Dynamic response of a pair of walls retaining a viscoelastic solid

Making use of a previously reported, simple, approximate method of analysis, a critical evaluation is made of the dynamic pressures and forces induced by horizontal ground shaking on a pair of infinitely long, parallel walls retaining a uniform viscoelastic solid. The walls are presumed to be rigid but elastically constrained against rotation at their base. The effects of both harmonic and earthquake-induced excitations are examined. The accuracy of the method is assessed by comparing its predictions for the special case of fixed-based walls with those obtained by an exact method, and comprehensive numerical data are presented which elucidate the underlying response mechanisms, and the effects and relative importance of the parameters involved. The parameters examined include the characteristics of the ground motion, the ratio of the distance between walls to the height of the contained material, and the flexibility of the rotational wall constraints. In addition to valuable insights into the responses of the systems investigated, the results presented provide a convenient framework for the analysis of more complex systems as well.     

Out‐of‐plane seismic response of vertically spanning URM walls connected to flexible diaphragms

A simplified numerical model was used to investigate the out‐of‐plane seismic response of vertically spanning unreinforced masonry (URM) wall strips. The URM wall strips were assumed to span between two flexible diaphragms and to develop a horizontal crack above the wall mid‐height. Three degrees of freedom were used to accommodate the wall displacement at the crack height and at the diaphragm connections, and the wall dynamic stability was studied. The equations of dynamic motion were obtained using principles of rocking mechanics of rigid bodies, and the formulae were modified to include semi‐rigid wall behaviour. Parametric studies were conducted that included calculation of the wall response for different values of diaphragm stiffness, wall properties, applied overburden, wall geometry and earthquake ground motions. The results of the study suggest that stiffening the horizontal diaphragms of typical low‐rise URM buildings will amplify the out‐of‐plane acceleration demand imposed on the wall and especially on the wall–diaphragm connections. It was found that upper‐storey walls connected to two flexible diaphragms had reduced stability for applied earthquake accelerograms having dominant frequency content that was comparable with the frequency of the diaphragms. It was also found that the applied overburden reduced wall stability by reducing the allowable wall rotations. The results of this study suggest that the existing American Society of Civil Engineers recommendations for assessment of vertically spanning walls overestimate the stability of top‐storey walls in multi‐storey buildings in high‐seismic regions or for walls connected to larger period (less stiff) diaphragms. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic analysis of flexible rectangular fluid containers subjected to horizontal ground motion

In this paper, an analytical method is proposed to determine the dynamic response of 3‐D rectangular liquid storage tanks with four flexible walls, subjected to horizontal seismic ground motion. Fluid–structure interaction effects on the dynamic responses of partially filled fluid containers, incorporating wall flexibility, are accounted for in evaluating impulsive pressure. The velocity potential in which boundary conditions are satisfied is solved by the method of separation of variables using the principle of superposition. The impulsive pressure distribution is then computed. Solutions based on 3‐D modeling of the rectangular containers are obtained by applying the Rayleigh–Ritz method using the vibration modes of flexible plates with suitable boundary conditions. Trigonometrical functions that satisfy boundary conditions of the storage tank such that the flexibility of the wall is thoroughly considered are used to define the admissible vibration modes. The analysis is then performed in the time domain. Moreover, an analytical procedure is developed for deriving a simple formula that evaluates convective pressure and surface displacements in a similar rigid tank. The variation of dynamic response characteristics with respect to different tank parameters is investigated. A mechanical model, which takes into account the deformability of the tank wall, is developed. The parameters of such a model can be obtained from developed charts, and the maximum seismic loading can be predicted by means of a response spectrum characterizing the design earthquake. Accordingly, a simplified but sufficiently accurate design procedure is developed to improve code formulas for the seismic design of liquid storage tanks. Copyright © 2013 John Wiley & Sons, Ltd.     

Evaluation and suggestions for improvement of seismic design procedures for R/C walls in dual systems

