**2. Numerical methods of direct integration of equation of motion and their application in seismic analysis**

This subchapter aims to compare the suitability of the explicit and implicit methods in the nonlinear earthquake analysis from the point of view of computational time and accuracy. We implemented these methods in RFEM [5] software and we used Ansys software [6] to compare and validate the results. In the following text let us introduce the numerical methods used in the study. The equation of motion of a discrete model of a structure subjected to a dynamic load can be written

$$\mathbf{M}\ddot{\mathbf{u}}(t) + \mathbf{C}\dot{\mathbf{u}}(t) + \mathbf{f}^{\text{int}}(t) = \mathbf{M}\mathbf{a}(t) + \mathbf{C}\mathbf{v}(t) + \mathbf{f}^{\text{int}}(t) = \mathbf{f}^{\text{ext}}(t) \tag{1}$$

where **M**,**C** are matrices of mass and damping respectively, **f** *ext* and **f** int are the vectors of the external and internal forces respectively and **u**, **v** ¼ **u**\_ , **a** ¼ **u**€ are displacement, velocity, and acceleration, respectively.

The numerical process of direct integration of the equation of motion (1) in a time interval *t*∈h i *t*0, *tm* is called transient analysis. Time is discretized to finite number of time instants *t*0,*t*1, … ,*tm*. The distance between individual time instants Δ*ti* ¼ *ti* � *ti*�<sup>1</sup> is called the time step. The lengths of time steps Δ*ti* influences the accuracy, stability and speed of the numerical solution. At the time *t* ¼ 0 an initial conditions **u**ð Þ¼ *t*<sup>0</sup> **u**0, **v**ð Þ¼ *t*<sup>0</sup> **v**<sup>0</sup> have to be defined. Eq. (1) then can be read as follows:

$$\mathbf{M}\ddot{\mathbf{u}}\_{i} + \mathbf{C}\dot{\mathbf{u}}\_{i} + \mathbf{f}\_{i}^{\text{int}} = \mathbf{M}\mathbf{a}\_{i} + \mathbf{C}\mathbf{v}\_{i} + \mathbf{f}\_{i}^{\text{int}} = \mathbf{f}\_{i}^{\text{ext}} \tag{2}$$

#### **2.1 Explicit method: Central differences**

In explicit methods, we introduce explicit assumption about the course of displacement in interval *ti* h i , *ti*þ<sup>1</sup> and the knowledge of the displacement vector **u***<sup>i</sup>* at time instant *ti*, and we calculate vectors **u***<sup>i</sup>*þ1, **v***<sup>i</sup>*þ1, **a***<sup>i</sup>*þ<sup>1</sup> from (2).

The numerical integration of differential equations uses the substitution of the derivative of the displacement with respect to time. in (2) by

$$\mathbf{v}\_{i} = \frac{1}{2\Delta t\_{i}} (\mathbf{u}\_{i+1} - \mathbf{u}\_{i-1}) \tag{3}$$

$$\mathbf{a}\_{i} = \frac{1}{\Delta t\_{i}^{2}} (\mathbf{u}\_{i+1} - 2\mathbf{u}\_{i} + \mathbf{u}\_{i-1}) \tag{4}$$

we obtain a recurrent formula for **u***<sup>i</sup>*

$$\left(\frac{1}{\Delta t\_i^2} \mathbf{M} + \frac{1}{2\Delta t\_i} \mathbf{C}\right) \mathbf{u}\_{i+1} = \mathbf{f}\_i^{\text{ext}} - \mathbf{f}\_i^{\text{int}} + \frac{2}{\Delta t\_i^2} \mathbf{M} \mathbf{u}\_i - \left(\frac{1}{\Delta t\_i^2} \mathbf{M} - \frac{1}{2\Delta t\_i} \mathbf{C}\right) \mathbf{u}\_{i-1} \tag{5}$$

The method has all the advantages of explicit methods as long as ½ �¼ **C** ½ � **0** or ½ �¼ **C** *α*½ � **M** , where *α* is the mass coefficient of the Rayleigh damping. The most effectivity is achieved when the mass matrix **M** is diagonal. The explicit methods are conditionally stable. The time steps must satisfy the following condition:

$$
\Delta t\_i \le \frac{T\_n}{\pi} \tag{6}
$$

where *Tn* is the smallest vibration period.

