Displacement Compatibility Research Articles

Drape simulation using solid-shell elements and adaptive mesh subdivision

In this paper, 4-node quadrilateral and 3-node triangular solid-shell elements are applied to drape simulations. With locking issues alleviated by the assumed natural strain method and plane-stress enforcement, static and dynamic drape problems are attempted by the quadrilateral element. If the drape is deep and the mesh density is inadequate, non-realistic sharp folds are predicted due to the non-physical interpenetration of top and bottom element surfaces. To avoid the interpenetration, a reversible adaptive subdivision based on the 1–4 splitting method is developed. To ensure displacement compatibility among elements at different subdivision levels, macro-transition elements are formed by quadrilateral and triangular solid-shell elements. To reduce the dynamic oscillation induced by newly inserted nodes, the discrete Kirchhoff condition is employed to determine the related nodal variables. Dynamic drape examples using adaptive meshing are presented. It can be seen that the predictions look realistic and deep drapes can be predicted with the interpenetration avoided yet the required number of nodes can be kept relatively small.

Free vibration of horizontally curved thin-walled beams with rectangular hollow sections considering two compatible displacement fields

ABSTRACTA finite element is presented for vibration analyses of horizontally curved thin-walled rectangular hollow beams. Eight cross-section deformation modes are employed to describe the mid-surface contour displacement field with the modal superposition method. Focused on the in-plane moment equilibrium condition and the displacement continuity condition, two compatible displacement fields are constructed to calculate the strain energy and the kinetic energy of the beam, respectively. With the application of Hamilton’s principle the dynamic governing equations are formulated, and then approximated for the finite element implementation. Finally, numerical examples are illustrated to verify the validity of the present theory.

Vertical vibration of a pile in transversely isotropic multilayered soils

A new method for the dynamic response of a vertically loaded single pile embedded in transversely isotropic multilayered soils is proposed in this paper. The dynamic response of the pile is governed by the one-dimensional (1D) vibration theory, and that of transversely isotropic multilayered soils is achieved by using an analytical layer-element method. Then, with the aid of the displacement compatibility and the contact forces equilibrium along the pile–soil contact surface, the dynamic pile–soil interaction problem is solved efficiently. The presented solution method is proved to be correct and efficient by comparing the obtained results with other existing solutions. Selected numerical results are presented to study the influence of mass density ratio, length–radius ratio, frequency of excitation, soil anisotropy and hard soil stratum on the pile vertical impedance.

Augmented Numerical Manifold Method with implementation of flat-top partition of unity

This paper formulates the flat-top partition of unity (PU)-based numerical manifold method (NMM). Through modifying the finite element PU into the flat-top PU, the linear dependence problem involved in finite element PU-based high-order polynomial approximations has been successfully alleviated. Moreover, the procedure to construct the flat-top PU is substantially simplified by taking full advantage of the unique features of the NMM, compared to that in other flat-top PU-based methods. The proposed method could also be treated as an improved version of the discontinuous deformation analysis (DDA) by completely avoiding the additional strategies previously used to enforce the displacement compatibility between adjacent elements. The formulations are presented in details for 1-D, 2-D and 3-D cases. The discrete equations of the flat-top PU-based NMM are derived based on the minimum potential energy principle. The simplex integration technique and its variation are employed to evaluate the integration of the matrices of the discrete equations over arbitrarily shaped manifold elements. Seven representative examples have validated the proposed method in the aspects of approximation accuracy and efficiency.

Predicting the influence of discretely notched layers on fatigue crack growth in fibre metal laminates

This paper presents an analytical model for fatigue crack growth prediction in Fibre Metal Laminates (FMLs) containing discretely notched layers. This model serves as a precursor in the development of a simplified prediction methodology for modelling the effect of load redistribution on a single crack in FMLs containing Multiple-site Damage (MSD) scenario. The model mainly focuses on capturing the influence of load distribution around discretely notched layers on the growth behaviour of an adjacent crack in a FML panel. The utilized approach in the model is the use of linear elastic fracture mechanics (LEFM) in conjunction with the principle of superposition and displacement compatibility. The proposed model is also validated using experimental data.

The long-term stability analysis of 3D creeping slopes using the displacement-based rigorous limit equilibrium method

The conventional limit equilibrium method is hardly applied to investigate the stability and the displacement of creeping slopes. In this paper, a novel displacement-based rigorous limit equilibrium method is proposed to analyze the stability and the displacement of three-dimensional (3D) creeping slopes. The relationship between the shear displacements of the slopes and the creep time is obtained based on the visco-elastoplastic creep model. According to the displacement compatibility among the columns, the shear displacements of all columns can be described by the vertical displacements at the key point. Combining the six equilibrium conditions of the discretized columns with the visco-elastoplastic creep model, the vertical displacement at the key point can be determined. With increase of creep time, the vertical displacement at the key point converges to a constant value. By introducing the concept of strength reduction technique into the displacement-based rigorous limit equilibrium method, the relationship between the reduction factor (RF) and the long-term displacement at the key point can be obtained. Then, the long-term factor of safety of 3D creeping slope is equal to the value of the reduction factor at the point of sharp increment of displacement. Two case studies are given to verify the robustness and precision of the proposed method.

