Equivalent Static Force Research Articles

On calculating equivalent static seismic forces in the 2005 National Building Code of Canada

Several major changes were introduced in the seismic design provisions of the 2005 edition of the National Building Code of Canada (NBCC). The lateral earthquake design force at the base and the lateral force distribution along the building height depend on the design spectra and on modification factors that, in most cases, require a large number of interpolations and calculations. This note presents a spreadsheet that facilitates determination of the 2005 NBCC seismic design forces from the equivalent static force procedure.

Disproportionate collapse analysis of cable-stayed steel roofs for cable loss

Disproportionate collapse has been identified lately as a real cause of failure for structural engineering projects. Rare and unexpected, the phenomenon of disproportionate collapse usually results to many fatalities and thus, its analysis and mitigation is deemed necessary. This work describes the analysis of a cable-stayed steel roof under the scenario of a cable loss. The event of a cable loss is assumed to be brittle, while relevant recent recommendations suggest the application of a scaled equivalent static force at the points of the anchorage of the cable but in the opposite direction of the original cable force. In this paper, three different conditions have been considered in order to study the effect of the cable loss into the overall structural response of a typical cable-stayed roof; the level of the equivalent nodal load in the opposite direction of the original cable force varies. The steel structure of the roof, in its complexity, closer to responding as a cable-stayed bridge rather than a steel roof provides a useful template for conclusions; several topics regarding disproportionate collapse and cable losses are discussed.

Determination of Behavior Coefficient of Prefabricated Concrete Frame ith Prefabricated Shear Walls

Increasing population in recent decades has made the necessity of quick low cost construction unavoidable. Meanwhile the use of prefabricated concrete frame with prefabricated concrete shear walls due to better quality of performance and reduction of manufacturing time is a good option for public multiple floor buildings. These structures in addition to resistance to seismic loads have more flexibility in methods of construction. There is little information about the exact behavior of prefabricated concrete frame with shear walls under earthquake and dynamic loads. Hence much research has been done by various researchers on the seismic behavior of these structures in the form of experimental and analytical models. Today, the main section of the seismic design of buildings is done based on equivalent static force method and calculating earthquake force of design from earthquake linear spectrum by applying a reduction coefficient called behavior coefficient of structure that embrace philosophy of design. So the necessity of determining behavior coefficient with respect to its importance in seismic design of structures seems essential. The behavior coefficient of structure is a coefficient that includes inflexible function of structure and indicates strength and hidden ductility of structure in inflexible stage. In this study various types of pre-fabricated concrete frames together with pre-fabricated 4 and 8-storey shear walls with 1 and 3 bays are studied. The effect of two kinds of uniform and triangular loading on behavior factor has also been taken into consideration. Non-linear static analysis method (Push-over) has been used in order to determine the behavior factor. On the whole, the results have shown that by increasing the bays and building storeys the structural behavior coefficient was increased.

EFFECTS OF RANDOMLY DISTRIBUTED INFILL ON COLUMNS OF REINFORCED CONCRETE FRAMES WITH SOFT GROUND STOREY

Reinforced concrete (RC) frames with an open ground floor and various infill distributions have been analyzed for seismic loadings by the finite element method. The infills have been modeled by diagonal struts. Focus is placed on the effects of infill distribution on various structural responses, including (i) the lateral deflection, (ii) the column axial forces, (iii) the column bending moments, (iv) the base shear, and (v) the natural period of the frame. The equivalent static force method (ESFM) and response spectrum method (RSM) for linear structures have been applied, and the results obtained have been compared. It was found that the structural responses do not change appreciably by the ESFM analysis for random infill distributions, while they increase noticeably in the RSM analysis. This manifests the inadequacy of using the ESFM for general purposes, for which modifications were proposed in this paper for the design of RC columns. As the natural period of the RC frame converges with the code equations only for higher amounts of infill, it is necessary to incorporate the amount and distribution of infill in the dynamic analysis of RC frames.

Static equivalent method for the kinematic interaction analysis of single piles

This paper presents a static equivalent approach to estimate the maximum kinematic interaction effects on piles subjected to lateral seismic excitation. Closed-form expressions are reported for the evaluation of the maximum free-field soil movements and for the computation of maximum pile shear force and bending moments. Firstly, modal analysis, combined with a suitable damped response spectrum, is used to evaluate the maximum free-field response. Secondly, the pile is schematised as a Winkler's beam subjected to equivalent static forces defined according to soil vibration modal shapes and amplitude. The method may be applied by using response spectra suggested by National Standards or those obtained with accelerograms. The procedure proposed may be conveniently implemented in simple spreadsheets or in commercial finite element programs and easily used by practicing engineers. Method accuracy is demonstrated by comparing the results with those obtained with a more rigorous model. Good results may be achieved by considering only the first soil vibration mode making the procedure straightforward for practical design purposes.

