Alfredo Reyes-Salazar,  Arturo López-Barraza, Luis A. López-López, Juan I. Velázquez-Dimas
Universidad Autónoma de Sinaloa, reyes@uas.uasnet.mx

Abstract

Several issues regarding multi-components seismic response analysis of structures are addressed in this paper.  The first issue is related to the accuracy of the commonly used rules to estimate the combined effect of the individual components of earthquakes. The rules are studied for normal and principal components, for local and global response parameters, elastic and inelastic behavior and two and three components. The critical orientation of the components is identified.  The second issue threats with the relative magnitude of the effect of the vertical component with respect to those of the horizontal components.  Finally, the accuracy of estimating the effect of the weaker horizontal component (R_Y) as 85% of that of the stronger horizontal component (R_X) is studied.  Results indicate that the total base shear is reasonable estimated by the rules.  However, they can underestimate the combined response in terms of axial loads for inelastic behavior. For the horizontal components the SRSS rule is, in general, less conservative than the 30% rule.  For the three components the 30% rule is less conservative.  It is concluded that if the percentile rule is used to estimate the combined response a value of 40% should be used instead of 30%.  It is also observed that the principal components give the maximum response and that the ratio of the effect of the vertical component to those of the horizontal components can be significant and consequently should explicitly be used in design.  Values larger than unity are observed in some cases.   Results also indicate that the ratio of the effect of the minor horizontal component to that of the major horizontal component is much smaller that the typical assumed value of 0.85.   The value obtained in this study is about 0.40. 

Introduction

After catastrophic damages during some recent earthquakes around the world, seismic analysis and design procedures have been significantly modified.  Several methods with different degrees of sophistication have been suggested in many codes (IBC 2003, RCDF 2004) including the equivalent lateral force procedure and several types of dynamic analysis procedures (modal response spectra analysis, linear time-history analysis, and nonlinear time-history analysis). 

Our understanding of the earthquake phenomenon has improved significantly during the last years; however, there are many issues that require our attention regarding its effects on the response of structures.  Energy released during an earthquake travels in the form of waves.  They are measured in the form of two horizontal and one vertical translational acceleration time histories.  Although, earthquakes can cause rotational excitations, they are not measured and are completely ignored in the analysis.  In addition, for far-source ground motions, the effect of the vertical component is usually smaller than those of the horizontal components and is consequently neglected.  Additional bases to neglect the vertical component effect are that building designs allow for gravity loads, which provides for a high factor of safety in the vertical direction.   Thus, when we analyze a structure, we generally apply either one assumed or recorded horizontal component in a specific way or two components along the two major axes of the structure, sometimes ignoring the orientation of maximum response.

Article 1620.2.10 of the IBC code (2003) states “The direction of application of seismic forces used in design shall be, that which will produce the most critical load effect in each component.  The requirement will be deemed satisfied if the design seismic forces are applied separately and independently in each of the two orthogonal directions.” Later, in Article 1620.3.2 for the design of common structures with various plan irregularities belonging to Seismic Design Category C and D, IBC states “The critical direction requirement of Section 1620.2.10 will be deemed satisfied if one hundred percent of the forces in one direction are added to the 30 percent of the forces in the perpendicular direction.  Alternatively, the effects of the two orthogonal directions are permitted to be combined on a square root of the sum of the squares (SRSS) basis. When the SRSS method of combining directional effects is used, each term computed shall be assigned the sign that will result in the most conservative result.” The two combination procedures discussed above will be denoted hereafter as the 30% and the SRSS combination rules, respectively. The relative magnitude of the effects of the components will be important in the relative accuracy of the rules.  Usually, the ratio of the effect of the minor horizontal component is estimated as 85% of that of the major horizontal component.   The RCDF (2004) code states similar requirements for the evaluation of the combined responses of the seismic components. The codes however, do not specify the type of effect or the type of structures to be considered.   

The above discussions clearly identify several issues that need our attention: (1) how to combine the separated responses in different directions to obtain the overall response?, (2) what is the critical orientation of the components?, (3) what is the relative importance of the vertical component?, and (4) what is the relative importance of the weak horizontal component compared to that of the strong horizontal component?  These issues are studied in this paper.  To study them comprehensively, one possible attractive option would be to estimate the response of structures as accurately as possible, preferably by using sophisticated three-dimensional nonlinear time history analysis, and compare the results obtained by the simplified practices.  The comparison may increase our understanding and may identify the limitations of the simplified approaches.

