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1外文资料Reliability of Frame and Shear Wall Structural Systems. I: Static LoadingAbstract: An efficient and accurate algorithm is developed to evaluate the reliability of a steel frame and reinforced concrete shear wall structural system subjected to static loading. In a companion paper, the algorithm is extended to consider dynamic loading, including seismic loading. The concept integrates the finite-element method and the first-order reliability method, leading to a stochastic finite element-based approach. In the deterministic finite-element representation, the steel frame is represented by beam-column elements and the shear walls are represented by plate elements. The stiffness matrix for the combined system is then developed. The deterministic finite-element algorithm is verified using a commercially available computer program. The deterministic algorithm is then extended to consider the uncertainty in the random variables. The reliability of a steel frame with and without the presence of reinforced concrete shear walls is evaluated for the strength and serviceability performance functions. The results are verified using Monte Carlo simulations. The algorithm quantitatively confirms the beneficial effect of shear walls, particularly when the steel frame is weak in satisfying the serviceability requirement of lateral deflection. The algorithm can be used to estimate the reliability of any complicated structural system consisting of different structural elements and materials when subjected to static loading. The procedure will be useful in the performance-based design guidelines under development by the profession. keywords: Limit states; Simulation; Shear walls; Static loads; Steel frames; Finite element method.IntroductionThe realistic reliability analysis of complicated structural systems consisting of different types of structural elements and materials is a major challenge to our profession. In most cases, the limit state or performance function (a functional relationship between the load- and resistance-related variables and the performance criterion) is implicit in evaluating the reliability of such systems. The analytical technique most frequently used to capture the mechanical behavior of complicated structural systems consisting of different materials appears to be the finite-element method (FEM)-based approach. Finite-element analysis is a powerful tool commonly used in many engineering disciplines to analyze simple or complicated structural systems. With this approach, it is straightforward to consider complicated geometric arrangements,various sources of nonlinearity, different materials, and the load path to failure. However, the deterministic finite-element method fails to consider the uncertainty in the variables, and thus cannot be used for reliability analysis. On the other hand, the available reliability methods fail to represent structures as realistically as possible. If the basic variables are uncertain, every quantity computed during the deterministic analysis is also uncertain. The currently available reliability methods can still be used if the uncertainty in the response can be tracked in terms of the variation of the basic variables at every step of the deterministic analysis. To capture the desirable features of these two approaches, they needed to be combined, leading to the concept of the stochastic finite-element method (SFEM) (Haldar and Mahadevan 22000b)。The SFEM algorithm for frame structures has been developed by several researchers. However, the main drawback of frame structures is their inability to transfer horizontal loads (e.g., wind, earthquake, and ocean waves) effectively. They are relatively flexible. To increase their lateral stiffness, bracing systems or shear walls are needed. Haldar and Gao (1997) Attempted to consider bracing systems in a steel frame structure. They used truss elements in their model. However, there has not been an attempt to consider shear walls, represented by two dimensional plate elements, in a frame in the context of SFEM.Shear walls have been used for a long time to increase the lateral stiffness of steel frames. The use of concrete shear walls is specifically addressed in this paper. It is not simple to capture the realistic behavior of a combined system consisting of steel frames represented by one-dimensional beam-column elements and concrete shear walls represented by two-dimensional plate elements. Furthermore, the consideration of uncertainty in modeling the combined system is expected to be challenging. A stochastic finite-element-based reliability analysis procedure for the combined system under a static loading condition is proposed in this paper. The companion paper (Lee and Haldar 2003) discusses the behavior of the same structural system in the presence of uncertainty under dynamic loading, including seismic loading.Deterministic Finite-Element MethodRepresentation of a frame and shear walls structural system by finite elements is the first essential step in the proposed algorithm. The basic frame is represented by two-dimensional (2D) beam-column elements and the shear walls are represented by four-node plane stress elements. The static governing equation for the combined system can be represented in incremental form astangent stiffness matrix of the frame, the global tangent stiffness matrix of the shear walls, the incremental displacement vector , and the external load vector at the nth iteration, respectively; and and =internal force vector of the frame andthe shear walls at the(n1)th iteration. Using the assumed stress-based finite-element method, the tangent stiffness matrix of the frame can be defined asWhere elastic property matrix and is a function of area , length, moment of inertia, and Youngs modulus of an element and and transformation matrix and the geometric stiffness matrix. The internal force vector in Eq. (1) corresponding to the frame can be defined aswhere =homogeneous part of the internal nodal force vector and =deformation difference vector. The detailed expressions for evaluating all the matrixes in Eqs.