POD-based representation of the alongwind Equivalent Static Force for long-span bridges

This paper develops and discusses a method by which it is possible to evaluate the Equivalent Static Force (ESF) of wind in the case of long-span bridges. Attention is focused on the alongwind direction. The study herein carried out deals with the classical problems of determining the maximum effects due to the alongwind action and the corresponding ESFs. The mean value of the maximum alongwind displacement of the deck is firstly obtained both by the spectral analysis and the Gust Response Factor (GRF) technique. Successively, in order to derive the other wind-induced effects acting on the deck, the Gust Effect Factor (GEF) technique is extended to long-span bridges. By adopting the GRF technique, it is possible to define the ESF that applied on the structure produces the maximum alongwind displacement. Nevertheless the application of the ESF so obtained does not furnish the correct maximum values of other wind-induced effects acting on the deck such as bending moments or shears. Based on this observation, a new technique is proposed which allows to define an ESF able to simultaneously reproduce the maximum alongwind effects of the bridge deck. The proposed technique is based on the GEF and the POD techniques and represents a valid instrument of research for the understanding of the wind excitation mechanism.

Seismic force modification factors for light-gauge steel-frame - wood structural panel shear walls

Design guidelines for laterally loaded light-gauge steel-frame – wood structural panel shear walls are not available in Canadian codes. A design method for the calculation of shear stiffness and strength has been developed for use with the 2005 National Building Code of Canada (NBCC), however. This method was based on the analysis, using an equivalent energy elastic–plastic (EEEP) approach, of over 180 single-storey shear wall tests of various configurations. Ductility-related (Rd) and overstrength-related (Ro) force modification factors also need to be defined to calculate equivalent static seismic forces following the 2005 NBCC. This paper describes the development of these two factors based on the EEEP analysis of the shear wall test results. To verify the "test-based" Rdand Rovalues, nonlinear time-history dynamic analyses of two representative buildings were carried out using a suite of 10 earthquake records scaled to the 2% in 50 year uniform hazard spectrum (UHS) for Vancouver, British Columbia. Preliminary values have been determined for the force modification factors, namely Rd= 2.5 and Ro= 1.7.Key words: shear wall, light-gauge steel, wood structural panel, seismic, R value.

Drift-controlled design of reinforced concrete frame structures under distant blast conditions—Part II: Implementation and evaluation

The model for the calculation of an equivalent static force (ESF) and the design procedure with ESF for single-degree-of-freedom (SDOF) systems presented in the first part of the two-part paper are extended into the design for a reinforced concrete (RC) frame structure under distant blast conditions. An empirical formula for the ESF factor involved in the ESF model is presented based on sample points obtained by comparing the nonlinear dynamic responses of frame structures under the blast loading with the corresponding nonlinear pushover analysis of the structure due to an ESF. The use of the method is demonstrated with two six-storey RC frame structures. Numerical verification of the method indicates that the maximum inter-storey drift ratios (MIDR) of the two designed frame structures in comparison to their respective targets are conservative to some extent. The reasons that may lead to the conservative designs are discussed.

Drift-controlled design of reinforced concrete frame structures under distant blast conditions—Part I: Theoretical basis

Proper control levels of lateral drifts anticipated for reinforced concrete (RC) frame structures within the predefined performance level become crucial when the frame structure is subjected to distant intense surface explosions. For this purpose, a new design method is presented in a two-part paper based on the transformation of a blast loading into an equivalent static force (ESF). The ESF is calculated in such a manner that the same maximum inter-storey drift ratio (MIDR) under the blast loading will be reproduced. The first part of the two-part paper focuses on the computational model of ESF for a single-degree-of-freedom (SDOF) system and the design method based on ESF with the requirement for controlling its maximum displacement response to achieve the specified target displacement. Numerical examples have been included to illustrate the method while the verifications of the dynamic responses of the designed SDOF system are performed with nonlinear dynamic analyzes. The numerical results indicate that the target displacement is well met for the designed SDOF system in resisting a given blast loading. Extension of the computational model of ESF and the corresponding design method with ESF for a SDOF system into a RC frame structure will be further discussed in the companion paper.

