Computer-Based Tools for Design by Strut-and-Tie Method Advances and Challenges

ACI Structural Journal , V. 99, No. 5, September-October 2002.MS No. 01-236 received August 6, 2001, and reviewed under Institute publication policies. Copyright © 2002, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion will be published in the July-August 2003 ACI Structural Journal if received by March 1, 2003.The strut-and-tie method (STM) is gaining recognition as a code-worthy and consistent methodology for the design of D- (discontinuity)regions in structural concrete. Unfortunately, the development of code provisions for the STM has been hampered by uncertainties in defining the strength and dimensions of the idealized load-resisting truss (or strut-and-tie model). In addition, the has been encumbered by an iterative and time-consuming design procedure in which many geometric details need to be considered. To overcome this problem, researchers are developing computer-based design tools,including the authors’ computer-aided strut-and-tie (CAST) design tool. CAST provides a graphical working environment for all aspects of the design process, including definition of the D-region,selection of the strut-and-tie model, truss analysis, member definitions, and creation of a design summary. This study reports on the STM, the barriers to its advancement, the capabilities of computer-based design tools, and the CAST program. It also makes suggestions for future STM research.Keywords : structural concrete; strut; tie.INTRODUCTIONIn selecting the appropriate design approach for structural concrete, it is useful to classify portions of the structure as either B- (beam or Bernoulli) regions or D- (disturbed or discontinuity) regions. B-regions are those parts of a structure in which it is reasonable to assume that there is a linear variation in strain over the depth of the section. D-regions are the remaining parts of the structure in which there is a complex variation in strain, occurring near abrupt changes in geometry (geometrical discontinuities) or concentrated forces (statical discontinuities). Based on St. Venant’s principle, the extent of a D-region spans approximately one section depth of the region on either side of the discontinuity. The distinction between B- and D-regions is illustrated in Fig. 1.Most design practices for B-regions are based on a model for behavior. For example, the design for flexure is based on conventional beam theory while the design for shear is based on the well-known parallel chord truss analogy. In contrast,the most familiar types of D-regions—such as deep beams,corbels, joints, and pile caps—are principally designed by empirical approaches, such as those given in ACI 318-99,1or by using common detailing practices. For most other types of D-regions, code provisions provide little guidance to designers. Not surprisingly, most structural problems occur in D-regions.The strut-and-tie method 2-4 (STM) is emerging as a code-worthy methodology for the design of all types of D-regions in structural concrete. Unfortunately, this conceptually powerful method can be complicated by the need to perform time-consuming calculations and graphical procedures. It is for this reason that computer-based graphical design aids are being developed.Beginning with a brief description of the STM, this paper discusses complications in the STM design process and the role that computer-based tools can serve in overcoming these obstacles. Then, a summary of the capabilities of a few computer-based STM design tools is presented, with emphasis on the features that distinguish one tool from another. This includes work at Purdue University, the Swiss Federal Institute of Technology (ETH), the University of Stuttgart, and others. Following this, the computer-aided strut-and-tie (CAST) design tool that is being developed by the authors is presented. The study concludes with a summary of the challenges that lie ahead for the STM and associated computer-based design tools.RESEARCH SIGNIFICANCEBecause of the inadequacy of traditional code provisions and detailing practices, most structural problems occur in D-regions. The STM has the potential to provide a consistent,well-founded, and widely applicable design methodology for D-regions, but it is marred by a cumbersome hand-based design process. To overcome this problem, computer-based design and analysis tools that bring simplicity and transparency to the STM design process, and thus can improve how D-regions are designed, are being developed. This paper presents the role and capabilities of these programs, summarizes uncertainties in the STM design methodology, and suggests directions for future research.STM FOR DESIGN OF D-REGIONSBackgroundThe idea of the STM came from the truss analogy method introduced independently by Ritter and Mörsch approximately 100 years ago for the shear design of B-regions. The truss analogy, or truss model, was used to idealize the flow of forces in a cracked concrete beam. In parallel with the increasing availability of experimental results and the development of limit analysis in the plasticity theory, the truss analogy method has been validated and improved considerably in the form of full member or sectional design procedures. The truss model has also been used as a basis for torsion-design methods. An excellent summary of the development of truss model for shear design of B-regions can be found in Reference 5. The STM was developed after Schlaich, Schäfer,and Jennewein 3 extended the use of a truss model to D-regions.Title no. 99-S60Computer-Based Tools for Design by Strut-and-Tie Method: Advances and Challengesby Tjen