Vertical Seismic Force Research Articles

Quantifying variation in maximum floor accelerations of modular buildings under earthquakes through stochastic nonlinear structural analysis

For earthquake-resistant design of modular buildings, reinforced concrete (RC) cores can be a viable choice as the vertical seismic force resisting system. The maximum floor acceleration determines the maximum inertial force that will be transferred by the connections in the horizontal diaphragm to the RC walls. However, the randomness in the structural properties of RC walls may result in fluctuations in the floor acceleration response. Given the limited number of horizontal inter-module connections in modular buildings, the failure of these connections can have severe consequences. To ensure adequate safety margins in the design of horizontal diaphragm connections, it is essential to account for the variations in maximum floor accelerations. To quantify these variations, a 9-story prototype building with RC walls was examined in this study. Monte Carlo simulations were conducted on sample structures with univariate and bivariate randomness in fc (concrete cylinder compressive strength) and Fy (yield strength of steel rebars) of RC walls. Nonlinear response history analyses were performed for each sample structure using 22 ground motions to account for uncertainty in earthquake excitation. The results demonstrated significant and highly complex variations in the maximum floor accelerations due to the randomness considered. A coefficient of variation (CoV) of 0.13 was determined for the maximum floor acceleration response resulting from the bivariate randomness in fc and Fy. Furthermore, the variations were found to increase as the correlation scale in the spatial variation of fc increased. Based on the findings, it is proposed to apply an increase ratio of 1.2 to the maximum floor acceleration to account for the randomness of RC wall properties to ensure adequate safety margins in the design of horizontal diaphragm connections.

Numerical investigation on the characteristics of dynamic response and damage of a RC tall chimney considering vertical ground motions

Although seismic risk assessment models for structures have become more sophisticated, they have typically disregarded the effects of vertical ground motions (VGMs). Nevertheless, this trend is gradually shifting due to recent recordings and field observations that have demonstrated the potentially destructive effects of high VGMs.In this paper, a nonlinear finite element (FE) model of a 234 m reinforced concrete (RC) chimney is developed, with a focus on investigating its dynamic response and damage characteristics under the effect of VGMs. The results are compared with the case of a combination of horizontal and vertical ground motions (HVGMs). First, a time history analysis (THA) is performed under various VGMs input conditions. The findings reveal that the vertical seismic force increases with the intensity of ground motion. However, the distribution of the seismic force remains unchanged, with the maximum vertical seismic force occurring at 85% of the chimney height. Then, the seismic response of the chimney under HVGMs input conditions is examine, revealing that the maximum vertical seismic force increased by 4%–6%, but its distribution does not change significantly. Furthermore, the effect of the additional mass of the flue is considered, resulting in a significant increase in the vertical seismic force at the suspension points, leading to a change in its distribution. An abrupt increase of about 40% at 1/3 height and an increase of about 45% in the horizontal displacement at the top are observed. The paper also highlights the potential stress concentration and resultants cracks or damage caused by the opening of the chimney barrel.

Damage features and dynamic formation mechanism of the Tangjiashan landslide triggered by the Wenchuan earthquake with a composite hypocenter

The Wenchuan earthquake triggered many landslides, which caused many casualties and led to considerable property losses. Due to the compositionality of the hypocenter mechanism of the Wenchuan earthquake, research on the effects of a composite hypocenter on the dynamic responses of landslides is urgently necessary. Using geological field investigations, a block discrete element method simulation, and related theoretical analysis, the dynamic formation mechanism and main controlling mechanical factors of the Tangjiashan landslide were revealed. The results demonstrated the following. Firstly, the dynamic responses of the slope involved progressive deformation, critical failure, disintegration, fragmentation, collision, climb, return, and accumulation. Secondly, when the earthquake lasted for approximately 23 s, the critical failure occurred, and the sliding mass near the shoulder reached a vertical acceleration of approximately 2.8 g at this time. Thirdly, the second and third subhypocenters contributed to the progressive deformation and critical failure, and vertical seismic forces were dominant contributions to these responses. Fourthly, the disintegration, fragmentation, collision, climb, return, and accumulation were mainly controlled by the self-gravity and local topography and were slightly influenced by subhypocenters. Based on these conclusions, the study of the dynamic formation mechanism of this type of landslide has increased the level of understanding.

