Curvature as a demand-capacity measure

By Dr. Graham H. Powell, Professor Emeritus, UC Berkeley

For strength design of beams the demand-capacity measure is bending moment. Hence, for deformation design it is natural to expect the demand-capacity measure to be curvature. Unfortunately, curvature is not a good choice because it is not possible to calculate reliable values for curvature demand. This note presents an example to illustrate the problem.

Example Beam

Consider the beam shown in Figure 1. This beam has a uniform cross section, no transverse load, and equal end rotations, as shown. Hence, the beam is loaded by equal and opposite end moments.

Figure 1 Example Beam

Assume that the actual moment-curvature relationship for the beam is known, as shown by the dashed line in Figure 2.

Figure 2 Moment-Curvature Relationships

Consider four different approximations for this relationship, as shown in the Figure. These are bilinear approximations with 4%, 2%, 1% and 0% strain hardening, respectively. For practical purposes they are all reasonable approximations of the actual relationship. 

For calculating demand-capacity ratios, let the curvature capacity be as shown in Figure 2. This is 10 times the yield curvature for the case with 0% strain hardening (the elastic-perfectly-plastic approximation). 

First consider the relationship between the end moment on the beam and the end rotation. Then consider the curvature demand and the curvature demand-capacity ratio. In both cases, consider the effects of the four different approximations. 

Moment-Rotation Relationship

For monotonically increasing end moments (i.e., no cyclic loading), it is easy to calculate the relationship between end moment and end rotation for the beam using "exact" beam theory (for example, the Moment-Area method). The result is shown in Figure 3. 

Figure 3 Moment-Rotation Relationships

The results show that all four moment-rotation relationships are similar. This indicates that the overall behavior of a beam, as represented by the relationship between end moment and end rotation, may not be very sensitive to approximations in the moment-curvature relationship. 

All four curves in Figure 3 pass through Point X. This is done deliberately, for the calculations in the next section. The four approximations to the moment-curvature relationship were chosen to make the moment-rotation relationship pass through this point. 

Demand-Capacity Ratio 

Consider the case where the beam is loaded to Point X in Figure 3, and consider all four moment-curvature relationships. At Point X the end rotation is 3 times the yield rotation for the e-p-p case, and the end moment is equal to the e-p-p moment. Hence, the analyses satisfy the following conditions:

1. The end rotations are the same in all cases. If the beam is part of a frame, this corresponds to similar story drifts.
2. The final bending moment diagrams are the same in all cases. If the beam is part of a frame, this corresponds to similar story shears.
3. Each bilinear moment-curvature relationship is a reasonable approximation of the actual relationship.

For each case, the maximum curvature in the beam can be calculated, using exact beam theory. This is the calculated curvature demand. The curvature capacity is known, corresponding to a curvature ductility ratio of 10 based on the e-p-p yield curvature. Hence, the curvature demand-capacity ratio can be calculated. If the maximum curvature is a useful deformation measure, the curvature demand-capacity ratios should be similar for all cases. If the ratios are substantially different, this indicates that the calculated demand is sensitive to the modeling assumptions (i.e., the strain hardening ratio), and hence that maximum curvature is not a useful deformation measure.

Table 1 summarizes the beam properties and loading, and shows the calculated curvature ductility ratios and demand-capacity ratios at point X for each of the four models. The calculated curvature variations along the beam length at point X are shown in Figure 4.

Table 1 Beam Behavior for Different Hardening Ratios: Theoretical Maximum Curvatures 

The maximum calculated curvature varies greatly as the strain hardening ratio is changed. Hence, the curvature demand-capacity ratio also varies greatly. For a curvature capacity equal to 10 times the e-p-p yield curvature, the demand-capacity ratio is smaller than 1.0 for two of the models, indicating that the beam design is OK at the chosen end rotation, and the ratio is greater than 1.0 for the other two, indicating that the beam is not OK. Hence, the maximum curvature is sensitive to the modeling assumptions, and is not a suitable demand-capacity measure for deformation-based design. 

If a finite element model is used, as the finite element mesh is progressively refined, the results get progressively closer to the exact results. If the maximum calculated curvature is used as the curvature demand, the D/C ratio is sensitive to the element mesh. Again, the maximum curvature is not a suitable demand-capacity measure.

Figure 4 Curvature Diagrams at Point X

It is useful to note that the curvature ductility ratios are much larger than the end rotation ductility ratio. The end rotation ductility ratio, based on the e-p-p yield rotation, is 3.0 in all cases, whereas the smallest corresponding curvature ductility ratio is 6.90, for the 4% hardening case. The curvature ductility ratio is larger because there is a concentration of plastic deformation near the ends of the beam. The curvature ductility ratio is equal to the rotation ductility ratio only if the plastic deformation is distributed proportionately over the full length of the beam.

Conclusions From This Example

For modeling inelastic beams, it is natural to expect that we can use nonlinear moment-curvature relationships, and use curvature as the deformation measure. However, there are two problems with this approach, as follows.

1. For a given moment-curvature relationship it is possible to calculate the moment-rotation relationship for a complete beam, and the calculation is not highly sensitive to approximations in the moment-curvature relationship. However, the calculation must account for spread of yield along the beam (i.e., for differences in effective stiffness at different points in the beam). This can require a complex model and be expensive computationally, especially for cyclic loading. 
2. For deformation based design it is not sufficient to calculate the moment-rotation relationship. For decision making it is necessary to calculate the curvature demand-capacity ratio. This example shows that the calculated curvature demand is sensitive to changes in the moment-curvature relationship, and hence that curvature is not a suitable demand-capacity measure.