N. Tjhin and Daniel A. KuchmaStrut-and-tie modelsIn the STM, the complex flow of internal forces in the D-region under consideration is idealized as a truss carrying the imposed loading through the region to its supports. This truss is called a strut-and-tie model. Like a real truss, a strut-and-tie model consists of struts and ties interconnected at nodes (nodal zones or nodal regions). A selection of strut-and-tie models for a few typical D-regions is illustrated in Fig. 2.Struts are the compression members of a strut-and-tie model and represent concrete stress fields whose principal compres-sive stresses are predominantly along the centerline of the strut. As shown in Fig. 2, struts are usually symbolized using a broken line. The actual shape of a strut, however, can be prismatic, bottle-shaped, or fan-shaped (Fig. 3). Struts can be strengthened by steel reinforcement and, when this is the case, they are called reinforced struts.Ties are the tension members of a strut-and-tie model.Ties mostly represent reinforcing steel, but they can occasionally represent prestressing steel or concrete stress fields with principal tension predominantly in the tie direction. Ties are usually denoted using a solid line.Nodes are analogous to joints in a truss, and are where forces are transferred between struts and ties. As a result,these regions are subject to a multidirectional state of stress.Nodes are classified by the types of forces being connected.Figure 4 shows basic types of nodes; in the figure, C is used to denote compression and T is used to denote tension.STM design processThe STM design process involves several steps that are illustrated in Fig. 5 using the design example of a dapped-ended beam, and are described as follows:1. Defining the boundaries of the D-region and then evaluating the concentrated, distributed, and sectional forces that act on the boundaries of this region;2. Sketching a strut-and-tie model and solving for the truss member forces;3. Selecting the reinforcing or prestressing steel that is necessary to provide the required tie capacity and ensuring that this reinforcement is properly anchored in the nodal zones (joints of the truss);4. Evaluating the dimensions of the struts and nodes such that the capacity of these components is sufficient to carry the design force values; and5. Providing distributed reinforcement to increase the ductility of the D-region.Because equilibrium of the truss with the boundary forces must be satisfied (Step 2) and stresses everywhere must be below defined code limits (Steps 3 and 4), the STM is a lower-bound (static or equilibrium) method of limit analysis.ACI member Tjen N. Tjhin is a doctoral candidate in the Department of Civil and Environmental Engineering at the University of Illinois at Urbana-Champaign. His research interests include nonlinear analysis and design of concrete structures.ACI member Daniel A. Kuchma is an assistant professor of civil and environmental engineering at the University of Illinois at Urbana-Champaign. He is a member of ACI Subcommittee 318-E, Shear and Torsion; and Joint ACI-ASCE Committee 445,Shear and Torsion.Fig. 2—Examples of strut-and-tie models.Fig. 1—Example of division of B- and D-regions in common structure.c(c)The most widely used guidance for selecting the appropriate truss so far is probably that suggested by Schlaich, Schäfer,and Jennewein,3 who proposed arranging the truss members within 15 degrees of the principal stress trajectories obtained from a linear elastic solution. With this approach, both service-ability and ultimate limit states are expected to be satisfied.Based on the minimum strain-energy principle, Schlaich,Schäfer, and Jennewein also suggest selecting a truss in which the total length of ties is a minimum. This guidance is very helpful, but elastic solutions or solutions for minimum length of ties are not always easy to obtain. The other method includes recent work by Ali and White.11 Also using the energy approach, they introduced the elastic strain compatibility error (SCER) concept to measure how well a selected strut-and-tie model deviates from the elastic solution; the lower the SCER value, the closer it is to the elastic stress distribution.Adjustment of truss geometry and dimensions The initially selected geometry of the truss, including the strut and node dimensions, must often be adjusted to satisfy stress limit criteria, to fit the struts and nodes within the D-region boundaries, to investigate other configurations,and to optimize the design. Refinement of the truss geometry sometimes needs to be undertaken for similar reasons. This can make hand-based solutions prohibitively time-consuming,particularly for the design of complex D-regions.Geometry and dimensions of ties and nodesTo determine the dimensions and shape of a node, it is necessary to know the widths of the incoming struts and ties.The width of a strut is typically selected so that a particular code-specified stress limit value is not exceeded. It is less clear, however, how to define the effective width of ties as well as the dimensions and shape of nodal zones.The classic method of dimensioning a node is by arranging the node shape so that the stresses on all sides of the node—from the truss member forces as well as from the boundary forces meeting at the node—are equal. The biaxial state of stress inside the node is hydrostatic; in other words, the in-plane stresses are isotropic, homogeneous, and equal to those on the sides. Arranging the node in this shape can be done by sizing the boundaries of the node so that they are proportional and perpendicular to the forces acting on them (Fig. 7). In