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

Abstract

The present paper deals with the influence of material variability on the seismic vulnerability assessment of reinforced concrete buildings. Existing r.c. buildings are affected by a strong dispersion of material strengths of both the base materials. This influences the seismic response in linear and nonlinear static analysis. For this reason, it is useful to define a geometrical parameter called “material eccentricity”. As a reference model, an analysis of a two storey building is presented with a symmetrical plan but asymmetrical material distribution. Furthermore, an analysis of two real buildings with a similar issue is performed. Experimental data generate random material distributions to carry out a probabilistic analysis. By rotating the vector that defines the position of the center of strength it is possible to describe a strength domain that is characterized by equipotential lines in terms of the Risk Index. Material eccentricity is related to the Ultimate Shear of non-linear static analyses. This relevant uncertainty, referred to as the variation of the center of strength, is not considered in the current European and Italian Standards. The “material eccentricity” therefore reveals itself to be a relevant parameter to considering how material variability affects such a variation.

1. Introduction

Reinforced Concrete buildings built during the 40–70s are affected by a large dispersion of mechanical properties, especially when concrete is considered. This is due to (1) the inadequate mix-design of concrete (with a huge dispersion of water-concrete ratio); (2) the use of unwashed or low quality aggregates; (3) the lack of automation and control of the concrete production; (4) ineffective procedures for the concrete casting batch and vibration [1,2]. This dispersion can be quantified through the definition of a geometrical parameter called material eccentricity [3,4]. Also, other parameters had a great relevance in the seismic evaluation of existing structures e.g., the geometry of the bearing elements, the aging of the concrete, corrosion, the type of subsoil [5,6,7,8].

The building characterization [9,10] has great relevance, in particular, to the seismic vulnerability assessment [11], in which non-structural elements also play a relevant role [12,13,14]. According to previous experiences on the monitoring and rehabilitation of existing buildings [4,15,16,17] it is possible to notice significant uncertainties in the mechanical properties of both concrete and steel bars. Innovative and traditional techniques—in particular, additional reinforcement [18,19], are not able to cover all these uncertainties. The tendency of the current Standards, e.g., [20], is to penalize the mean value of the materials’ strength with a specific “confidence factor” depending on the available information. This approach does not take into account the real distribution of the mechanical values, together with their standard deviation in material tests [21].

Nevertheless, the tendency to carry out a probabilistic analysis of existing buildings through the assessment of the uncertainties modelled with random and epistemic variables might imply analytical complications. A significant analysis can be the assessment of a material eccentricity, which induces two different effects: eccentricity of stiffness and eccentricity of strength. In the present paper both aspects are analysed.

The effects of spatial variability of concrete and steel strength are considered in this paper for typical Italian r.c. buildings. Firstly, the effect of material variability on the strength domains of the usual failure mechanisms is evaluated in Section 2, according to [22]. Those domains are related to eccentric axial force and shear for beams and columns [23]. After that, the effects of changing strength and stiffness are investigated through two simple benchmark buildings (Section 3). Section 4 then illustrates the possible failure mechanisms that can occur in r.c. buildings affecting material variability. This issue is applied to a real structure in Section 5: the school of Don Bosco in an Italian city. This structure was subjected to a set of experimental tests addressed to seismic retrofitting. The material eccentricity is evaluated in non-linear static analyses. Random distributions of strength are considered to obtain fragility curves. This method highlights the way to achieve a probabilistic estimation of the structural safety level for similar buildings.

2. Centre of Strength and Centre of Stiffness

2.1. Generalities

The European seismic code [20] generally considers in-plan irregularity throughout the eccentricities of the equivalent seismic forces. Also, non-uniform distribution of the columns material in r.c. buildings can produce torsional effects [24,25,26,27]: hence the definition of “material eccentricity” e_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

In a linear elastic analysis the coordinates of the center of stiffness are [28]:

$$x_{s,j}=\frac{{\displaystyle {\sum }_i^n{x}_i{K}_{yi}}}{{\displaystyle {\sum }_i^n{K}_{yi}}}$$

(1)

$$y_{s,j}=\frac{{\displaystyle {\sum }_i^n{y}_i{K}_{xi}}}{{\displaystyle {\sum }_i^n{K}_{xi}}}$$

(2)

where:

• $x_i,y_i$ 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]:

  $$E_c=22,000{\left(\frac{f_c}{10}\right)}^{0.3}$$

  (3)

The stiffness eccentricity in x and y directions $e_{sx,j}$ and $e_{sx,j}$. is given by these ratios:

$$e_{sx,j}=\frac{x_{s,j}-x_{c,j}}{L_x}$$

(4)

$$e_{sy,j}=\frac{y_{s,j}-y_{c,j}}{L_y}$$

(5)

• $x_c,y_c$ 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 collapse of R.C. columns depends on:

• 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.

