# Effect of Material Variability and Mechanical Eccentricity on the Seismic Vulnerability Assessment of Reinforced Concrete Buildings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Centre of Strength and Centre of Stiffness

#### 2.1. Generalities

_{m}. Conventional analyses on existing buildings generally consider uniform materials by assuming an average value obtained from an experimental test. Taking into account the material variability, even just for columns, produces a sensitive variation in the response of the model both in nonlinear and linear analyses. In this paragraph, the eccentricities of stiffness and of strength, deriving from the material eccentricity are illustrated.

#### 2.2. Center of Stiffness and Eccentricity of Stiffness

- ${x}_{i},{y}_{i}$ coordinates of the i-th column;
- ${K}_{xi}\left(E\right),{K}_{yi}\left(E\right)$ stiffness of the i-th column in the coordinate directions, the function of the Young modulus E. Because of this definition, the center of the stiffness position is affected by material variability through E values. The elastic modulus is related to the concrete compressive strength with the following expression [20]:$${E}_{c}=22,000{\left(\frac{{f}_{c}}{10}\right)}^{0.3}$$

- ${x}_{c},{y}_{c}$ coordinates of centroid of the j-th floor;
- ${L}_{x},{L}_{y}$ building length in x and y directions.

#### 2.3. Centre of Strength and Eccentricity of Strength

- The strength of the concrete;
- The strength of steel bars;
- The position of the columns in the building;
- The geometry (length, restraints, and inertial characteristics) of the columns.

- ${N}_{u,i}$ compressive axial resistance of the i-th columns;
- ${x}_{i},{y}_{i}$ coordinates of the i-th columns;
- $n$ number of columns at j-th level.

- ${M}_{u,i}$ failure bending moment of the R.C. section of the i-th column.
- The corresponding strength eccentricity is:$${e}_{Rx,j}=\frac{{x}_{R,j}-{x}_{c,j}}{{L}_{x}}$$$${e}_{Ry,j}=\frac{{y}_{R,j}-{y}_{c,j}}{{L}_{y}}$$

- ${x}_{Rj},{y}_{Rj}$ coordinates of the center of strength, computed with Method 1 or Method 2;
- ${x}_{cj},{y}_{cj}$ coordinate of the j-th column;
- ${L}_{x},{L}_{y}$ building length in x and y directions.

## 3. A Benchmark Case-Study

#### 3.1. Description of the Benchmark Building

^{2}and n. 4 steel bars with a diameter of 20 mm for each one. The beams have a cross section of 40 × 50 cm

^{2}. The following assumptions are made to highlight the effect of material variability:

- Uniform material distribution (u.m.d.);
- Non-uniform material distribution (n.u.m.d.);
- Flexible slab, simulated by assuming equivalent bracing of axial stiffness ${K}_{B}=3000\mathrm{kN}/\mathrm{cm}$;
- Rigid slab, obtained through kinematic restraints at the diaphragm level.

#### 3.2. Calculation of the Risk Index

_{R}than the non-linear one, as also shown in the benchmark model. It is also relevant to point out that irregular material distribution in the plan also produces torsional effects also for a symmetric building such as the benchmark building; that occurs in both analyses, due to strength eccentricity. Moreover, it is important to observe that I

_{R}and the ultimate shear strength of the columns are not influenced by the local variation of strength in LA or NLA as suggested by Eurocode 8 [20]. It is a consequence of their dependence only on the average mechanical properties of the columns. In other terms, material eccentricity involves torsional modes [27,30] as well as a geometric eccentricity but this aspect is not discussed in the current Standards [20]. The multimodal and adaptive pushover appear the most suitable methods to consider torsional effects here induced [31,32]. A comparison of the traditional model with uniform (u.m.d.) and the model with non-uniform material distributions (n.u.m.d.) in the benchmark example highlights that:

- In-plane displacements and rotations are greater for (n.u.m.d.), inducing additional stresses on the external columns;
- The maximum ratio between capacity and demand decreases for (n.u.m.d.);
- The ultimate shear does not have relevant changes because it is related to the mean resistance of the structure, that does not sensitively vary in average terms;
- The ultimate displacement decreases for (n.u.m.d.) compared to (u.m.d.).