This paper aims to shed some further light on the seismic behaviour and design of reinforced concrete (R/C) walls which form part of dual (frame + wall) structures. The significance of post‐elastic dynamic effects is recognized by most seismic codes in the definition of the design action effects on walls, i.e. bending moments and shear forces. However, the resulting envelopes are not always fully satisfactory, particularly in the case of medium‐to‐high‐rise buildings. The relevant provisions of modern seismic codes are first summarized and their limitations discussed. Then an extensive parametric study is presented which involves typical multi‐storey dual systems that include walls with unequal lengths, designed according to the provisions of Eurocode 8 for two different ductility classes (M and H) and two effective peak ground acceleration levels (0.16 and 0.24g). The walls of these structures are also designed according to other methods, such as those used in New Zealand and Greece. The resulting different designs are then assessed by subjecting the structures to a suite of records from strong ground motions, carrying out inelastic time history analysis, and comparing the results with the design action effects. It is found that for (at least) the design earthquake intensity, the first two modes of vibration suffice for describing the seismic response of the walls. The bending moment envelope, as well as the base shear of each wall, is found to be strongly dependent on the second mode effect. As far as the code‐prescribed design action effects are concerned, only the NZ Code was found to be consistently conservative, whereas this was not always the case with EC8. A new method is then proposed which focuses on quantifying in a simple way the second mode effects in the inelastic response of the walls. This procedure seems to work better than the others evaluated herein. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamic soil pressures on rigid cylindrical vaults

A critical evaluation is made of the dynamic pressures and the associated forces induced by ground shaking on an upright, circular, rigid vault that is embedded in a uniform viscoelastic stratum of constant thickness and infinite extent in the horizontal plane. Both the vault and the stratum are presumed to be supported on a non-deformable base undergoing a space-invariant, uniform horizontal motion. The effects of both harmonic and earthquake-induced excitations are examined. Simple approximate expressions for the responses of the system are formulated, and comprehensive numerical data are presented which elucidate the underlying response mechanisms and the effects and relative importance of the various parameters involved. The parameters investigated include the height to radius ratio for the vault, the conditions at the vault-medium interface, and the material properties of the stratum. In addition to valuable insights into the response of the particular system being examined, the results presented provide a conceptual framework for the analysis and interpretation of solutions for more involved systems as well.     

Seismic pressures on rigid cantilever walls retaining elastic continuously non-homogeneous soil: An exact solution

The dynamic response of an elastic continuously nonhomogeneous soil layer over bedrock retained by a pair of rigid cantilever walls to a horizontal seismic motion and the associated seismic pressure acting on these walls are determined analytically–numerically. The soil non-homogeneity is described by a shear modulus increasing nonlinearly with depth. The problem is solved in the frequency domain under conditions of plane strain and its exact solution is obtained analytically. This is accomplished with the aid of Fourier series along the horizontal direction and solution of the resulting system of two ordinary differential equations with variable coefficients by the method of Frobenius in power series. Due to the complexity of the various analytical expressions, the final results are determined numerically. These results include seismic pressures, resultant horizontal forces and bending moments acting on the walls. The solution of the problem involving a single retaining wall can be obtained as a special case by assuming the distance between the two walls to be very large. Results are presented in terms of numerical values and graphs using suitable dimensionless quantities. The effect of soil non-homogeneity on the system response is assessed through comparisons for typical sets of the parameters involved.     

Out-of-plane shaking table tests of full-scale historic adobe corner walls retrofitted with timber elements

Historic adobe structures pose a high seismic risk mainly because of the poor out-of-plane bending response of their walls that may produce fatalities and significant economic, cultural, and heritage losses. In this paper, we propose a retrofitting technique that increases the wall strength for both in-plane and out-of-plane directions. This technique consists of vertical and horizontal timber elements symmetrically installed on each face of the wall to form a confining wood frame, supplemented with vertical tensors that pre-compress the wall. This study evaluates the performance of this retrofitting technique with a two-set experimental program on full-scale historic adobe walls. On the first set, four specimens were subjected to a static overturning test with boundary conditions representing the confinement effect at both ends by orthogonal walls. On the second set, three full-scale specimens, one unretrofitted and two retrofitted, were subjected to four ground motion records on a shaking table to assess the out-of-plane dynamic behavior of typical corner walls. The unretrofitted specimen collapsed during the second motion (peak ground acceleration [PGA] = 0.39 g), while both retrofitted walls survived all four motions (maximum PGA of 0.75 g) proving the high effectiveness of the proposed retrofitting. The addition of base anchors as a variation of the retrofitting technique significantly reduced the rocking effects and the residual drifts of the system, thus improving its overall seismic performance. Further research is needed to develop guidelines for seismic retrofit of heritage buildings including multistory full-scale tests of specimens with various types of openings and retrofitting strategies that minimize their architectural impact.     