Two possible criteria to define, in a simple way, the center of strength are here described: the first one is related to collapse caused by compression force (method 1), the second one by bending moment (method 2).

Method 1. The center of compressive strength of n columns at the j-level is calculated by a weighted average of the axial compression resistance of the columns with respect to their in plan positions.

$$x_R^C=\frac{{\displaystyle {\sum }_i^n{N}_{u,i}{x}_i}}{{\displaystyle {\sum }_i^n{N}_{u,i}}}$$

(6)

$$y_R^C=\frac{{\displaystyle {\sum }_i^n{N}_{u,i}{y}_i}}{{\displaystyle {\sum }_i^n{N}_{u,i}}}$$

(7)

• $N_{u,i}$ 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.

Method 2. The center of bending strength is calculated by a weighted average of the ultimate bending moment of the columns with respect to their in-plan positions.

$$x_{R,j}=\frac{{\displaystyle {\sum }_i^n{M}_{u,i}{x}_i}}{{\displaystyle {\sum }_i^n{M}_{u,i}}}$$

(8)

$$y_{R,j}=\frac{{\displaystyle {\sum }_i^n{M}_{u,i}{y}_i}}{{\displaystyle {\sum }_i^n{M}_{u,i}}}$$

(9)

where:

• $M_{u,i}$ failure bending moment of the R.C. section of the i-th column.
• The corresponding strength eccentricity is:

  $$e_{Rx,j}=\frac{x_{R,j}-x_{c,j}}{L_x}$$

  (10)

  $$e_{Ry,j}=\frac{y_{R,j}-y_{c,j}}{L_y}$$

  (11)

where:

• $x_{Rj},y_{Rj}$ 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

A simply two storey R.C. frame (Figure 1) is analyzed here as a benchmark example. The in-plan dimension of the building is 7.35 m × 6.45 m. This model is composed of six columns for each floor of cross section 40 × 40 cm² and n. 4 steel bars with a diameter of 20 mm for each one. The beams have a cross section of 40 × 50 cm². 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.

The assumed R.C. mechanical properties are listed in

Table 1. Mechanical characteristics of the benchmark model.

Material Number	fc (MPa)	Ec (MPa)	fy (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

. Linear and nonlinear static analyses are compared to evaluate the seismic response. All the analyses are performed with FEM Software Midas GEN (ver. 2016 release 2.1, MIDAS Information Technology, Gyeonggi-do 13487, Korea).

The assigned strength distribution causes material eccentricity.

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.

are calculated through the criterion of bending collapse (Equations (8) and (9)).

Both linear and non-linear analyses are performed by applying the seismic action in the X direction Ultimate Limit State ULS for the models of Figure 1. The results in terms of elastic displacements and rotations are reported in

Table 3. Summary of results.

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

.

The symbols of Table 3 are so defined:

$D_u-D_y$ Elastic displacement ULS (Ultimate Limit State) for Linear Analysis (LA) or target displacement for Non Linear Analysis (NLA).

${\phi }_u-{\phi }_y$ Elastic rotation corresponding to ULS for LA, target rotation for NLA.

$C/D$ Max. Capacity/Demand ratio for bending mechanism. For LA it is the ratio $M_{Ed}/M_{Rd}$, while for NLA the ratio is $\theta /{\theta }_u$. ${\theta }_u$ is the floor rotation corresponding to the achievement of the ultimate condition. $M_{Rd}$ is defined in Section 3.2.

3.2. Calculation of the Risk Index

The Risk Index is calculated according to [29] as:

$${\mathrm{I}}_{\mathrm{R}}=\frac{\mathrm{P}\mathrm{G}{\mathrm{A}}_{\mathrm{C}}}{\mathrm{P}\mathrm{G}{\mathrm{A}}_{\mathrm{D}}}$$

(12)

$\mathrm{P}\mathrm{G}{\mathrm{A}}_{\mathrm{C}}$ is the collapse capacity derived from the push-over analysis; $\mathrm{P}\mathrm{G}{\mathrm{A}}_{\mathrm{D}}$ is the demand in terms of peak ground acceleration as mentioned in the Eurocode 8 [20]. For the models with flexible slab, displacements and rotations of the plan are evaluated as the average value at the top of each column.