## 4. Effects of Material Variability on Collapse Mechanisms

#### 4.1. Input Parameters and Failure Mechanisms for Parametric Analysis

_{Rd}is:

_{i}is:

- ${h}_{i}$ is the height of the i-th column.
- $\alpha $ is taken equal to 2 for slender elements and 1 for the squat ones (see Table 6). The slenderness is defined as:$$\lambda =\frac{{h}_{i}}{{h}_{s}}$$

^{2}; (b) 40 × 70 cm

^{2}; (c) 40 × 180 cm

^{2}(Figure 2) varying once the concrete and the steel strength, considering both flexural and shear failures as in Table 6. The curves are also related to different values of axial force ${N}_{Ed}$ consistent with the design criteria adopted during the 1960s. A set of reasonable “historical” strength values (Table 7) for concrete and steel are then assigned to the structural elements.

#### 4.2. Results of Parametric Analysis

#### 4.2.1. Concrete

_{ed}acting on the columns of r.c. buildings.

#### 4.2.2. Steel

_{c}, while in the case of the variation of the steel resistance, the relation is linear and the lines are parallel (Figure 6a). Therefore, in Figure 6a, only the diagram of the section 40 × 40 (cross sect. A) is displayed. Also, the variation of the steel resistance is shown for the graphs $\left({M}_{Rd,B};{M}_{Rd,V}-{f}_{y}\right)$ of Figure 6b which, in this case, assumes a linear trend with respect to the case of variable concrete. The ultimate curvature and the initial stiffness do not change by varying the steel strength, affecting only the yield values (Figure 7a) [34,35,36]. By contrast, in the case of the variation of the compressive strength of the concrete, there is an increase of the ductility of the curvature.

_{Ed}(Figure 7b). Further parametric analyses were carried out by considering the simultaneous variation of steel and concrete strength; they did not change the main results here described, and are not reported for the sake of brevity.

## 5. A Real Case-Study

#### 5.1. Description of the Buildings

^{2}(72% of the total number of columns), 40 × 70 cm

^{2}(7%), and 40 × 180 cm

^{2}(21%), irregularly distributed. The structure is also irregular in elevation (with a mass reduction of 40%).

^{2}(45%), 40 × 50 cm

^{2}(25%), 25 × 40 cm

^{2}(20%) and 40 × 70 cm

^{2}(10%). Each building has an independent strip foundation system placed at a different level. The two structures are divided by a separating joint with insufficient width from a seismic point of view (Figure 10). Several experimental tests were made on the concrete columns of two buildings: rebound hammer, sonreb and crushing test (made on cylindrical coring samples). In particular, the experimental data of the investigations have been 11 sclerometric tests, 12 sonreb tests, and eight crushing tests on cylindrical samples of the real structure. These tests are made on the same structural elements to have more reliable values. The resulting mean data are presented in Table 8. These data are used in two different ways: (1) extreme values (min and max) are calculated to determine the strength domain in Section 4.2.2; (2) mean value and standard deviation are used to generate random distributions of resistances (compatible with those found in situ) in order to derive fragility curves.

#### 5.2. Parametric Analysis

#### 5.2.1. Extreme Strength Distribution

#### 5.2.2. Strength Domain

_{R}.

_{R}, given by Equation (12). I

_{R}values are reported in the histograms of Figure 14a,b.

#### 5.2.3. Fragility Curves

- -
- Non-Linear static analysis for each random case;
- -
- PGA for each random case;
- -
- Mean value $\mu PGA$ and standard deviation $\sigma PGA$ of all calculated PGA;
- -
- Determination of the probability cumulative function (PCF) as:$$\left(x\right)=\Phi \left[\frac{PGA-\mu PGA}{\sigma PGA}\right]$$

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

- $\mathsf{\beta}$ coefficient equal to: 0.81;
- $\mathsf{\alpha}$ coefficient equal to: 0.83;
- $b$ width of the section;
- ${A}_{s}$ area of the longitudinal steel reinforcement;
- $h$ height of the section;
- $c$ filler tiles;
- ${C}_{Rd,c}$ $\frac{0.15}{{\mathsf{\gamma}}_{c}}$
- $k$ $1+\sqrt{\frac{200}{d}}$;
- ${\mathsf{\rho}}_{l}$ ratio of longitudinal steel reinforcement;
- ${k}_{1}$ coefficient equal to: 0.15;
- ${\mathsf{\sigma}}_{cp}$ mean compressive stress in the section due to axial force;
- ${b}_{w}$ minimum width of the section;
- $d$ internal height of the section;
- ${A}_{sw}$ transverse area of the stirrups;
- $s$ spacing between the stirrups;
- $z$ moment arm;
- ${f}_{ywd}$ strength of the stirrups;
- $\mathsf{\theta}$ inclination of the concrete struts;
- ${\mathsf{\alpha}}_{cw}$ coefficient equal to: 1;
- ${\mathsf{\nu}}_{1}$ $0.5\left(1-{f}_{lck}/250\right)$.