Toward more rational criteria for determination of design earthquake forces

The preliminary design of most buildings is based on equivalent static forces specified by the governing building code. The height wise distribution of these static forces seems to be based implicitly on the elastic vibration modes. Therefore, the employment of such a load pattern in seismic design of normal structures does not guarantee the optimum use of materials. This paper presents a new method for optimization of dynamic response of structures subjected to seismic excitation. This method is based on the concept of uniform distribution of deformation. In order to obtain the optimum distribution of structural properties, an iterative optimization procedure has been adopted. In this approach, the structural properties are modified so that inefficient material is gradually shifted from strong to weak areas of a structure. This process is continued until a state of uniform deformation is achieved. It is shown that the seismic performance of such a structure is optimal, and behaves generally better than those designed by conventional methods. By conducting this algorithm on shear-building models with various dynamic characteristics subjected to 20 earthquake ground motions, more adequate load patterns are introduced with respect to the period of the structure and the target ductility demand.

Seismic Performance of Concentrically Braced Steel Frames in Multistory Buildings with Mass Irregularity

The influence of mass irregularity on building seismic response is examined for an eight-story concentrically braced steel frame with different setback configurations resulting in sudden reductions in plan dimensions and seismic weight along the height of the structure. Three locations of mass discontinuity were considered (25, 50, and 75% of the building height), together with two ratios of seismic weight (200 and 300%). A reference regular structure was also considered for comparison. The design of each structure was performed according to the proposed 2005 National Building Code of Canada NBCC provisions using two analysis methods: The equivalent static force procedure and the response spectrum analysis method. Although severe, the mass irregularity conditions considered in this study were found to have a limited negative impact on the seismic performance of the structures designed with the static analysis method. The performance of irregular structures exhibiting lower performance could be improved by...

Application of the equivalent static force procedure for the seismic design of multistorey buildings with vertical mass irregularity

The seismic forces and deformations obtained from the equivalent static force procedure and the dynamic analysis method proposed for the 2005 National Building Code of Canada are compared for 4-, 8-, 12-, and 16-storey buildings with structural mass irregularity. Setbacks resulting in 200% and 300% mass discontinuities located at 25%, 50%, or 75% of the building height were considered. The buildings are located in Vancouver and Montréal, and lateral loads are resisted by concentrically braced steel frames. The storey shear forces, overturning moments, and storey drifts obtained from static analysis were found to exceed the values determined from dynamic analysis for all irregular structures except at the base of "podium-type" buildings located at the Vancouver site.Key words: mass, static, dynamic, seismic, shear, overturning moment, deflections, braced steel frames.

Dynamic analysis of buildings for earthquake-resistant design

The proposed 2005 edition of the National Building Code of Canada specifies dynamic analysis as the preferred method for computing seismic design forces and deflections, while maintaining the equivalent static force method for areas of low seismicity and for buildings with certain height limitations. Dynamic analysis procedures are categorized as either linear (elastic) dynamic analysis, consisting of the elastic modal response spectrum method or the numerical integration linear time history method, or nonlinear (inelastic) response history analysis. While both linear and nonlinear analyses require careful analytical modelling, the latter requires additional considerations for proper simulation of hysteretic response and necessitates a special study that involves detailed review of design and supporting analyses by an independent team of engineers. The paper provides an overview of dynamic analysis procedures for use in seismic design, with discussions on mathematical modelling of structures, structural elements, and hysteretic response. A discussion of the determination of structural period to be used in association with the equivalent static force method is presented.Key words: dynamic analysis, earthquake engineering, elastic analysis, fundamental period, hysteretic modelling, inelastic analysis, National Building Code of Canada, seismic design, structural analysis, structural design.

General tendencies and classification of vertical structures under gust buffeting

The 3-D gust effect factor (GEF) technique is an efficient method to determine the maximum displacements and internal forces of vertical structures subjected to alongwind, crosswind and torsional gust-excited vibrations. All these effects and the related equivalent static forces may be evaluated through one non-dimensional quantity referred to as the 3-D GEF, whose expression was found in closed form. This property makes very attractive the use of functional analysis to establish general tendencies and classification strategies. Confirming some previous results limited to alongwind vibrations, this paper shows that such tendencies are mainly related to the shape of the fundamental modes of vibration. Structures are thus classified into homogeneous categories and general rules are provided to assess a priori the distribution of internal forces. Errors inherent the determination of internal forces by the classical gust response factor technique are also discussed. Among several aspects of engineering interest, those concerning wind actions on the foundation and the stress state for wind-induced fatigue deserve particular attention.

3-D gust effect factor for slender vertical structures

Original studies on gust factor buffeting dealt with the alongwind displacement of structures. Research on this topic carried out since the nineties followed two distinct lines: the first determines the maximum effects due to the alongwind response; the second extends the original method from the alongwind response to crosswind and torsional responses. This paper represents the junction point of these research lines with reference to cantilever slender vertical structures. It derives the most relevant effects associated with the three-dimensional (3-D) wind-excited response of this structural type and shows that a suitable definition of one non-dimensional quantity, referred to as the 3-D gust effect factor, provides such effects at any level through a wide set of experimental, numerical and analytical procedures. A new definition of a 3-D equivalent static force consistent with this method is also introduced and critically compared with previous analogous statements.

