Failures of Structures due to Inappropriate Assessment of Critical Force Paths

 Figure 1 Primary strut-and-tie model.
It is possible that a heavily loaded nib may require more than one load path to transfer a load safely. Strut-and-tie models can demonstrate alternative methods for reinforcing. However, it should be realized that the least direct paths will cause the most distortion and cracking, and should not be used for the serviceability state.

Figure 1 shows the most direct strut-and-tie (primary) model. The force paths are closest to that of an elastic model and will create the least internal distortion to achieve equilibrium. Figure 2 shows a secondary strut and tie model. This may be accompanied by distortion and cracking of the concrete before it can achieve equilibrium.

 Figure 2 Secondary strut-and-tie model.
If the forces on the nib are too great for the primary model, it is reasonable to superimpose the secondary model to provide sufficient resistance for the total ultimate loads. However, it is important to ensure that the primary model is sufficient to resist the serviceability loads and provide crack control.

Comment: This approach was used for a major viaduct. The combination of the two models (see Figure 3) enabled all the reinforcement to fit—just!

SHEAR WALL WITH HOLES AND CORNER SUPPORTS

A multi-storey shear wall required so many openings (windows, doors, etc.) that the load path became very complicated. The designer assumed that the load would flow to the corners and then track vertically down the edge of the wall (see Figure 4a). In fact, since the wall was built in situ as a homogeneous structure, strain compatibility caused the load to flow back into the full width of the wall. The result was that several storeys of load were supported by a deep beam that transferred the load to its end supports (see Figure 4b).

 Figure 3 Example of use of the two strut-and-tie models. (Courtesy of Gill Brazier.)
The limiting height of the natural arch of a deep beam (0.6 × span) was not considered (see Figure 4b) and this resulted in the omission in the design of much of the reinforcement needed for the bottom tie. Construction had reached several floors up by the time the mistake was recognised and this led to a redesign of the wall during construction and heavy remedial work. Each part of the wall required careful re-appraisal.
 Figure 4a Incorrect simple modelling | Figure 4b Correct simple modellingFigure 4 Multi-storey shear wall

This led to the requirement of much more reinforcement at each floor level. The bottom corner reinforcement details required special attention to ensure that the junction between the tie and compression struts was adequately designed.

Figure 5 shows in a simple diagrammatic form how the force paths automatically flow out and back again. The assumed force path down the edges would not require ties at top and bottom, but without these the actual force path would cause large cracks to open up from the top and bottom surfaces. Even after cracking, the angle struts would still exist and so would the consequential horizontal component. Without sufficient tie force to resist, the support joint would move outward and eventually failure would follow.

Comment: The consequence of missing this simple principle of deep beam behavior before construction reached such an advanced state meant that it required the redesign of the structure and reprogramming of construction which were extremely costly.

DESIGN OF BOOT NIBS

Where nibs are attached to the bottom of a beam it is important to understand the load path of the forces. Figure 5.6 shows a typical section of such a nib.
 Figure 5 Modelling deem beams
The conventional assumption for a short cantilever of dc and zc (shown in red in Figure 6) is unsafe for such a nib. The design compression zone for such a model would be close to the bottom face of the beam and likely to fall outside the beam reinforcement (both the links and main reinforcement). Strut-and-tie modelling is helpful to explain why this is so. The strut (shown in red in Figure 6) would just cause the cover to the reinforcement to spall off. The strut must be supported mechanically by the reinforcement of the supporting beam (shown in black in Figure 6). The effective lever arm becomes much smaller and the tension force in the nib top reinforcement much larger than assumed by the short cantilever approach.

It should also be noted that the force in the supporting links of the beam, Ft2d, is likely to be much greater than the applied load on the nib, FEd, to satisfy equilibrium. For the situation shown in Figure 6, it is conservative to assume the compression acts at the centroid of a triangular compression stress block. Hence the force in the link, in addition to any shear, may be calculated as follows:

 Figure 6 Nib attached to bottom of a beam
Comment: There are probably many nibs of this type that have been designed incorrectly and survive because of built-in safety factors and the fact that the load assumed in the design has not occurred. The error described in this case study was found in the design of a nib for a very prestigious project. It was very fortunate that it was discovered before construction started.