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**Figure 1.**Main views of the benchmark finite element models: rigid slabs (

**a**,

**c**); flexible slabs (

**b**,

**d**).

**Figure 3.**Bending moment of collapse (M

_{Rd,B}) vs. concrete strength (f

_{c}) and axial force (N

_{ed}). (

**a**) Cross section A (40 × 40); (

**b**) Cross section B (40 × 70).

**Figure 4.**(

**a**) Bending moment of collapse (M

_{Rd,B}) vs. concrete strength (f

_{c}) and axial force (N

_{ed})-cross section C (40 × 180); (

**b**) bending (M

_{Rd,B}) and shear mechanism (M

_{Rd,V}) vs. (f

_{c}) with N

_{ed}= 400 kN-cross section (A–C), with Ned = 400 kN.

**Figure 5.**Bending moment (M

_{Rd,B}) vs dimensionless curvature χ(h

_{s}) varying concrete strength (f

_{c})-cross section A (40 × 40), N

_{ed}= 400 kN (

**a**); curvature ductility μχ vs. concrete strength (f

_{c}) and axial force (N

_{ed})-cross section A (40 × 40) (

**b**).

**Figure 6.**Bending moment of collapse (M

_{Rd}) varying steel strength (f

_{yd}) and axial force (N

_{ed})-cross section A (40 × 40) (

**a**); cross section (40 × 40) Ductile and brittle failure mechanism of the analysed cross sections (40 × 40, 40 × 70, 40 × 180), assuming N

_{ed}= 400 kN (

**b**).

**Figure 7.**Bending moment (M

_{Rd}) vs. dimensionless curvature χ(h

_{s})varying steel strength (f

_{yd})-cross section A-(40 × 40), N

_{Ed}= 400 kN (

**a**); curvature ductility μχ vs. steel strength f

_{y}and axial force (N

_{Ed})-cross section A-(40 × 40) (

**b**).

**Figure 12.**Linear regression ${V}_{u}-{e}_{m}$ for building A. x direction (

**a**); y direction (

**b**). The orange square with the label u.m.d. indicates the uniform material distribution case.

**Figure 13.**Linear regression V

_{u}- e

_{m}for building B. x direction (

**a**); y direction (

**b**). u.m.d. indicates the uniform material distribution case.

**Figure 14.**Strength domain of building A (

**a**) and B (

**b**). At the top and at the bottom is the change rate of the risk index I

_{R}in x and in y directions (units in cm). C is the centre of mass of each storey.

Material Number | f_{c} (MPa) | E_{c} (MPa) | f_{y} (MPa) | Color |
---|---|---|---|---|

1 | 10 | 22,000 | 230 | Ciano |

2 | 20 | 27,085 | 290 | Blue |

3 | 30 | 30,590 | 350 | Green |

4 | 20 | 13,542 | 290 | Magenta |

**Table 2.**Material eccentricity of the benchmark model with percentage of the total building length in the corresponding direction.

Material Eccentricity | x | y | Units |
---|---|---|---|

${e}_{m}$ | 61 | 108 | cm |

8.3 | 16.7 | % ^{1} |

^{1}Material eccentricity ratio main dimensions of the building.

Types of Analysis | Rigid Slab | Flexible Slab | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{D}}_{\mathit{u}}-{\mathit{D}}_{\mathit{y}}$ | ${\mathit{\phi}}_{\mathit{u}}-{\mathit{\phi}}_{\mathit{y}}$ | $\mathit{C}/\mathit{D}$ | ${\mathit{I}}_{\mathit{R}}$ | ${\mathit{D}}_{\mathit{u}}-{\mathit{D}}_{\mathit{y}}$ | ${\mathit{\phi}}_{\mathit{u}}-{\mathit{\phi}}_{\mathit{y}}$ | $\mathit{C}/\mathit{D}$ | ${\mathit{I}}_{\mathit{R}}$ | ||