The Seismic Provisions of the 1997 Uniform Building Code

Currently the most widely accepted code regulations in the United States for seismic design of structures and nonstructural components are those found in the Uniform Building Code ( UBC). The UBC seismic requirements were significantly revised in the 1997 edition. Among the issues addressed in the UBC revisions are near-source effects and ground acceleration dependent soil site amplification factors for both short- and long-period structures. Also, the design force levels in the 1997 UBC are based on strength design rather than allowable stress design, as had been used previously. Other significant changes include introduction of a redundancy/reliability factor, a more realistic consideration of story drift and deformation compatibility, and new equations for equivalent static forces for both structural and nonstructural components. This paper traces the recent history of the code development and describes the major elements of the 1997 UBC seismic provisions.

Optimal design of steel frames subject to gravity and seismic codes' prescribed lateral forces

Allowable stress design of two-dimensional braced and unbraced steel frames based on AISC specifications subject to gravity and seismic lateral forces is formulated as a structural optimization problem. The nonlinear constrained minimization algorithm employed is the feasible directions method. The objective function is the weight of the structure, and behaviour constraints include combined bending and axial stress, shear stress, buckling, slenderness, and drift. Cross-sectional areas are used as design variables. The anylsis is performed using stiffness formulation of the finite element analysis method. Equivalent static force and response spectrum analysis methods of seismic codes are considered. Based on the suggested methodology, the computer program OPTEQ has been developed. Examples are presented to illustrate the capability of the optimal design approach in comparative study of various types of frames subjected to gravity loads and seismic forces according to a typical code.

Seismic vulnerability evaluation of a 32-story reinforced concrete building

Seismic evaluation of a 32-story reinforced concrete framed tube building is performed by checking damageability, safety, and toughness limit states. The evaluation is based on Standard 2800 (Iranian seismic code) which recommends equivalent lateral static force, modal superposition, or time history dynamic analysis methods to be applied. A three dimensional linearly elastic model checked by ambient vibration test results is used for the evaluation. Accelerograms of three earthquakes as well as linearly elastic design response spectra are used for dynamic analysis. Damageability is checked by considering story drift ratios. Safety is evaluated by comparing demands and capacities at the story and element force levels. Finally, toughness is studied in terms of curvature ductility of members. The paper explains the methodology selected and various aspects in detail.

NON-LINEAR DYNAMICS OF THE FOLLOWER-LOADED DOUBLE PENDULUM WITH ADDED SUPPORT-EXCITATION

The partially follower-loaded elastic double pendulum subjected to excitation of the support, parallel to the straight upright pendulum position, is studied. The effect of small-amplitude off-resonant (high-frequency) excitation on the linear stability and non-linear behaviour of the pendulum, is examined. By use of the method of direct partition of motion (DPM) [1] (Blekhman, 1994, Vibrational Mechanics), the model equations are transformed into autonomous equations with the high-frequency excitation approximated by equivalent static forces. Linear stability analysis shows that the support-excitation has a stabilizing effect for most system parameters, but can also destabilize the upright pendulum position in certain situations. Local post- and pre-critical non-linear behaviour is analyzed by using centre manifold reduction and normal forms. Support-excitation is seen to change the bifurcational behaviour qualitatively: e.g., supercritical bifurcations may change to become subcritical. Chaotic behaviour of the pendulum is shown to exist for a wider range of system parameters and initial conditions with added support-excitation, compared to the case of a fixed support.

Reinforcement of Masonry Walls Subjected to Seismic Loading Using Steel X-Bracing

A design criterion is proposed for the seismic reinforcement of masonry stuctures by joining shear walls with steel X-bracing. The seismic response of the coupled system was investigated in the time domain by a step-by-step integration. Two suitable mechanical models for the shear walls and the X-bracing were used. A wide parametric analysis was performed varying the mechanical characteristics of the masonry walls as well as the X-bracing. Using 20 simulated ground motions the uncertainties related to the definition of seismic loading were taken into account. The results obtained established a structural design criterion based on the equivalent static lateral force method. Given this criterion, the structural factor of the steel X-bracing had to be reduced. This reduction was necessary due to the mixed matched interaction of the two different seismic behaviors. The reduction was based on a suitable coefficient, named the “coupling parameter.” This coefficient is given in specific graphs proposed for design purposes.