Freedom Model Research Articles

Nonlinear vibrations of four-degrees of freedom for piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular plate with PPF control strategy

The nonlinear vibration suppression of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever (PFG-GRLCC) rectangular plate with positive position feedback (PPF) control strategy is investigated firstly. The material properties of the graphene-reinforced structure are calculated through the Halpin–Tsai micromechanical model. Considering the transverse external excitation and the converse effect of piezoelectricity, the governing equations of motion are formulated through von Karman large deformation theory, the classical laminated plate theory, and Hamilton principle. After adding the PPF controllers, a four-degrees of freedom model for the close-loop vibration control system is achieved via Galerkin truncation technique. The average equations in the case of the primary resonance and 1:1:3:3 internal resonance can be obtained using the multiple scale perturbation (MSP) method. The amplitude–frequency response curves are studied by the numerical continuation algorithm. The detailed parametric analyses show that the PPF controller can effectively reduce the nonlinear vibration response amplitudes of the PFG-GRLCC rectangular plate. In addition, the results reveal the energy transform between the host system and the PPF controller. This work is expected to provide theoretical guidance for nonlinear large amplitude vibration reduction of graphene-reinforced structure.

Mixed [Formula: see text]-synthesis tracking control and disturbance rejection in a robotic digit of an impaired human hand for anthropomorphic coordination.

In a partially impaired anthropomorphic hand, maintaining the movement coordination of the robotic digits with the central nervous system (CNS) and natural digits is crucial for robust performance. A challenge in the control perspective of movement coordination of a human hand is finding methods robust to the disturbances in a well-posed control problem of a biomechanical model. We use visco-elastic dynamics in the human palm frame of reference to explore the biomechanics of movement coordination to solve this control problem. Our biomechanical model incorporates the time delay due to actuation force, parametric uncertainty, exogenous disturbances, and sensory noise to constitute a 21-degree of freedom model. A mixed [Formula: see text]-synthesis controller, considering the real parametric uncertainty, represents the CNS in the control paradigm. We consider the robotic finger's flexion movement when perturbed from the initial equilibrium. The controller provides feedback force at the joints to regulate the robotic finger movement. The index finger follows a reference trajectory of the joint angular position profile and stabilizes at a flexion angle of 1 rad/s at a time of 1 s. The main control objective is to keep the angular displacement of the finger joint constant when a disturbance force acts. We simulate the modeling scheme in MATLAB/ Simulink. The results demonstrate that our controller scheme is robust against the worst-case disturbance and achieves the desired performance value. Developing a biologically inspired neurophysiological controller with robust performance has many applications, including assistive rehabilitation devices, hand movement disorder diagnosis, and robotic manipulators.

Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

This work aims to study the effect of uncertainties and noise on the nonlinear global dynamics of a micro-electro-mechanical arch obtained from an imperfect microbeam under an axial load and electric excitation. An adaptative phase-space discretization strategy based on an operator approach is proposed. The Ulam method, a classical discretization of flows in phase-space, is extended here to nondeterministic cases. A unified description is formulated based on the Perron–Frobenius, Koopman, and Foias linear operators. Also, a procedure to obtain global structures in the mean sense of systems with parametric uncertainties is presented. The stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. For parameter uncertainty cases, the phase-space is augmented with the corresponding probability space. The microarch is assumed to be shallow and modeled using a nonlinear Bernoulli–Euler beam theory and is discretized by the Galerkin method using as interpolating function the linear vibration modes. Then, from the discretized multi degree of freedom model model, an accurate single degree of freedom reduced order model, based on theory of nonlinear normal modes, is derived. Several competing attractors are observed, leading to different (acceptable or unacceptable) behaviors. Extensive numerical simulations are performed to investigate the effect of noise and uncertainties on the coexisting basins of attraction, attractors’ distributions, and basins boundaries. The appearance and disappearance of attractors and stochastic bifurcation are observed, and the time-dependency of stochastic responses is demonstrated, with long-transients influencing global behavior. To consider uncertainties and noise in design, a dynamic integrity measure is proposed via curves of constant probability, which give quantitative information about the changes in structural safety. For each attractor, the basin robustness as a function of a stochastic parameter is investigated. The weighted basin area can quantify the integrity of nondeterministic cases, being also the most natural generalization of the global integrity measure. While referring to particular micro-electro-mechanical system, the relevance of the dynamical integrity analysis for stochastic systems to quantify tolerances and safety margins is underlined here.

