Experimental investigation of strut-and-tie layouts in deep RC beams designed with hybrid bi-linear topology optimization

Highlights

• Experimental investigation of Strut-and-Tie Modelled (STM) reinforced concrete beams.

• Includes two designs that are automatically generated with topology optimization.

• Stiffness and ultimate strength are improved for the topology-optimized STMs.

• High complexity of the reinforcement layout is not required for improved performance.

Abstract

This paper presents the experimental study of deep reinforced concrete beams designed with strut-and-tie models. Three different reinforcement layouts are designed with the same material quantities and for the same loading and boundary conditions. Two of the layouts are automatically generated using a topology optimization framework in which obtaining increasingly complex steel layouts is allowed. The experimental performance of these are compared to a beam designed with a conventional strut-and-tie model. Two samples are constructed and experimentally tested of each of the three designs. To limit post-processing of the reinforcement layouts, the topology-optimized designs are obtained using a hybrid bi-linear approach and an alternative reinforcement method has been used; the layouts have been waterjet cut from steel plates. The experimental study reveals that the topology-optimized reinforcement layouts have an increase in stiffness and strength compared to the conventional design. Interestingly, all specimens with topology-optimized reinforcement exhibit a similar behavior regardless of the topological complexity. This suggests that significant performance improvements are obtained when using topology optimization to generate strut-and-tie models without necessarily requiring the resulting layout to be highly complex and costly to construct.

Introduction

In the past decades there has been an increasing interest in automating the engineering design for civil structures. Within civil structures, Reinforced Concrete (RC) is a widely used composite construction material. It consists of a concrete phase that is strong in compression and a reinforcing phase that compensates concrete’s low tensile strength. Steel bars are most often used as reinforcement. Within RC design, structures are commonly divided into two regions; B (Bernoulli or Beam) regions and D (Disturbed or Discontinuity) regions. B regions are governed by bending and have linear stress fields, whereas D regions are typically parts of the structure where stress distributions are nonlinear. D regions are e.g. located where there is an opening or other type of discontinuity within the structure or where concentrated loads are applied. The nonlinear stress fields in D regions complicates the design in these areas of the structure. Instead of relying solely on the past experience of the design engineer, Strut-and-Tie Models (STMs) can be used for D region design.

The foundation of STM design is the truss analogy proposed by Ritter [1] and Mörsch [2], and formalized by Schlaich et al. [3]. The basic assumption is that a cracked concrete beam behaves like a truss. This notion was later supported by Marti [4] that connected the truss model to lower bound plasticity theory. The general STM is therefore a truss model with compressive concrete struts and tensile reinforcing ties. Ideally the struts are placed along the elastic stress trajectories and a higher structural performance is achieved with a stiffer truss. At current, STM is a basic tool for the design of D regions and implemented in a range of design codes (e.g. EC 2 [5], ACI 318-08 [6], and AASHTO [7]). The starting point for STM design is for the design engineer to select the stiffest truss connectivity or layout, which is typically not a trivial task. Decisions are often based on the design engineers previous experience and iterations can be required. This is e.g. demonstrated in [8], where two example designs for a hammerhead pier are provided and shown to require different levels of transverse and web reinforcement. Several researchers have suggested methodologies, such as topology optimization, for automatizing the generation of STMs.

Topology optimization is a free-form tool for engineering design that has been known to lead to new solutions that typically outperform conventional low weight designs. The design engineer is not required to have an initial notion of the final design layout, but must only define a design domain $\mathrm{\Omega }$ with applied loads and boundary conditions on it. Fig. 1 gives an example of a design domain $\mathrm{\Omega }$ defined for a simply supported deep beam design problem. The key idea behind most of topology optimization approaches is to distribute the available material within the design domain such that a chosen performance objective is optimized. The design problem is formulated as a formal optimization problem and most rigorously solved using a mathematical program. To computationally evaluate the performance and update the design, the domain is discretized. The discretization is typically done by populating the domain with nodes and defining connected finite elements. The design engineer chooses what element type that is used for the discretization. The choice of discretization type will affect the layout of the final topology. This is shown in Fig. 2 where a simply supported deep beam (design problem from Fig. 1) is designed with different finite element types populating the design domain. The designs in Fig. 2 are created herein using existing topology optimization frameworks as described in detail below. The left column of the figure shows the initial material distributions and the optimized topologies are given in the right column of Fig. 2.

Several researchers have suggested using truss topology optimization to design STMs (e.g. [9], [10], [11]). Most frameworks use the so-called ground structure approach where a dense connectivity of potential bars are defined. Through the optimization the potential elements are sized where the bar area is allowed to approach zero. Fig. 2a shows a ground structure truss element discretization of the design domain ($\mathrm{\Omega }={\mathrm{\Omega }}_t$). Since nodes and elements cannot be moved or added, the chosen ground structure inherently restricts the force flow. The solution in Fig. 2b is therefore highly dependent on the node spacing and the potential connectivity within the initial ground structure. With the choice of a coarse mesh as in Fig. 2a, the design space is significantly reduced. Although it can be desirable for the design engineer to maintain the discrete nature of the reinforcement bars, this reduction notably alters the force flow within the concrete phase compared to a freeform solution. As a result, better performing global minima with finer initial ground structures cannot be identified.