(1) through (3)for the frame are given by Gao and Haldar(1995) and are not repeated here due to lack of space.A four-node plane stress element is used to incorporate the presence of shear walls 3in the frame. An explicit expression of the stiffness matrix of the plate elements is necessary for efficient reliability analysis. To achieve this, the shape of the shear wall is restricted to be rectangular. Two displacement (horizontal and vertical) dynamic degrees of freedom are used at each node point. These are plane stress elements. Based on an extensive literature review and discussions with experts on finite-element methods, it was concluded that the rotation at a node point could be overlooked. The rotation of the combined system at the node point is expected to be very small and was independently verified using a commercially available computer program discussed later. To bring the shear wall stiffness into the frame structure, the components of the shear wall stiffness are added to the corresponding frame stiffness components in Eq. (1). The explicit form of a stiffness matrix of a four-node plane stress element can be obtained as (Lee 2000)where 2a and 2blong and short dimensions of the rectangular shear wall, respectively, tthickness of the wall, gthe ratio of b and a; i.e., gb/a. The matrixes A, B, C, and E in Eq. (4)can be represented asandwhere =modulus of elasticity and Poissons ratio of shear walls.Different types of shear walls are used in practice, but the reinforced concrete(RC)shear wall is the most commonly used and is considered in this study. Thus, two additional parameters ,namely, the modulus of elasticity and the Poissons ratio of concrete ,are necessary in the deterministic formulation as in Eq.(8). The tensile strength of concrete is very small compared to its compressive strength. Cracking may develop at a very early stage of loading. The behavior of a RC shear wall before and after cracking can be significantly different and needs to be considered in any realistic evaluation of the behavior of shear walls. There has been extensive research on cracking in RC panels. It was observed that the degradation of the stiffness of the shear walls occurs after cracking and can be considered effectively 4by reducing the modulus of elasticity of the shear walls. Based on the experimental research reported by Lefas et al. (1990), the degradation of the stiffness after cracking can vary from 40 to 70% of the original stiffness depending on the amount of reinforcement and the intensity of axial loads. In this study, the behavior of a shear wall after cracking is considered by introducing the degradation of the shear wall stiffness based on the observations made by Lefas et al. (1990). The shear wall is assumed to develop cracks when the tensile stress in concrete exceeds the prescribed value. The rupture strength of concrete , according to the American Concrete Institute (ACI, 1999) is assumed to be =7.53 , where =the compressive strength of concrete. Once the explicit form of the stiffness matrix of shear walls is obtained using Eq. (4), the information can be incorporated in Eq.(1) to study the static behavior of the combined system. The finite-element representation of the RC shear walls is kept simple in order to minimize the number of basic random variables present in the SFEM formulation. More sophisticated methods can be attempted in future studies, if desired. One of the main objectives of this study is to demonstrate the advancement of the reliability evaluation of complicated structural systems, and in that context, the method is appropriate. Reliability evaluation procedures are emphasized in this paper. The governing equation of the combined system consisting of the frame and shear walls, i.e., Eq. (1), is solved using the modified Newton-Raphson method with arc-length procedure.Verification of Deterministic Finite-Element Method FormulationThe success of any reliability method depends on the accuracy and efficiency of the deterministic method used. The basic deterministic method used in this study was discussed briefly in the previous section. A very sophisticated assumed stress-based nonlinear FEM algorithm was used to represent the steel frame. The shear walls are represented using information from experiments. The adequacy and accuracy of the FEM representation are necessary at this stage. A two-story two-bay frame structure with shear walls in each floor is considered, as shown in Fig. 1. All columns are made of a W1258 section and all beams are made of a W1860 section. The compressive strength of concrete in shear walls and its Poissons ratio are assumed to be 20.68 Mpa and 0.17, respectively. Principal stresses are calculated at node points. When the stress in the principal direction exceeds the prescribed tensile stress, the reduced modulus of elasticity of concrete is assumed to be 40% of for the element. All material and sectional properties required to analyze the combined system are given in Table 1. 