### **2.2 Implicit method: Newmark**

Implicit methods in transient dynamics are based on the equation of motion (2) at the time instant *ti*. The numerical solution of the system is performed step by step by the following text.

$$\mathbf{a}\_{i} = f(t\_{i}, \ \mathbf{f}\_{i}, \ \mathbf{u}\_{i-1}, \ \mathbf{v}\_{i-1}, \ \mathbf{a}\_{i-1}) \tag{7}$$

with the necessity to evaluate the acceleration at the start of the motion at time *t*<sup>0</sup> from the equation of motion (2)

$$\mathbf{M}\mathbf{a}\_0 + \mathbf{C}\mathbf{v}\_0 + \mathbf{K}\mathbf{u}\_0 = \mathbf{f}\_0 \tag{8}$$

The basic formula of the Newmark method that specifies the relations between displacement, velocity, and acceleration vectors have the following form:

$$\mathbf{u}\_{i} = \mathbf{u}\_{i-1} + \Delta t\_{i} \mathbf{v}\_{i-1} + \left(\frac{\mathbf{1}}{2} - \beta\right) \Delta t\_{i}^{2} \mathbf{a}\_{i-1} + \beta \Delta t\_{i}^{2} \mathbf{a}\_{i} \tag{9}$$

$$\mathbf{v}\_{i} = \mathbf{v}\_{i-1} + (\mathbf{1} - \boldsymbol{\chi})\Delta t\_{i}\mathbf{a}\_{i-1} + \boldsymbol{\chi}\Delta t\_{i}\mathbf{a}\_{i} \tag{10}$$

where *β* and *γ* are what is termed Newmark's parameters. As formulas.

**u***<sup>i</sup>* ¼ **u***<sup>i</sup>*�<sup>1</sup> þ Δ**u***i*, **v***<sup>i</sup>* ¼ **v***<sup>i</sup>*�<sup>1</sup> þ Δ**v***i*, **a***<sup>i</sup>* ¼ **a***<sup>i</sup>*�<sup>1</sup> þ Δ**a***<sup>i</sup>* (11)

holds we can write the following formula for the vector of acceleration increments and velocity increments:

$$\begin{aligned} \Delta \mathbf{a}\_{i} &= \Delta \overline{\mathbf{a}}\_{i} + \frac{1}{\beta \hbar} \Delta \mathbf{u}\_{i}, \Delta \mathbf{v}\_{i} = \Delta \overline{\mathbf{v}}\_{i} + \frac{\underline{r}}{\beta \hbar} \Delta \mathbf{u}\_{i} \text{ (12)} \text{where} \\ \Delta \overline{\mathbf{a}}\_{i} &= \left( -\frac{1}{\beta \Delta i} \mathbf{v}\_{i-1} - \frac{1}{2\beta} \mathbf{a}\_{i-1} \right), \Delta \overline{\mathbf{v}}\_{i} = \left( \mathbf{1} - \frac{\underline{r}}{2\beta} \right) \Delta t\_{i} \mathbf{a}\_{i-1} - \frac{\underline{r}}{\beta} \mathbf{v}\_{i-1} \text{ (13)} \\ \text{The total increments of the vector of displacement can be written as:} \end{aligned}$$

$$
\Delta \mathbf{u}\_i = \sum\_{k=1}^{n\_{ITER}} {}^k \Delta \Delta \mathbf{u}\_i \tag{11}
$$

By substitution according to formulas above into the equation of motion (2) and modifying the obtained relation, we get:

$$\left(\frac{1}{\beta \Delta t\_i^2} \mathbf{M} + \frac{\gamma}{\beta \Delta t\_i} \mathbf{C} + {}^k \mathbf{K}\_{\Gamma, i}\right) \cdot {}^k \Delta \mathbf{u}\_i = {}^k \Delta \mathbf{f}\_i - \mathbf{M} \Delta \overline{\mathbf{a}}\_i - \mathbf{C} \Delta \overline{\mathbf{v}}\_i \tag{12}$$

where *<sup>k</sup>***K**T,*<sup>i</sup>* is the tangent stiffness matrix for the kth iteration of the ith time step.