Literature Review

The critical orientation of the components as well as the ways of combining their individual effects have been of interest to the civil engineering profession. Penzien and Watabe (1975) stated that the three components of an earthquake are uncorrelated along a set of axes generally denoted as principal axes.  The major principal axis is horizontal and directed toward the epicenter, the intermediate axis is horizontal and perpendicular to the orientation of the major component, and the minor principal axis is vertical.  The critical response could be obtained when these components are applied.  López and Torres (1996) proposed a method to estimate the critical angle of incidence. Smeby and Der Kiureghian (1985) observed that, for response spectra analysis of linear structures, when the two horizontal principal components are not along the structural principal axes, the effect of correlation is small and that if the two horizontal components have identical or nearly identical intensities, then the effect of correlation disappears.  Newmark (1975) and Rosenblueth and Contreras (1977) proposed the Percentage Rule to approximate the combined response as the sum of the 100% of the response resulting from one component and some percentage (λ) of the responses resulting from the other two components. To combine the two horizontal components, Newmark (1975) suggested λ to be 40% and Rosenblueth and Contreras (1977) suggested λ to be 30%.  Wilson et. al. (1995) concluded that the Percentage Rule could underestimate the design forces in certain members.  More recently, many other studies attempted to combine the seismic responses due to two or three components (Correnza and Hutchinson 1994, Wilson et. al. 1981).  These studies are limited in scope. They used simple one-story or two degree-of-freedom concrete structures.

Der Kiureghian (1981) and Wilson et. al. (1995) proposed the Complete Quadratic Combination (CQC) rule to combine modal responses due to a single seismic component.  Smeby and Der Kiureghian (1985) proposed an extension of the CQC rule, known as the CQC3 rule to combine modal responses due to the three seismic components.  Smeby and Der Kiureghian (1985) and Lopez and Torres (1996) verified the application of the CQC3 rule by considering building-type structures with rectangular geometry.  Menun and Der Kiureghian (1981) extended these studies by considering more complex three-dimensional curved bridge structures subjected to two horizontal components. They compared the results of the CQC3 rule with those of the SRSS, the 30% (λ = 0.3), and the 40% (λ = 0.4) rules and examined the shortcomings of these three rules.  López  et. al. (2001) conducted a similar study to combine the two horizontal components with a range of one-story systems with symmetrical and unsymmetrical plan, and two multi-story buildings.  Hernández and López (2003) extended the work of López et. al. (2001) by considering the effect of the vertical component. They observed that if a principal component does not coincide with the vertical direction, the critical response could be underestimated.

Most of the previous studies on combination rules were limited to elastic analysis applied to simplified plane concrete frames connected by rigid diaphragms and/or only few stories high. They did not consider the inelastic behavior of the structural elements or the appropriate energy dissipation mechanisms.  In a typical strong-column weak-beam (SCWB) moment resisting steel frame (MRSF), plastic hinges are expected to be formed in the weak elements. Reyes-Salazar et. al. (2000) found that SCWB steel frames are very efficient in dissipating earthquake-induced energy.  If a SCWB is modeled as a frame with rigid diaphragms, one of the most important sources of energy dissipation, i.e. dissipation of energy at plastic hinges will be lost and the structural behavior will be different.  Numerical studies by Wang and Wen (2000) showed that the response of SCWB steel buildings would be significantly underestimated if they were modeled by shear-beams assuming rigid diaphragms.  Recently, Reyes-Salazar et al. (2004) observed that both the 30% and the SRSS rules could underestimate the combined response and the energy dissipation mechanism should be considered as accurately as possible. Thus, it is essential that the SCWB steel frames should be analyzed as complex multi-degree of freedom (MDOF) systems. 

