1 Technion Israel Institute of Technology The Faculty of Civil Engineering Simultaneous Shape and Topology Optimization of Prestressed Concrete Beams Emad Shakour Oded Amir

2 Background and motivationTopology optimization has been rarely used in the field of civil engineering. The implementation of topology optimized concrete structures is a complex task comparing with traditional non optimal structures. Concrete is a highly non-linear material with almost no tensile strength. Manufacturing constraints. Economical issues. Motivation concrete … Ancient Greek building, small beam spans.

3 Background and motivationPre-stressed concrete advantages: Section remains un-cracked under service loads Increase in durability. The full section is utilized. High span-to-depth ratios. Reduction in self weight . More aesthetic appeal due to slender sections. More economical sections. Reinforced Concrete Prestressed concrete Dead load (self-weight) Prestressing cable pre-stressing force Service load 𝝈 𝒕 < 𝝈 𝒕, 𝒂𝒍𝒍𝒐𝒘𝒆𝒅 Reinforcing bars Un-cracked with likely small upwards deflection under dead load and pre-stress. Cracked beam. Deflection under dead and full service load. Service load = D.L + L.L Prestressed concrete concept. pre-stressing force pre-stressing force No tension → No cracks. very small deflection.

4 Background and motivation𝑃𝑟𝑒𝑠𝑡𝑟𝑒𝑠𝑠𝑖𝑛𝑔 𝑐𝑎𝑏𝑙𝑒 Fig. 1 - The shape of the cable correlated to the moment diagram. Fig. 2 - The shape of the cable is restricted to a straight line in the bottom. Impossible cable shape. Possible cable shape. Topology optimization whitout prstress

5 Modeling the pre-stressed structureThe concrete domain is discretized into simple 4 node finite element with bi-linear shape functions. Linear elastic material model. 𝑻 𝒑𝒓𝒆 The continuous cable is divided into piecewise linear segments. 𝑻 𝒑𝒓𝒆 Boundary conditions Structural Model. The shape of the cable obtained by knowing the y coordinates at each junction between two adjacent segments. 𝑻 𝒑𝒓𝒆 −Cable tension force 𝑷−Equivalent prestress forces, due to different inclination angle between two adjacent cable segments.

6 Pre-stressing forces (1) 𝑃 𝑖 - equivalent prestressing load.(2) 𝑃 𝑖,𝑥 , P i,y calculated through equilibrium equation. 𝒅 𝟐 𝒅 𝟏 Structural Model. (3) The equivalent prestressing load is transferred to the joints through linear interpolation. 𝑻 𝒑𝒓𝒆 −Cable tension force 𝑷−Equivalent prestress forces, due to different inclination angle between two adjacent cable segments.

7 Formulation – shape optimization for the cable only𝑃 𝑒𝑥𝑡 Only for verification purposes min 𝑦 𝜙 𝑦 = 𝒇 𝑒𝑥𝑡 𝑇 𝒖 𝑇𝑜𝑡𝑎𝑙 𝑠.𝑡: 𝑦 𝑖 −ℎ+𝑐≤ 𝑖=1,…, 𝑁 𝐶𝑎𝑏𝑙𝑒 𝑛𝑜𝑑𝑒𝑠 − 𝑦 𝑖 +𝑐≤ 𝑖=1,…, 𝑁 𝐶𝑎𝑏𝑙𝑒 𝑛𝑜𝑑𝑒𝑠 𝑤𝑖𝑡ℎ: 𝑲 𝒖 𝑇𝑜𝑡𝑎𝑙 = 𝒇 𝑒𝑥𝑡 + 𝒇 𝑝𝑟𝑒 𝒖 𝑇𝑜𝑡𝑎𝑙 = 𝒖 𝑓 𝑒𝑥𝑡 + 𝒖 𝑓 𝑝𝑟𝑒 ℎ −𝐵𝑒𝑎𝑚 ℎ𝑒𝑖𝑔ℎ𝑡 𝑐 −𝐶𝑜𝑛𝑐𝑟𝑒𝑡𝑒 𝑐𝑜𝑣𝑒𝑟 Expected solution 𝑓 𝑒𝑥𝑡 𝛿 + = 0 Expected solution 𝑐 Formulation for cable only. 𝑭𝒆𝒂𝒔𝒊𝒃𝒍𝒆 𝒍𝒐𝒄𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒂𝒃𝒍𝒆 𝑦 𝑖 ℎ−2𝑐 𝑦 𝑐 𝑥