Pushover Analysis of Multi-Story Reinforced Concrete Frame Structure

Pushover analysis is a non-linear static procedure (NSP) used to examine the structural response of a building during the inelastic limit before it reaches the edge of partial or total collapse. In this study a model of multi-story reinforced concrete building along with applied loads is analyzed in powerful non-linear static procedure using computer software ETABS. A limited number of horizontal forces are then applied to imitate the impact of ground movement. The building is supposed as a 6-story building, no shear walls used in the model as the level of earthquake hazard assumed very low. Moreover, deformations are determined. The forces would then be increased gradually, NSPs convert multi-degree of freedom models to single-degree of freedom models utilizing a pushover analysis to determine capacity curve or pushover curve which is horizontal roof displacement vs base shear, This curve plot reveals the highest base motion that the structure can withstand, and then estimate the maximum global displacement by combining a ground response spectrum with the capacity curve.

Adaptive chaos control of a humanoid robot arm: a fault-tolerant scheme

Abstract. Safety in human–robot physical interaction and cooperation is of paramount interest. In this work, a human–humanoid interaction and cooperation scenario is considered. The robot arm is controlled by a proportional derivative (PD) controller in combination with an inherently fault-tolerant sliding-mode controller. During normal operation, if any of the joints of the robot arm develops a fault, the robot arm end effector may go into chaotic and dangerous motion. If such a scenario occurs, it poses a serious danger to the human in the loop as well as the robot. In this paper, an adaptive chaos sliding-mode controller is proposed to recover the tracking of the end effector when a fault occurs in any of the actuators. This tracking restoration is very important to complete a safety-critical task. The proposed scheme may help in addressing some safety issues arising from a joint failure, allowing it to finish the task at hand and stop it from going into a dangerous situation. The scheme is tested in simulation on the four degrees of freedom (DOF) model of the Bristol Robotics Laboratory (BRL) robot arm. Simulation results show efficient tracking recovery after a joint actuator failure. The controller demonstrated good performance in terms of tracking and stability when dealing with different joint actuator failures.

Effect of wind barriers on the aeroelastic instabilities of a hinged-deck cross-section cable-stayed bridge

The main idea of a hinged-deck cross-section cable-stayed bridge appears with the need of increasing the capacity of a simple bridge by adding to lateral extension decks to the main deck. An experimental wind tunnel campaign to analyse the aeroelastic behaviour of a hinged-deck cross-section cable-stayed bridge has been performed. The bridge cross-section under study is formed by three parts: a central deck and two lateral extension decks joined to the central one using hinges and supported by their respective cable systems. The experimental model used is a reduced 6 degrees of freedom (DOFs) model for the elastic problem, which has previously proved to give good results concerning the bridge behaviour. The number of DOFs induces a new complexity in the oscillating motion of the bridge deck, compared to classical single deck 2 DOFs configurations.In this work, the aeroelastic tests performed at the UPM/IDR ACLA-16 wind tunnel facility to address the effect of wind barriers on the aeroelastic characteristics of the bridge section are reported. Nominal barriers are considered and compared with the case with no barriers. The effect of barrier porosity (wind barriers of the central deck and lateral decks) is also analysed. Stability ranges are reported in terms of reduced wind speed. The results show the large influence of the barriers in controlling the aeroelastic characteristics of the bridge. In some configurations, a large increase in bridge stability has been found due to the relevant effect of the barriers in changing the flow around the bridge.

Trajectory tracking for agricultural tractor based on improved fuzzy sliding mode control.

Trajectory tracking is one of the key technologies for tractor automatic navigation. Its main purpose is to adjust the steering mechanism of the tractor to follow the planned trajectory. Thus, in this paper a trajectory tracking control system is designed for an agricultural tractor with the electric power steering mechanism. A DC brush motor is added on the steering column of the tractor and the hardware circuits for the steering controller are designed to control the front wheel angel. The three degrees of freedom model of the tractor is established, and a trajectory tracking control system is proposed including a fuzzy sliding mode controller and a steering angle tracking controller designed according to the internal mode control and minimized sensitivity theory. The effectiveness of the designed trajectory tracking control system is demonstrated by simulation analyses in reference to the planed trajectory.