Research on the Seismic Isolation Effect of the Ring Spring–Friction Pendulum Bearing in the Dakai Underground Subway Station

During strong earthquakes, the vertical seismic force becomes an essential factor affecting shallow underground structures that cannot be ignored. In this situation, it is proposed that ring spring–friction pendulum two-way seismic isolation bearings be set up at the bottom of such structures. Targeting the Dakai underground station and the Kobe earthquake, a soil-structure force model was established based on the damage to the underground structure and dynamic simulation was carried out using Abaqus software. The advantages of the ring spring–friction pendulum bearing over the ring spring bearing alone and the friction pendulum bearing alone were compared. The simulation results showed that the structure displayed a good seismic isolation effect in both horizontal and vertical directions after setting the ring spring–friction pendulum bearing. The deflection in the midspan of the roof reduced to less than 5 cm, the horizontal relative displacement of the structure reduced to less than 3 cm, there was no obvious damage to the structure, the axial pressure ratio of the mid-pillar reduced to less than 0.5, the axial force on the mid-pillar reduced by more than 50%, and the shear force reduced by more than 30%. In addition, by comparing the damage to the structure after setting the annular spring bearing and the friction pendulum bearing, we found that a vertical seismic isolation device is more crucial than a horizontal seismic isolation device for shallow underground structures. Setting a vertical damping device can make the structure retain part of the ductility and reduce the damage to the structure, so the ground vertical seismic mitigation design should be strengthened when a shallow underground structure is designed.

Experimental, numerical, and theoretical studies on vertical stiffness reduction of lead thick rubber bearings

Lead thick rubber bearings (LTRBs) are capable of mitigating vertical subway-induced vibration and horizontal seismic force as well as adding supplementary damping. During earthquakes, LTRBs would be accompanied by lateral displacements, which would cause vertical stiffness reduction. However, much uncertainty still exists about quantifying this reduction of vertical stiffness. This research intends to investigate the influence factors of the vertical stiffness reduction of LTRBs and to develop a formulation for accurate prediction. Three full-scale LTRBs with different first and second shape factors (S1 and S2) were designed and subjected to combined lateral displacement and compressive load for determining their vertical stiffness. Based on material and isolator level tests, a finite element (FE) model of LTRBs was calibrated. Then, parametric FE analyses were performed to explore the effects of various parameters on the normalized vertical stiffness over a wide range of lateral offset. Experimental and numerical studies quantified the reduction of vertical stiffness for different values of vertical pressure, S1, and S2. Besides, the test and simulated results were compared to the results from existing formulations and the inaccuracy of these formulations was indicated. Finally, a new formulation was developed and it showed good agreement with experimentally and numerically determined vertical stiffness of LTRBs.

Upper bound analysis of the anti-seismic stability of slopes considering the effect of the intermediate principal stress

Considering the intermediate principal stress effect of soils, the unified strength theory was applied to correct the shear strength index of soils. Based on a mechanical model of the anti-seismic stability of slopes, the analytical formula of the stability coefficient was deduced by the upper bound method and the quasi-static method. Then the optimal solution of the stability coefficient was figured out via the Matlab software. The result shows that, if the intermediate principal stress is ignored, the shear strength of soils would obviously be underestimated; this is also the case with the slope stability coefficient, the relative error of which could reach 39.87%. The horizontal and vertical seismic forces significantly affect slope stability. When the horizontal seismic force is considered, the slope stability coefficient Ns is reduced by 79.82%. Similarly, if the seismic effect is not neglected, the stability of the slope would be seriously overestimated. Slope cutting can significantly improve slope stability. When the slope angle is reduced from 90° to 50°, the stability factor increases by 279.82%. The suitable design angle of the slope is between 50° and 60° without taking into account additional elements like groundwater level and stratum structure.

