Optimum Column Layout Design of Reinforced Concrete Frames Under Wind Loading

Abstract

The geometric layout optimization of a structure is a significant stage in a design process, and selecting an appropriate geometric layout can impact all the subsequent stages of the design procedure and the relevant costs. This study presents a heuristic approach for the optimum layout design of two-dimensional reinforced concrete frames in order to optimize the total cost and controls the application under wind loadings. The aim is to find the optimum column layout design for 2D frames under wind loadings considering the involved cost elements. A heuristic methodology is developed in order to achieve a new design space and an objective function for the cost and layout optimization problem. The proposed method has the capability to make use of action effects of the structure as alternative design variables in place of the commonly used cross-sectional ones. Such a feature provides the method the ability to be easily employed in large and realistic structural optimization problems, and helps the optimization algorithms to take less time, in an iterative optimization process. Then, an Ant System based algorithm is proposed to solve the presented optimization problem. Examples are included to illustrate the robustness of the methodology.

Appendices

Appendix

Mathematical Calculations for (33.5) and (33.9)

33.2.1 A.1. Beams: Calculations for (33.5)

Any changes in the longitudinal steel $A_{{sl}}^{{(b)}}$ results in changes in the capacities of a beam section as follows:

$$ \left\{ \begin{array}{lllllll} \frac{{\Delta\, M_u^{{ - (b)}}}}{{\Delta\, A_{{sl}}^{{(b)}}}} \cong {{f}_{{yl}}}({{d}^{{(b)}}} - d_c^{{(b)}}) \to \Delta\, A_{{sl}}^{{(b)}} = {{(\,{{f}_{{yl}}}\,\,{{d}^{{(b)}}}(1 - 0.5\gamma \,{{k}_u}))}^{{ - 1}}}\Delta\, M_u^{{ - (b)}} = {{K}_1}\,\Delta\, M_u^{{ - (b)}} \hfill \\\frac{{\Delta\, M_u^{{ + (b)}}}}{{\Delta\, A_{{sl}}^{{(b)}}}} \cong {{f}_{{yl}}}({{d}^{{(b)}}} - d_c^{{(b)}}) \to \Delta\, A_{{sl}}^{{(b)}} = {{(\,{{f}_{{yl}}}\,\,{{d}^{{(b)}}}(1 - 0.5\gamma \,{{k}_u}))}^{{ - 1}}}\Delta\, M_u^{{ + (b)}} = {{K}_1}\,\Delta\, M_u^{{ + (b)}} \hfill \\\frac{{\Delta\, V_u^{{(b)}}}}{{\Delta\, A_{{sl}}^{{(b)}}}} \cong 0 \to \Delta\, V_u^{{(b)}}\,and\,\Delta\, {{{\rm A}}_{{sl}}}\,\,are\,\,independent\,\,of\,\,each\,\,other \end{array} \right. $$

(A.1)

If $A_{{sv}}^{{(b)}}/s$ changes the variation of the beams bearing capacities are

$$ \left\{\begin{array}{llllllllll} \Delta\, M_u^{{ - (b)}}and\,\frac{{\Delta\, A_{{sv}}^{{(b)}}}}{s}are\,\,independent\,of\,each\,\,other \hfill \\ \Delta\, M_u^{{ + (b)}}\,\,and\,\frac{{\Delta\, A_{{sv}}^{{(b)}}}}{s}are\,\,independent\,of\,each\,other \hfill \\ \frac{{\Delta\, V_u^{{(b)}}}}{{\frac{{\Delta\, A_{{sv}}^{{(b)}}}}{s}}} \cong {{f}_{{yv}}}d \to \frac{{\Delta\, A_{{sv}}^{{(b)}}}}{s} \cong {{(\,{{f}_{{yv}}}\,d)}^{{ - 1}}}\,\Delta\, V_u^{{(b)}} = {{K}_2}\,\,\Delta\, V_u^{{(b)}} \end{array}\right. $$

(A.2)

