Strut-and-Tie Models Using Multi-Material and Multi-Volume Topology Optimization: Load Path Approach

Tuo Zhao, Ammar A. Alshannaq, David W. Scott, Glaucio H. Paulino

Research output: Contribution to journalArticlepeer-review


The development of strut-and-tie models (STMs) for the design of reinforced concrete (RC) deep beams considering a general multi-material and multi-volume topology optimization framework is presented. The general framework provides flexibility to control the location/inclination/length scale of the ties according to practical design requirements. Optimality conditions are applied to evaluate the performance of the optimized STM layouts. Specifically, the Michell number Z (or load path) is used as a simple and effective criterion to quantify the STMs. The experimental results confirm that the layout with the lowest load path Z achieves the highest ultimate load. Moreover, significantly reduced cracking is observed in the optimized layouts compared to the traditional layout. This observation implies that the optimized layouts may require less crack-control reinforcement, which would lower the total volume of steel required for the deep beams.

Original languageEnglish (US)
Pages (from-to)7-21
Number of pages15
JournalACI Structural Journal
Issue number6
StatePublished - 2023

All Science Journal Classification (ASJC) codes

  • Civil and Structural Engineering
  • Building and Construction


  • Michell number
  • load path
  • multi-material topology optimization
  • reinforced concrete (RC) deep beam
  • strut
  • tie


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