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This paper presents a comparison of P-Delta effects on the nonlinear seismic behavior of the steel moment-resisting frame structures (MRFs) subjected to near-fault and far-fault ground motions. The 3-, 9- and 20-story MRFs designed for the American SAC Phase II Steel Project are used as benchmark models. The 40 near-fault ground motions with large velocity pulses, as well as ten typical far-fault ground motions, are selected and scaled for the nonlinear time-history analysis. The P-Delta effect is quantified based on peak inter-story drift ratio (PIDR) demands. The displacement demands of the whole structure and the distortion of the structural components are compared and analyzed. It was found that, at each floor, the P-Delta effect under near-fault ground motions is more significant than that under the far-fault ground motions. The P-Delta effect under near-fault ground motions also increases more rapidly with decreasing structure height even for low-rise structures or low earthquake intensity. It was also found that the P-Delta effect cause the PIDR demands to increase by 10% for all three structures subjected to far-fault ground motions. In contrast, considering the P-Delta effect, the PIDR demands rapidly increase by 45% for the high-rise building subjected to near-fault ground motions. Note that the increasing PIDR demands occur at the weakest floor and with the stronger earthquake intensity. However, the P-Delta effect does not change the location of the weakest floor and the yield sequence of components. The seismic behaviors under far-fault and near-fault ground motions are significantly different, because near-fault ground motions not only have velocity pulse but also possibly trigger structural higher vibration modes. In addition, the P-Delta effect may change the distortion direction of the components so that the prediction of the structural collapse direction may be affected. In addition, it was found that if the structure’s period is near the pulse period, the P-Delta effect becomes more significant with the increase of earthquake intensity, and accordingly, it should not be ignored. Moreover, the P-Delta effect cannot be neglected either for the structures susceptible to near-fault ground motions, even if those structures are not tall or the earthquake intensity is not strong.

This paper presents a detailed investigation into the seismic response of non-deteriorating and deteriorating single degree-of-freedom systems controlled by P-Δ effects, with due account for the influence of earthquake duration. In order to isolate the effect of duration from other ground motion characteristics, 77 pairs of records with equivalent spectral shapes are considered in the study. The structural characteristics examined include the structural period, applied gravity loading, post-yield stiffness, viscous damping, material hysteretic behaviour, as well as the level of cyclic deterioration within the pinching systems. Detailed incremental dynamic analyses are carried out, considering an intensity measure corresponding to the spectral acceleration at the structural period of vibration of the system. Based on the incremental dynamic analysis results, predictive relationships are proposed for determining the structural collapse capacity, accounting for the influence of key parameters including instability and duration effects. The median and dispersion of the collapse capacity distribution embedded in the predictive models are also presented. The effect of duration is shown to increase with longer structural periods and to decrease with higher P-Δ levels. The more rapid instigation of dynamic instability in relatively stiff systems is also shown to reduce their comparative sensitivity to variations in ground motion characteristics. Overall, it is indicated that disregarding the influence of duration could lead to over-estimations of up to 50% in the collapse capacity. The paper concludes with a discussion of other sources of structural damage that instigate collapse when using records with equivalent spectral shape but without especial consideration for duration effects.

This paper explains the mathematical foundations of a method for modelling semi-rigid unions. The unions are modelled using rotational rather than linear springs. A nonlinear second-order analysis is required, which includes both the effects of the flexibility of the connections as well as the geometrical nonlinearity of the elements. The first task in the implementation of a 2D Beam element with semi-rigid unions in a nonlinear finite element method (FEM) is to define the vector of internal forces and the tangent stiffness matrix. After defining the formula for this vector and matrix in the context of a semi-rigid steel frame, an iterative adjustment of the springs is proposed. This setting allows a moment–rotation relationship for some given load parameters, dimensions, and unions. Modelling semi-rigid connections is performed using Frye and Morris’ polynomial model. The polynomial model has been used for type-4 semi-rigid joints (end plates without column stiffeners), which are typically semi-rigid with moderate structural complexity and intermediate stiffness characteristics. For each step in a non-linear analysis required to adjust the matrix of tangent stiffness, an additional adjustment of the springs with their own iterative process subsumed in the overall process is required. Loops are used in the proposed computational technique. Other types of connections, dimensions, and other parameters can be used with this method. Several examples are shown in a correlated analysis to demonstrate the efficacy of the design process for semi-rigid joints, and this is the work’s application content. It is demonstrated that using the mathematical method presented in this paper, semi-rigid connections may be implemented in the designs while the stiffness of the connection is verified.