Ride Behaviour of Three-Wheel Vehicle using Semi-Active Suspension using Skyhook Control Strategy

This paper analyses the ride behaviour of a 2 degrees of freedom model of three-wheel vehicle subjected to bounce and pitch velocity inputs. This paper also demonstrates the isolation properties of semi-active suspension using skyhook dampers. The results show semi-active suspension can provide isolation near resonance as compared with passive suspension when subjected to bounce and pitch velocity inputs. Below and beyond the resonating frequencies, the behaviour of passive and semi-active suspension is identical for all damping values.

Computational and experimental studies of wave–structure interaction: Wave attenuation by a floating breakwater

Interactions between surface gravity waves and a floating rigid body are complex, as waves may reflect from, break on, and be transmitted behind the body. Studies of these phenomena are critically important in improving the safety and functional efficiency of offshore structures. Here, the wave attenuation performance and motions of a type of floating breakwater (FB) are studied through numerical and experimental approaches. A numerical wave tank (NWT) is developed based on the software OpenFOAM and properties of wave channel from a laboratory. In the NWT, the air–water interface is captured by the volume of fluid method. The motions of FB are tracked by the six degrees of freedom model. A mooring system model is developed to simulate the constraints of the FB. Large eddy simulation turbulence modeling is implemented for the wave breaking processes. A model FB with a scale of 1:20 is tested in both the experimental and numerical wave channel. Wave heights at the back/front of the FB and the constraint forces of the mooring wires are measured. The numerical models are validated by comparing the results with experimental measurements. The variations of transmission/reflection coefficients, energy dissipation rate, and maximum mooring force are calculated. Changes of the response amplitude operators with the ratio of FB width to wavelength ( B / L w) and wave steepness are analyzed. The wave transmission coefficient will drop below 0.8 if the value of B / L w is larger than 0.3, but will go over 0.95 if B / L w is less than 0.1. Wave steepness has a large influence on FB motions and the mooring system. The effect of Stokes drift is observed by the shift of position of the FB.

Frequency domain transfer function based multimodal design method for seismic mitigation of building structures

Multiple tuned mass dampers (MTMD) are widely used to mitigate structural seismic response. With the emergence of more and more high-rise buildings, various MTMD design methods applicable to multi degree of freedom (MDOF) model have been developed. However, the existing MTMD design methods for MDOF models often involve complex optimization algorithms. In order to simplify the MTMD design while effectively reducing the seismic response of the structure, this paper proposes an explicit frequency domain transfer function, theoretically showing the mechanism of decomposing the seismic mitigation of MDOF model into simultaneous seismic mitigation of multiple modes, and developed a theoretical design method of MTMD, so that the dynamic parameters and position of each TMD can be directly obtained based on the developed explicit design formula. A benchmark model was used as an example to verify the seismic mitigation performance. It is shown that the developed multimodal control MTMD design method achieves 22.25% reduction rate of acceleration and 18.5% reduction rate of displacement with 3% mass ratio.

Dynamic modeling and vibration control of barge offshore wind turbine using tuned liquid column damper in floating platform

This study investigated the vibration suppression of a barge-type floating offshore wind turbine (FOWT) using a passive tuned liquid damper (TLCD) in the floating platform. The fully coupled aero-hydro–servo-elastic model was established using OpenFAST, and it was used to evaluate the suppression performance of the TLCD in the time and frequency domains. To optimally design the TLCD parameters, a 3-degree of freedom (DOF) model was proposed based on the Lagrange equation for a barge-type FOWT with a platform TLCD, which accounts for the DOF of platform pitching, tower bending in the fore–aft direction, and vertical-liquid column motion of TLCD. The unknown parameters of the 3-DOF model were estimated and validated to ensure result accuracy. The exhaustive search method was implemented to obtain the optimized TLCD parameters of the barge-type FOWT, and the simulation results showed that TLCD can effectively mitigate the structural vibration of barge-type FOWT. When applying a large TLCD mass to the barge platform for structural vibration control, replacing part of the ballast with a TLCD of equal mass to maintain the overall mass of the barge-type FOWT was prioritized.