If the effective area of the beam section $\Delta\, A_c^{{(b)}}$, i.e. bd varies:

$$ \Delta\, A_c^{{(b)}} = \Delta\, (bd) = b \;\Delta\, d + d \;\Delta\, b + \Delta\, b \;\Delta\, d = b \;\Delta\, d $$

(A.3)

$\Delta\, M_u^{{ - (b)}}$ is not a function of b, and based on the second assumption, the dependence of $\Delta\, M_u^{{ + (b)}}$ on the changes of b is neglected. Moreover, the first term of $V_u^{{(b)}}$ is not dependent on b as well. So, the variations of the section capacities due to variation of cross-section area are

$$ \frac{{\Delta\, M_u^{{(b)}}}}{{\Delta\, A_c^{{(b)}}}} = \frac{{\Delta\, M_u^{{(b)}}}}{{\Delta\, (bd)}} \cong \frac{{\Delta\, M_u^{{(b)}}}}{{b(\Delta\, d)}} $$

(A.4)

$$ \left\{\begin{array}{cccccccccccc} \frac{{\Delta\, M_u^{{ - (b)}}}}{{\Delta\, (b{{d}^{{(b)}}})}} \cong \frac{{{{f}_{{yl}}}A_{{sc}}^{{(b)}}}}{b}(1 - 0.5\gamma {{k}_u}) \to \Delta\, A_c^{{(b)}} = \Delta\, (b{{d}^{{(b)}}}) \cong {{\left[ {{{f}_{{yl}}}\frac{{A_{{sc}}^{{(b)}}}}{b}(1 - 0.5\gamma {{k}_u})} \right]}^{{ - 1}}}\Delta\, M_u^{{ - (b)}} = {{K}_3}\,\Delta\, M_u^{{ - (b)}} \hfill \\\frac{{\Delta\, M_u^{{ + (b)}}}}{{\Delta\, (b{{d}^{{(b)}}})}} \cong \frac{{{{f}_{{yl}}}A_{{st}}^{{(b)}}}}{b}(1 - 0.5\gamma {{k}_u}) \to \Delta\, A_c^{{(b)}} = \Delta\, (b{{d}^{{(b)}}}) \cong {{\left[ {{{f}_{{yl}}}\frac{{A_{{st}}^{{(b)}}}}{b}(1 - 0.5\gamma {{k}_u})} \right]}^{{ - 1}}}\Delta\, M_u^{{ + (b)}} = {{K}_4}\,\Delta\, M_u^{{ + (b)}} \hfill \\\frac{{\Delta\, V_u^{{(b)}}}}{{\Delta\, (b{{d}^{{(b)}}})}} \cong \frac{{{{f}_{{yv}}}A_{{sv}}^{{(b)}}}}{{bs}} + \beta {{(\,f{{\prime}_c})}^{{1/2}}} \to \Delta\, A_c^{{(b)}} = \Delta\, (b{{d}^{{(b)}}}) \cong {{\left[ {\frac{{{{f}_{{yv}}}A_{{sv}}^{{(b)}}}}{{bs}} + \beta {{{(\,f{{\prime}_c})}}^{{0.5}}}} \right]}^{{ - 1}}}\Delta\, V_u^{{(b)}} = {{K}_5}\,\Delta\, V_u^{{(b)}} \end{array} \right. $$

(A.5)

The variation of the perimeter of a rectangular beam section, which determines the variation of the beam formwork, affects the section capacity as follows:

$$ \Delta\, P_f^{{(b)}} = \Delta\, b + 2\Delta\, {{d}^{{(b)}}} $$

(A.6)