Upper Bound Analysis of the Stability of 3D Slopes in the Saturated Soft Clay Subjected to Seismic Effect

In order to study the three-dimensional stability problem of the saturated soft clay slope under earthquake loads, based on the three-dimensional rotation failure model, the seismic force was introduced into the calculation by the quasi-static method. The work rate of external loads and the internal energy dissipation rate of the saturated soft clay slope were calculated using the upper bound method of limit analysis, and the analytical solution of stability coefficient of saturated soft clay slopes was derived based on the fictitious power principle. By virtue of the exhaust algorithm, the optimal solution of stability coefficient of saturated soft clay slopes was obtained. The influence of the slope angle and the horizontal and vertical seismic forces on the stability coefficient of saturated soft clay slope was analyzed. The results show that the slope angle has a great influence on the stability coefficient, and the relative difference is up to 35.7%. Therefore, the stability coefficient of saturated soft clay slopes can be effectively increased by a proper slope setting. The horizontal and vertical seismic forces also have a significant influence on the stability of saturated soft clay slopes. The relative differences of the stability coefficient under horizontal and vertical seismic forces are as high as 41 and 14.7%, respectively. If they are ignored, the stability coefficient of saturated soft clay slopes will be seriously overestimated. It is suggested that the effects of horizontal and vertical seismic forces must be considered simultaneously in the seismic design of saturated soft clay slopes.

Theoretical Study on Relaxed Surrounding Rock Pressure on Shallow Bias Neighborhood Tunnels under Seismic Load

To study the distribution of relaxed surrounding rock pressure on the shallow bias neighborhood tunnels under the combined action of horizontal and vertical earthquake force, finite element software was used for failure mode analysis. Moreover, with the pseudo-static method, the calculation formula for the relaxed pressure on the shallow bias neighborhood tunnels was derived and used to analyze the variation of the rupture angle of these tunnels under the action of the seismic force. The study shows that: shallow bias neighborhood tunnels basically follow a “W” failure pattern under the combined action of horizontal and vertical seismic force, and the failure scope of the surrounding rock is controlled by four rupture angles. Rupture angles β2 and β3 between the deep and shallow tunnels of the shallow bias neighborhood tunnels are not affected by the surface slope. For tunnels with the same grade of the surrounding rock, the greater the seismic intensity, the smaller the value of β2, and the greater the value of β3. While at the same seismic intensity, the higher the grade of the surrounding rock, the smaller the β2 and β3. Ruptures angles β1 and β4 are influenced by the surface slope, seismic intensity and surrounding rock grades. A steeper surface slope leads to a smaller β1 and a greater β4; β1 increase and β4 decrease with increasing seismic intensity; while, β1 and β4 both show a decreasing trend with an increasing rock grade.

Reliability analysis of anti-seismic stability of 3D pressurized tunnel faces by response surfaces method

The limit analysis and response surfaces method were combined to investigate the reliability of pressurized tunnel faces subjected to seismic force. The quasi-static method was utilized to introduce seismic force into the tunnel face. A 3D horn failure mechanism of pressurized tunnel faces subjected to seismic force was constructed. The collapse pressure of pressurized tunnel faces was solved by the kinematical approach. The limit state equation of pressurized tunnel faces was obtained according to the collapse pressure and support pressure. And then a reliability model of pressurized tunnel faces was established. The feasibility and superiority of the response surfaces method was verified by comparing with the Monte Carlo method. The influence of the mean of soil parameters and support pressure, variation coefficients, distribution type and correlation of c-phi on the reliability of pressurized tunnel faces was discussed. The reasonable safety factor and support pressure required by pressurized tunnel faces to satisfy 3 safety levels were presented. In addition, the effects of horizontal seismic force, vertical seismic force and correlation of kh-kv on the reliability of pressurized tunnel faces were also performed. The method of this work can give a new idea for anti-seismic design of pressurized tunnel faces.