defining the width of a tie, the tie force is treated as a compressive force acting from behind the node (Fig. 7(b)through (d)). This type of node is called a hydrostatic ying out nodes in this manner can be very laborious,especially for nodes bounded by more than three truss members (Fig. 8(a)), as the centerlines of truss members framing into hydrostatic nodes of more than three sides are unlikely to coincide (Fig. 8(b)).A more simplified method that is applicable for nodes with typical configurations was proposed by Schlaich and Schäfer.4The shape of a node is simply formed by the intersection of actual dimensions of struts and ties whose centerlines coincide at that node (Fig. 8(c)). The in-plane stresses acting on all sides of the node do not need to be equal, but the stress on each side must be constant and below the node stress limit.This convenient method, however, must be used discrim-inately for complex node configurations.Statically indeterminate strut-and-tie modelsThere is little guidance available for evaluating the relative stiffness of members in a statically indeterminate strut-and-tie model. As a result, the designer is unsure how to determine the distribution of forces in these types of trusses.The classic way to handle a statically indeterminate case is to employ the so-called plastic truss method. In this method,the most heavily loaded ties in the truss are assumed to have yielded so that the tire forces become known and the truss becomes statically determinate. This method, however, must be used with caution because of strain compatibility require-ments and limited ductility in concrete. Another method,suggested by Schlaich and Schäfer,4 is to decompose the statically indeterminate truss into several statically determi-nate trusses. Again, as a result of the strain compatibility re-quirements, this technique requires a reasonable estimation of the stiffness and the imposed-loading distribution of each stat-ically determinate truss.Multiple load cases and load combinationsThe need to consider multiple load cases and load com-binations can increase the time required to complete a design using the STM by several times. This is so because different strut-and-tie models may need to be prepared to handleFig. 6—Two admissible strut-and-tie models for squat structural wall under horizontal force: (a) workable truss;and (b) less feasible truss due to excessive ductility demands.Fig. 7—Examples of hydrostatic nodes: (a) CCC; (b) CCT:(c) CTT; and (d) TTT.each different loading situation. In addition, load cases usually cannot be superimposed directly to form load com-binations as a result of strain compatibility requirements.To summarize the aforementioned complications, the STM design process can be encumbered by the challenge of selecting the initial truss, the need to iteratively adjust and refine truss geometry and member dimensions, the lack of guidance for the design of statically indeterminate trusses,and the need to consider multiple load cases and load combina-tions. Additionally, the STM does not yet address structural performance under service loading. Computer-based strut-and-tie design programs, as described in the following section,can overcome many of these challenges.ADVANCES IN COMPUTER-BASED STMMost of the work on computer-based STM in Europe was done at the Swiss Federal Institute of Technology (ETH)under the supervision of Anderheggen 12,13 and at the University of Stuttgart under the supervision of Schlaich and Schäfer.14,15 In North America, a series of computer-based STM tools were developed by Ramirez and his students at Purdue University 16-20 as well as by Ali and White 11 at Cornell University.In general, all of the tools that have been developed in those institutions offer interactive analyses of STM. The con-struction and modification of the boundaries of the structure under consideration as well as the selected strut-and-tie models are done graphically using a mouse and keystrokes.The main thing that distinguishes one tool from another is the approach they use to obtain valid STM solutions, as well as other features that they provide. Some of the noteworthy capabilities provided by these STM programs are presented as follows.Tools for strut-and-tie model constructionAs discussed previously, principal stress trajectories obtained from a linear elastic solution can provide guidance in selecting an appropriate strut-and-tie model. For this purpose, many computer-based STM tools are equipped with a linear elastic finite element analysis that can be used for generating plots of principal stress trajectories.17,18,21 These plots are then used as a backdrop for constructing a strut-and-tie model.Programs that include nonlinear finite-element analysis capabilities have also been created.19,20 The latter analysis may be particularly useful when the capacity of a D-region using a strut-and-tie model is to be assessed.The use of principal stress trajectories for guiding the construction of strut-and-tie models has also been extended to the automatic generation of strut-and-tie models. An example of this can be seen in the work by Harisis and Fardis,22who used statistical analysis of principal stress data obtained from linear finite element analysis to identify locations of struts and ties. The strut-and-tie models generated using this approach consist of triangles. Another example is the work by Rückert.14 Based on the fact that regions bounded by the mesh of principal stress trajectories represent finite elements subjected only to normal stress on each side, an strut-and-tie model can be formed using these finite elements. Ali and White 11recently