$$ \left\{\begin{array}{llllllll} \frac{{\Delta\, M_u^{{ - (b)}}}}{{\Delta\, b + 2\Delta\, {{d}^{{(b)}}}}} \cong {{f}_{{yl}}}A_{{sc}}^{{(b)}} \to \Delta\, P_f^{{(b)}} = \Delta\, b + 2\Delta\, {{d}^{{(b)}}} \cong 2{{\left[ {{{f}_{{yl}}}A_{{sc}}^{{(b)}}(1 - 0.5\gamma {{k}_u})} \right]}^{{ - 1}}}\Delta\, M_u^{ - } = {{K}_6}\,\Delta\, M_u^{ - } \hfill \\\frac{{\Delta\, M_u^{{ + (b)}}}}{{\Delta\, b + 2\Delta\, {{d}^{{(b)}}}}} \cong {{f}_{{yl}}}A_{{st}}^{{(b)}} \to \Delta\, P_f^{{(b)}} = \Delta\, b + 2\Delta\, {{d}^{{(b)}}} \cong 2{{\left[ {{{f}_{{yl}}}A_{{st}}^{{(b)}}(1 - 0.5\gamma {{k}_u})} \right]}^{{ - 1}}}\Delta\, M_u^{ + } = {{K}_7}\,\Delta\, M_u^{ + } \hfill \\\frac{{\Delta\, V_u^{{(b)}}}}{{\Delta\, b + 2\Delta\, {{d}^{{(b)}}}}} \cong \frac{1}{{\frac{{{{f}_{{yl}}}A_{{sv}}^{{(b)}}}}{s} + \beta b{{{(\,f{{\prime}_c})}}^{{1/2}}}}} + \frac{1}{{\beta d{{{(\,f{{\prime}_c})}}^{{1/2}}}}} \to \Delta\, P_f^{{(b)}} = \Delta\, b + 2\Delta\, {{d}^{{(b)}}} \cong \left[ {\frac{2}{{\frac{{{{f}_{{yv}}}A_{{sv}}^{{(b)}}}}{s} + \beta b{{{(\,f{{\prime}_c})}}^{{1/2}}}}} + \frac{1}{{\beta d{{{(\,f{{\prime}_c})}}^{{1/2}}}}}} \right]\Delta\, {{V}_u} = {{K}_8}\;\Delta\, {{V}_u} \end{array} \right. $$

(A.7)

Now, that the reciprocal relationships between the variations of cross-sectional variables and strength capacity parameters of a beam section are obtained, multiplying both sides of (A.1) by csl/2, (A.2) by csv, (A.5) by cc/3 and (A.7) by cf/3 and then adding them up result in:

$$ \begin{array}{lllllllll} {{c}_c}\Delta\, A_c^{{(b)}} + {{c}_{{sl}}}\,\Delta\, A_{{sl}}^{{(b)}} + {{c}_{{sv}}}A_{{sv}}^{{(b)}} + {{c}_f}\Delta\, P_f^{{(b)}} = \left( {\frac{1}{3}{{c}_c}{{K}_4} + \frac{1}{2}{{c}_{{sl}}}{{K}_1} + \frac{2}{3}{{c}_f}{{K}_7}} \right)\Delta\, M_u^{{ + (b)}} \hfill \\ + \left( {\frac{1}{3}{{c}_c}{{K}_3} + \frac{1}{2}{{c}_{{sl}}}{{K}_1} + \frac{2}{3}{{c}_f}{{K}_6}} \right)\Delta\, M_u^{{ - (b)}} \hfill \\ + \left( {\frac{1}{3}{{c}_c}{{K}_5} + {{c}_{{sv}}}{{K}_2} + \frac{2}{3}{{c}_f}{{K}_8}} \right)\Delta\, V_u^{{(b)}}\end{array} $$

(A.8)

33.2.2 A.2. Columns: Calculations for (33.9)

Any changes in the longitudinal steel $A_{{sl}}^{{(c)}}$ results in changes in the capacities of a column section as follows:

$$ \left\{ \begin{array}{llllllllll} \frac{{\Delta\, N_u^{{(c)}}}}{{\Delta\, A_{{sl}}^{{(c)}}}}\, \cong \,\,{{E}_s}\sum\nolimits_i^n {{{\varepsilon }_i} \to } \,\Delta\, A_{{sl}}^{{(c)}} = {{\left( {{{E}_s}\sum\nolimits_i^n {{{\varepsilon }_i}} } \right)}^{{ - 1}}}\,\Delta\, N_u^{{(c)}} = {{K}_9}\,\,\Delta\, N_u^{{(c)}} \hfill \\\frac{{\Delta\, M_u^{{(c)}}}}{{\Delta\, A_{{sl}}^{{(c)}}}}\, \cong \,{{E}_s}\,\sum\nolimits_i^n {{{\varepsilon }_i}\,{{e}_i} \to } \Delta\, A_{{sl}}^{{(c)}} = {{\left( {{{E}_s}\sum\nolimits_i^n {{{\varepsilon }_i}} \,{{e}_i}} \right)}^{{ - 1}}}\Delta\, {\rm M}_u^{{(c)}} = {{K}_{{10}}}\,\,\Delta\, M_u^{{(c)}} \hfill \\\frac{{\Delta\, V_u^{{(c)}}}}{{\Delta\, A_{{sl}}^{{(c)}}}} \cong 0 \to \Delta\, V_u^{{(c)}}\,\,and\,\Delta\, A_{{sl}}^{{(c)}}\,are\,\,independent\,\,of\,each\,\,\,other\end{array}\right. $$

(A.9)

With respect to $A_{{sv}}^{{(c)}}/s$, the variations of the column capacities are

$$ \left\{\begin{array}{llllllllll}\Delta\, N_u^{{(c)}}\,\,and\,\,\,\frac{{\Delta\, { A}_{{sv}}^{{(c)}}}}{s}\,\,\,are\,\,independent\,\,of\,\,each\,\,other \hfill \\ \Delta\, M_u^{{(c)}}\,and\,\,\,\frac{{\Delta\, { A}_{{sv}}^{{(c)}}}}{s}\,\,\,are\,\,independent\,\,of\,\,each\,\,other \hfill \\ \frac{{\Delta\, V_u^{{(c)}}}}{{\frac{{\Delta\, A_{{sv}}^{{(c)}}}}{s}}} \cong {{f}_{{yv}}}\,{{d}^{{(c)}}} \to \frac{{\Delta\, A_{{sv}}^{{(c)}}}}{s} \cong {{(\,{{f}_{{yv}}}\,{{d}^{{(c)}}})}^{{ - 1}}}\,\Delta\, V_u^{{(b)}} = {{K}_{{11}}}\,\,\Delta\, V_u^{{(b)}} \end{array} \right. $$

(A.10)

If the cross-section area of the column varies:

$$ \left\{ \eqalign{ \frac{{\Delta\, N_u^{{(c)}}}}{{\Delta\, A_c^{{(c)}}}} \cong 0.85\gamma {{k}_u}f{{\prime}_c} \to \Delta\, A_c^{{(c)}} \cong {{\left( {0.85\gamma {{k}_u}\,f{{\prime}_c}} \right)}^{{ - 1}}}\Delta\, N_u^{{(c)}} = {{K}_{{12}}}\,\,\Delta\, N_u^{{(c)}} \hfill \\\frac{{\Delta\, M_u^{{(c)}}}}{{\Delta\, A_c^{{(c)}}}} \cong 0.85\gamma {{k}_u}f{{\prime}_c}{{e}_c} \to \Delta\, A_c^{{(c)}} \cong {{\left( {0.85\gamma {{k}_u}f{{\prime}_c}{{e}_c}} \right)}^{{ - 1}}}\Delta\, M_u^{{(c)}} = {{K}_{{13}}}\,\,\Delta\, M_u^{{(c)}} \hfill \\\frac{{\Delta\, V_u^{{(c)}}}}{{\Delta\, A_c^{{(c)}}}} \cong \frac{{{{f}_{{yv}}}A_{{sv}}^{{(c)}}}}{{2hs}} + \beta {{{(}f_c^{\prime}{)}}^{{0.5}}} \to \Delta\, A_c^{{(b)}} = \Delta\, {(}{{h}^{{(c)}}}{{d}^{{(c)}}}{)} \cong {{\left( {\frac{{{{f}_{{yv}}}A_{{sv}}^{{(c)}}}}{{2hs}} + \beta {{{(\,f_c^{\prime})}}^{{0.5}}}} \right)}^{{ - 1}}}\Delta\, V_u^{{(c)}} = {{K}_{{14}}}\,\,\Delta\, V_u^{{(c)}} \hfill \\}<!endgathered> \right. $$