Applied element modelling of unreinforced and reinforced concrete masonry walls under blast loading

Masonry walls are the first defense line against exterior explosions in buildings; however, no adequate guidelines are currently available for the analysis and design of these walls under such extreme loading. In this paper, a comprehensive applied element model (AEM) is developed and validated to investigate the behaviour of both reinforced (RMWs) and unreinforced (URMWs) concrete masonry walls under blast loading. A single degree of freedom (SDOF) model is also developed to estimate their pressure-impulse diagram for concrete masonry walls. The developed models are utilized to conduct a parametric study by analyzing 30 walls with different design parameters (boundary conditions, dimensions, compressive strength, as well as reinforcement ratio) under various blast loadings. Crack pattern, fundamental period, support rotation, as well as pressure–displacement and pressure-impulse diagrams for the analyzed walls were compared. The boundary conditions and compressive strength had the most and least significant influence on the behaviour of walls. The two-sides-supported walls experienced severe cracks compared to the four-sides-supported walls. Fundamental period decreased by 74% when the wall thickness increased from 100 mm to 250 mm for either URMWs or RMWs. Resistance of a wall with dimensions of 3 × 7 m was less than that of a 5 × 2 m wall by 81% and 77% for URMW and RMW, respectively. The SDOF model showed that URMWs and RMWs with dimensions of 5 × 2 m had the maximum pressure-impulse response limit compared to the other walls. The average increase in the support rotation was 200% and 380% when the charge weight increased from 5 kg to 15 kg for URMWs and from 15 kg to 150 kg for RMWs, respectively. Using reinforcement ratio greater than 0.85% considerably improved the performance of RMWs.

Effect of Spring-Mass-Damper Pedestrian Models on the Performance of Low-Frequency or Lightweight Glazed Floors

For structural design purposes, human-induced loads on pedestrian systems can be described by several simplified (i.e., deterministic equivalent-force models) or more complex computational approaches. Among others, the Spring-Mass-Damper (SMD), Single Degree of Freedom (SDOF) model has been elaborated by several researchers to describe single pedestrians (or groups) in the form of equivalent body mass m, spring stiffness k and damping coefficient c. For all these literature SMD formulations, it is proved that the biodynamic features of walking pedestrians can be realistically reproduced, with high computational efficiency for vibration serviceability assessment of those pedestrian systems mostly sensitive to human-induced loads (i.e., with vibration frequency f1 < 8 Hz). Besides, the same SMD proposals are characterized by mostly different theoretical and experimental assumptions for calibration. On the practical side, strongly different SMD input parameters can thus be obtained for a given pedestrian. This paper focuses on a selection of literature on SMD models, especially on their dynamic effects on different structural floor systems. Four different floors are explored (F#1 and F#2 made of concrete, F#3 and F#4 of glass), with high- or low-frequency, and/or high- (>1/130th) or low- (1/4th) mass ratio, compared to the occupant. Normal walking scenarios with frequency in the range fp = 1.5–2 Hz are taken into account for a total of 100 dynamic simulations. The quantitative comparison of typical structural performance indicators for vibration serviceability assessment (i.e., acceleration peak, RMS, CREST) shows significant sensitivity to input SMD assumptions. Most importantly, the sensitivity of structural behaviours is observed for low-frequency systems, as expected, but also for low-mass structures, which (as in the case of glazed floor solutions) can be characterized by the use of lightweight modular units with relatively high vibration frequency. As such, major attention can be required for their vibrational analysis and assessment.

On the nonlinear dynamics of a piezoresistive based mass switch based on catastrophic bifurcation

