**3. Plasticity and damage response of structure**

The aim of the article is to compare responses of a building structure for two different nonlinear material models. A wall of a six-story building (see **Figure 1**) *On the Nonlinear Transient Analysis of Structures DOI: http://dx.doi.org/10.5772/intechopen.108446*

**Figure 5.**

*Implicit method—The time course of the horizontal displacement of the upper right corner for the time step 0.002 s [2].*

**Figure 6.**

*Implicit method—The time course of the horizontal displacement of the upper right corner for the time step 0.005 s [2].*

exposed to seismic load according to an accelerogram from Umbro-Marchigiana, Italy, was used to compare the two material models.

Three different material models were used: the linear elastic model, the Drucker-Prager plasticity model, and the Mazars continuous damage model [4]. Geometric nonlinearity was used for all the calculations but its influence is fairly small due to the relatively small displacement and rotation values involved.

#### **3.1 Elasto-plastic material model**

The Drucker-Prager model with isotropic hardening was executed to include different behavior of material in tension and compression which is necessary for concrete. The yield surface has the following form [3]:

$$\Phi(\mathfrak{o}, \overline{\mathfrak{e}}^p) = \sqrt{f\_2(\mathfrak{o})} + c\_1 \frac{1}{3} I\_1(\mathfrak{o}) - c\_2 \text{coh}(\overline{\mathfrak{e}}^p) \tag{14}$$

where *J*<sup>2</sup> is the second deviatoric stress invariant, *I*<sup>1</sup> is the first stress invariant, *coh* is the cohesion that is dependent on accumulated plastic strain *εp*, and the relevant coefficients *c*1, *c*<sup>2</sup> are chosen according to the required approximation to the Mohr-Coulomb criterion or can be calculated using the stress–strain diagram for uniaxial stress state and determining two conditions for the relevant yield points in tension and compression.

A standard procedure for elastic prediction and plastic correction with an implicit algorithm was used for the analysis of this elasto-plastic model. After the iterative calculation (general return-mapping update formula) of the plastic multiplicator Δ*γ*, the resultant stress tensor can be calculated [3]:

$$
\sigma\_{n+1} = \sigma\_{n+1}^{trial} - \Delta \gamma \mathbf{D}^{\epsilon} : \mathbf{N}\_{n+1} \tag{15}
$$

In this case, the updated stress, **σ***<sup>n</sup>*þ1, obtained by the implicit return mapping is the projection of the trial stress **σ***trial <sup>n</sup>*þ<sup>1</sup> onto the updated yield surface along the direction of the tensor **<sup>D</sup>***<sup>e</sup>* : **<sup>N</sup>***<sup>n</sup>*þ1. Note that, since the definition of the flow vector in the smooth portion of the cone differs from that at the apex singularity, two possible explicit forms exist for the return-mapping algorithm.

#### **3.2 Elasto-damage material model**

To account for the different nonlinear performances of concrete under tension and compression, and to explicitly reproduce the dissimilar effects of the tensile and shear damage mechanisms, a decomposition of the effective stress tensor into positive and negative components with the fourth-order projection tensors **P***<sup>t</sup>* and **P***<sup>c</sup>* . The Mazars isotropic damage model is modified to consider the various tensile and compressive behaviors of concrete in a more effective manner. The calculation of the resultant stress tensor is carried out with the aid of linear elastic stress estimates **σ***trial* and the damage parameters *dt* and *d<sup>c</sup>* in the following way [4]:

$$\begin{split} \boldsymbol{\sigma} &= \left(\mathbf{1} - d^t\right) \boldsymbol{\sigma}^t + \left(\mathbf{1} - d^t\right) \boldsymbol{\sigma}^t = \left(\mathbf{1} - d^t\right) \mathbf{P}^t : \boldsymbol{\sigma}^{trial} + \left(\mathbf{1} - d^t\right) \mathbf{P}^t : \boldsymbol{\sigma}^{trial} \\ &= \left[\left(\mathbf{1} - d^t\right) \mathbf{P}^t + \left(\mathbf{1} - d^t\right) \mathbf{P}^t\right] : \boldsymbol{\sigma}^{trial} \end{split} \tag{16}$$

where **<sup>σ</sup>***trial* <sup>¼</sup> **<sup>D</sup><sup>e</sup>** : **<sup>ε</sup>**.