The bracket on the left-hand side of (15) represents what is termed a modified stiffness matrix, which can be denoted **K**^ *<sup>i</sup>*. The presented formula can be written in a similar form as:

$$\begin{aligned} \mathbf{^k \Delta \Delta \mathbf{u}\_i} : \begin{cases} \hat{\mathbf{K}}\_i^k \Delta \Delta \mathbf{u}\_i = {}^k \Delta \Delta \mathbf{f}\_i - \mathbf{M} \Delta \overline{\mathbf{a}}\_i - \mathbf{C} \Delta \overline{\mathbf{v}}\_i \ \text{; for } k = 1 \\\ \hat{\mathbf{K}}\_i^k \Delta \Delta \mathbf{u}\_i = {}^k \Delta \Delta \mathbf{f}\_i \ \text{; for } k > 1 \end{cases} \end{aligned} \tag{13}$$

Eq. (16) enable us to calculate partial increments of displacement.

### **2.3 Comparison of the explicit and implicit method in seismic analysis**

For this numerical study, a wall of six-floor building was used. The structure was subjected to seismic load due to the accelerogram from Umbro-Marchigiana, Italy.

This sub-chapter aims to decide what numerical method of the transient analysis of buildings exposed to earthquake is most suitable from the point of view of accuracy and processing time. The earthquake should be strong enough to cause a highly nonlinear response of the analyzed building so the modal analysis cannot be used and the usage of a numerical method for direct time integration of the equation of motion is justified.

It is widely known that the explicit method (see [7–9]) is proper for analyzing processes with very short duration, such as explosions or the collisions of vehicles, i.e., processes generally studied via transient dynamic analysis.

Because the explicit method is conditionally stable, the time step must fulfill the inequality (6). Only in this case is it possible to use the diagonal form of matrices **M**,**C**,**K** to enable conditions for maximal performance. Small time steps imposed by the relation (6) are not a substantial disadvantage in the case of earthquake analysis as the earthquake accelerograms also require very short time steps.

Implicit methods of solving a set of differential equations (see [7, 10–12]) are characterized by the fact that a system of linear equations must be solved at each time step. These methods do not require such a short time step as is needed in an explicit method. Therefore, the implicit methods are mostly used for the solution of dynamical analyses of a duration longer than several seconds.

To choose what numerical method is the most suitable for transient seismic analysis of a typical building structure (see **Figure 1**), a real accelerogram from Italy (see **Figure 2**) was used along with the Drucker-Prager material model. The accelerogram of quite strong earthquake was chosen to be strong enough to cause huge nonlinear behavior with damage and plastic yielding, where spectral analysis cannot be applied and a transient analysis using the direct integration of the equation of motion has to be used. The comparison of the explicit and implicit methods method was focused on accuracy and computational performance. A time step of 0.0001 s was used for the explicit method due to stability requirements. The same time step was also chosen for the implicit method to compare the time of processing of both methods. Another reason was to obtain a very accurate reference solution for accuracy comparisons.

The following graphs (**Figures 3**–**6**) show the time course of the horizontal displacement of the upper right corner of the building. The graphs show very good concordance between the explicit and implicit method. It can be also seen that RFEM and ANSYS programs provide practically identical results.

The calculation performed by the implicit method for the same time step was approximately five times slower than the explicit method. Increasing the time step for the implicit method showed that this method is accurate enough until the time step was twenty times greater than that for the explicit method.

The implicit method provides good results for a computational time four times lower than that of the explicit method. But because of the difficulty to estimate the highest acceptable time step from the point of view of accuracy, it can be concluded that explicit and implicit methods are comparable for seismic analysis, but the implicit Newmark's method is slightly preferred.

An interesting partial observation of this study is the finding that both basic numerical methods for the transient analysis, namely the explicit method and Newmark's implicit method, are competitive for seismic analyses.

*On the Nonlinear Transient Analysis of Structures DOI: http://dx.doi.org/10.5772/intechopen.108446*

#### **Figure 1.**

*The analyzed concrete wall with 21 GPa of initial elasticity modulus, elements size of 1 meter, thickness of 0.2 meters [2].*


#### **Figure 2.**

*Accelerograms Umbro-Marchigiana, station Colfiorita-Casermette [2].*

Results of the numerical analysis showed excellent concordance between the results of the ANSYS and RFEM programs for the implicit method. The comparison in explicit methods showed a bit worse concordance between the above-mentioned programs. The RFEM program gives more accurate results than ANSYS.

**Figure 3.**

*Explicit method—The time course of the horizontal displacement of the upper right corner for the time step 0.0001 s [2].*

**Figure 4.** *Implicit method—The time course of the horizontal displacement of the upper right corner for the time step 0.0001 s [2].*
