Topology optimisation of concrete structures considering stress constraints
Creators: name: Chen, Anbang
Publisher: name: RMIT University
Year: 2024
Subjects: classificationCode: 400510
subject: Structural engineering

description: Concrete is one of the most widely used construction materials in civil engineering. Due to the brittle behaviour of plain concrete, steel bars are often used to reinforce concrete structures so that both ductility and tensile resistance can be significantly improved. In practice, steel-reinforced concrete (SRC) structures may have both Bernoulli (or beam) regions (‘B-regions’) and disturbed (or discontinuity) regions (‘D-regions’). As opposed to B-regions, which can be well designed by the traditional beam theory, D-regions usually have nonlinear deformation, and thus cannot be designed using the same method. To deal with this challenge, the strut-and-tie model (STM) has been developed to analyse the D-regions in a structure. However, the detailed stress states in concrete and steel bars are not fully considered in the STM, and a practical method for the application of the STM in structural design is still lacking. In recent years, both continuum and truss topology optimisations have been employed in the design of SRC structures to reduce the consumption of material while maintaining the performance of the structure. In the design of SRC structures, the tensile stress level of concrete must be controlled to prevent concrete from cracking. However, the ‘singularity’ problem is usually encountered when optimising truss layouts using the density-based topology optimisation methods, where elements with low densities may have high-stress values, leading to failure in topology optimisation. Besides, the stress level in local elements may also result in a large number of constraints. At present, few topology optimisation techniques have been reported with the consideration of stress constraints in SRC structures. Therefore, it is essential to conduct a systematic investigation on the topology optimisation of concrete structures to achieve efficient and lightweight designs. In this thesis, new topology optimisation methods are proposed to deal with the structural design considering stress constraints of concrete structures.

The bi-directional evolutionary structural optimisation (BESO) is a classic discrete topology optimisation method, which can avoid the ‘singularity’ problem of stress constraint. Therefore, an effective approach (BESO-FPS) based on the BESO technique is firstly developed to control the maximum first principal stress (FPS) in concrete. In the proposed method, the p-norm function is adopted to relax the local stress constraint into a global one. The optimisation objective function is modified under stress constraint by introducing a Lagrangian multiplier, and the sensitivity is deduced by the adjoint method. A simply-supported deep beam structure is used to study the effect of the norm parameter p. The results show that 3 &lt; p &lt; 10 is generally suggested, and the sharp reentrant corners in an initial design can be well smoothed by the present method with a higher value of p. The optimisation examples of a 2D rectangle and a 2D L-bracket structures reveal that, by constraining the stress level in concrete, distinctly different layouts are generated compared to the designs without stress constraint. Moreover, a 3D L-bracket example is also presented to demonstrate the effectiveness of the proposed algorithm. All numerical examples show that the optimised design obtained by the proposed method has slightly higher compliance but a significantly lower stress level than the solution without the FPS constraint.

To optimise the layout of steel reinforcement in SRC structures, a modified method (M-BESO) is developed based on the BESO, where the tensile stress level of host concrete is considered. A hybrid truss-continuum finite element model is firstly developed, in which reinforcing steel bars inclined at different angles are considered. To make full use of the tensile strength of steel and the compressive strength of concrete, a modified sensitivity formulation is proposed by multiplying the FPS value of the host concrete. To obtain a more reasonable layout trajectory of reinforcement,  filtering operations of the sensitivity are conducted separately for steel bars with different directions. In this method, the steel elements in the SRC structure are gradually removed when the host concrete exhibits compressive stress or very low tensile stress. An optimisation algorithm that incorporates a volume constraint on steel bars is implemented and tested using four classic examples in civil engineering, i.e., a cantilever beam, a simply supported beam, a corbel, and a wall with a window. For comparison, the SRC structures with uniform steel bar layouts are also analysed. The numerical results show that clear reinforcement patterns can be obtained from both BESO and M-BESO algorithms. Compared with the uniform layout and the layout from the traditional BESO solution, the design obtained by the M-BESO method has a significantly smaller area of critical tensile stress in the host concrete. The proposed method has been demonstrated to be able to generate reasonable steel reinforcement layouts in SRC structures.

By combining the stress constraint method and the hybrid truss-continuum finite element model, a new topology optimisation formulation of the SRC structure is proposed. Here the design variables include the location and orientation of reinforcing bars, and the constraint is the maximum FPS in concrete. The adjoint method of the p-norm function is integrated into the compliance minimisation of SRC structures to deal with the maximum FPS constraint in host concrete. The developed topology optimisation method is then compared with the BESO method without FPS constraint in four optimisation examples. The optimised design obtained by the proposed method shows a slightly higher compliance but a much lower maximum FPS in concrete. Compared to the solutions from the traditional BESO method, the rearranged reinforcement distribution obtained from the proposed method can significantly reduce the tensile stress concentration in concrete. Numerical results demonstrate that the technique can provide an effective steel reinforcement layout in an SRC structure considering the maximum FPS constraint.
 
By using the topology optimisation techniques developed in this thesis, reasonable designs of concrete structures and steel reinforcement layouts can be achieved. The proposed methods enable lower tensile stress levels and lightweight designs, leading to more efficient concrete structures. The techniques developed in this thesis will be useful for various applications in civil engineering and can be extended to other fields.