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NSF CAREER Award Project 0092668

Tools and Research to Advance the Use of Strut-and-Tie Models
in Education and Design

1.1

B- (Beam) and D- (Discontinuity) Regions

B- (Beam) Regions are those parts of the structure in which there is a linear variation in strain over the depth of the member, while D- (Discontinuity) Regions are those parts of a structure in which there is a complex variation in strain. Based on St. Venant’s principle, D-Regions are those parts of a structure within a distance equal to the depth of the member from a concentrated force (load or reaction point), change in section depth, an opening, or another discontinuity. As Figure 1 illustrates, a large portion of even common structures are D-Regions. This CAREER proposal is concerned with the behavior and design of D-Regions.

Figure 1   Example of D-Regions in Common Structures
(Click here to view a larger image)

1.2

Shortcoming of Common Design Practice for D-Regions

Empirical code provisions and/or unstandardized detailing practice are used for designing the most familiar types of D-Regions, such as deep beams, corbels, joints, and pile caps. These procedures are unacceptably inexact, which leads to deficiencies or inefficiencies in the design of these commonly occurring and often critical parts of structures. To illustrate this point, Figure 2 demonstrates the inability of the ACI 318-99 [1] provisions to reasonably well estimate the shear strength of deep beams [2, 3]. If provisions were adequate and efficient, the V_test/V_ACI ratio for the vast majority of the results would lie between about 1 and 1.25. Provisions and practices for the design of other types of D-Regions are unlikely to be better and are probably less accurate than those for the comparatively simple and heavily-tested deep beam. Another shortcoming of current design provisions is that engineers are provided with little to no guidance for the design of less common or unique D-Regions. Due to the inadequacies in common practice, coupled with the unlimited variety of D-Region shapes and loading conditions, it is not surprising that most structural problems occur in D-Regions.

Figure 2   Shortcomings of Existing Provisions
(Click here to view a larger image)

1.3

The Strut-and-Tie Method (STM) for the Design of D-Regions

An emerging methodology for the design of all types of D-Regions is to envision and design an internal truss, consisting of concrete compressive struts and steel tension ties that are interconnected at nodes, to support the imposed loading through to the boundaries of the discontinuity region. This design methodology is called the Strut-and-Tie Method (STM) [4-9]. The design process involves the steps described below. In Figure 3, these steps are illustrated using a variety of D-Region designs examples including a corbel, a corner joint, a dapped-ended beam, and a deep beam.

(i) Define the boundaries of the D-Region and determine the imposed local and sectional forces.

(ii) Sketch the internal supporting truss, determine equivalent loadings, and solve for truss member forces.

(iii) Select reinforcing or prestressing steel to provide the necessary tie capacity and ensure that this reinforcement is properly anchored in the nodal zone (joint of the truss).

(iv) Evaluate the dimensions of the struts and nodes, such that the capacity of these components (struts and nodes) is sufficient to carry the design forces values.

(v) Provide distributed reinforcement to ensure ductile behavior of the D-Region.

Figure 3   Strut-and Tie Models and Steps in Design
(Click here to view a larger image)

The STM is based on the lower bound theory of plasticity. Therefore, the actual capacity of the structure is considered to be equal to or greater than that of the idealized truss. This suggests that if Truss A (Cut-Away Truss shown in Figure 4) can support a load of PA, then the capacity PB of Deep Beam B (equivalent to Truss A + three concrete fills) is at least equal to PA. This statement is almost true. In the “filled-in” structure, the forces may spread out along the length of the strut resulting the strut failing by splitting at a lower load than it would have failed by crushing at had the stress trajectories been parallel. Such effects can, however, be easily accounted for in provisions by reducing ultimate stress limit values.

Figure 4   Illustration of “Cut-Away” and “Filled-In” Truss
(Click here to view a larger image)

STM design provisions consist of rules for defining the maximum dimensions and ultimate stress limit capacities of struts and nodes, as well as reinforcement anchorage and distribution requirements. Existing and proposed code provisions differ substantially due to uncertainties in what these rules should be. This situation is created by a lack of sufficient and detailed experimental research. Guidelines [10-11] for design by the STM have been developed for European practice. A version of the STM was incorporated in the Canadian Concrete Design Code [12] in 1984 and in the AASHTO LRFD [13] code in 1994. Another specific set of provisions has been developed to include as an alternative design procedure in the 2002 ACI code. These provisions were submitted to the full ACI 318 committee as CE49 [14], and at the time of submission of this proposal, were under revision.

1.4

Complications and Barriers to Design by the STM

While the STM is a conceptually simple design tool, there are numerous uncertainties and complications that can encumber the five-step design procedure. A few of these are briefly described below:

Strut and Node Capacity: The ultimate stress at failure in struts and nodal zones is influenced by several factors including shape, state of strain/cracking, and the level of confinement. The influence of these factors is poorly understood and this leads to uncertainties in the design method. Additionally, designers are not able to take advantage of factors that they believe would increase capacity or improve behavior.

Geometry of Struts and Nodal Zones: It is unclear how to define the effective dimensions of struts and nodal zones. This is particularly difficult for configurations in which more than three members intersect. An example of such a complex strut-and-tie model is illustrated in Figure 5. Since the capacity of the struts and nodes are directly proportional to their effective widths, this creates uncertainties in the design process.

Figure 5   Radial Walls of Skydome, Toronto: Designed using the STM
(Click here to view a larger image)

Anchorage of Tie Reinforcement: In the cut-away truss, the transfer of forces between members and the anchorage of tension ties occurs entirely in the nodal zone. In the full structure (“filled-in” truss), this force transfer is more broadly distributed. There are uncertainties about anchorage requirements, the need to distribute reinforcement throughout the nodal region, and the factors that influence these requirements.

Truss Geometry and Dimensions: The initially selected geometry of the truss, including strut and nodal zone dimensions, must often be adjusted in order to satisfy stress limit criteria, to investigate other configurations, and to optimize the design. This can make hand-solutions prohibitively time consuming, particularly for the design of complex structures for which there is the need to consider multiple load cases.

Statically Indeterminate Trusses: The non-linear axial stiffness characteristics of struts and ties are poorly understood. Consequently, the designer has little guidance for determining the distributions of loads in statically indeterminate strut-and-tie (truss) models.

1.5

Objectives of the CAREER Plan

The objective of the proposed work is to help overcome the identified barriers through experimental and analytical research (Section 2), the development of the CAST education and design tool (Section 3), and through the creation of additional educational resources (Section 4). The team of collaborators and the previous experience of the PI are considered vital to accomplishing these objectives (Section 5).

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