developed algorithms for the automatic generation of strut-and-tie models. Based on user-selected locations of nodes,an optimal truss solution can be generated following the elastic,minimum reinforcement volume, or composite criterion.Approaches in nodal-zone constructionOne simple computer-based approach to construct a nodal zone is to define the node shape as the intersection of the dimensions of truss members whose centerlines coincide,as illustrated in Fig. 8(c). The adequacy of the node is checked using linear elastic finite element analysis with the Coulomb failure criterion 18 or nonlinear finite element analysis with failure criteria determined from experimental test data of two-dimensional plain concrete.19,20Another approach is to use modified hydrostatic node construction.12 In this approach, a node with more than three members intersecting is handled by breaking down the node into several hydrostatic nodes of triangular shapes connected by short prismatic struts. Figure 9 shows how the CCC node of Fig. 8(a) is arranged using this approach.Trapezoidal transition stress zones between a node and the intersecting truss members were also formulated using this approach, allowing different stress intensities of truss members to act on the node. The length of these transition zones is determined using separation failure criteria of modified Coulomb material.Fig. 9—Arrangement of CCC node of Fig. 8(a) that comprises two hydrostatic nodes of triangular shape and short prismatic strut.Fig. 8—Example of CCC node with four truss members intersecting: (a) forces acting on node; (b) node shape arrangement causing hydrostatic stress state; and (c)simplified arrangement.(c)f c 4In determining the width of a tie, both of these approaches treat the tie as a compressive force that acts from behind the node.Procedures for determining truss member forces Several computer-based procedures have been developed to determine truss member forces, especially for statically indeterminate cases. The simplest procedure is to use an elastic truss analysis. This analysis is not very convenient,because the relative stiffness of truss members, which is very difficult to determine, must be supplied.Anderheggen and Schlaich 13 developed two procedures based on a lower-bound solution: rigid-plastic optimal design and shakedown optimal design. In the first procedure, the member forces of a statically indeterminate strut-and-tie model are determined in such a way that minimum tie resistances,corresponding to the minimum weight of steel ties, are obtained.This is done through linear programming. Determining the distribution of forces in this manner is possible because strain compatibility equations are ignored in the lower-bound solution. Based on the static shakedown theorem, the second procedure is similar to the first, except that the elastic truss solutions have to be considered. Because the computational time is reduced, this procedure is useful when several load cases need to be considered. The obtained solution lies between the elastic solution and the rigid-plastic solution.Prediction of load-deformation response and capacityWork on computer-based STMs has been advanced to predict load-deformation response and capacity using strut-and-tie models.14,15 In this work, the load-deformation response of struts and ties must be specified, and the minimum energy criterion is employed. For each loading stage, the geometry of the strut-and-tie model is adjusted so that the internal energy of the system is at a minimum. The predictions of these methods are reported by the authors of these programs to show good agreement with experimental data.CAST DESIGN TOOLOverviewThe development of the CAST program was motivated by the desire to bring efficiency and transparency to the STM design process. To this end, the goal was to create a computer-based graphical working environment in which the designer is able to readily sketch the boundaries of the D-region, draw a proposed internal load-resisting truss,seamlessly solve for member forces, readily select the dimen-sions of struts and reinforcement for ties, and then generate a printout that summarizes the design. Other necessary features included the ability to make on-screen adjustments of all key variables, tailor the program for specific design code provisions,and handle multiple load cases and combinations. In developing these capabilities, it was considered essential that CAST functioned as a transparent tool that helped the designer to explore and evaluate potential solutions. CAST is believed to be well on its way to achieving these objectives and thereby enabling the STM to achieve its potential of providing engineers with an efficient, consistent, and well-founded methodology for the design of D-regions in structural concrete.Basic operation of CASTA user begins by defining the boundaries of the D-region using a mouse, by numerical entry, or by selecting an object from a group of templates. The position of all of the nodes that define the shape of the object can be easily adjusted at any stage in the design process. The applied loading, bearing plates, material properties, and support conditions can then be selected from toolbars or specified in dialogue boxes. Next,the proposed load-resisting truss can be drawn as a series of straight lines using the mouse. The truss can then be analyzed and the results displayed alongside the truss members. Using a toolbar, the user can then select and position the estimated tie reinforcement as well as select the widths of the struts.The capacity of the struts and ties as well as the applied stress on the faces of the nodes can then be displayed. The designer can use the nodal zoom-edit tools to change features of the node that influence node and truss geometry, and therebyFig. 