5.3.1 Uncertain Parameters Considered
Typical sources of uncertainty are those in ground motion and structural properties, known as aleatoric and epistemic uncertainties, respectively. The present study focusses mainly on sensitivity of response to various material, geometric, and loading properties i.e., the parameters responsible for epistemic uncertainties only.
However, the sensitivity analyses are carried out for seven different ground motions to understand if there is any influence of different ground motion parameters on sensitivity of structural response. Ten structural parameters are considered as uncertain variables, namely, width of column (Bc), depth of beam (Db), equivalent viscous damping (ξ), concrete compressive strength (fck), yield stress of steel (fy), masonry prism strength (fm′), unit weight of infill (γm) or infill load (IL), weight density of concrete (γc), width of equivalent diagonal strut used for modelling masonry infill walls (Ws), and ultimate strain at failure in masonry (εm). Width of the strut is different for ground (Ws-gr) and upper (Ws-up) stories. The statistical characteristics of each of the random input variables are presented in Table 5.1. The parameters related to the characteristics of masonry infill walls are considered only for FI frames, and not for the OGS frames, since their lateral load behavior is not affected by presence of infills in the upper floors (Choudhury and Kaushik 2017a). Random samples of each variable are generated based on their statistical characteristics.
As suggested in the literature (Table 5.1), a normal distribution is considered for the majority of the random variables, with the exception of the yield stress of reinforcing steel for which lognormal distribution is recommended (Ranganathan 1999). For each of the random variable, data is generated based on their corresponding standard deviations.
Building mass and member dimensions are uncertain variables as the actual member dimensions (Bc, Db) vary from those shown in the design documents, and unit weights (γc,γm) are imperfectly known. Moreover, material properties differ from those assumed in the analysis, e.g., actual stress-strain behavior at the element-fiber level differs from engineering idealizations. These independent material and geometric properties affect other dependent structural properties, for example, the concrete compressive strength affects the elastic modulus of concrete (Ec), and masonry prism strength affects the elastic modulus of masonry (Em). Similarly, the parameters Db and Bc affect the moment- curvature (or force-deformation) relationship of the structural members. A comprehensive discussion on uncertainty in viscous damping is presented by Porter et
5.3 Sensitivity Analysis
al. (2002) and Celik and Ellingwood (2010). Uncertainties in structural strength and stiffness are considered at the stress-strain level. In constitutive models of reinforced concrete and masonry infill, three independent parameters are considered as random input variables: fck, fy, and fm′ for strength uncertainty. The stiffness uncertainty is taken care of by using the dependent variables Ec and Em. The masonry infill walls are modelled as diagonal struts for which the additional uncertain input variables considered are Ws and εm. Although, only the independent variables are monitored in the sensitivity analysis, the corresponding dependent variables are also given the due importance by varying their values appropriately in the analytical models.
Table 5.1 Considered uncertain input variables and their statistical characteristics.
Sr. Variable Probability Distribution
Mean COV Standard Deviation
Reference 1 fck (Ec) N 25 MPa 0.124 3.11 Ranganathan (1999)
2 fy LN 450 MPa 0.038 17.26 Ranganathan (1999)
3 fm' (Em) N 4.1 MPa 0.24 0.98 Kaushik et al. (2007)
4 ξ LN 5% 0.76 3.8 Celik and Ellingwood (2010)
5 Bc N 0.3 m 0.026 0.0079 Ranganathan (1999)
6 Db N 0.45 m 0.021 0.0094 Ranganathan (1999)
7 γc N 25 kN/m3 0.1 2.5 Ranganathan (1999)
8 γm (or IL) N 18 kN/m3 0.1 1.8 Kaushik et al. (2007) 9 Ws-gr N 1.05 m 0.394 0.41 Basha and Kaushik (2015)
10 εm N 0.0015 0.43 0.0006 Kaushik et al. (2007)
11 Ws-up N 0.84 m 0.394 0.331 Basha and Kaushik (2015) Note: N = Normal/Gaussian distribution, LN = Lognormal distribution, COV = Coefficient of variation, Ec =5000 fck and Em=550fm′
In the present study, the output is considered a known deterministic function of a variety of input variables. The sensitivity analyses are carried out based on the probability distribution of each of the input variables. Each variable is treated as uncertain but the simulation is controlled so that all the variables except one are taken at their median (50th percentile) value, and the response is evaluated establishing a baseline output. Then the output for a set of random input variables are determined, the input being the values based on their probabilistic distribution and coefficient of variation. The absolute value of the difference between the output from the two extreme cases (e.g., 16th and 84th fractile values) is a measure of the sensitivity of the output to that input variable.
This difference is also called the swing (Porter et al. 2002). The process is then repeated, to determine the swing associated with the variability of all the other input variables. One can then determine the order of importance of the input variables to the sensitivity of the
output according to their swing. A larger swing in the seismic response to a particular input variable reflects more importance of that input uncertainty on the sensitivity of the seismic response.
5.3.2 Structural Modelling
A three-bay four-storey (3B-4S) reinforced concrete frame is considered as the base model (or mean model) where all the material and geometric properties and loading details are set to their median values. Three variants of the RC frame typologies (as discussed in Chapter 3) have been considered, namely, the bare frame (BF), open ground storey (OGS) frame, and fully infilled (FI) frame. The sectional details and member dimensions of the frame members are as shown in Fig. 3.1. The RC members are modelled as two-noded line elements and the masonry infill walls are idealized as single equivalent diagonal struts in SAP2000 (CSI 2015). The RC members are assumed to fail in flexural failure mode only as shear reinforcement with ductile detailing is provided in all the frame members as per the relevant Indian codes (BIS 2016b). Therefore, the RC members are provided with nonlinear flexural hinges defined using fiber modelling of the sections.
The single equivalent diagonal struts are modelled with nonlinear axial hinges. The concrete compressive cube strength is taken to be 25 MPa (Elastic Modulus: 25000 MPa) and the reinforcing bars have expected yield strength of 450 MPa. Masonry prism strength is considered to be 4.1 MPa and elastic modulus is 2255 MPa. Weight of the masonry infill walls is considered as uniformly distributed load on the beams (weight density of masonry = 18 kN/m3).
Mander’s model (Mander et al. 1988) is used to characterize the stress-strain curve of concrete. The Mander model gives the stress-strain envelop in each of the concrete fibers in a section, and its hysteresis characteristics, i.e., the strength and stiffness degradation for every loading and unloading, is defined by the Takeda hysteresis model (Takeda et al. 1970). The hysteresis model is based on kinematic hardening behavior which is commonly observed in metals, is used for the reinforcing bars (Hahn et al. 1990).
The idealized stress-strain model proposed by Kaushik et al. (2007) is used to model the material nonlinearity in masonry infill and the Pivot hysteretic model developed by Cavaleri and Trapani (2014) is used for modelling the hysteretic behavior of the equivalent struts for masonry infill.