8 Formulation – shape optimization combined with topology optimizationmin 𝜌,𝑦 𝜙 𝜌,𝑦 = 𝒇 𝑒𝑥𝑡 𝑇 𝒖 𝑇𝑜𝑡𝑎𝑙 𝑠.𝑡: 𝑒=1 𝑁 𝑒𝑙𝑒𝑚 𝜌 𝑑 (𝜌) 𝑁 𝑒𝑙𝑒𝑚 − 𝑉 ∗ ≤0 (Concrete volume fraction) 0< 𝜌 𝑚𝑖𝑛 ≤𝜌≤1 𝑦 𝑖 −ℎ+𝑐≤ 𝑖=1,…, 𝑁 𝐶𝑎𝑏𝑙𝑒 𝑛𝑜𝑑𝑒𝑠 − 𝑦 𝑖 +𝑐≤ 𝑖=1,…, 𝑁 𝐶𝑎𝑏𝑙𝑒 𝑛𝑜𝑑𝑒𝑠 𝑤𝑖𝑡ℎ: 𝑲 ( 𝜌 𝑒 , 𝑦 𝑖 )𝒖 𝑇𝑜𝑡𝑎𝑙 = 𝒇 𝑒𝑥𝑡 + 𝒇 𝑝𝑟𝑒 Design variables: 𝒖 𝑇𝑜𝑡𝑎𝑙 = 𝒖 𝑓 𝑝𝑟𝑒 + 𝒖 𝑓 𝑒𝑥𝑡 ℎ −𝐵𝑒𝑎𝑚 ℎ𝑒𝑖𝑔ℎ𝑡 𝑐 −𝐶𝑜𝑛𝑐𝑟𝑒𝑡𝑒 𝑐𝑜𝑣𝑒𝑟 Concrete element densities according to SIMP. Y coordinates of the cable. The volume constraint is added. Formulation for cable an concrete. 𝑭𝒆𝒂𝒔𝒊𝒃𝒍𝒆 𝒍𝒐𝒄𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒂𝒃𝒍𝒆 𝑐 ℎ−2𝑐 𝑥 𝑦 𝑦 𝑖 ? ? (Bendsøe 1989, Sigmund and Torquato 1997)

9 Filters and projections (1)→ 𝜌 𝜌 𝑒 𝜌 𝑑 Density filter Cable filter Dilated/Eroded Design variable 1. Density filter 𝜌 𝑖 = 𝑗∈ 𝑁 𝑖 𝑤 𝜌 𝑗 𝜌 𝑖 𝑗∈ 𝑁 𝑖 𝑤 𝜌 𝑗 2. Cable filter - filter that insures covering the cable. 𝜌 𝑖 = 𝜌 𝑖 + 1− 𝜌 𝑖 𝑒 − 𝑑 𝑒𝑗 𝛽 𝑓𝑖𝑙 6 If an element is close enough to the cable his density is modified to 1. If not, the density remains the same. 2 𝛽 𝑓𝑖𝑙 𝑑 𝑒𝑗 𝜌 𝜌 =0.8 𝛽 𝑓𝑖𝑙 =4 Filters. To decrease the font size. To prepare beautiful graphs in MATLAB. Finish this slide. (Bruns and Tortorelli 2001; Bourdin 2001; Guest et al. 2004; Xu et al. 2010)