Reliability analysis for seismic stability of tunnel faces in soft rock masses based on a 3D stochastic collapse model

A new horn failure mechanism was constructed for tunnel faces in the soft rock mass by means of the logarithmic spiral curve. The seismic action was incorporated into the horn failure mechanism using the pseudo-static method. Considering the randomness of rock mass parameters and loads, a three-dimensional (3D) stochastic collapse model was established. Reliability analysis of seismic stability of tunnel faces was presented via the kinematical approach and the response surface method. The results show that, the reliability of tunnel faces is significantly affected by the supporting pressure, geological strength index, uniaxial compressive strength, rock bulk density and seismic forces. It is worth noting that, if the effect of seismic force was not considered, the stability of tunnel faces would be obviously overestimated. However, the correlation between horizontal and vertical seismic forces can be ignored under the condition of low calculation accuracy.

Analysis of Seismic Stability of an Obsequent Rock Slope Using Time–Frequency Method

Obsequent rock slopes are often thought to be more stable than consequent slopes and passive sliding is unlikely to occur under earthquake loading. However, failures in obsequent rock slopes were indeed observed in recent large earthquakes. This paper presents a time–frequency solution to the seismic stability of obsequent rock slopes fully considering the time–frequency characteristics of earthquake waves. Large-scale shaking table tests were conducted to illustrate the application of the time–frequency method to an obsequent rock slope containing multiple weak layers with a small dip angle. The seismic stability of the obsequent rock slope is analyzed combining the time–frequency method, outcomes from the shaking table test, and conventional pseudo-static and dynamic numerical analyses. The results show that passive sliding can develop in the obsequent rock slope when taking the time–frequency components of the earthquake waves and the vertical seismic force into account. The middle–upper part of the obsequent rock slope is more vulnerable to seismic damage. The slope bulges under the earthquake loading; the maximum permanent surface displacement occurs at the middle–upper part of the slope, rather than the slope crest. Additionally, the response of seismic safety factor lags behind the responses of acceleration and surface displacement.

Taylor's slope stability chart for combined effects of horizontal and vertical seismic coefficients

Design standards and codes of practice on earth slope stability often recommend the pseudo-static method of analysis for determining the factor of safety of a slope subjected to seismic forces. In most pseudo-static methods of analysis, the horizontal seismic force is considered without due weightage to vertical seismic force. In the past, Taylor's stability chart for a homogeneous cohesive-frictional soil slope has been extended to consider the effect of horizontal seismic force only. In this paper, an attempt is made to develop an analytical formulation considering both horizontal and vertical seismic forces in order to estimate the factor of safety of the homogeneous, cohesive-frictional soil slopes with simple profiles using Taylor's stability chart. The analytical formulation is based on the friction circle method, which is one of the methods of static slope stability analysis. Several field cases have been analysed considering slope geometry, soil properties and seismic loading conditions so that Taylor's stability chart can be routinely used by practising engineers considering the effects of both horizontal and vertical seismic forces. An illustrative example is included in order to explain how practising engineers can use the graphical presentations developed in this paper as the design charts for stability analysis. This illustrative example has also been solved using Plaxis 2D, a commercially available finite-element software, as a comparison.

Three-dimensional limit analysis of seismic stability of tunnel faces with quasi-static method

Based on the existing research results, a three-dimensional failure mechanism of tunnel face was constructed. The dynamic seismic effect was taken into account on the basis of quasi-static method, and the nonlinear Mohr-Coulomb failure criterion was introduced into the limit analysis by using the tangent technique. The collapse pressure along with the failure scope of tunnel face was obtained through nonlinear limit analysis. Results show that nonlinear coefficient and initial cohesion have a significant impact on the collapse pressure and failure zone. However, horizontal seismic coefficient and vertical seismic proportional coefficient merely affect the collapse pressure and the location of failure surface. And their influences on the volume and height of failure mechanism are not obvious. By virtue of reliability theory, the influences of horizontal and vertical seismic forces on supporting pressure were discussed. Meanwhile, safety factors and supporting pressures with respect to 3 different safety levels are also obtained, which may provide references to seismic design of tunnels.