(uniform) | Linear | 0.722 | 0.000 | - | 0.45 | 0.72 | 0.000 | - | 0.45 |

n-Linear | 7.840 | 0.000 | 0.82 | 0.76 | 7.96 | 0.000 | 0.82 | 0.76 | |

(n-uniform) | Linear | 0.680 | 0.005 | - | 0.40 | 0.71 | 0.030 | - | 0.38 |

n-Linear | 6.840 | 0.020 | 0.77 | 0.76 | 7.05 | 0.020 | 0.79 | 0.76 |

**Table 4.**Yield strength of steel type AQ42, AQ50 and AQ60 [33].

Steel Strength | AQ42 | AQ50 | AQ60 | Units |
---|---|---|---|---|

f_{y,min} | 265.0 | 282.4 | 353.7 | N/mm^{2} |

f_{y.med} | 325.4 | 369.9 | 432.6 | N/mm^{2} |

f_{y.max} | 397.4 | 530.0 | 560.8 | N/mm^{2} |

σ | 23.17 | 29.45 | 36.59 | N/mm^{2} |

Cubic Compression Strength | Test A ^{1} | Test B ^{2} | Test C ^{2} | Units |
---|---|---|---|---|

R_{c,min} | 7.0 | 21.9 | 29.6 | N/mm^{2} |

R_{c.mean} | 26.0 | 37.3 | 38.2 | N/mm^{2} |

R_{c.max} | 45.0 | 40.1 | 46.7 | N/mm^{2} |

^{1}secondary school Don Bosco in Francavilla in Sinni (Potenza–Italy);

^{2}secondary school Francesco Carrara in Lucca (Lucca–Italy).

Parameters | A | B | C |
---|---|---|---|

${h}_{i}/{h}_{s}$ | 8.75 | 5.00 | 1.95 |

$\alpha $ | 2.00 | 1.55 | 1.00 |

Strength | 01 | 02 | 03 | 04 | 05 | Unit |
---|---|---|---|---|---|---|

${f}_{c}$ | 10 | 15 | 20 | 25 | 30 | MPa |

${f}_{y}$ | 230 | 260 | 290 | 320 | 350 | MPa |

TEST | Hammer | Sonreb | Coring | Units |
---|---|---|---|---|

1 | 29.0 | 14.9 | 17.2 | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

2 | 29.0 | 17.2 | 13.9 | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

3 | 22.0 | 9.4 | 10.4 | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

4 | 25.0 | 8.6 | 9.6 | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

5 | 33.0 | 11.7 | 12.3 | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

6 | 42.0 | 23.2 | 28.7 | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

7 | 45.0 | 28.9 | 31.3 | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

8 | 43.0 | 29.0 | - | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

9 | 38.0 | 27.5 | - | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

10 | 40.0 | 17.4 | - | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

11 | 43.0 | 22.8 | - | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

12 | - | 21.7 | 19.5 | $\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}$ |

Average Value and Standard Deviation of PGA | Low Damage (LD) | Severe Damage (SD) | |||
---|---|---|---|---|---|

PGAx | PGAy | PGAx | PGAy | ||

Building A | ${\mu}_{PGA}$ | 0.064 | 0.064 | 0.117 | 0.093 |

${\sigma}_{PGA}$ | 0.003 | 0.002 | 0.003 | 0.002 | |

Building B | ${\mu}_{PGA}$ | 0.062 | 0.006 | 0.133 | 0.093 |

${\sigma}_{PGA}$ | 0.0007 | 0.002 | 0.134 | 0.003 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Puppio, M.L.; Pellegrino, M.; Giresini, L.; Sassu, M.
Effect of Material Variability and Mechanical Eccentricity on the Seismic Vulnerability Assessment of Reinforced Concrete Buildings. *Buildings* **2017**, *7*, 66.
https://doi.org/10.3390/buildings7030066

**AMA Style**

Puppio ML, Pellegrino M, Giresini L, Sassu M.
Effect of Material Variability and Mechanical Eccentricity on the Seismic Vulnerability Assessment of Reinforced Concrete Buildings. *Buildings*. 2017; 7(3):66.
https://doi.org/10.3390/buildings7030066

**Chicago/Turabian Style**

Puppio, Mario Lucio, Martina Pellegrino, Linda Giresini, and Mauro Sassu.
2017. "Effect of Material Variability and Mechanical Eccentricity on the Seismic Vulnerability Assessment of Reinforced Concrete Buildings" *Buildings* 7, no. 3: 66.
https://doi.org/10.3390/buildings7030066