System Reliability and New Measure of Robustness of Truss Structure in Progressive Collapse

Abstract

Progressive collapse is an important system failure mode for structures. It may be initiated by local damage due to accidental loads or extreme design loads. It is necessary to find out the robustness of a structure against progressive collapse. The robustness index should (i) give a measure of how well a structure can absorb the initial damage without collapsing, (ii) account for uncertainties in loads, material properties and models, and (iii) help compare different designs and repair strategies. Material and geometric nonlinearities can play an important role in progressive collapse. This paper applies a new measure of robustness for coherent systems on an indeterminate truss structure subject to progressive collapse. Robustness corresponding to different member losses are compared which can in turn be used to select individual members for strengthening and thus improve system reliability.

Appendices

Appendices

${C}_{i}$: Strength of the ith member, ${\sigma }_{i}^{j}$: Stress of the i-th member after j-th member failure. Now the sequence

$F_{{11}}^{{9 - 8}} \; = \;\{ Member\;8\;fails\;2nd\; \cap \;9\;fails\;1st\;|\;Member\;11\;has\;failed\}$

$$\begin{aligned} & \{ (C_{1} > \sigma _{1}^{{11 - 9}} ,\;C_{2} > \sigma _{2}^{{11 - 9}} ,\;C_{3} > \sigma _{3}^{{11 - 9}} ,\;C_{4} > \sigma _{4}^{{11 - 9}} ,\;C_{5} > \sigma _{5}^{{11 - 9}} ,\; \\ & C_{6} > \sigma _{6}^{{11 - 9}} ,\;C_{7} > \sigma _{7}^{{11 - 9}} ,\;C_{8} > \sigma _{8}^{{11 - 9}} ,\;C_{{10}} > \sigma _{{10}}^{{11 - 9}} ) \\ & \cap \;(C_{1} > \sigma _{1}^{{11}} ,\;C_{2} > \sigma _{2}^{{11}} ,\;C_{3} > \sigma _{3}^{{11}} ,\;C_{4} > \sigma _{4}^{{11}} ,\;C_{5} > \sigma _{5}^{{11}} ,\;C6 > \sigma _{6}^{{11}} ,\; \\ & C_{7} > \sigma _{7}^{{11}} ,\;C_{8} > \sigma _{8}^{{11}} ,\;C_{9} < \sigma _{9}^{{11}} ,\;C_{{10}} > \sigma _{{10}}^{{11}} )\} \\ \end{aligned}$$

$$\begin{aligned} & = \;\{ (C_{1} > \max (\sigma _{1}^{{11 - 9}} ,\;\sigma _{1}^{{11}} )),\;(C_{2} > \max (\sigma _{2}^{{11 - 9}} ,\;\sigma _{2}^{{11}} )),\; \\ & (C_{3} > \max (\sigma _{3}^{{11 - 9}} ,\;\sigma _{3}^{{11}} )), (C_{4} > \max (\sigma _{4}^{{11 - 9}} ,\;\sigma _{4}^{{11}} )),\;\\ & (C_{5} > \max (\sigma _{5}^{{11 - 9}} ,\;\sigma _{5}^{{11}} )),\;(C_{6} > \max (\sigma _{6}^{{11 - 9}} ,\;\sigma _{3}^{{11}} )), \\ & (C_{7} > \max (\sigma _{7}^{{11 - 9}} ,\;\sigma _{7}^{{11}} )),\;(\sigma _{8}^{{11}} < C_{8} < \sigma _{8}^{{11 - 9}} ),\;\\ & (C_{9} < \sigma _{9}^{{11}} ),\;(C_{{10}} > \;\max (\sigma _{{1o}}^{{11 - 9}} ,\;\sigma _{{10}}^{{11}} ))\} \\ \end{aligned} .$$

$$\begin{aligned} & = \;P\{ (C_{1} > \max (\sigma _{1}^{{11 - 9}} ,\;\sigma _{1}^{{11}} )),\;(C_{2} > \max (\sigma _{2}^{{11 - 9}} ,\;\sigma _{2}^{{11}} )),\;\\ & (C_{3} > \max (\sigma _{3}^{{11 - 9}} ,\;\sigma _{3}^{{11}} )), (C_{4} > \max (\sigma _{4}^{{11 - 9}} ,\;\sigma _{4}^{{11}} )),\; \\ & (C_{5} > \max (\sigma _{5}^{{11 - 9}} ,\;\sigma _{5}^{{11}} )),\; (C_{6} > \max (\sigma _{6}^{{11 - 9}} ,\;\sigma _{3}^{{11}} )), \\ & (C_{7} > \max (\sigma _{7}^{{11 - 9}} ,\;\sigma _{7}^{{11}} )),\;(\sigma _{8}^{{11}} < C_{8} < \sigma _{8}^{{11 - 9}} ),\;\\ & (C_{9} < \sigma _{9}^{{11}} ),\;(C_{{10}} > \;\max (\sigma _{{1o}}^{{11 - 9}} ,\;\sigma _{{10}}^{{11}} ))\} \\ \end{aligned}$$