Hello everyone. In the previous sequence, you already studied how nonlinearities can affect the response of a structure. More particularly, we have showed how nonlinearities could be described by means of a simple models. In this sequence, I represent the main concepts of a well-known approach in earthquake engineering, the Pushover Analysis. It is based on the comparison between the structural demand and the structural offer. Let us start by introducing what the structural demand is. We shall consider one degree of freedom oscillator, characterized by a fundamental frequency F a viscous damping ratio, XI0, and a mass M. We will assume it is subjected to an acceleration time history, W2UG in blue, and that is displacement response, U, is computed at the top here, represented in red. Given the displacement time evolution, the corresponding spectra is SD that is the maximum displacement of each single degree of freedom of natural frequency F can be computed straightforwardly. From the displacement response spectra we can deduce a pseudo acceleration response spectra SPA. To do this, it is sufficient to multiply the spectral displacement by the square of the frequency OMEGA. The way to represent a structural response is commonly used by design engineers because of pseudo acceleration is proportional to a force. Indeed, to estimate the maximum resisting force or the reaction at the basement, we just have to multiply the pseudo acceleration by a mass. Now, as you already understood, we have the two ingredients to represent the structural demand. For a given frequency we represent the couple pseudo displacement pseudo acceleration
on the graph. It is the acceleration displacement response spectrum, also known as the ADRS format. An important property is the fact that in this plane, the ISO period curves t, are displaced as straight line starting from the origin. This property originates from the fact that both ingredients are linked to each other by the square of the frequency. We will see in the following how this property can be very useful. Now, we will expose how to obtain the structural offer. Always keep in mind that the pushover methodology lies in comparing it with the structural demand. In the pushover approach, the non-linear behavior of a structure is investigated for a monotonic quasi-static loading. This allows to assess the structural behavior up to failure. To account for nonlinearities, two way exist. The first one lies in using non-linear constitutive laws, and the second one lies in modifying the elastic response. The first way is usually time-consuming. That is why engineers often prefer the second one. More precisely, non-linear constitutive laws allow to obtain a fully non-linear structural response. When the loading is low, the force displacement relationship stays elastic
linear, damage does not appear. As the displacement d increases, the reaction force f, tends to an asymptote and also nonlinearities grow. The structural offer, or more classically called the structural capacity, can be represented by this curve. As I have mentioned before, a second way to describe the capacity curve is to modify the elastic response of the structure. The main advantage of this approach is that it will save much of computational time. Using an elastic linear law, the reaction force is proportional to the displacement FE and UE are the maximum elastic reaction force and the maximum displacement. However, because the structure should exhibit nonlinearities when the elastic domain is overcome, the threshold that separates the linear and non-linear domains and characterized by FY and UY exists. In a consistent way with the capacity curve obtained by implementing non-linear constitutive laws, When the threshold is overcome, the reaction force is assumed to be constant up to a maximum displacement UM. Now it is necessary to define the link between the black curve and the red one. To do so, two quantities are introduced, the behavioral coefficient Q, defined as the ratio between the elastic force, FE, and the threshold force, FY, and the ductility MU equal to the ratio between the maximum displacement, UM, and the threshold displacement, UY. Knowing these coefficients allows to switch from the elastic response to a simplified non-liner one. In order to derive the red capacity curve from the black one, it is sufficient to modify the seismic demand, the behavior coefficient is used to divide the initial seismic demand. This coefficient is always higher or equal to one by convention, then the modified seismic demand is reduced. In addition, the spectral displacement is multiplied by MU to account for the fact ductility is activated. Why? Because remember, the behavior is linear elastic. In order to allow the comparison between the demand expressed in the acceleration displacement diagram and the capacity, it is necessary to express the capacity curves in the same format. The reaction forces are converted in pseudo accelerations by dividing them using the coefficient Gamma f, in green, having the same dimension, as a mass. Regarding the displacements, they are converted into an equivalent displacement through the coefficient Gamma u, in red, that does not have any unit. The coefficients I have just mentioned can be defined in several ways and I will not go into details here. The final step of the pushover methodology is to compare the structural demand with the capacity in the acceleration displacement response spectrum space. Let us assume that we have already determined the initial demand and the capacity according what we have seen from the beginning. Both curves intersect each other at a specific point called functioning point. Depending on the nonlinearity level for which a functioning point appears, the structural demand should be adjusted. An updated demand is determined, leading to a new functioning point. Several methodologies exist in the literature to carry out this updating. I will not go into details in the sequence. The knowledge of this point is very useful because it allows to quantify how far ductility has to contribute to withstand the seismic forces. To sum up this sequence, we introduced the pushover methodology. We showed that it lies in comparing the structural demand with the capacity curve in the pseudo acceleration, pseudo displacement plane. We ended by illustrating the functioning point which is a key concept in the pushover analysis in order to assess if a structure, can withstand seismic forces.