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

Table 2.  Earthquake models

Case 1, the structures are simultaneously excited by the three normal components; the first component is acting along the strong horizontal structural direction, the second one along the weak horizontal structural direction and the third along the vertical direction (X_n, Y_n, Z_n).

Case 2, same as Case 1, but the horizontal components are interchanged (Y_n, X_n, Z_n).  This is another possibility of applying the normal components.

Case 3, the structures are simultaneously excited by the two horizontal normal components; the first component is acting along the strong horizontal structural direction and the second one along the weak horizontal structural direction (X_n, Y_n, 0).

Case 4, same as Case 3, but the horizontal components are interchanged (Y_n, X_n, 0). 

Case 5, the structures are excited only by the first normal horizontal component applied to the strong horizontal structural direction (X_n, 0, 0)

Case 6, the structures are excited only by the second normal horizontal component applied to the weak horizontal structural direction (0, Y_{n, 0})

Case 7, the structures are excited only by the third normal horizontal component in the vertical direction (0, 0, Z_n)

Case 8, same as Case 5, but the second component is applied along the strong direction (Y_n, 0, 0).

Case 9, same as Case 6, but the first component is applied along the weak direction (0, X_n, 0).

Similarly, another nine cases of analysis are considered when the principal components are applied.  They are: Case 10 (X_p, Y_p, Z_p); Case 11 (Y_p, X_p, Z_p); Case 12 (X_p, Y_p, 0); Case 13 (Y_p, X_p, 0); Case 14 (X_p, 0, 0); Case 15 (0, Y_p, 0); Case 16 (0, 0, Z_p), Case 17 (Y_p, 0, 0) and Case 18 (0, X_p, 0).  Cases 1 and 2 will give the actual response for normal components since the three components are simultaneously applied.  Cases 10 and 11 will give the actual response for principal components.  Thus, for four structures, twenty earthquakes, eighteen cases, and considering the responses to be elastic and inelastic, a total of 2880 analyses were required.

The accuracy of the combination rules for the normal horizontal components are specifically addressed in this section of the paper.  The structural responses obtained according to the SRSS and the 30% rules are estimated and compared to that given by the actual solution.  For the normal horizontal components the actual response will be given by Cases 3 and 4 while the combined response according to the rules will be obtained by combining Cases 5, 6, 8 and 9.  In addition, the combined effect is calculated by using λ = 40% instead 30%This additional combination rule will be referred hereafter as the 40% rule.  For comparison purposes, the following error terms are defined:


 

where the terms, 30% value, actual value, SRSS value, and 40% value represent the combined effect according to l = 30%, actual response, SRSS rule and l = 40%, respectively.  A negative error in the above equations implies that the combination rule under consideration underestimates the combined effect of both components.  The errors are calculated for the axial loads at ground level columns and for the total base shear for all the models.  Both, elastic and inelastic behavior are considered.  Results for elastic analysis are discussed first.  The errors for the axial load on columns of Model 1 are shown in Figs 2a through 2d.  It is observed that both the 30% and the SRSS rules reasonable overestimate the response for most of the earthquakes.  The combined response is underestimated only in a few cases.  In general, the curve for the 30% rule is over the corresponding curve for the SRSS rule.  In other words, the 30% rule is more conservative than the SRSS rule.  The errors for the axial load on columns of Models 2, 3 and 4, are similarly estimated but are not shown.  The major observations made for Model 1, also apply to these models.

The errors in terms of the total base shear for elastic behavior are shown in Figure 3.  As for the case of axial loads in columns, both the 30% and the SRSS combination rules accurately estimate the combined effect for the total base shear.  However, the errors are in general smaller for the total base shear.  The maximum underestimation error is about –5% for a few cases.  It is observed that the scatter in the error values tends to increase with the fundamental period of the frames.   As for the case of axial loads, the underestimation is more for the SRSS rule than for the 30% rule.

Fig. 2. Errors for axial load, Model 1, elastic behavior, normal horizontal components
 

Fig. 4 shows the errors for the inelastic axial loads on columns of Model 1.  As for the case of elastic axial loads, both the 30% and SRSS rules may underestimate the combined response. However, the number of cases and the magnitude of the underestimation are larger for the inelastic case.  The underestimation error is larger than –15% in many cases.  This indicates that the results for elastic analysis may be quite different from those of inelastic analysis. The implication of this is that the results obtained from elastic analysis of steel frame structures subjected to strong motions may be a very crude approximation. The errors in terms of axial load on columns of Models 2, 3 and 4 are also estimated.  The results are similar to those of Model 1.

Fig. 3.  Errors for  total base shear, elastic behavior, normal horizontal components

The errors for the total inelastic base shear are shown in Figs 5 for all the models.  It is observed that, as for the case of elastic base shear, the 30% and SRSS rules, in general estimate the combined effect very well.  The errors are positive or relatively small in most of the cases. The underestimation is always more for the SRSS than for the 30% rule.  In summary, whether elastic or inelastic analysis is used the combined total base shear is reasonably overestimated by the rules.  However, the combined response in terms of axial loads can be underestimated for inelastic behavior.  The errors when the frames are excited for the normal components have been considered so far.   Similar plots to those of Figs 2 through 5 were also developed for the principal components. But are not shown.  The major conclusion made for normal component also apply to principal components.  Only the average errors are presented as discussed below.

