frac{{\left( {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right)^{2} }}{{\left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|^{3} }}} \right)\frac{{\partial {\mathbf{J}}_{1}^{{{\text{tendon}}}} }}{{\partial y_{i} }}$$

(44)

Likewise, the tangential direction ${{\partial {\mathbf{n}}^{*} }}/{{\partial y_{i} }}$ can be written as:

$$\frac{{\partial {\mathbf{n}}^{*} }}{{\partial y_{i} }} = \left( {\frac{1}{{\left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|}}\frac{{\partial {\mathbf{J}}_{1}^{{{\text{tendon}}}} }}{{\partial y_{i} }} - \frac{{{\mathbf{J}}_{1}^{{{\text{tendon}}}} }}{{\left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|^{2} }}\frac{{\partial \left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|}}{{\partial y_{i} }}} \right)$$

(45)

where ${\mathbf{J}}_{1}^{{{\text{tendon}}}}$ is the tendon elemental Jacobian matrix from physical space to parametric space is calculated by:

$${\mathbf{J}}_{1}^{{{\text{tendon}}}} = \sum\limits_{j \in e} {\frac{{\partial R_{j}^{{{\text{ten}}}} \left( e \right)}}{\partial \xi }{\mathbf{P}}_{j}^{{{\text{refine}}}} }$$

(46)

and the corresponding derivatives term ${{\partial {\mathbf{J}}_{1}^{tendon} }}/{{\partial y_{i} }}$ is:

$$\frac{{\partial {\mathbf{J}}_{1}^{{{\text{tendon}}}} }}{{\partial y_{i} }} = \sum\limits_{j \in e} {\frac{{\partial R_{j,p}^{{{\text{tendon}}}} \left( e \right)}}{\partial \xi }\frac{{\partial {\mathbf{P}}_{j}^{{{\text{refine}}}} }}{{\partial y_{i} }}}$$

(47)

According to Eq. (14), the term ${{\partial {\mathbf{P}}_{j}^{{{\text{refine}}}} }}/{{\partial y_{i} }}$ in Eq. (47) can be given as:

$$\frac{{\partial {\mathbf{P}}_{j}^{{{\text{refine}}}} }}{{\partial y_{i} }} = \frac{{\partial {\mathbf{P}}_{j}^{{{\text{refine}}}} }}{{\partial y_{j}^{{{\text{refine}}}} }}\frac{{\partial y_{j}^{{{\text{refine}}}} }}{{\partial y_{i} }} = W_{j,i}^{{{\text{refine}}}} \frac{{\partial {\mathbf{P}}_{j}^{{{\text{refine}}}} }}{{\partial y_{j}^{{{\text{refine}}}} }}$$

(48)

where the corresponding derivative is set as $({{\partial {\mathbf{P}}_{j}^{{{\text{refine}}}} }}/{{\partial y_{j}^{{{\text{refine}}}} }}) = [0,1,0]$. Furthermore, the derivative term ${{\partial \left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|}}/{{\partial y_{i} }}$ is calculated by:

$$\frac{{\partial \left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|_{e} }}{{\partial y_{i} }} = \frac{{\left( {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right)_{e} }}{{\left| {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right|_{e} }}\frac{{\partial \left( {{\mathbf{J}}_{1}^{{{\text{tendon}}}} } \right)_{e} }}{{\partial y_{i} }}$$

(49)