地震作用下夯土城墙动力响应分析——以山丹明长城为例

2022年1月8日门源M6.9地震造成山丹明长城局部破坏。为研究此次地震作用下夯土城墙的动力响应与破坏特征,基于地震现场考察结果,采用振幅等效处理后的记录地震波为输入地震动,开展双向地震荷载作用下夯土城墙的动力响应数值分析,研究不同位置测点的最大位移、峰值加速度与墙体应力分布特征,探讨地震导致夯土城墙破坏的主要内因。研究结果表明：双向地震荷载作用下,墙体位移和峰值加速度(PGA)随着高度的增加逐渐增加,但距墙体底部0.5 m高度范围内PGA放大效应不明显,最大位移、加速度均出现在墙体顶部裂缝位置处;水平地震荷载作用下墙体的地震动响应更为显著;墙体的最大主应力、最大剪应力均出现在有裂缝处的底端掏蚀悬空部位,墙体裂缝、夯筑搭接、掏蚀悬空处应力集中明显;裂缝对夯土城墙的地震动放大效应在一定高度范围内表现为弱化作用,但随高度增加逐渐过渡为强化作用;裂缝可显著增强墙体顶部地震动响应,可能是本次地震诱发城墙破坏的主要内因。研究成果可为古城墙遗址的加固修缮提供科学指导。     

Influence of boundary conditions on the out‐of‐plane response of brick masonry walls in buildings with RC slabs

In modern unreinforced masonry buildings with stiff RC slabs, walls of the top floor are most susceptible to out‐of‐plane failure. The out‐of‐plane response depends not only on the acceleration demand and wall geometry but also on the static and kinematic boundary conditions of the walls. This paper discusses the influence of these boundary conditions on the out‐of‐plane response through evaluation of shake table test results and numerical modelling. As a novum, it shows that the in‐plane response of flanking elements, which are orthogonal to the wall whose out‐of‐plane response is studied, has a significant influence on the vertical restraint at the top of the walls. The most critical configuration exists if the flanking elements are unreinforced masonry walls that rock. In this case, the floor slabs can uplift, and the out‐of‐plane load‐bearing walls loose the vertical restraint at the top. Numerical modelling confirms this experimentally observed behaviour and shows that slab uplift and the difference in base and top excitation have a strong influence on the out‐of‐plane response of the walls analysed. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic response of rigid tanks with inhomogeneous liquids

A study of the response to horizontal ground shaking of a rigid cylindrical tank containing an inviscid liquid with a continuous vertical variation in density is presented. In addition to the free vibrational sloshing characteristics of the liquid, the responses examined include the vertical displacements at the free surface, and the impulsive and convective components of the hydrodynamic wall pressures and associated tank forces. The equations of motion for the system are formulated for an arbitrary variation in liquid density but the solutions presented are for a density that increases exponentially from top to bottom. Comprehensive numerical data are included which elucidate the underlying response mechanisms and the effects and relative importance of the various parameters involved. The solution for the continuous density variation considered herein is also compared with a previously reported solution in which the liquid was modelled as a multi-layered, discrete system.     

Dynamic pressures on a pair of rigid walls retaining poroelastic soil

This work deals with the evaluation of the dynamic pressures and the associated forces on a pair of rigid vertical cantilever walls retaining a uniform, fully saturated poroelastic layer of soil. Hysteretic damping in the soil skeleton may also be present. Wall pressures and forces are induced by horizontal ground shaking harmonically varying with time and spatially invariant. The problem is solved analytically under conditions of plane strain. The governing partial differential equations of motion, after separation of variables and the simplifying assumptions of zero vertical normal stresses and zero horizontal variation of vertical displacements, reduce to a system of two ordinary differential equations for the amplitudes of the solid skeleton horizontal displacement and the pore water pressure, which are easily solved. The parameters examined include the ratio of the distance between walls to the height of the retained soil material and the soil material properties such as porosity, permeability and damping. The comprehensive numerical data presented indicate that the displacements, wall pressures and resultant forces are highly dependent on the distance between the walls for any values of porosity and permeability.     

Dual‐plastic hinge design concept for reducing higher‐mode effects on high‐rise cantilever wall buildings

This paper explores the notion of detailing reinforced concrete structural walls to develop base and mid‐height plastic hinges to better control the seismic response of tall cantilever wall buildings to strong shaking. This concept, termed here dual‐plastic hinge (DPH) concept, is used to reduce the effects of higher modes of response in high‐rise buildings. Higher modes can significantly increase the flexural demands in tall cantilever wall buildings. Lumped‐mass Euler–Bernoulli cantilevers are used to model the case‐study buildings examined in this paper. Buildings with 10, 20 and 40 stories are designed according to three different approaches: ACI‐318, Eurocode 8 and the proposed DPH concept. The buildings are designed and subjected to three‐specific historical strong near‐fault ground motions. The investigation clearly shows the dual‐hinge design concept is effective at reducing the effects of the second mode of response. An advantage of the concept is that, when combined with capacity design, it can result in relaxation of special reinforcing detailing in large portions of the walls. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional beam‐truss model for reinforced concrete walls and slabs – part 1: modeling approach,validation, and parametric study for individual reinforced concrete walls