This paper presents a comparison of near-fault and far-fault ground motion effects on geometrically nonlinear earthquake behavior of suspension bridges. Boğaziçi (The First Bosporus) and Fatih Sultan Mehmet (Second Bosporus) suspension bridges built in Istanbul, Turkey, are selected as numerical examples. Both bridges have almost the same span. While Boğaziçi Suspension Bridge has inclined hangers, Fatih Sultan Mehmet Suspension Bridge has vertical hangers. Geometric nonlinearity including P-delta effects from self-weight of the bridges is taken into account in the determination of the dynamic behavior of the suspension bridges for near-fault and far-fault ground motions. Near-fault and far-fault strong ground motion records, which have approximately identical peak ground accelerations, of 1999 Chi-Chi, 1999 Kocaeli, and 1979 Imperial Valley earthquakes are selected for the analyses. Displacements and internal forces of the bridges are determined using the finite element method including geometric nonlinearity. The displacements and internal forces obtained from the dynamic analyses of suspension bridges subjected to each fault effect are compared with each other. It is clearly seen that near-fault ground motions are more effective than far-fault ground motion on the displacements and internal forces such as bending moment, shear force and axial forces of the suspension bridges.

This study proposes an “optimal” spectral acceleration based intensity measure (IM) to assess the collapse capacity of generic moment frames vulnerable to the P-delta effect. The IM is derived from the geometric mean of the spectral pseudo-acceleration over a certain period interval. The optimized IM includes for first time a flexible lower limit for the period interval, corresponding to the structural period associated with the exceedance of 95% of the total effective mass. This flexible lower limit bound provides an efficient IM, independently of the contribution of higher modes to the total response. The upper bound period is 1.6 times the fundamental period to account for period elongation due to inelastic deformations and gravity loads. In a parametric study on generic frames, structural parameters are varied to quantify the performance of this IM compared to classical benchmark IMs. The “optimal” IM provides minimum, or close to the minimum, dispersion for the entire set of frames with different fundamental periods of vibration, number of stories, and P-delta vulnerability.

The study evaluates two alternative seismic intensity measures (IMs) that reduce the collapse capacity dispersion of inelastic non-degrading single-degree-of-freedom (SDOF) systems vulnerable to the P-delta effect. This dispersion of collapse capacity is caused by record-to-record variability, which refers to frequency content variation of the ground motions used in the dynamic analyses. This reduction (of dispersion) is achieved utilizing efficient elastic pseudo-spectral acceleration based IMs. The first set of evaluated IMs is based on the spectral pseudo-acceleration averaged in a certain period interval between the structural period and an elongated period. The “optimal” lower bound of the period interval corresponds to the structural period of vibration, since naturally in an SDOF system no higher modes effects do exist. The “optimal” upper bound of the period interval for averaging, referred to as elongated period, is found to be 1.6 times the system period. The second IM considered in the study is the 5 % damped spectral pseudo-acceleration at the system period in the presence of gravity loads, which is a single target IM. The most widely accepted IM, the 5 % damped pseudo-spectral acceleration at the system period without P-delta, serves as the benchmark IM. The results show that both proposed IMs lead to a reduction of the collapse capacity dispersion compared to the benchmark IM outcomes. The IM based on the averaged spectral acceleration of the “optimal” period interval is more efficient up to a negative post-yield stiffness ratio of 0.45, while the single target IM based on the system period in the presence of gravity loads is superior for extreme negative post-yield stiffness ratios larger than 0.45. Additionally, the sufficiency and the scaling robustness property of the considered IMs with respect to the natural logarithm of the record-dependent individual collapse capacities is discussed for a wide range of structural configurations.