Effect of dynamic soil properties and frequency content of harmonic excitation on the internal stability of reinforced soil retaining structure

The effect of dynamic soil properties and frequency content of harmonic excitation on the internal stability of reinforced soil retaining structure is investigated. Arc of a log-spiral is considered as the failure surface in the present limit equilibrium analysis. Backfill and reinforced soil is modeled as a visco-elastic material. The whole structure is considered to be resting on a rigid stratum. Backfill soil and the reinforced soil retaining structure are subjected to harmonic shaking at the base. Present methodology satisfies the stress boundary condition at the ground surface. In the present study, amplitude and phase of the horizontal and vertical seismic accelerations change with depth and the variation of accelerations along the depth is found to be time dependent and nonlinear. All the four possible combinations of horizontal and vertical seismic inertia force directions are considered to determine the total reinforcement force and critical length of the reinforcement. In the present study, amplification of accelerations towards the ground surface depends on the dynamic soil properties and frequency content of input excitation. Detailed parametric study is done to understand their implications on the solution. An algorithm is proposed at the end of this paper which uses strain dependent equivalent linear values of shear wave velocity (Vs) and damping ratio (ξ) to compute the total reinforcement force and critical length of the reinforcement. The limitation of equivalent linear based approach is that it only considers vertically propagating shear wave. Comparison of present method with other theories is also presented showing the merit of the present study.

The Effects of Vertical Stress on the Liquefaction Potential Originated from Buildings in The Urban Areas

Main purpose of this paper is to study the influence of vertical stress on soil liquefaction in urban areas. The literature provides limited information on vertical stress analysis of multiple footings, and, as a result, there is no accurate way to account for the effect of the foundation depth on liquefaction. Additionally, practical methods do not exist for considering the interaction between the neighboring foundations vertical stress and seismic forces in the urban area. Vertical stress distribution was calculated in examining the soil liquefaction potential exhibited by building foundations as a case study. The vertical stresses were chosen randomly for some buildings with foundation depths of 3.00 m; 4.50 and 6.00 m at the Burkent site (Burdur-Turkey). The influence of 5-storey buildings on the liquefaction potential of sandy soils was evaluated in terms of the safety factor (FS) against liquefaction along soil profile depths for different earthquakes. Standard Penetration Test (SPT) results were used based on simplified empirical procedure.

Surrounding rock pressure of shallow-buried bilateral bias tunnels under earthquake

By means of finite element numerical simulation and pseudo-static method, the shallow-buried bilateral bias twin-tube tunnel subject to horizontal and vertical seismic forces are researched. The research includes rupture angles, the failure mode of the tunnel and the distribution of surrounding rock relaxation pressure. And the analytical solution for surrounding rock relaxation pressure is derived. For such tunnels, their surrounding rock has sliding rupture planes that generally follow a W shape. The failure area is determined by the rupture angles. Research shows that for shallow-buried bilateral bias twin-tube tunnel under the action of seismic force, the load effect on the tunnel structure shall be studied based on the relaxation pressure induced by surrounding rock failure. The rupture angles between the left tube and the right tube are independent of the surface slope. For tunnels with surrounding rock of Grade IV, V and VI, which is of poor quality, the recommended reinforcement range for the rupture angles is provided when the seismic fortification intensity is VI, VII, VIII and IX respectively. This study is expected to provide theoretical support regarding the ground reinforcement range for the shallow-buried bilateral bias twin-tube tunnel under seismic force.