In this study, the four questions raised earlier, i.e., the issues related to the combination of the individual effects of the components, the critical orientation, the ratio of the effect of the vertical component to those of the horizontal components and the ratio of the effect of the minor horizontal component to that of the major horizontal component, are comprehensively addressed. The responses of MRSF are specifically studied. To study the directionality issue, the normal and principal components of an earthquake are considered.  Recorded horizontal time histories will be denoted as normal components.  When they are transformed to uncorrelated components following the procedure suggested by Penzien and Watabe (1975), they will be denoted as principal components.

Combination Rules

The combination rules are formally defined in this section.  For the ease of discussion, R_X will represent hereafter the maximum absolute load effect at a particular location when the structure is excited by the horizontal X component of a given earthquake.  Similarly, R_Y and R_Z will denote the corresponding maximum absolute load effect at the same location when the structure is excited by the horizontal Y and the vertical component of the earthquake, respectively.  The load effects produced by each component can be calculated using many simplified methods including the equivalent lateral load procedure, modal analysis, and time history analysis.  Then, the combined effect can be calculated as the most unfavorable of:

The above Eqs. represent the Percentage Rule.   If λ = 0.3 is used, it represents the 30% rule for three components.  According to the SRSS rule the combined response is given by

Despite these rules seem to be simple to apply; there is no indication in the codes on how to select the critical directions for the seismic components nor the types of effects to be considered.   As stated earlier, Penzien and Watabe (1975) observed that during the strong motion phase of an earthquake the ground components are uncorrelated along a well-defined orthogonal system of axes defined as the principal axes of the motion.  It could represent the critical condition.  The basic assumption of the SRSS rule is that there is no correlation between the orthogonal components.

Mathematical Formulation

The authors and their associates (Gao and Haldar 1995, Reyes-Salazar 1997) developed an efficient finite element-based time-domain nonlinear analysis algorithm, which is used to estimate the effect of the seismic components on the overall structural response.  The procedure estimates nonlinear seismic responses of steel frames considering all major sources of energy dissipation.  Material nonlinearity and geometric nonlinearity are considered.  Considering its efficiency, particularly for steel frame structures, the assumed stress-based finite element method (Kondo and Atluri 1987) is used.  Using this approach, an explicit form of the tangent stiffness matrix is derived without any numerical integration.  Fewer elements can be used in describing a large deformation configuration without sacrificing any accuracy.  Furthermore, information on material nonlinearity can be incorporated in the algorithm without losing its basic simplicity.  It gives very accurate results and is very efficient compared to the displacement-based approach.  The procedure has been studied and verified with existing theoretical and experimental results.

The geometric and material nonlinearities are considered in the tangent stiffness matrix.  The mathematical details of the derivation are not shown here, but can be found in the literature (Kondo and Atluri 1987).  The material is considered to be linear elastic except at plastic hinges.  Concentrated plasticity behavior is assumed at plastic hinge locations.  In the past, several analytical procedures were proposed to predict the deformation of elasto-plastic frames under increasing seismic and static loads.  However, most of these formulations were based on small deformation theory.  In this study, each elasto-plastic beam-column element can experience arbitrary large rigid deformations and small relative deformations.  Thus, in addition to the elastic stress-strain relationships, the plastic stress-strain relationships need to be incorporated into the constitutive equations if a given yield condition is satisfied.  Several yield criteria have been proposed in the literature in terms of stress components or nodal forces.  Since the nodal forces can be obtained directly from the proposed method, the yield criteria used here is expressed in terms of nodal forces.  When the combined action of the resultant stresses satisfies a prescribed yield function at a given end of an element, a plastic hinge is assumed to occur instantaneously at that location.  Plastic hinges are considered to form at the ends of the beam-columns elements.  The yield function depends on both, the type of section and loading acting on the beam-column element (Mahadevan and Haldar 1991). 

The yield function for three-dimensional beam-column elements has the following general form: where P is the axial force, M_x and M_y are the bending moments with respect to the mayor and minor axis, respectively, M_z is the torsional moment, s_y is the yield stress, and l_p is the location of the plastic hinge. For the W-type sections of the models used in this study, this equation has the following particular form:

where P_n is the axial strength, M_nx and M_ny are the flexural strength with respect to the major and minor axis, respectively and M_nz is the torsional strength.