This research investigates the feasibility of mass sensing in piezoresistive MEMS devices based on catastrophic bifurcation and sensitivity enhancement due to the orientation adjustment of the device with respect to the crystallographic orientation of the silicon wafer. The model studied is a cantilever microbeam at the end of which an electrostatically actuated tip mass is attached. The piezoresistive layers are bonded to the vicinity of the clamped end of the cantilever and the device is set to operate in the resonance regime by means of harmonic electrostatic excitation. The nonlinearities due to curvature, shortening and electrostatic excitation have been considered in the modelling process. It is shown that once the mass is deposited on the tip mass, the system undergoes a cyclic fold bifurcation in the frequency domain, which yields a sudden jump in the output voltage of the piezoresistive layers; this bifurcation is attributed to the nonlinearities governing the dynamics of the response. The partial differential equations of the motion are derived and discretized to give a finite degree of freedom model based on the Galerkin method, and the limit cycles are captured in the frequency domain by using the shooting method. The effect of the orientation of the device with respect to the crystallographic coordinates of the silicon and the effect of the orientation of the piezoresistive layers with respect to the microbeam length on the sensitivity of the device is also investigated. Thanks to the nonlinearity and the orientation adjustment of the device and piezoresistive layers, a twofold sensitivity enhancement due to the added mass was achieved. This achievement is due to the combined amplification of the sensitivity in the vicinity of the bifurcation point, which is attributed to the nonlinearity and maximizing the sensitivity by orientation adjustment of the anisotropic piezoresistive coefficients.

Investigation of rocking mechanism of shallow foundations on sand via PIV technique

AbstractAdoption of the rocking isolation concept in foundation design is considered as an effective method in reducing seismic loads of structures. Despite much research conducted on the rocking behavior of foundations, concerns regarding the practical application of the rocking foundation design concept still prevail. The concerns primary stem from the lack of accurate understanding and knowledge associated with the soil‐foundation deformation mechanisms during rocking motions. Determination of deformation mechanisms and soil mass failure modes during foundation rocking helps to gain insight into the rocking behavior, recognize influential factors on foundation settlement or uplift, and propose appropriate improvement strategies to control foundation deformations. The primary objective of this study is to investigate the deformation behavior and failure modes of the soil mass beneath a rocking foundation with various initial static vertical factors of safety (F.S.v) and embedment depths. To this end, a series of reduced‐scale slow‐cyclic tests under 1g condition have been conducted using a single degree of freedom (SDOF) model. To determine soil deformation mechanisms, soil displacement fields have been measured using Particle Image Velocimetry (PIV) technique. Soil deformation mechanisms during loading and unloading paths, as well as causes leading to the foundation settlement or uplift have been evaluated and discussed. Additionally, changes in the effective soil‐foundation contact area in different loading cycles have been examined based on the PIV results. Based on the results, soil deformation mechanisms in the loading portions include soil densification, wedge deformation and scoop deformation. As the footing embedment depth increases, wedge deformation mechanism becomes limited, while scoop deformation mechanism becomes stronger. Additionally, as the foundation F.S.v increases, the depth of influenced zone of the rocking foundation decreases and the soil displacement occurs to a more limited depth.

A Comparative Study of Integrated Vehicle–Seat–Human Models for the Evaluation of Ride Comfort

In the literature the value of the driver’s head acceleration has been widely used as an objective function for the modification of the suspension and/or the seat characteristics in order to optimize the ride comfort of a vehicle. For these optimization procedures various lumped parameter Vehicle–Seat–Human models are proposed. In the present paper a Quarter Car model is integrated with three Seat–Human models with different levels of detail. The level of detail corresponds to the number of degrees of freedom used to describe the Seat–Human system. Firstly, the performance of the Quarter Car model, used as a basis, is analyzed in six excitations with different characteristics. Then, the performance of the three lumped parameter Vehicle–Seat–Human models are monitored in the same excitations. The results indicated that in the case of single disturbance excitations the Quarter Car model provided 50–75% higher values of acceleration compared with the eight degrees of freedom model. As far as the periodic excitation is concerned, the Vehicle–Seat–Human models provided values of acceleration up to eight times those of the Quarter Car model. On the other hand, in stochastic excitations the Vehicle–Seat–Human model with three degrees of freedom produced the closest results to the Quarter Car model followed by the eight degrees of freedom model. Finally, with respect to the computational efficiency it was found that an increase in the degrees of freedom of the Vehicle–Seat–Human model by one caused an increase in the CPU time from 2.1 to 2.6%, while increasing the number of the degrees of freedom by five increased the CPU time from 7.4 to 11.5% depending on the excitation.