Effects of vertical seismic force on initiation of the Daguangbao landslide induced by the 2008 Wenchuan earthquake

This paper analyses the effects of vertical seismic force on initiation mechanism of near-fault large-scale landslides by using the Daguangbao landslide induced by the 2008 Wenchuan earthquake as an example. Field investigations showed that tension failure plays an important role in failure mechanism, and the tension failure is due to the effect of a large vertical seismic motion that occurred in the meizoseismal area during the earthquake. Firstly, two key issues in seismic analysis of near-fault large-scale landslide are proposed: i) how to select and correct the severe near-fault excitations with significant co-seismic displacements and ii) how to consider the effect of the landslide scale on estimation of material strength. Then, in order to investigate the effect of the severe vertical seismic force on the initiation of the huge Daguangbao landslide, five cases, 1) static; 2) pseudo-static only-horizontal; 3) pseudo-static horizontal-and-vertical; 4) dynamic only-horizontal; 5) dynamic horizontal-and-vertical, are performed using the finite difference program FLAC3D. The simulation results are presented in terms of tension failure area, factor of safety, and displacement. The results confirm the significant effects of vertical seismic force on seismic slope failure mechanism as the tension failure areas and the displacements observed in the case of horizontal-and-vertical are larger than those observed in the case of only-horizontal.

Lateral force distribution on base-isolated structures under near-field earthquakes

In the current paper the accuracy of the International Building Code equivalent static method for base-shear distribution is evaluated to determine the response of seismically isolated structures under near-field earthquakes. For this purpose two-, three-, four- and six-storey building models are designed according to the equivalent lateral-response procedure for different elastomeric isolation systems of 2 s, 2·5 s and 3 s for periods of isolated structures and 6%, 8% and 11% for their damping ratio. Different lateral-force resisting systems such as ordinary steel moment frame, ordinary steel concentrically braced frame and ordinary reinforced concrete shear wall are employed as a superstructure and the effects of 13 near-field ground motions on vertical seismic force distribution, and the linear and non-linear properties of isolation systems are investigated. The results of the equivalent static procedure in vertical distribution of base shear are compared with results obtained from dynamic linear and non-linear analysis and the efficiency of their application is studied. Based on linear and non-linear dynamic analysis results of isolated structures, two new formulae are proposed for vertical distribution of base shear for the isolated structures under near field earthquakes. The proposed linear formula is based on the first mode shape of an isolated structure and the non-linear formula follows three first mode shapes which implies contribution of higher modes in more realistic non-linear analyses.

Providing a Model for Assessment of Horizontal and Vertical Earthquake Effects on the Double-layer Barrel Vaults in Space Structures

In this study, the effect of place soil and seismic vulnerability has not been considered in determining the earthquake force, although all results have been based on scale accelerographs. Moreover, the importance of structures has not been rated based on small or large spans, decorative or structural structure and the effect of these cases is not considered. In this regard, double-layer barrel vaults have been studied and effect of horizontal and vertical seismic force on it reviewed and equations are presented in order to evaluate the equivalent static force of horizontal and vertical earthquake. Presented equations facilitate determining the equivalent static force due to earthquake for space structures, but they also have deficiencies which solving them can be useful for researchers who want to continue the way.

Failure Mechanisms of the Landslides Triggered by the 2008 Wenchuan Earthquake, China

Taking the landslides triggered by the 2008 Wenchuan Earthquake as examples, their dynamic responses with different epicenter distances due to single and combined action with regionality and spatial heterogeneity of the Primary and Secondary waves were simulated by applying the Universal Distinct Element Code software. The results shows that the slope suffered from the combined action between P and S waves appears instability prior to the slope under single action of P wave. With the epicenter distance increasing, the key controlling factor resulting in the slope failure varies from the combined seismic action between P and S waves to the single seismic action of the P wave. As for the formation mechanism of slope instability, coupled action between the vertical and horizontal seismic forces results in the slope dynamic failure with key action varying from the vertical to the horizontal one. Finally, the initial instability originates always at slope shoulder due to the peak ground acceleration amplification effect and the variation trend of the slope mechanical parameters on its fracturing of the seismic action.