Strength and Ductility of Frame

Consideration of the behavior of reinforced concrete frames at and near the ultimate load is necessary to determine the possible distributions of bending moment, shear force, and axial force that could be used in design. It is possible to use a distribution of moments and forces different from that given by linear elastic structural analysis if the critical sections have sufficient ductility to allow redistribution of actions to occur as the ultimate load is approached. Also, in countries that experience earthquakes, a further important design aspect is the ductility of the structure when subjected to seismic-type loading, since present seismic design philosophy relies on energy dissipation by inelastic deformations in the event of major earthquakes.

Both these aspects of behavior at ultimate load depend on the deformation characteristics of the members, which for frames depend mainly on the relationship between moment and curvature. Figure 11.1 gives a typical moment-curvature curve for a section in which the tension steel is at the yield strength at the ultimate moment. The curve is marked to indicate points at which the concrete starts to crack, the tension steel begins to yield, and spalling and crushing of the concrete commences. A ductile section is capable of maintaining moment capacity at near the ultimate value for large curvatures beyond the curvature at first yield.

1. Moment redistribution and plastic hinge rotation

It is evident that the nonlinear nature of the moment-curvature relationship for reinforced concrete sections will cause some adjustment to the relative values of the bending moments if the structure is loaded into and beyond the service load range. In particular, because of plastic rotations at some sections, it is possible for the bending moments to assume a pattern different from that derived from linear elastic structural analysis, and for all the critical positive and negative moment sections to reach their ultimate moments of resistance at the ultimate load. Thus moment redistribution can have a marked influence on the ultimate load of a statically indeterminate structure.

Therefore, if sufficient rotation capacity of the plastic hinges is available, the bending moment distribution at the ultimate load may be quite different from that calculated using elastic theory and will depend on the ultimate moments of resistance of the sections. In reinforced concrete structures, the ductility at the first plastic hinges to form may be insufficient to enable full redistribution of moments to take place with the ultimate moment at each critical section. Thus if moment redistribution is to be relied on, the availability of sufficient ductility at the plastic hinges must be ensured.

2. Calculation assumptions

All sections were assumed to have the same constant flexural rigidity EI up to the ultimate moment. This assumption is only accurate at low loads before cracking of the concrete commences. When the beam cracks, the flexural rigidity reduces in the cracked regions and the variation of flexural rigidity along the member causes the distribution of bending moments to change from that calculated by elastic theory using a constant flexural rigidity. With further loading the extent of cracking increases and the distribution of flexural rigidity, hence bending moment, will be again modified. This effect is particularly noticeable when members contain different amounts of negative and positive moment steel, it is even more noticeable in T beams because cracking of the flange in the negative moment region reduces the flexural rigidity there much more than cracking of the web in the positive moment region. This variation of the flexural rigidity along the beam will affect the amount of plastic rotation required for full moment redistribution at the ultimate load. Strictly, the effect of cracking on the flexural rigidity EI of the sections needs to be taken into account in the determination of the plastic hinge rotation at the ultimate load.

The moment-curvature relationship chosen was assumed to have a horizontal branch beyond yield, with the moment remaining constant at the ultimate value. This assumption only approximates the actual moment curvature relationship after first yielding, for this curve has an ascending portion to the ultimate moment after first yield of the tension steel. Therefore, both the negative and positive moment critical sections cannot develop the ultimate moments simultaneously because the curvatures at those sections will be at different points on the moment curvature curves. It is evident that the assumption that the ultimate moment exists at all critical sections simultaneously will give a nonconservative value for the ultimate load. If, for example, the moment capacity at first yield is My =0.9Mu ,where Mu is the ultimate moment, the error in ultimate load calculated, may be about 5% Clearly, if the attainment of yield(My)at the last hinge to form is taken as the ultimate moment, and if My is significantly less than the ultimate moment Mu, the error in calculating the ultimate load may be significant.

It is difficult, as we have seen, to calculate accurately the required plastic hings rotation in reinforced concrete frames for full moment redistribution and the ultimate load. However, if moment redistribution is to be relied on in design, we need assurance that the ductility available at the critical sections is in excess of the ductility demand calculated from theoretical considerations such as those just discussed.

3. Complete analysis of frames

The bending moments, shear and axial forces, and deflections of reinforced concrete frames at any stage of loading from zero to ultimate load can be determined analytically using the conditions of static equilibrium and geometric compatibility, if the moment-curvature relationships of the sections are known. However, difficulties are caused by the nonlinearity of the moment-curvature relationships, and a step-by-step procedure, with the load increased increment by increment, is generally necessary. Also, the moment-curvature relationship of sections carrying moment and axial force is dependent not only on the section geometry and the material properties but also on the level of axial force. This interdependence means that the moment-curvature relationship for each section must be recomputed at each increment of loading. A successive linear approximation method based on the stiffness method of analysis can be used to follow through the behavior of the frame from zero to ultimate load. In this method the members of the frame are divided along their length into small elements. At each load level the flexural rigidity(EI=M/φ),corresponding to the particular bending moment and axial force at each element, is obtained from the appropriate point on the moment-curvature relationship. Members are assumed to be uncracked for the initial load increments, and the deformations are determined using the uncracked section flexural rigidity. The elements are searched at each load increment to ascertain whether the cracking moment has been reached. When it is found that the cracking moment has been reached, the flexural rigidity of the element is recomputed on the basis of the cracked section, and the actions in the frame are recomputed. This procedure is repeated at the load level until all flexural rigidities are correct. At higher loads, when the stresses at the elements enter the inelastic range, the flexural rigidity of each element is adjusted to that corresponding to the appropriate point of the moment-curvature curve calculated for that moment and axial force level. Eventually, with further increments, plastic hinges spread throughout the frame, and the ultimate load is reached when a mechanism forms and no further load can be carried.