10 Filters and projections (2)→ 𝜌 𝜌 𝑒 𝜌 𝑑 Density filter Cable filter Dilated/Eroded Design variable 3. Projection function: 0.5 𝜌 𝑖,𝑒 = 𝑡𝑎𝑛ℎ 𝛽 𝜂 𝑒 + 𝑡𝑎𝑛ℎ 𝛽 𝜌 𝑖 − 𝜂 𝑒 𝑡𝑎𝑛ℎ 𝛽 𝜂 𝑒 + 𝑡𝑎𝑛ℎ 𝛽 1− 𝜂 𝑒 𝜌 𝑖,𝑑 = 𝑡𝑎𝑛ℎ 𝛽 𝜂 𝑑 + 𝑡𝑎𝑛ℎ 𝛽 𝜌 𝑖 − 𝜂 𝑑 𝑡𝑎𝑛ℎ 𝛽 𝜂 𝑑 + 𝑡𝑎𝑛ℎ 𝛽 1− 𝜂 𝑑 𝜂 𝑒 =0.6 𝜂 𝑑 =0.4 Using the dilated/eroded formulation, allows determining a minimum length scale in the optimal design. Filters. To decrease the font size. To prepare beautiful graphs in MATLAB. Finish this slide. (Wang et al. 2011; Lazarov et al. 2016)

11 𝜙 = 𝑓 𝑒𝑥𝑡 𝑇 𝑢 2 +𝜆 𝑲𝒖− 𝒇 𝑒𝑥𝑡 − 𝒇 𝑝𝑟𝑒 =𝜙Sensitivity analysis 𝜙 𝜌,𝑦 = 𝒇 𝑒𝑥𝑡 𝑇 𝒖 𝑇𝑜𝑡𝑎𝑙 2 𝑔 𝑥 = 𝑒=1 𝑁 𝑒𝑙𝑒𝑚 𝜌 𝑑 (𝜌) 𝑁 𝑒𝑙𝑒𝑚 − 𝑉 ∗ ≤0 → 𝑑𝑔 𝑑𝜌 , 𝑑𝑔 𝑑𝑦 The derivatives of volume constraint are straightforward using the chain rule. The derivatives of the objective function 𝜙 are computed using the adjoint method. 𝜙 = 𝑓 𝑒𝑥𝑡 𝑇 𝑢 2 +𝜆 𝑲𝒖− 𝒇 𝑒𝑥𝑡 − 𝒇 𝑝𝑟𝑒 =𝜙 → 𝑑 𝜙 𝑑𝜌 , 𝑑 𝜙 𝑑𝑦 𝑋 𝐶𝑀 , 𝑌 𝐶𝑀 𝑦 𝑗,2 𝑦 𝑗,1 𝑗 𝑑 𝑒𝑗 𝜌 𝑖 = 𝜌 𝑖 + 1− 𝜌 𝑖 𝑒 − 𝑑 𝑒𝑗 𝛽 𝑓𝑖𝑙 6 𝑑 𝑒𝑗 =𝑑𝑖𝑠𝑡 𝑋 𝐶𝑀 , 𝑌 𝐶𝑀 , 𝑦 𝑗,1 , 𝑦 𝑗,2 𝑑 𝜌 𝑖 𝑑 𝑦 𝑗,𝑘 = 𝑑 𝜌 𝑖 𝑑 𝑑 𝑒𝑗 𝑑 𝑒𝑗 𝑑 𝑦 𝑗,𝑘 Topology design variables Shape design variables The derivative of the topology according to the distance The derivative of the distance according to the shape → 𝜌 𝜌 𝑒 𝜌 𝑑 Density filter Cable filter Dilated/Eroded Design variable 𝜌 𝑖 Here because of the cable filter, we have a correlation between the topology and the shape design variables trough the distance dej which is a function of the concrete element centroid and the y coordinates of the cable. An then we can calculate the derivative of the topology according to the shape by the chain rule using this expression.

