1.

The restraint (support) system is not sufficient

The solution to this problem is to provide a sufficient restraint system.

In general, a sufficient restraint system for a CAST's strut-and-tie (truss) model must consist of at least three non-concurrent supports on the D-Region Boundary. When more than three non-concurrent restraints are provided in a truss model, the model becomes externally statically indeterminate.

Figure 9.2(a) shows an example of a CAST's truss model in which the restraint (support) system consists of only two supports. The model is lack of restraint to prevent movement in horizontal direction. Figure 9.2(b) is one of many solutions to make restraint system is sufficient. As shown in the figure, a short STM Element with a horizontal support is added on the lower left STM Node so that the support system has a minimum three non-concurrent supports.

Figure 9.2(a)   Example of CAST's Strut-and-Tie Model Lacking Restraint in Horizontal Direction

Figure 9.2(b)   Revised CAST's Strut-and-Tie Model of Figure 9.2(a)

Figure 9.3 shows another strut-and-tie model example where there are three supports on the D-Region Boundaries, but they are concurrent. The support system of this truss is not sufficient because it is free to rotate about an axis that passes through the meeting point of the supports (Point A).

Figure 9.3   Example of a CAST's Strut-and-Tie Model with Concurrent Supports

2.

The number of STM Elements and/or the configuration of the truss system is not adequate

The solution to this problem is to provide a sufficient number of STM Elements in the truss system and make sure that the truss configuration is adequate for stability.

In general, a minimum number of STM Elements for a stable truss system is twice the number of STM Nodes that are located within the D-Region Boundaries plus the number of nodes that are located on the D-Region Boundaries minus the number of supports. If the number of STM Elements exceeds the minimum required, the truss system becomes internally statically indeterminate.

An example of CAST's strut-and-tie model that lacks STM Elements is shown in Figure 9.4(a). The truss has a total of nine STM Nodes, nine STM Elements, and three supports. Four STM Nodes are located within the D-Region Boundaries, and five nodes are located on the D-Region Boundaries. A stable truss needs at least 2(4) + 5 - 3 = 10 STM Elements, but we have only nine STM Elements. Therefore, we need to add one STM Element to make it stable. Figure 9.4(b) shows an example of how to properly place the required STM Element to fix the problem.

Figure 9.4(a)   Example of a CAST's Strut-and-Tie Model in Which the Number of Elements are Not Enough

Figure 9.4(b)   Revised CAST's Strut-and-Tie Model of Figure 9.4(a)

Figure 9.5(a) shows a CAST's strut-and-tie model which has enough number of STM Elements, but its configuration is not adequate; the top part of the truss system has an extra diagonal member whereas the middle part is lack of a diagonal member. Figure 9.5(b) shows how to fix the problem.

Figure 9.5(a)   Example of an Inadequate Truss Configuration

Figure 9.5(b)   Corrected Truss Configuration of Figure 9.5(a)

3.

The STM Elements are not properly connected to each other at STM Nodes

The solution to this problem is to simply check if STM Elements framing into STM Nodes are properly connected.

An example of this problem is shown in Figure 9.6(a). The STM Node shown in the figure appears to properly connect STM Elements E18 and E40. As clearly shown in Figure 9.6(b), the right end of STM Element E18 and the left end of STM Element E40 are not actually properly connected; there are two STM Nodes (N23 and N47), which are the ends of STM Element E18 and E40, respectively, occupy the same place. Further, the brief information shown in the same figure indicates that STM Node N23 has one element connectivity, which is not what it is expected. This problem can be fixed easily by gluing STM Nodes N23 and N47.

Figure 9.6(a)

Figure 9.6(b)