A three‐dimensional beam‐truss model for reinforced concrete (RC) walls developed by the first two authors in a previous study is modified to better represent the flexure–shear interaction and more accurately capture diagonal shear failures under static cyclic or dynamic loading. The modifications pertain to the element formulations and the determination of the inclination angle of the diagonal elements. The modified beam‐truss model is validated using the experimental test data of eight RC walls subjected to static cyclic loading, including two non‐planar RC walls under multiaxial cyclic loading. Five of the walls considered experienced diagonal shear failure after reaching their flexural strength, while the other three walls had a flexure‐dominated response. The numerically computed lateral force–lateral displacement and strain contours are compared with the experimentally recorded response and damage patterns for the walls. The effects of different model parameters on the computed results are examined by means of parametric analyses. Extension of the model to simulate RC slabs and coupled RC walls is presented in a companion paper. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical analysis of reinforced soil walls with granular and cohesive backfills under cyclic loads

Novel approaches to the dynamic analysis of the reinforced soil walls have been reported in the literature. Use of marginal soils reduces the cost of geosynthetic reinforced soil walls if proper drainage measures are taken. Therefore the affect of using cohesive marginal soils as backfill in geosynthetic reinforced retaining structures were investigated in this research. The dynamic response of reinforced soil walls was investigated in a similar focus, using finite element analysis. The results obtained from walls with cohesive backfill were compared to the results obtained from walls with granular backfill. The height of the wall was chosen as 6 m in the two-dimensional plane strain finite element model and the base acceleration was chosen to be a harmonic motion. The effects of various parameters like the backfill type, facing type, reinforcement stiffness, and peak ground acceleration on the cyclic response of reinforced soil retaining walls were investigated. After analyzing the wall response for end of construction and dynamic excitation phases, it was determined that the deformations and reinforcement tensile loads increased during the cyclic load application and that the amount of additional deformation that occurred during cyclic load application was strongly related to backfill soil type, facing type, reinforcement type and peak ground acceleration. It was determined that a cohesive backfill and geotextile reinforcement was a good combination to reduce the deformations of geosynthetic reinforced walls during cyclic loading for medium height walls.     

Rocking instability of free-standing statues atop slender cantilevers under ground motion

This paper deals with the dynamic response of free-standing statues on the top surface of slender elastically supported cantilevers subjected to horizontal ground motion. Given that there is no link between the base of the statue and the top surface of the monolithic cantilever the statue is in equilibrium in the vertical direction under its own weight. Attention is focused on the determination of the minimum amplitude ground acceleration which leads to the rocking (overturning) instability of the statue whose mass and rotatory inertia are a priory known. It is assumed that the friction between the base of the statue and the top surface of the cantilever is sufficiently large to prevent sliding so that rocking prevails. After simulating the statue by a rigid block freely supported on the top surface of the elastically restrained monolithic cantilever, a theoretical dynamic analysis of the cantilever–rigid block system under horizontal ground motion is comprehensively presented. Two modes of overturning instability of the free standing rigid block are discussed: instability without or with impact. Criteria for overturning instability of the rigid block associated with the minimum amplitude ground acceleration which leads through the vanishing of the angular velocity to an escaped motion in the phase-plane portrait, are properly assessed.     

Seismic behavior of an inverted T-shape flexible retaining wall via dynamic centrifuge tests

In the design procedure for a retaining wall, the pseudo-static method has been widely used and dynamic earth pressure is calculated by the Mononobe–Okabe method, which is an extension of Coulomb’s earth pressure theory computed by force equilibrium. However, there is no clear empirical basis for treating the seismic force as a static force, and recent experimental research has shown that the Mononobe–Okabe method is quite conservative, and there exists a discrepancy between the assumed conditions and real seismic behavior during an earthquake. Two dynamic centrifuge tests were designed and conducted to reexamine the Mononobe–Okabe method and to evaluate the seismic lateral earth pressure on an inverted T-shape flexible retaining wall with a dry medium sand backfill. Results from two sets of dynamic centrifuge experiments show that inertial force has a significant impact on the seismic behavior on the flexible retaining wall. The dynamic earth pressure at the time of maximum moment during the earthquake was not synchronized and almost zero. The relationship between the back-calculated dynamic earth pressure coefficient at the time of maximum dynamic wall moment and the peak ground acceleration obtained from the wall base peak ground acceleration indicates that the seismic earth pressure on flexible cantilever retaining walls can be neglected at accelerations below 0.4 g. These results suggest that a wall designed with a static factor of safety should be able to resist seismic loads up to 0.3–0.4 g.     