(A.11)

The variation of the perimeter of a square column section, which determines the variation of the column formwork, affects the section capacity as follows:

$$ \left\{ \begin{array}{llllllllll} \frac{{\Delta\, N_u^{{(c)}}}}{{\Delta\, P_f^{{(c)}}}} \cong 0.85\gamma {{k}_u}\;f{{\prime}_c}\frac{{{{h}^{{(c)}}}}}{2} \to \Delta\, P_f^{{(c)}} \cong {{\left( {0.85\gamma {{k}_u}\;f{{\prime}_c}\frac{{{{h}^{{(c)}}}}}{2}} \right)}^{{ - 1}}}\Delta\, N_u^{{(c)}} = {{K}_{{15}}}\,\,\Delta\, N_u^{{(c)}} \hfill \\\frac{{\Delta\, M_u^{{(c)}}}}{{\Delta\, P_f^{{(c)}}}} \cong 0.85\gamma {{k}_u}\;f{{\prime}_c}\frac{{{{h}^{{(c)}}}}}{2}{{e}_c} \to \Delta\, P_f^{{(b)}} \cong {{\left( {0.85\gamma {{k}_u}\;f{{\prime}_c}\frac{{{{h}^{{(c)}}}}}{2}{{e}_c}} \right)}^{{ - 1}}}\Delta\, M_u^{{(c)}} = {{K}_{{16}}}\,\,\Delta\, M_u^{{(c)}} \hfill \\\frac{{\Delta\, V_u^{{(c)}}}}{{\Delta\, P_f^{{(c)}}}} \cong \frac{{{{f}_{{yv}}}A_{{sv}}^{{(c)}}}}{{4s}} + \frac{1}{2}\beta {{h}^{{(c)}}}{ {(\,f_c^{\prime})}^{{0.5}}} \to \Delta\, P_f^{{(c)}} \cong {{\left( {\frac{{{{f}_{{yv}}}A_{{sv}}^{{(c)}}}}{{4s}} + \frac{1}{2}\beta {{h}^{{(c)}}}{{{(\,f_c^{\prime})}}^{{0.5}}}} \right)}^{{ - 1}}}\Delta\, V_u^{{(c)}} = {{K}_{{17}}}\,\,\Delta\, V_u^{{(c)}}\end{array}\right. $$

(A.12)

The reciprocal relationships between the variations of cross-sectional variables and those of strength capacity variables of a column section are obtained by multiplying both sides of (A.9) by csl/2, (A.10) by csv, (A.11) by cc/3 and (A.12) by cf/3 then adding them up :

$$ \begin{array}{lllllllll} {{c}_c}\,\,\Delta\, A_c^{{(c)}} + {{c}_{{sl}}}\,\,\,\Delta\, A_{{sl}}^{{(c)}} + {{c}_{{sv}}}\,\,A_{{sv}}^{{(c)}} + {{c}_f}\,\,\,\Delta\, P_f^{{(c)}} = \left( {\frac{1}{3}{{c}_c}{{K}_{{12}}} + \frac{1}{2}{{c}_{{sl}}}{{K}_9} + \frac{1}{3}{{c}_f}{{K}_{{15}}}} \right)\,\Delta\, N_u^{{(c)}} \hfill \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \left( {\frac{1}{3}{{c}_c}{{K}_{{13}}} + \frac{1}{2}{{c}_{{sl}}}{{K}_{{10}}} + \frac{2}{3}{{c}_f}{{K}_{{16}}}} \right)\Delta\, M_u^{{(c)}} \hfill \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \left( {\frac{1}{3}{{c}_c}{{K}_{{14}}} + {{c}_{{sv}}}{{K}_{{11}}} + \frac{2}{3}{{c}_f}{{K}_{{17}}}} \right)\Delta\, V_u^{{(c)}} \end{array} $$

(A.13)