10—User interface of CAST education and design tool.affect the stresses on the face of the nodes and design strengths. Once the user is satisfied with the design, the detailed design—including member forces, capacities, stresses on the faces of nodal zones, and reinforcement details—can be printed. Figure 10 provides an image of CAST’s working environ-ment. The interior images show a view of the entire structure with applied loads and support conditions, a detail of the labeled forces, the selection of tie reinforcement, and the detail of a node.REMAINING CHALLENGES TO BERESOLVED IN STMDespite the advances in computer-based STM tools, a number of uncertainties still remain in the STM. These uncer-tainties must be addressed in future research so that CAST or other computer-based tools can make the strut-and-tie design process as efficient and transparent as the STM is conceptually powerful. A few of the main challenges are described as follows.Capacity of strutsThere is still much debate over the effective compressive strength of a strut. This is reflected in the different strength values specified in codes and guidelines6-10 and proposed by researchers.2-4,23 Nevertheless, it is generally agreed that the strut strength is a fraction of the uniaxial concrete compressive strength as obtained from cylinder tests. The following factors have been identified as influences to the ultimate compressive stress capacity of struts:1. Shape of strut—If the stress trajectories in a strut are parallel as shown in Fig. 3(a) and 11(a) and (f), the strength of the strut is close to that of the compressive strength of a concrete cylinder. If the strut is within the core of the D-region, however, the compressive stresses in the strut tend to spread out as they move away from the nodes (Fig. 3(b) and 11(d) and (e)). This bottle-shaped stress path can lead to the splitting of a strut at a compressive stress that is consid-erably less than the cylinder compressive strength;2. Disturbances in strut—Other factors that influence the splitting strength of a strut include initial cracks parallel or inclined to the strut axis and tensile transverse stress or strain induced by a crossing tie or another effect (Fig. 11(b) and (c));3. Distributed reinforcement—The use of distributed rein-forcement can control splitting of the strut due to spreading or disturbances (Fig. 11(d)). It may also increase the overall structural ductility and thus help improve the performance ofa D-region designed using a less-than-ideal truss selection;4. Confinement—The performance of the strut can also be enhanced by confinement provided by either a specially designed confining reinforcement or by mass reinforced concrete that surrounds the strut (Fig. 11(f)); and5. Angle of strut—It is well documented24 that the com-pressive strength of concrete decreases as the level of transverse tensile strain increases. The implication for strut-and-tie modeling is that, as the angle between the strut and a tie in a nodal region decreases, the compressive strength of the strut is expected to decrease. Thus, in the design of a deep beam, a steeper strut is expected to be stronger than a horizontal strut.While many of the factors that can influence the compressive strength of concrete struts have been identified, the effect of these factors has yet to be adequately quantified. This is due to the very limited amount of experimental research25 that has been conducted to understand how to define the shape, stiffness, and strength characteristics of struts.Load-deformation response of struts and ties Another major challenge is to accurately estimate the load-deformation response of struts and ties. This is important for the following reasons:1. If the stiffness characteristics of the struts and ties are known, the load distribution in statically indeterminate strut-and-tie models, as mentioned previously, may be predicted. This is illustrated in Fig. 12, where the point load is considered to be transferred to the support by two distinct load paths: (i) a direct strut from the point of loading to the support; and (ii) a path consisting of two steeper struts connected by a steel tie. The portion of the load taken by each load path will be in proportion to the relative stiffness of these paths;2. If the stiffness characteristics of the struts and ties at service loads are known, the equivalent deflection response may be predicted for the evaluation of serviceability limit state; and3. As illustrated in Fig. 12, a tie may cross the path of a strut. The straining and cracking induced in this situation may influence the strength of the strut. By examining the factors that influence the tension-stiffening effect and distribution of cracking in ties, the capacity and response of struts can be better understood.Anchorage and distribution of tie reinforcementIn the STM, the selected steel ties can fully develop tensile forces and transfer the forces in the nodes only if the ties are properly anchored in the nodes. There are still uncertainties, however, about the anchorage requirements and the need to distribute the tie reinforcement throughout the nodal region.Fig. 11—Types of struts in discontinuity region: (a) prismatic in uncracked field; (b) prismatic in cracked field where struts are parallel to cracks; (c) prismatic in cracked field where struts are not parallel to cracks; (d) bottle-shaped with crack control reinforcement; (e) bottle-shaped without crack control rein-forcement; and (f) confined strut.。