Damage parameters are derived from the stress–strain diagram where equivalent strains calculated according to [4] are used:

$$
\varepsilon\_t = \frac{I\_\varepsilon}{2(1 - 2\nu)} + \frac{\sqrt{f\_\varepsilon}}{2(1 + \nu)},
\tag{17}
$$

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

$$\varepsilon\_{\varepsilon} = \frac{I\_{\varepsilon}}{5(1 - 2\nu)} + \frac{6\sqrt{f\_{\varepsilon}}}{5(1 + \nu)}\tag{18}$$

### **3.3 Comparison of the material models**

The Drucker-Prager plasticity model parameters were selected so as to ensure that there is an adequate level of plasticization in the most exposed areas of the structure for the given accelerogram. The aim of the investigation is not to execute the calculations required for a real building, but merely to compare the suitability for use of various numerical methods, as well as the behavior of various material models. While the earthquake was taking place, plasticization occurred in some parts of the structure. This demonstrated itself through increased damping and a change in the distribution of tension with regard to the linear material. It can be seen from the results that at a time of 10 s (i.e., practically straight after seismicity ended) the shape of the building had undergone permanent deformation. This was in accordance with expectations. There was an overall increase in the height of the building, as well as in its width in the plasticized area. On higher floors, where plasticization did not occur, the width of the building remained the same. The increase in height and the partial expansion in terms of width might seem surprising at first sight, but these phenomena can be explained by the fact that only plasticization in tension occurred, and that the rocking of the building from one side to the other caused plasticization to take place on both sides of the structure. This increased the height of the whole building, and not only in its tilting. It also became wider for similar reasons.

The Mazars damage model expects the structure to fail due to micro cracks. The cracks are not localized but it is expected that they occur continuously (smeared cracking model). According to the Mazars model, it can be concluded from the stress– strain curve for the material that micro-cracks close after stress disappears, and that deformation also disappears in a similar way, as took place in the case of the linear elastic model. However, failure did occur. It demonstrated itself via the fact that the stress–strain curve followed a different trajectory during unloading and thus energy dissipation occurred, which resulted in a damping in oscillation. After the end of seismicity, no permanent deformation or strain remained, unlike in the case of plastic material, but there was a permanent decrease in the stiffness of the material and a change in the response of the structure to loading as a result (in this case, the loading was from continuing seismic effects). The decrease in stiffness will have an impact on dynamics in terms of a decrease in natural oscillation frequencies. The following images show the differences in the response of the structure to the same seismic loads for various material models. The corresponding results of both material models have been placed next to each other, e.g., resultant deformations and strain, with the same being shown for the time of maximum displacement (see **Figure 7**).

The main aim of the study was to compare the response to seismic loading of various material models, specifically the Drucker-Prager plasticity model and Mazars damage model. At first sight, the results appeared somewhat surprising but on closer investigation, it became clear that they are correct, and that they correspond to the relevant material models. In the case of Drucker-Prager, the building remained slightly tilted after the end of the earthquake (see **Figure 8**, which was expected) and was also higher and wider as a whole (which was surprising at first). This somewhat remarkable result was obtained because plasticization took place only in tension and, because of the oscillation of the building from one side to the other, plasticization due

**Figure 7.** *The resulting shapes of the building a) Drucker-Prager plasticity model and b) the Mazars damage model.*

#### **Figure 8.**

*Drucker-Prager plasticity model—Time diagram of the horizontal displacement of the highest and the lowest nodes.*

to tension gradually occurred on both sides of the building. The result was an overall increase in the height of the building. For similar reasons, the building was also wider after the earthquake. In the case of the Mazars model, the geometry of the building remained the same after the end of the earthquake even though material failure occurred. However, the building was damaged. The micro-cracks did close after loading, though, and the deformation and stress from seismicity remained zero. The damage only demonstrated itself through lowered stiffness of the material in the damaged areas, which is also demonstrated in **Figure 9** by the decreasing frequency. The energy loss in the form of warmth appeared as increased damping in the numerical solution.

**Figure 9.** *Mazars damage model—Time diagram of the horizontal displacement of the highest and the lowest nodes.*