The averaged of the errors are presented in Table 3 for normal and principal components.  Elastic behavior is first discussed.  It is observed that, for low-rise frames (Models 1 an 2), the 30% rule is conservative practically in all the cases. The SRSS rule is conservative in most of the cases. For taller frames (Models 3 and 4) these two rules are either conservative or slightly underestimate the response.  The level of overestimation or underestimation varies from one column to another.  The 40% rule on the average practically overestimates the response for all models and columns.  Results also indicate that for inelastic behavior, the SRSS and 30% rules underestimate the response in many cases.  The corresponding errors can be up to -11% for the SRSS rule.  The results are quite similar for normal and principal components.

Accuracy of the rules for the three components

In this section of the paper the accuracy of the combination rules for the effects of the three components or earthquakes is studied.  The structural responses obtained according to the SRSS and the 30% rules are estimated and compared to that given by the actual response. 

For the three components of the earthquakes, Cases 1 and 2 will give the actual response for normal components while the combined response according to the rules is obtained by combining Cases 5, 6, 7, 8 and 9, which were defined in Section 6.  Similar plots to those of Section 6 for two components are developed for the three components but are not shown; only the mean values of the errors are presented.   The results for all models and columns are presented in Table 4.  It is observed that, for low-rise frames (Models 1 an 2) and elastic behavior, the 30% and SRSS rules are conservative practically in all the cases.  For taller frames (Models 3 and 4) these two rules are either conservative or slightly underestimate the response.  The level of overestimation is similar for all the columns.  The 40% rule on the average practically overestimates the response for all models and columns.  Results also indicate that for inelastic behavior the SRSS and 30% rules underestimate the response in many cases.  The corresponding errors can be up to -11% for the SRSS rule.   By comparison of Tables 3 (two components) and Table 4 (three components) it is observed that, in general, the 30% rule is more conservative than the SRSS rule when only the horizontal components are considered.  For the three components however, the SRSS is more conservative.  If the criteria to say that a rule is acceptable is that, on the average, it should give values of the error close to but greater than zero, these two commonly used rules (30% and SRSS) would be acceptable for elastic behavior but they would not be for inelastic behavior.  Results clearly indicate that the 40% rule gives error values, which, in general, are larger than and close to zero.  Consequently, if the percentile rule is used to estimate the combined response, in general, a value of 40% should be used for l. 

Table 4.  Average errors (%) for axial loads three components

Critical components

In order to identify the type of components that produce the maximum response, the ratio (P) of the actual response for principal components (Cases 10 and 11) and the actual response for normal components (Cases 1 and 2), is introduced.  The values of P for the axial loads acting at the base of the four column locations are estimated for the twenty earthquake time histories.   Only the statistics are presented.

The mean value and standard deviation of P (μ_P, σ_P) for different cases are summarized in Table 5 for elastic (actual earthquakes) and inelastic (scaled earthquakes) behavior.  Results indicate that la mean values of P vary from one column to another and from one model to another.  In general, the larger values of μ_P occur for interior columns and elastic behavior while the larger values of σ_P are in general for interior columns and inelastic behavior.  The most important observation that can be made is that μ_P is larger than unity practically for all the cases indicating that the principal components produce greater axial loads than the normal components, although the difference may be significant only in a few cases (up to 21%).


Effect of the vertical component

In order to evaluate the relative importance of the effect of the vertical component, the structural responses produced by this component acting alone is estimated and compared to those of the horizontal components.  Typical results are shown in Fig. 6 for Model 2, elastic behavior and normal components.  It is observed that, for lateral columns, the maximum of the two effects produced by each horizontal component [(X_n, 0, 0) or (0,Y_n, 0)] will correspond to the case where the orientation of the component is the same than that of the maximum eccentricity of the column with respect to the structural center of mass (see Fig. 1b).  It is illustrated in Fig. 6 where for lateral columns located on X direction, the load case (0, Y_n,, 0) produces the maximum response while for lateral column on Y direction the maximum effect is for (X_n,0, 0).  The most important observation that can be made is that the relative effect of the vertical component with respect to those of the horizontal components, represented by the Z ratio, is significant. In some cases this parameter is  larger than unity.

Results for the other models, type of components and inelastic behavior were also estimated but are not shown. The major observation made before also apply to these models:  the effect of the vertical component may be significant.  In total 64 plots (four models, four columns, two type of components and two levels of deformation) similar to those of Figure 6 were developed.   Only the statistics of Z are presented below.