5The combined system is subjected to dead, live, and horizontal load applied statically to represent wind or seismic load. The pattern of loading is shown in Fig. 1 and the intensities are given in Table 1. A computer program denoted hereafter as Shdyn is specifically developed to implement the concept. The program provides the structural responses at each node in terms of displacement and force. For the verification of the deterministic algorithm considered in the study, the lateral displacements at locations a and f, as shown in Fig. 1, are evaluated. Member forces are also estimated in terms of axial force and bending moments of a beam(nodes d and e ) and a column (nodes e and g), as shown in Fig. 1. The results are summarized in Table 2. Similar information was evaluated for the combined structural system using ABAQUS (Hibbit et al. 1998), a commercially available computer program. The results are shown in Table 2. The numerical procedures used in Shdyn and ABAQUS are different, but the structural responses are very similar. The example clearly demonstrates that the deterministic 6algorithm used in the study is accurate in predicting the behavior of a combined system consisting of a frame and shear walls. This verified deterministic algorithm is extended to consider the presence of uncertainty in the following sections.Reliability AnalysisStochastic Finite-Element FormulationHaldar and Mahadevan (2000b)proposed a stochastic finite element-based algorithm to estimate the reliability of a complicated structural system where the limit-state function or the performance function is implicit. It is based on the first-order reliability method (Haldar and Mahadevan 2000a). This concept is extended here to evaluate the reliability of a complicated structural system under static loading conditions.Without losing any generality, the limit-state function g can be expressed in terms of the set of basic random variables x (e.g., loads, material properties, and structural geometry), the set of displacements u and the set of load effects s(except the displacements, such as internal forces). The displacement uQD, where D is the global displacement vector and Q is a transformation matrix. The limit-state function can be expressed as g(x,u,s) 0. For reliability computation it is convenient to transform x into the standard normal space yy(x) such that the elements of y are statistically independent and have a standard normal distribution. An iteration algorithm is used to locate the design point (the most likely failure point)on the limit-state function using first-order approximation. During each iteration, the structural response and the response gradient vectors are calculated using finite-element models. The following iteration scheme can be used for finding the coordinates of the design point:whereTo implement the algorithm and assuming the limit-state equation has a general form of g(x,u,s)=0, the gradient of the limit-state function in the standard normal space can be derived aswhere =Jacobians of transformation (e.g., ). Once the coordinates of the design point y* are evaluated with a preselected convergence criterion, the reliability index b can be evaluated asThe evaluation of Eq. (11) will depend on the problem underconsideration and the performance functions used. The essential numerical aspects of SFEM were just discussed in the evaluation of the three partial derivatives and four Jacobians in Eq. (11). They are evaluated in the following sections in the context of a frame and RC shear wall structural system.Performance Functions7The safety or reliability of a structural system is always estimatedwith respect to predetermined performance criteria. The performance criteria are usually expressed in the form of limit-state functions. Two types of limit-state functions are commonly used in the engineering profession: strength and serviceability.Strength Performance FunctionsAccording to the American Institute of Steel Constructions (AISO) Load and Resistance Factor Design (LRFD) Design guidelines, the strength performance criteria for 2D steel frame members can be defined asand where =required tensile and compressive strength; =nominal tensile and compressive strength; =required flexural strength; and =nonminal flexural strength. and in Eqs. (13) and (14) are unfactored load effects. From theAISCs LRFD manual (1994), these parameters can be evaluated As=AFcr(compression) or =AFy(tension) (15)andwhereandwhere A=gross area of a member (in.2); Fy=specified yield stress (ksi); E=modulus of elasticity(ksi); k=effective length factor; lu=unbraced length of a member(inch); and r=governing radius of gyration about the plane of buckling (inch). Not all the membersin the frame are connected to the shear walls. The shear walls are expected to prevent local and lateral torsional buckling of steel members, thus improving their strength. To consider the strength failure probability of the weakest steel members, this study considers the failure of steel members where shear walls are mnot present.Serviceability Performance FunctionsFor the serviceability criterion, the limit-state function is represented aswher calculated displacement component and limitprescribed maximum value of the displacement component. As will be elaborated later, the vertical deflection at the midspan of beams and the lateral displacements at the top of the frame are considered to be the two serviceability performance functions in this study.8Implementation of Proposed SFEM to the Combined SystemTo implement the concept, the three partial derivatives and four Jacobians in Eq. (11) need to be evaluated in terms of random variables x, u, and s for all the performance functions to be considered.Evaluation of Partial DerivativesThe three partial derivatives in Eq. (11), namely, and for the strength limit states are evaluated first. Neither g function in Eqs. (13) and (14) contains any explicit displacement component, therefore 0. In order to evaluate the basic random variables in the limit-state functions need to be defined. The Youngs modulus E, area A, yield stress Fy , plastic modulus Zx , and the moment of inertia of a cross-section I along with the external force F are considered to be basic random variables. Therefore, it can be shown thatThus, substituting Eqs. (15)and (16)into Eqs. (13) and (14), each term of Eq.(21) can be evaluated. can also be derived by taking the partial derivatives withrespect to Pu and Mu asAs discussed earlier, only steel members where RC shear walls are not present are checked for the strength limit states. The steel members are expected to be weaker in strength in this case. Thus, although the parameters in Eqs. (13) and (14) are expected to be influenced by

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