Simplified Life Cycle Cost Estimation of Low-Rise Steel Buildings Using Fundamental Period

In the current study, a simplified seismic life cycle cost (LCC) estimation procedure is proposed utilizing the mean values of the structure’s main input variables. The main input variables of the building are used for constructing a relationship between the structural fundamental period (T) and an average estimation of the LCC (LCCavg). Using the actual building properties related to damage probability, the T–LCCavg relationship is used to obtain the final LCC (LCCfin). The equivalent single degree of freedom (ESDOF) model and SAC-FEMA framework are utilized for damage probability calculation. The dispersion measure in demand is approximately calculated based on the mean plus one standard deviation of the seismic hazard response spectrum, and, then, verified through nonlinear time history (NLTH) analyses of the original structure. Five and three-story steel buildings are used as case studies for verification of the proposed method. The analysis results indicate that the proposed procedure provides reasonable LCC estimations for low-rise buildings dominated by the fundamental mode of vibration.

Numerical study on dynamic response of hollow and cavity type clay brick masonry infill panels subjected to blast loading

Laboratory and prototype study of masonry panel subjected to earthquake, wind and impact is relatively easy however, the experimental study of masonry panel under the action of blast phenomenon is quite a risky and restrictive work area to perform. But computational advancement has made it within our capabilities and is deemed to yield acceptable results. Therefore, this study focuses on investigating the dynamic blast response of novel hollow and cavity type clay brick masonry (HBM and CBM) infill panels enclosed in reinforced concrete (RC) frames using explicit finite element (FE) method. The properties of HBM and CBM required for defining the numerical FE model were derived from the experimental tests. Moreover, due to lack of experimental study on HBM and CBM subjected to the action of blast, the FE peak displacement observed at mid-height of masonry panels were agreeably validated with the peak displacement obtained from an equivalent non-linear single degree of freedom (SDOF) model wherein, a link between positive phase duration of blast and natural time-period of structure is established. Further, the effects of nature of explosion, standoff distance, weight of explosive and height of detonation from ground level were investigated for HBM and CBM walls. Thereafter, HBM panel was strengthened using a low-cost technique, making use of steel bars for enhancing the out-of-plane stiffness of the wall against blast loading. Reinforced hollow brick masonry (RHBM) panel was found to be an excellent alternative for blast load mitigation.

Can a nonlinear quasi-zero-stiffness spring improve the ride quality of a vehicle?

The paper examines the possibility of using a nonlinear Quasi-Zero-Stiffness (QZS) spring in a vehicle suspension. The response of a Single Degree of Freedom (SDOF) model to harmonic base excitations is obtained which may be unstable and unbounded depending on the excitation level and the damping ratio. This is followed by obtaining the response of the SDOF model to harmonic base excitation with an amplitude that is varying according to a road profile spectrum. Such dependency of the excitation amplitude to the frequency changes the qualitative behaviour of the nonlinear system and the responses would be always bounded. The QZS spring also improves the isolation of the system. Transient responses of a quarter car model with a QZS suspension to a road hump and random road profile are investigated. The maximum acceleration of the vehicle with the QZS suspension passing over the speed hump is considerably lower than a vehicle with a conventional linear suspension. Wb weighted RMS of acceleration (BS 6841-1987) is also lower by as much as 14% for a vehicle with the QZS suspension travelling at 30 km/h on a class E road compared to its linear counterpart.

TDOF model for UHPC members under lateral low-velocity impact

The dynamic behaviors of ultra-high performance concrete (UHPC) members under lateral low-velocity impact loading have been intensively studied by experiment and numerical simulation, while the corresponding structural design approach is still limited and mainly concerned in this study. Firstly, through comprehensively considering the compression arch action, strain rate effect, and unloading stiffness, the dynamic resistance function of UHPC members, e.g., reinforced UHPC (R-UHPC) and UHPC-filled steel tube (UHPC-FST) beams/columns, are built and verified. Then, a two degrees of freedom (TDOF) model of the drop hammer test on UHPC members is improved by considering the P-δ effect and stress wave propagation effect. Furthermore, the impact force- and mid-span deflection-time histories of the existing five series of drop hammer tests on UHPC members are applied to verify the validity and applicability of the established TDOF model. The influence of considering the stress wave propagation effect, strain rate effect, and steel fibers in UHPC for predictions of the TDOF model are discussed by parametric analyses. Finally, based on the energy balance, the explicit formulae are proposed to predict the maximum and residual mid-span deflections of the flexural critical UHPC members under the lateral low-velocity impact, which can be helpful for the design of UHPC structures under lateral impact loadings.