The mean values of Z are presented in Table 6 for all the models, type of components and levels of deformation. Since the interior columns are close to the center of mass, the axial load produced by the horizontal components is small and consequently the Z ratio for these columns is not considered. Results in the table indicate that the μ_Z values significantly vary from one model to another and from one column to another without showing any trend.  Mean values of up to 0.80 are observed indicating the relative importance of the effect of the vertical component.   These members are expected to be designed as beam columns.   According to the AISC LRFD code the following equations should be satisfied:

where P_u is the required axial strength, M_ux and M_uy are the required flexural strength with respect to the major and minor axis, respectively; f_c is the resistance factor for compression and f_b the resistance factor for bending.  Other symbols were defined before.  The implication of this is that if the total axial load is not properly considered it will produce a detrimental effect on the behavior of the columns.  Thus, the effect of the vertical component not only should not be overlooked in design but also it should be explicitly considered no matter how complicated the analysis process becomes.

Accuracy of Estimating R_y as 0.85 of R_x.

As stated earlier, the ratio of the effect of the minor horizontal component to that of the major horizontal component is usually assumed to be 0.85.  As discussed below, the value of this ratio is important in the relative accuracy of the rules in the evaluation of the combined response.  Let us assume that R_X is the larger of the two effects and consequently R_Y will be the smaller one.  If Q denotes the ratio of the smaller to the larger effect, that is Q = R_Y /R_X, then for the horizontal components the SRSS rule can be expressed as:


 

The values of R are plotted in Fig. 7 for several values of Q and λ.  It is observed from this figure that for λ= 0.3, R_C2 and R_C1 are close each other.  From Q = 0 to ≈ 0.3, the 30% rule gives larger response that the SRSS rule (R< 1).  From Q » 0.3 to 0.65 the R ratio is still smaller than unity however, the SRSS combined value increases with respect to the 30% combined value.  For Q » 0.65 to 1.0 the R parameter is larger than unity and consequently the SRSS rule give values larger than the 30% rule.

If the SRSS combination rule were the exact solution, the minimum introduced error by using λ= 0.3 would be zero for Q »0.65, the maximum underestimation error would be about 4% for Q » 0.3 and the maximum overestimation error would be about 8% for Q = 1.0.  Thus, depending on the Q ratio the SRSS rule can give greater or smaller combined response than the 30% value.

The actual values of the Q parameter are estimated for the earthquakes and models considered in this study.  Only the columns (corner and interior) where the individual effects may be comparable are studied.  The mean values and standard deviation of Q are presented in Table 7.   It is observed that m_Q values vary from one model to another and from one column to another.  The values range from 0.21 to 0.69.  In general the values are smaller for corner columns than for interior columns. This difference, however, tend to decrease as the fundamental period of the models increases.   For a given column, the results in general are similar for both types of component.   Also, for a given column and type of components the results are similar for both levels of deformation (elastic and inelastic).  The most important observation that can be made from the table is the mean values of Q  (which range from 0.31 to 0.46 when averaged over all models and columns) obtained in this study are significantly smaller than the usually assumed value of 0.85.  For this reason the SRSS rule, when used for the horizontal components, is less conservative than the 30% rule in the estimation of the combined response, as shown in Section 6 of the paper.

Conclusions

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 rules to estimate the combined effect of the individual components of earthquakes. The rules are studied for normal and principal components, for local (axial loads) and global (total base shear) response parameters, elastic and inelastic behaviour and two and three components.  The critical orientation of the components is identified. Several models of Moment Resisting Steel Frames (MRSF) are used in the study.  The second issue threats with the relative magnitude of the effect of the vertical component (R_Z) con respect to those of the horizontal components (R_X and R_Y).  Finally, the accuracy of estimating the intensity of the weaker component (R_Y) as 85% of that of the stronger component (R_X) is studied. 

Results indicate that whether elastic or inelastic behavior occurs the combined total base shear is reasonably overestimated by all the rules.  The 30% and the SRSS rules, however, can underestimate the combined response in terms of axial loads for inelastic behavior.  This indicates that the results for elastic analysis may be quite different from those of inelastic analysis. The implication of this is that the results obtained from elastic analysis of steel frame structures subjected to strong motions may be a very crude approximation.  For the horizontal components the SRSS rule is less conservative than the 30% rule.  For the three components, the 30% rule is less conservative.  If the criteria to say that a rule is acceptable is that, on the average, it should give values of the error close to but greater than zero, these two commonly used rules would be acceptable for elastic behavior but they would not be for inelastic behavior.  Results clearly indicate that the 40% rule gives error values, which, in general, are larger than and close to zero practically in all cases.  Consequently, if the percentile rule is used to estimate the combined response, in general, a value of 40% should be used for l.  It is also observed that the principal components give the critical 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.  Mean values of up to 0.80 are observed.   Results also indicate that the ratio of the effect of the minor horizontal component to that of the major horizontal component is much smaller than the typical assumed value of 0.85.   The value obtained in this study is about 0.40. 

Acknowledgements

This paper is based on work supported by El Consejo Nacional de Ciencia y Tecnología (CONACyT) under grant 50298-J and by La Universidad Autónoma de Sinaloa (UAS) under grant PI-PROFAPI-06-06.  Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsors.

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