In the past two decades, researchers have proposed topology optimization frameworks for STM generation that uses continuum elements. Suggested algorithms include evolutionary methods (ESO) that use heuristic updating schemes for solving the optimization problem [12], [13], [14], [15], and frameworks based on the density-based approach to topology optimization [16] where rigorous mathematical programs are used to solve the optimization problem [17], [18]. In Fig. 2c–d the simply supported deep beam is designed using a continuum density-based topology optimization approach ($\mathrm{\Omega }={\mathrm{\Omega }}_c$) where the domain is discretized with 2D plane stress four node quadrilateral elements. Initially, all elements in the design domain are assigned an intermediate ‘gray’ material as shown in Fig. 2c. Through the optimization the densities of all elements are guided towards selecting either to place material (black) or void (white). For STM layout generation, the resulting connectivity of solid elements visualize the force flow within the RC structure. The main advantage of a continuum-based topology optimization framework is its free-form nature. It does not require the design engineer to define node locations and connectivity a priori and can therefore lead to higher performing solutions. However, for the purpose of RC design, the discrete nature of the reinforcing bars is lost and post-processing of the continuum solution is required before the bars can be sized.

Few works have diverged from the assumption of the RC exhibiting isotropic material behavior. Using a continuum discretization, Victoria et al. [19] implemented different moduli in tension and compression and solved the design problem with a heuristic optimality criteria. Recently, Du et al. [20] presented a density-based two-phase bi-modulus framework with rigorous sensitivity analysis and demonstrated its application for generating STMs. However, the proposed approach still requires post-processing of the continuum solution to design the reinforcement. To reduce the need for post-processing, Gaynor et al. [21] suggested a density-based hybrid mesh framework where the design domain is discretized with both truss and continuum elements with different bi-linear material behaviors. This eliminates the post-processing step as the truss members within the hybrid mesh are tension-only whereas the continuum elements visualize the compressive force flow. Yang et al. [22] extended the hybrid mesh approach to generate 3D STMs. Fig. 2e–f show the initial- and final deep beams designed with the bi-linear hybrid topology optimization approach [21], [22] where the design domain is discretized using both truss- and continuum elements such that $\mathrm{\Omega }={\mathrm{\Omega }}_t\cup {\mathrm{\Omega }}_c$. This provides the design engineer with control over the complexity and hence constructability of the reinforcement layout and allows for straightforward sizing. It should be noted that both the truss- and continuum topology optimization approaches presented in Fig. 2a–d are based on linear elastic isotropic material assumptions. Since the hybrid mesh approach requires the use of (at least) bi-linear material models, it has been shown to capture transverse tensile stresses in the concrete phase caused by force spreading [21], [22].

Experimental investigations have confirmed the improved performance of RC beams designed with STMs. However, little work has focused on experimentally evaluating beams designed with automated STM generation. Perera and Vique [23] presented a database of experimental results for simply supported deep beams and proposed a framework for generating STMs using a stochastic genetic algorithm as the optimizer. For topology-optimized STMs, the pattern observed in an experimental test has been compared to designs obtained design by Guan [24], [25] with a heuristic evolutionary method. Recently Oviedo et al. [26] experimentally evaluated optimized STMs for a dapped beam, including specimens with topology-optimized STMs. However, significant post-processing was performed to the continuum solutions before sizing the reinforcement. Heuristic post-processing of topology-optimized results is well known to potentially introduce significant alterations to the optimized design (see e.g. the experimental and numerical investigation of topology-optimized plain concrete beams in [27]). Still, an increased ultimate strength and better crack growth control was observed.

This paper experimentally investigates the performance of increasingly complex topology-optimized STMs for a deep RC beam design problem. Two 2D STMs are generated with topology optimization, the beams are constructed, tested and compared to the experimental behavior of a beam with a conventionally designed STM. The amount of steel within the three RC beam designs is held constant to allow for a comparison of how the performance is affected by the complexity of the STM. Efforts are made to construct experimental specimens with as little post-processing as possible. Therefore the bi-linear hybrid topology optimization approach from Gaynor et al. [21] is used as the back bone of the current work and will be described for completeness in Section 2. To reduce the post-processing need further, an alternative reinforcement method is used; steel plates have been cut using waterjet. This reinforcement method has allowed the use of continuous area sizes of the reinforcing bars, eliminates imperfections caused by bar bending and allows for precise placement of the reinforcement during construction. Section 3 details the design, construction and test set-up. The experimental results are presented and discussed in Section 4.