It is well known that linear analysis furnishes more severe Risk Indexes I_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.).

This leads to a reduction of the risk index in linear analysis, which is more noticeable for the models with flexible slabs. The same effects are also evaluated in Section 4 on a real case-study.

4. Effects of Material Variability on Collapse Mechanisms

4.1. Input Parameters and Failure Mechanisms for Parametric Analysis

The recurring mechanical properties of the typical r.c. existing structures built in Italy in the 1960s are listed below. The yield stress of steel bars used in the present analysis are reported in

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

Steel Strength	AQ42	AQ50	AQ60	Units
fy,min	265.0	282.4	353.7	N/mm2
fy.med	325.4	369.9	432.6	N/mm2
fy.max	397.4	530.0	560.8	N/mm2
σ	23.17	29.45	36.59	N/mm2

and are taken from [33].

Similarly, the concrete cubic compression strength used here is listed in

Table 5. Concrete cubic compression strength from experimental in situ tests.

Cubic Compression Strength	Test A 1	Test B 2	Test C 2	Units
Rc,min	7.0	21.9	29.6	N/mm2
Rc.mean	26.0	37.3	38.2	N/mm2
Rc.max	45.0	40.1	46.7	N/mm2

¹ secondary school Don Bosco in Francavilla in Sinni (Potenza–Italy); ² secondary school Francesco Carrara in Lucca (Lucca–Italy).

.

The failure mechanism, due to transverse forces, often involves collapse of the columns at the ground floor. In fact, in the common practice without considering the seismic action, as done for many existing R.C. structures built in the 1960s, the columns were loaded by axial forces due to vertical loads, together with bending and shear due to transverse wind loads. In most cases, no variation in geometric section along the height of the buildings was considered. The collapse bending moment in r.c. columns is given by Eurocode 2 (for symbols see Appendix A or BS EN 1992-1-1 2004):

$$M_{Rd,B}=N_{Ed}\left(\frac{h}{2}-\frac{k}{\beta \cdot b\cdot \alpha \cdot f_{cd}}{N}_{Ed}\right)+A_s(h-2\cdot c)f_{yd}$$

(13)

The corresponding shear strength is:

$$V_{Rd,c}=\left[C_{Rd,c}k{\left(100{\rho }_l{f}_{ck}\right)}^{1/3}+k_1{\sigma }_{cp}\right]b_{w}d$$

(14)

$$V_{Rd,s}=\frac{A_{sw}}{s}z{f}_{ywd}\cot \theta$$

(15)

$$V_{Rd,\max }={\alpha }_{cw}{b}_{w}z{\nu }_1{f}_{cd}/\left(\cot \theta +\tan \theta \right)$$

(16)

The ultimate shear resistance of the section V_Rd is:

$$V_{Rd}=\min \left(V_{Rd,c},V_{Rd,s},V_{Rd,\max }\right)$$

(17)

From which, the collapse bending moment of a column with height h_i is:

$$M_{Rd,T}=\frac{V_{Rd}{h}_i}{\alpha }$$

(18)

where:

• $h_i$ 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. Value of $\alpha$ depending on slenderness ratio.

  Parameters	A	B	C
  $h_i/h_s$	8.75	5.00	1.95
  $\alpha$	2.00	1.55	1.00

  ). The slenderness is defined as:

  $$\lambda =\frac{h_i}{h_s}$$

  (19)

  being $h_s$ the height of the cross section.

The collapse occurs when the minimum value between ${\mathrm{M}}_{\mathrm{R}\mathrm{d},\mathrm{B}}$ and ${\mathrm{M}}_{\mathrm{R}\mathrm{d},\mathrm{T}}$ is attained.

In the following case, three types of columns are analysed: (a) 40 × 40 cm²; (b) 40 × 70 cm²; (c) 40 × 180 cm² (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. Discretization of steel and concrete strength.

Strength	01	02	03	04	05	Unit
$f_c$	10	15	20	25	30	MPa
$f_y$	230	260	290	320	350	MPa

) for concrete and steel are then assigned to the structural elements.

In case of concrete strength variation, a steel strength of 290 MPa is fixed. In case of steel strength variation, the concrete cubic compression strength of 20 MPa is assumed.

