Constant Displacement Iteration Algorithm for Non-Linear Static Pushover Analysis

Abstract

The paper presents formulation and implementation of a new static push-over analysis algorithm for the seismic rehabilitation of structure, in accordance with “NEHRP Guidelines for the Seismic Rehabilitation of Buildings”. The concept of non-linear pushover analysis is described and one problem of steel structure with piping racks are examined. Earlier work contributed pushover analysis by controlled node method. Here we go with the different method with new problem statement. The sample structure is steel framed structure with piping racks with the G+10 story building. The final results give us the target displacement.

Introduction

I. INTRODUCTION

The method of nonlinear analysis is used to test structures under gravity loading and a lateral load pattern with monotonic displacement control, and it is also known as pushover analysis.

Pushover analysis is one of the streamlined approaches for non-linear technique to evaluate seismic structural deformation for existing structures. It is carried out for the base fixed structure to examine the yield displacement and succeeding inelastic behaviour.

In several current guidelines for retrofit seismic design, pushover analysis is applied mainly to evaluate the seismic capacity of existing structures.

According to FEMA (NEHRP guidelines for the seismic rehabilitation of buildings) nonlinear analysis is a valuable method for determining the inelastic strain and deformation needs of a structure as well as for identifying design flaws in the structure. Because nonlinear analysis is primarily concerned with elastoplastic materials and large displacements, the superposition effect cannot be used in this case.

When performing a pushover study, a sequence of consecutive elastic analyses is performed and then superimposed to approximate the force-displacement curve of the entire structure.

II. NSP (NONLINEAR STATIC PROCEDURE)

Under the Nonlinear Static Procedure (NSP), a model directly incorporating inelastic material response is displaced to a target displacement, and resulting internal deformations and forces are determined.

The nonlinear load-deformation characteristics of individual components and elements of the building are modelled directly. The building's mathematical model is subjected to lateral forces or displacements that increase monotonically until it either the target displacement gets exceeded or the structural failure.

The target displacement is intended to represent the maximum displacement likely to be experienced during the design earthquake. The target displacement may be calculated by any procedure that accounts for the effects of nonlinear response on displacement amplitude; one rational procedure is presented in FEMA 273Section 3.3.3.3.

A. Target Displacement

The target displacement δt for a building with rigid diaphragms (Section 3.2.4) at each floor level shall be estimated using an established process that accounts for the likely nonlinear response of the building. From roof displacement and base shearwe get capacity curve as shown in Fig. 1 which gives target displacement.

III. PROBLEM STATEMENT

Nonlinear static analysis has two methods to do the pushover analysis from that one is the displacement coefficient method which is very long and heavy method to do. To calculate target displacement, have to go through the FEMA 273 formulas.

For existing structure, it is the one of the convenient methods to calculate the target displacement.it is very important to find at which load the structure can be collapse or get the cracks. Loads coming on the structure is different at every time so the displacement of the structure is also different at each time here, we find the solution on that problem by finding the target displacement of the structure. So, we can get the maximum loads that structure can bear.

IV. REVIEW OF DISPLACEMENT COEFFICIENT METHOD

Many compromises were required to transform research results into the FEMA 273/356 nonlinear static procedure (Krawinkler). The C1 factor as defined in FEMA 356 is smaller than research indicates, as noted in FEMA 274.

Miranda (2001) points out that the C1 term should be derived from oscillator response values and not from the R-µ-T relations that are based on these responses, to avoid statistical bias in the results.

MacRae and Tagawa (2001) note that the coefficient C2 should approach unity as the strength of the pinched system approaches the strength required for elastic response.

Song and Pincheira (2000) find that the FEMA 273 recommendations provide conservative estimates of the displacement amplification factors for degrading oscillators with periods greater than 0.3 sec on firm soils, and are unconservative at shorter periods.

Lew and Kunnath (2000) compare demands computed using the LSP, LDP, NSP, and NDP of FEMA 273 with the acceptance criteria of the document for an instrumented 7-story reinforced concrete frame building (the Holiday Inn, Van Nuys, California) subjected to ground motions having a 10% probability of exceedance in 50 years, as developed for the FEMAfunded SAC1 project. A triangular load pattern was used in the pushover analysis, and member plastic rotations were calculated from chord rotations as suggested in FEMA 273.

The beam plastic rotation demands determined in this way were similar to the mean beam plastic rotations determined by nonlinear dynamic analysis, with pushover analysis underestimating the plastic rotation demands in the columns relative to those determined by nonlinear dynamic analysis, particularly in the upper stories. As a part of the current work for the ATC-55 project, the Displacement Coefficient Method as presented in FEMA 356 was evaluated for a wide range of parametric values.

