Hello everyone. In the previous sequence, we have explored the concept of the pushover analysis, whose the principle is to compare the structural demand with the capacity. Especially we understood why to determine the capacity curves, nonlinearities should be taken into account either in a simplified or in a more complex way. Finally, we saw that the pushover methodology is non-linear, and is intrinsically static. Why? Remember that the capacity curves is computed from a static analysis. Depending on the complexity of the structure and their analysis, static analysis may not be sufficient. In this case, time history analysis have to be performed eventually by taking into account nonlinearities. In this sequence, I will present you the concept of a time history analysis and I will illustrate it on simple systems. We saw previously that the spatial discretization of the equations of the motion lead to this non-linear algebraic system. Note that the term related to viscous damping forces is assumed to be null. We can further simplify this equation by assuming that the behavior is linear or elastic. In this situation, we end up with this new equation. By comparing both equations, we can note that the initial term and the external loading are the same. The only difference is related to the term expressing the back forces highly related to the constitutive equations. It is important to keep in mind that the algebraic system is discrete in space but still continuous in time. Therefore, time integration schemes should be used in order to obtain an approximation of the displacement at each time. To apply time integration schemes, the time domain is split into a discrete set of small intervals. More precisely, if this arrow represents the full-time domain between T0 and TN, then the discrete times are introduced to define the small intervals. The length of the small intervals can be constant or variable and is called the timestep. For the sake of simplicity, we will assume that they are constant. The discrete system I have presented previously is solved on each small interval. To do this, time integration schemes are used. Time integration schemes lie in assuming a relationship between the displacement and the first derivative of the displacement that is to say the velocity. Time integration schemes are characterized by several key features. Among them we have the stability and the order. The scheme is stable if when the timestep is perturbed or modified, then the solution is not disproportionally perturbed. To illustrate the concept of stability, let us consider a displacement time solution obtained with the timestep of DELTA_T here in green. Now, if the timestep is modified, then new solution will be obtained here in red. If this highly perturbed, then the time integration scheme is termed to be unstable. Now, let us illustrate the concept of order for a given scheme. The error can be defined as the difference between the true solution here in green and its approximation here in red. When the timestep tends to zero, the error tends to DELTA_P + 1. The scheme will be termed of p order. In other words, the order traduces the velocity with which you see error tends to zero. This illustration allows us to understand why a high-order scheme is more precise than a low-order one. To illustrate the stability and the order of a scheme let us consider a very simple discrete system. A first-order homogeneous differential system. We also add initial conditions which allow to define the value of the displacement as the initial time. In the following, the initial displacement is assumed to be equal to one. To build the numerical solution of this equation, we will consider three first-order numerical schemes, known as the Euler schemes. Here we use the Euler forward integration scheme. The quantity at the timestep TN + 1 can be obtained from the quantities known under the timestep TN. As we can see on this video, the displacement amplitudes increases with respect to the time. The scheme is unstable. It is interesting to look at the solution obtained from the Euler backward integration scheme. With this scheme, the solution at time step TN + 1 is obtained from the values of the quantities at times TN and TN + 1. We note that the displacement amplitudes drastically decreases. The scheme is stable but dissipative. The last scheme is known as a trapezoidal rule. Here, the quantity at time step TN + 1 is computed from the quantity at time TN and TN + 1. In this case, displacement oscillations are unchanged with respect to the time. The scheme is stable and conservative. These properties can be checked analytically by studying the eigenvalues of the characteristic matrix of the set of differential equations. If the absolute values of the eigenvalues are lower than one, then the scheme is stable and dissipative. If they are higher than one, then the scheme is unstable, and if they are equal to one, then the scheme is stable and conservative. Now, since we understood the main principle of the first order time integration schemes, we will wonder what scheme is usually used in structural dynamics in the context of earthquake engineering. So, as you know, we have to solve a second-order system of differential equations. A widely used scheme is a well-known Newmark scheme, it is a second-order scheme. The displacement is approximated as a function of the velocity and of the acceleration. The velocity is approximated as a function of the acceleration. In both cases, we can note that these three parameters can be setup as a user. The time step here in red, and two numerical parameters, GAMMA and BETA in blue and green respectively. The values affected to these three parameters will have consequences and the stability of the Newmark scheme, as we will see later on. Considering this scheme, we can note similarities between displacement and the velocity approximations. In both cases, solutions at the previous time steps and first order terms appear. They are proportional to the time step. However, in case of the displacement, a second-order term is used to take into account the effect of acceleration on the displacement. To start using the scheme, initial values of the displacement and of the velocity should be known. These values are the initial conditions of the problem. As you saw, two numerical parameters are included and three values are chosen in order to ensure scheme is stable and conservative. If GAMMA is lower or equal to 0.5, then the scheme will be unstable. If GAMMA is higher or equal to 0.5 and BETA lower or equal to GAMMA over 2, then the scheme will be conditionally stable. This means that the stability will be a function of the time step. Last, if GAMMA is higher or equal to 0.5 and BETA higher or equal to GAMMA over 2, then the scheme will be unconditionally stable. This is the best situation because stability is ensured for any time step. As a consequence, large time steps can be chosen. Let us illustrate the case of GAMMA equal to 0.25. According to what we saw previously, the scheme should be unstable. We simulated the free vibration test with two time steps, 0.5 second and 0.1 seconds. As you see, a higher time step leads to perturbations on the solution. Let us illustrate the best situation, GAMMA equal to 0.5 and BETA equal to 0.25. The same case as the previous one has been simulated. As you see, the value of the time step does not have any influence on the solution. This is the unconditional stability. The Newmark scheme can be used to solve highly complex problems. To illustrate this, we took the example of the simulation of a nuclear power plant. The finite element mesh has not been plotted because it is rather fine. The displacement are amplified by a factor of 100. To sum up this sequence, we introduced the main concept of time integration schemes. We showed key properties such as stability and we illustrate it on a simple discrete system. We ended by introducing a well-known integration scheme, the Newmark scheme, and presenting their basic properties.