Soil-structure interaction effects for laterally excited liquid storage tanks

A comprehensive study is made of the effects of soil-structure interaction on the response of liquid containing, upright, circular cylindrical tanks subjected to a horizontal component of ground shaking. A simple, physically motivated method of analysis is employed which elucidates the effects and relative importance of the principal actions involved. Both the impulsive and convective actions of the liquid are examined. The interrelationship of the tank responses to horizontal and rocking actions of the foundation is established, and the well known mechanical model for laterally excited, rigid tanks supported on a non-deformable medium is generalized to permit consideration of the effects of tank and ground flexibilities and base rocking. Critical responses are evaluated for harmonic and seismic excitations over wide ranges of tank proportions and soil stiffnesses, and the results are presented in a form convenient for use in practical applications. In addition to a precise method of analysis, an approximate, hand-computation method is presented with which the effects of the primary parameters may be evaluated readily. The soil-structure interaction effects in the latter approach are provided for by modifying the natural frequency and damping of the tank-liquid system and evaluating its response to the prescribed free-field ground motion considering the tank to be rigidly supported at the base. The requisite modifications may be determined from information presented herein. It is shown that soil-structure interaction may reduce significantly the impulsive components of response but that it has a negligible effect on the convective components.     

Static and seismic limit equilibrium analysis of sliding retaining walls under different surcharge conditions

The static and seismic sliding limit equilibrium condition of retaining walls is investigated, and analytical solutions for the angle of the active slip surface, the critical acceleration coefficient and the coefficient of active earth pressure are provided for different surcharge conditions. In particular, walls retaining a horizontal backfill without surcharge, walls supporting an extended uniform surcharge applied at different distances from the wall and walls supporting a limited uniform surcharge or linear uniform surcharge parallel to the wall are considered in the analysis.The solutions have been derived in the framework of the limit equilibrium approach, considering the effect of the wall through its weight, and accounting for the shear resistance at the base of the wall and the inertia force arising in the wall under seismic conditions.For the wall without surcharge the effect of the vertical component of the seismic acceleration as well as the effects of the inclination of the wall internal face and of the soil–wall friction were also investigated.The angle of the slip plane, the critical seismic acceleration coefficient and the coefficient of active earth pressure are given as functions of dimensionless parameters and the boundary conditions for the applicability of each solution are specified. The influence of soil weight, surcharge conditions and inertia forces on the active earth pressure coefficient is analysed.     

Seismic design and response evaluation of unbonded post‐tensioned hybrid coupled wall structures

This paper presents an analytical investigation on the seismic design and response of coupled wall structures that use unbonded post‐tensioned steel coupling beams. Both monolithic cast‐in‐place reinforced concrete wall piers and precast concrete wall piers are considered. Steel top and seat angles are used at the coupling beam ends for energy dissipation. The seismic design of prototype structures to achieve target displacement‐based performance objectives is evaluated based on nonlinear static and dynamic time history analyses. Additional recommendations are provided on shear design. Comparisons with ‘conventional’ structures that use embedded steel coupling beams as well as isolated walls with no coupling are provided. The results indicate that while the peak lateral displacements of unbonded post‐tensioned coupled wall structures are larger than the peak displacements of structures with embedded beams, the residual displacements are significantly reduced as a result of the restoring effect of the post‐tensioning steel. Copyright © 2008 John Wiley & Sons, Ltd.     

A linear mathematical model for the seismic inplane behaviour of brick masonry walls part 2: Determination of model parameters through optimization using experimental data

The parameters appearing in the mixture and effective modulus models proposed in Part 1 are determined through optimization by matching theoretical and experimental responses. The optimization analysis is performed in frequency space. The response quantities chosen to be matched are the complex frequency response functions (experimental and theoretical) relating the Fourier transforms of top and base accelerations of the wall. Computations in optimization analysis are carried out by introducing an object (error) function and minimizing it using the Gauss-Newton method. The results show that the mixture model is capable of predicting accurately the dynamic response of masonry walls up to a frequency which is well above the second modal frequency, whereas the effective modulus model describes the wall behaviour only up to the first modal frequency. Furthermore, it is shown that the mixture model is still valid when micro cracks, which may exist between the mortar and brick constituents, are present.