4.2. Results of Parametric Analysis

4.2.1. Concrete

The main results of the parametric analysis, by varying concrete strength (Table 7), are displayed in Figure 3, Figure 4 and Figure 5. The parameters of the curves are common values of design normal force N_ed acting on the columns of r.c. buildings.

Figure 3a,b and Figure 4a show a nonlinear $\left(M_{Rd,B}-f_c\right)$ behavior depending on $\left(N_{Ed}\right)$.

For the considered sections, the value of the dimensionless axial stress is variable, so the slope of the respective diagrams $\left(M_{Rd,B}-f_c\right)$ varies in the ranges of resistance. Figure 4b shows that bending and shear mechanisms depend on concrete strength and the slenderness ratio.

Because of the different slope of continuous and dashed curves, a different type of failure can occur depending on the concrete strength. Figure 5a,b illustrates how strength affects stiffness and curvature ductility.

4.2.2. Steel

In the second set of analyses, the steel strength is varied. The main results are reported in Figure 6 and Figure 7. It can be observed that the variation of the collapse bending moment depends linearly on the steel strength $\left(f_y\right)$.

In case of variation of the concrete resistance, the resistant moment of the section has a nonlinear relationship with f_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.

A decreasing trend of curvature ductility with steel strength also occurs by varying N_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

In this Section, the effects of material variability are investigated for a real R.C. building, taking into account the failure mechanisms illustrated in Section 4.

5.1. Description of the Buildings

The secondary school “Don Bosco” in Francavilla in Sinni (Potenza, Italy) is made up of two R.C. framed buildings (A and B) built in the 1970s (Figure 8 and Figure 9).

Building A has a rectangular shape (45.0 m × 16.5 m), composed of two floors with an inter-storey height of 3.52 m. Column cross sections are of 40 × 40 cm² (72% of the total number of columns), 40 × 70 cm² (7%), and 40 × 180 cm² (21%), irregularly distributed. The structure is also irregular in elevation (with a mass reduction of 40%).

Building B has an approximately squared shape with maximum in-plan dimensions of 26.0 m × 21.0 m, composed of three floors, one semi-basement, and two completely above the ground, with an inter-storey height of 3.52 m (with the exception of a central part with taller columns (5.24 m)). The main frames are in both directions with r.c. and hollow-core concrete slabs with different heights (16 + 4 cm for the offices, 42 + 8 for the roof and the central corridor). Building B is also irregular, both in terms of its plan and height. Column cross sections are of 40 × 40 cm² (45%), 40 × 50 cm² (25%), 25 × 40 cm² (20%) and 40 × 70 cm² (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. Experimental test carried out on the school “Don Bosco”.

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	$\mathrm{N}/\mathrm{m}{\mathrm{m}}^2$

. 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.

The original drawings show an improper use of stirrups, especially in the R.C. columns, and inadequate bonded lengths and overlapping of bars. All these deficiencies in structural details may cause the local brittle shear mechanism and global collapse mechanism of “weak columns”[37,38].

5.2. Parametric Analysis

5.2.1. Extreme Strength Distribution

To obtain the in-plan limit distributions of strength (that may include all the real ones), extreme values of strength are assigned to the model, and studied analogously to the models in Section 3. A vertical plane passing through the centroid of each floor subdivides it into two parts. Each distribution is obtained by rotating a vertical plane by 45° around the centroid C (Figure 11), thus getting eight limit distributions of strength for the columns (Figure 11).

The variable parameters are the concrete compression strength and the yielding strength of the steel bars. Each pair of values is assumed in both uniform and non-uniform in-plan distribution. Firstly, the two variables are considered separately. The material strengths assigned are the extreme values of Table 4 and Table 5.

In Figure 11 the subscript i assumes the values 1 or 2 according to whether the variable strength is of concrete or steel, taking the other parameter constant and equal to the mean value. Cases 1.i 2.i, and 3.i are not represented in Figure 11 but are representative respectively of uniform mean, and the minimum and maximum value of resistance. For each configuration of strength nonlinear static analyses (NLSA) are performed.

Figure 12 and Figure 13 show the regression between ultimate shear and eccentricity of material. The two seismic directions are separately considered and the ultimate shear in x-direction ($V_{Ux}$) is related to the eccentricity and vice versin y-direction ($e_{my}$) (and vice versa).