12 Examples – Shape Optimization OnlySimply supported beam Nelx=416; Nely=40; 52 Cable segments Prestressing force 𝑇 𝑝𝑟𝑒 = 𝑇 𝑝𝑟𝑒 ( 𝜎 𝑡 =0); 𝑃 𝑒𝑥𝑡 𝑃𝑜𝑖𝑛𝑡 𝑙𝑜𝑎𝑑 𝑓 𝑒𝑥𝑡 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑 𝑙𝑜𝑎𝑑 𝐼𝑡𝑒𝑟. 𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒 𝑃 𝑒𝑥𝑡 𝑃𝑜𝑖𝑛𝑡 𝑙𝑜𝑎𝑑 𝑓 𝑒𝑥𝑡 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑 𝑙𝑜𝑎𝑑 25-30 Iterations to zero displacement Parabolic Nelx=416; Nely=40; Linear Nelx=416; Nely=40;

13 Examples – Shape Optimization Onlyoptimal shapes for different prestressing force 𝑇 𝑝𝑟𝑒 0 =1.0 𝑇 𝑝𝑟𝑒 ( 𝜎 𝑡 =0); 1.5 𝑇 𝑝𝑟𝑒 0 1.0 𝑇 𝑝𝑟𝑒 0 0.75 𝑇 𝑝𝑟𝑒 0 0.5 𝑇 𝑝𝑟𝑒 0 𝑇 𝑝𝑟𝑒 =1.5 𝑇 𝑝𝑟𝑒 ( 𝜎 𝑡 =0); 𝑇 𝑝𝑟𝑒 =1.0 𝑇 𝑝𝑟𝑒 ( 𝜎 𝑡 =0); 𝑇 𝑝𝑟𝑒 =0.75 𝑇 𝑝𝑟𝑒 ( 𝜎 𝑡 =0); 𝑇 𝑝𝑟𝑒 =0.5 𝑇 𝑝𝑟𝑒 ( 𝜎 𝑡 =0);

14 Examples – Shape and topology Optimization𝑓 𝑒𝑥𝑡 𝑃𝑟𝑒𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑖𝑛𝑔 𝑐𝑎𝑏𝑙𝑒 𝑛𝑒𝑙𝑦=40 𝑛𝑒𝑙𝑥=416 𝑛𝑒𝑙𝑦 𝑛𝑒𝑙𝑥 ≅1:10 𝐼𝑡𝑒𝑟. 𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒 Cons. convergence Implemented Continuation Schemes 𝑉𝑜𝑙𝑓𝑟𝑎𝑐.=0.55 Iter. 𝛽 𝛽 𝑓𝑖𝑙𝑙 𝑝 𝐸 1-40 1 40-70 2 70-110 4 3 𝑇 𝑝𝑟𝑒 0.55 =0.2 𝑇 𝑝𝑟𝑒 1 ( 𝜎 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 =0); 𝑇 𝑝𝑟𝑒 𝑉𝑜𝑙𝑓𝑟𝑎𝑐

15 Examples – Shape and topology Optimizationoptimal prestressing design Fig. 1 normal stresses distribution only 𝑓 𝑒𝑥𝑡 is applied 𝑓 𝑒𝑥𝑡 + 𝑓 𝑝𝑟𝑒 is applied – final design Compression normal stress Tensile normal stress Fig. 2 No tension stress Fig. 3

16 Prestressed Concrete – More ExamplesStatically indetermined structures ` 𝑛𝑒𝑙𝑥1=208 𝑛𝑒𝑙𝑦=40

17 Prestressed Concrete – More ExamplesMulti-span bridge ` Distributed load Design domain Covered pre-stressing cable Volfrac=0.55 Zero deflection 𝑛𝑒𝑙𝑥1=300 𝑛𝑒𝑙𝑥2=370 𝑛𝑒𝑙𝑥3=340 𝑛𝑒𝑙𝑥2=370 𝑛𝑒𝑙𝑥1=300 𝑛𝑒𝑙𝑦=50

18 Summary An new procedure for top. opt. of prestressed concrete structures is proposed. Simultaneous shape optimization of the cable and topology optimization of the concrete. Special filter is developed to insure cover of the prestressing cable with concrete. Focus on prestressed concrete allows the use of linear-elastic modelling. The obtained designs represent efficient and aesthetic structures, which can be used for conceptual design by architects and engineers. Possible practical applications: prestressed concrete bridges, slabs, bracing systems. Future work: Extend to 3-D Constrain tensile cracking stresses Embed code requirements: interaction, relaxation, creep

19 Acknowledgments Thank youThis research was funded by the European Commission Research Executive Agency, grant agreement PCIG12-GA Thank you