V. PROBLEM SOLUTION

The peak displacement in a nonlinear system is estimated as the peak displacement in an elastic system (Keff = Kinitial) multiplied by a series of coefficients. Of primary interest here are the coefficients C1, the ratio of the peak displacement in the inelastic system and the peak displacement in the elastic system having the same period of vibration; C2, which accounts for the effect of pinching in the loaddeformation relation; and C3, which accounts for second-order (P-Delta) effects. FEMA 356 is the primary source of documentation for the Displacement Coefficient Method (DCM).

For calculating target displacement some terms are considered which is given in FEMA 273.

A. Control Node

The NSP requires definition of control node in a building. These Guidelines consider the control node to be the centre of mass at the roof of a building; the top of a penthouse should not be considered as the roof. The displacement of control node is compared with the target displacement a displacement that characterizes the effects of earthquake shaking.

???????B. Lateral Load Patterns

Lateral loads shall be applied to the building in profiles that approximately bound the likely distribution of inertia forces in an earthquake. For three-dimensional analysis, the horizontal distribution should simulate the distribution of inertia forces in the plane of each floor diaphragm. For both two- and three-dimensional analysis, at least two vertical distributions of lateral load shall be considered.

The first pattern, often termed the uniform pattern, shall be based on lateral forces that are proportional to the total mass at each floor level. The second pattern, termed the modal pattern in these Guidelines, should be selected from one of the following two options:

1. A lateral load pattern represented by values of Cvx given in FEMA 273, which may be used if more than 75% of the total mass participates in the fundamental mode in the direction under consideration; or
2. A lateral load pattern proportional to the story inertia forces consistent with the story shear distribution calculated by combination of modal responses using (1) Response Spectrum Analysis of the building including a sufficient number of modes to capture 90% of the total mass, and (2) the appropriate ground motion spectrum.

???????C. Period Determination

The proposed direction's effective fundamental period Te shall be calculated using the force-displacement relationship of the NSP. The nonlinear relation between base shear and displacement of the target node shall be replaced with a bilinear relation to estimate the effective lateral stiffness, Ke, and the yield strength, Vy, of the building. The effective lateral stiffness shall be taken as the secant stiffness is calculated at a base shear of 60% of the yield strength. The effective fundamental period Te shall be calculated as:  

  ???????D. Analysis of three-dimensional Models

Static lateral forces shall be imposed on the three-dimensional mathematical model corresponding to the mass distribution at each floor level. Independent analysis along each principal axis of the three-dimensional mathematical model is permitted unless multidirectional evaluation is required.

???????E. Analysis of two-dimensional Models

Mathematical models describing the framing along each axis (axis 1 and axis 2) of the building shall be developed for two-dimensional analysis. If multidirectional excitation effects are to be considered, component deformation demands and actions shall be computed for the following cases: 100% of the target displacement along axis 1 and 30% of the target displacement along axis 2; and 30% of the target displacement along axis 1 and 100% of the target displacement along axis 2.

VI. DETAIL DESCRIPTION OF PROPOSED ALGORITHM

The assemble algorithm in the figure is the NSP solution algorithm where the ∂t is the target displacement, lateral load F, Ki is the elastic lateral stiffness of building in the direction under consideration, Ke is the effective lateral stiffness of building in the direction under consideration. QG is the gravity load which is equal to 0.9 of dead load. Initially Ke, Ki and F is on the zero and the load vector Q is equal to QG.

???????A. First Incremental Step

The first step of the algorithm is the incremental step which is significantly different from subsequent steps. In this step the analysis of structure is done on the only gravity load QG. Before this step all the factors are set to be zero. If the Q < F then the increment in the Q is 0.6Q which is new F. such iterative step is first step in nan linear static analysis.

???????B. Successive Step

In the incremental step whatever the F gets after increment in it this will be added in QG and get the value of the Q if it is greater than F then the iterative scheme is started. The total external load is used to calculate effective lateral stiffness.

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VIII. NOTATIONS

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Conclusion

The presented algorithm is based upon requirements of the Nonlinear Static Procedure specified by the NEHRP Guidelines for Seismic Rehabilitation of Buildings. It is formulated for two and three-dimensional analysis, with the three-dimensional analysis initially applied and tested in this work.An algorithm successfully implements displacement coefficient method, which gives us the target displacement up to the collapse mode and thus agree with the nature of push-over strategy. It involves the number of iterative steps which gives us the displacement up to the collapse mode. The displacement that we get is the target displacement which shows when lateral load collapse on the structure then the structure can displace up to that target displacement. Beyond the target displacement it has chances to fail the structural members or has chances to collapse the whole structure. Science existing structures instability is eliminated and also if any performance design require it can be done further after the pushover analysis.

References

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Copyright

Copyright © 2022 Prachi Dattatray Kaldhone. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.