The elongated in-plan shape of building A produces the differences in the slope of the line that represent the relationship between $V_{Ux}-e_{my}$ and $V_{Uy}-e_{mx}$ (Figure 12). In the case of building B, characterized by a compact shape, the difference between the two-coordinate direction is less remarkable.

5.2.2. Strength Domain

With Equations (6)–(9) it is possible to calculate the position of the resistance centers referring to the limit distributions of Figure 11. For each of these points a F.E.M. model is subjected to non-linear static (pushover) analysis to determine the risk index. It is seen that the risk index does not significantly differ from one case to another so the points represent equipotential lines for parameter I_R.

Figure 14a,b shows the domains of the center of strength for each building, obtained from the set of extreme strength distributions. They represent the material eccentricity due to extreme values of concrete strength (grey domain) or steel strength (yellow domain) by rotating the vertical plan that divides the two regions in Figure 14.

In this figure, the numbers in the strength domain j.i mean the following: subscript j the eight cases of Figure 11, starting from 4 to 11. Cases 1-2-3 are referred to uniform distributions and are not reported for the sake of brevity. Subscript i assumes the values 1 or 2 according to whether the variable strength is of concrete or steel. NLSA are carried out for both buildings. Each analysis evaluated the structural capacity relating to the border points of each domain. The main results are the variation of the index of seismic risk I_R, given by Equation (12). I_R values are reported in the histograms of Figure 14a,b.

The results show that the risk index has a sensitive change in the shorter direction of the building, which is lower in the compact in-plan building (B), whereas along the longer direction the variation is negligible. The strength domains are greater considering the variation of concrete rather than steel. The material eccentricities are much more relevant in the irregular building, for which the strength domain does not include the centre of mass of the storey C.

5.2.3. Fragility Curves

Finally, random material distributions are numerically generated to obtain fragility curves. The random distribution considers variation both in steel and in concrete strength. To ensure that material distribution could be realistic, a probability density function (PDF) is defined starting from the mean value and the standard deviation of strength. Two strength values (for concrete and steel) are obtained and randomly assigned to each column.

ULS and DLS (Damage Limit States) are examined. The ULS limit state is achieved for the value of PGA that determines the collapse of the structure while the DLS state is achieved for the maximum interstorey drift [13].

The steps to determine the fragility curves are [39,40]:

-

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:

$$\left(x\right)=\Phi \left[\frac{PGA-\mu PGA}{\sigma PGA}\right]$$

(20)

Table 9. PGA values for building A and B.

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

contains the P(x) values for both buildings. The corresponding fragility curves are shown in Figure 15. The probability cumulative function PCF related to low damage is represented with the grey line; meanwhile, the one related to severe damage is represented with the black line.

The fragility curves of Building A had a greater slope than that related to Building B. This means that Building B had a greater dispersion of the PGA values and therefore its seismic capacity is more sensitive to the material variability. The comparison of the curves (a) and (b) of Figure 15 reveals the vulnerability of building A that collapses for lower PGA values.

6. Conclusions

In this paper, a simplified method to consider material variability is proposed for the seismic vulnerability assessment of existing R.C. buildings. From the consideration that concrete and steel variability can produce additional torsional effects, a material eccentricity is defined.

The illustrated methodology is based on the idea of adding a material eccentricity to those already considered in the Standards for seismic analysis. In the structural design of rehabilitation, as shown in the examples, it is easy to take into account the variability of the mechanical parameters.

Several nonlinear analyses are performed on a benchmark example and on two real case-studies of R.C. frames. The effects on the risk indexes and on collapse mechanisms are discussed. Extreme strength distributions are considered leading to the definition of a strength domain, varying concrete and steel strength separately. These domains can be considered as contour lines of the seismic risk index.

For the benchmark model, the results showed that in the case of non-uniform material distribution the ratio of capacity to demand in terms of PGA and the ultimate displacement capacity decreased. For the real case study, the risk index was observed to have a sensitive change in the shorter direction of the building, whereas along the longer direction the variation is negligible. The strength domains are greater considering the variation of concrete rather than steel. The material eccentricities are much more relevant in the irregular building, for which the strength domain do not include the centre of mass of the storey.

Further investigations could be carried out on similar structures by considering random and/or real strength distributions, evaluating the changes in the strength domain. Further studies can be addressed to evaluating which types of buildings can be more sensitive to material variability, with the aim of properly taking this into account for the evaluation of the seismic capacity.