The seismic responses of structures are always influenced by P-Δ effects. The significance of this effect may be negligible when the structure responds elastically but is very important when the structure responds into inelastic range. The P-Δ effect usually increases the displacement response of structures. It may even cause dynamic instability when the structure subjected to severe earthquake ground motions. In conventional seismic design codes, the P-Δ effect usually leads to an increase in design base shear. The same approach is used in direct displacement-based design. Various researchers have proposed different relationships for base shear increase which can be used in different seismic design approaches such as force-based design and performance-based design. In this paper, the proposed expressions are reviewed extensively and their adequacy is evaluated by detail. Then two main purposes are pursued: (1) the development of new expressions for strength amplification factor due to P-Δ effect for current force-based design seismic codes; and (2) the modification of equivalent viscous damping or the required additional base shear considering the P-Δ effect in direct displacement-based design procedure. For this purpose, a new algorithm based upon elasto-plastic SDOF system analyses is presented. The algorithm is implemented 102,500 times overall for different periods, ductility levels, stability indices and different earthquake ground motions that each implementation needs a large amount of trial and error process in linear and nonlinear SDOF systems. The results seem to present a good development of the P-Δ effect relations for the seismic design procedures.

Present study introduces two concepts for direct estimation of P-delta effect in both, strength based, and displacement based design methods. Although various previously conducted studies focused on inclusion of P-delta effect into the aforementioned design methods, development of reliable procedures is still attractive. The major argument of present study is that: treatment of P-delta effect can be enhanced by using an alternative response/design spectrum. To this end, and based on period-dependence feature of stability coefficient (SC), the “stability coefficient response spectra” (SCRS) is introduced. The SCRS, plots spectral acceleration versus SC, instead of period, for a pendulum with known height. To facilitate implementation of SCRS on multi-degree-of-freedom systems, and by considering some special features of the first-storey, the concept of “first-storey-single-degree-of-freedom” (FSSDOF) system is introduced. The FSSDOF system permits setting the minimum necessary lateral stiffness, conforming to a pre-selected SC limit, and a given ductility level, at very early stages of design process. Moreover, it is shown that implementation of SCRS and FSSDOF system can be extended to account for drift limits. This is done by introducing a modified version of the “yield-point-spectra” method in which period-dependence feature of SC is recognized. Several numerical examples are included as part of the presentation.

An innovative displacement-based seismic design procedure for regular planar framed structures considering sidesway collapse prevention is presented. The method proposed, based on the characterization of a multiple degree of freedom system by means of a single degree of freedom system, allows the design of structures with a P-Delta induced negative stiffness to satisfy an interstorey drift threshold and/or prevent dynamic instability. The design procedure relies on the use of elastic analysis of simplified models and collapse or constant ductility spectra, hence, non-linear dynamic analyses are not required. To illustrate the potential of the method proposed, example applications and validation of the results obtained via incremental dynamic analysis are shown.

A modified energy-balance equation accounting for P-delta effects and hysteretic behavior of reinforced concrete members is derived. Reduced hysteretic properties of structural components due to combined stiffness and strength degradation and pinching effects, and hysteretic damping are taken into account in a simple manner by utilizing plastic energy and seismic input energy modification factors. Having a pre-selected yield mechanism, energy balance of structure in inelastic range is considered. P-delta effects are included in derived equation by adding the external work of gravity loads to the work of equivalent inertia forces and equating the total external work to the modified plastic energy. Earthquake energy input to multi degree of freedom (MDOF) system is approximated by using the modal energy-decomposition. Energy-based base shear coefficients are verified by means of both pushover analysis and nonlinear time history (NLTH) analysis of several RC frames having different number of stories. NLTH analyses of frames are performed by using the time histories of ten scaled ground motions compatible with elastic design acceleration spectrum and fulfilling duration/amplitude related requirements of Turkish Seismic Design Code. The observed correlation between energy-based base shear force coefficients and the average base shear force coefficients of NLTH analyses provides a reasonable confidence in estimation of nonlinear base shear force capacity of frames by using the derived equation.