The additional axial deformations and relative rotations produced by the presence of plastic hinges are taken into account in the stiffness matrix and the internal force vector of the plastic stage. Explicit expressions for the elasto-plastic tangent stiffness matrix and the elasto-plastic internal force vector are also developed.  The mathematical derivations can be found in the literature (Kondo and Atluri 1987). Depending on the level of earthquake excitation, in a typical structure, all the elements may remain elastic, or some of the elements may remain elastic and the rest yield.  The structural stiffness matrix and the internal force vector can be explicitly developed from the individual elements and the particular state (elastic or plastic) they are in.  Based on an extensive literature review, it is observed that viscous Rayleigh-type damping is commonly used in the profession and is used in this study (Clough and Penzien 1993).  The consideration of both the tangent stiffness and the mass matrices is a rational approach to estimate the energy dissipated by viscous damping in a nonlinear seismic analysis.  The mass matrix is assumed to be concentrated-type. The step-by-step direct integration numerical analysis procedure and the Newmark β method (Bathe 1982) are used to solve the nonlinear seismic governing equation of the problem.  

A computer program has been developed to implement the solution procedure.  The program was extensively verified using information available in the literature.  The structural response behavior in terms of members' forces (axial load, shear force and bending moment), total base shear and interstory displacements, can be estimated using this computer program.

Structural Models

To study the issues raised earlier, four three-dimensional MRSF structures as shown in Fig. 1a, with different dynamic characteristics, are considered. They are one, three, eight, and fifteen story structures and will be denoted hereafter as Models 1, 2, 3 and 4, respectively. The locations of the columns are shown in Fig. 1b.  For each model, four plane frames are used as shown in Fig. 1a: two interior (M_xi and M_yi) and two exterior (M_xe and M_ye). The plane frames in both directions were designed according to the UBC standards and then modified following the strong column-weak beam (SCWB) concept.  The seismic design load was computed for seismic zone 4.  The dead and live loads were 5.8 and 2.9 kN/m² (≈ 120 and 60 psf), respectively. The member sizes are summarized in Table 1.  The story height for all the models is a constant of 3.66 m and their bay width is 7.32 m in both directions.  The fundamental periods of the models in the major direction (X direction) are 0.20, 0.47, 1.06 and 1.55 sec., respectively. The corresponding values for the minor direction are 0.33, 0.73, 1.64 and 2.28 sec., respectively.  In all these frames, the columns are assumed to be made of Grade-50 steel and the girders are of A36 steel.  The columns are assumed to be fixed at the base and the connections fully restrained (FR).  

The frames are modeled as MDOF systems.  Each column is represented by one element and each girder is represented by two elements, having a node at the mid-span.  Each node is considered to have six degrees of freedom.  Thus, the total number of degrees of freedom for Models 1, 2, 3 and 4 are 240, 720, 1920 and 3660, respectively.  The four models are excited by twenty recorded earthquake motions, as listed in Table 2, in time domain. They are denoted hereafter as Earthquakes 1 to 20.  Epicentral distance (ED), Magnitude (M), peak ground acceleration (PGA) and predominant period (T) are given in the table. The damping is considered to be 5% of the critical damping; the same damping is used in the codified approaches.  As will be discussed later in detail, the real earthquakes are applied first and later they are scaled up in such way that produce approximately 1.5% interstory displacement in all frames to study the inelastic response.

Accuracy of the Rules for the Horizontal Components

For the ease of discussion, the following notations will be used in the remainder of the paper.  Numerically, an earthquake excitation will be represented by three acceleration time histories: two in the horizontal directions and one in the vertical direction. The first component will be denoted as X.  The second and third components will be denoted as Y and Z, respectively.  They will be denoted as X_n, Y_n and Z_n when normal components are used to excite structures, and as X_p, Y_p and Z_p, if principal components are used instead.  Hence, the notations (X_n, Y_n, Z_n) indicate that the first, second and third normal components are simultaneously applied to the major horizontal, minor horizontal, and vertical directions of the structure, respectively.  The notations (Y_p, 0, 0) indicate the structure is excited by only the second principal component acting along the major structural axis.  In order to fulfill the objectives of the study, the following cases of analysis are considered for each structure, earthquake and elastic and inelastic behavior: 

Table 1. Member sizes