Potential fracture propagation into the caprock induced by cold [formula omitted] injection in normal faulting stress regimes

Thermal effects are an important component in the analysis of geologic carbon storage because the injected CO2 reaches the storage formation at a lower temperature than that of the reservoir rock. The main fear is related to the possibility that the shear slip that may occur within the reservoir due to cooling could propagate into the caprock, which could result in CO2 leakage. We model a baserock–reservoir–caprock system in a normal faulting stress regime using an axisymmetric model in which we inject cold CO2 and use an elasto-plastic constitutive model to simulate inelastic deformation. CO2 forms a cold region around the injection well of a significantly smaller extension than the CO2 plume. Within this cold region, inelastic strain is yielded due to the thermal stress reduction caused by the thermal contraction of the rock. This inelastic strain occurs only within the reservoir and does not propagate into the caprock. Actually, the stability of the lower portion of the caprock improves in the cooled region as a result of a stress redistribution that occurs to satisfy stress equilibrium and displacement compatibility caused by the thermal stress reduction of the reservoir. This stress redistribution tightens the caprock and prevents CO2 leakage. Thus, thermally induced stresses are not likely to generate fracture propagation into the caprock in normal faulting stress regimes.

Deformability of units of a planetary gear and its effect on load distribution in gear meshes

A method is considered to determine the axis compliance of a planetary pinion subject to the deformation of the carrier web. It is revealed that the deformability of the axis, its mating parts, and sun gear affects the load distribution over power flows and rims of a two-row pinion. Variation factors for the load distribution in gear meshes are determined by solving of sets of displacement compatibility equations including the compliance of gear units and by its preparation error.

Impact coefficient and reliability of mid-span continuous beam bridge under action of extra heavy vehicle with low speed

To analyze the dynamic response and reliability of a continuous beam bridge under the action of an extra heavy vehicle, a vehicle-bridge coupled vibration model was established based on the virtual work principle and vehicle-bridge displacement compatibility equation, which can accurately simulate the dynamic characteristics of the vehicle and bridge. Results show that deck roughness has an important function in the effect of the vehicle on the bridge. When an extra heavy vehicle passes through the continuous beam bridge at a low speed of 5 km/h, the impact coefficient reaches a high value, which should not be disregarded in bridge safety assessments. Considering that no specific law exists between the impact coefficient and vehicle speed, vehicle speed should not be unduly limited and deck roughness repairing should be paid considerable attention. Deck roughness has a significant influence on the reliability index, which decreases as deck roughness increases. For the continuous beam bridge in this work, the reliability index of each control section is greater than the minimum reliability index. No reinforcement measures are required for over-sized transport.

Floor diaphragms and a truss method for their analysis

Floor diaphragms form a critical component of seismic resistant buildings, but unfortunately, in the main their analysis and design in New Zealand leaves much to be desired. No worse example exists than the CTV Building in Christchurch. Despite the critical importance of diaphragms, there is a paucity of code provisions and design guidance relating to them.
 Using generic examples, the author describes a number of common diaphragm design deficiencies. These include diaphragms where valid load paths do not exist; diaphragms where the floors are not properly connected to the lateral load resisting elements, diaphragms that lack adequate flexural capacity and where re-entrant corners are not properly accounted for, and transfer diaphragms into which the reactions from the walls above cannot be properly introduced or transmitted.
 Three main types of seismic diaphragm action are discussed – ‘inertial,’ ‘transfer’ and ‘compatibility.’ These are, respectively, the direct inertial load on a floor that must be carried back to the lateral load resisting elements, the transfer forces that occur when major changes in floor area and lateral load resisting structure occur between storeys, and the compatibility forces that must exist to force compatible displacements between incompatible elements, such as shear walls or braced frames and moment frames, or as a result of redistribution.
 The author presents a simple Truss Method that allows complex diaphragms to be analysed for multiple load cases, providing accurate force distributions without the multiple models that conventional strut and tie methods would require. Being a type of strut and tie method, the Truss Method is compliant with requirements in NZS3101:2006 [1] to use strut and tie models for the analysis and design of certain aspects of diaphragm behaviour.

Vulnerability assessment and retrofit solutions of precast industrial structures

The seismic sequence which hit the Northern Italian territory in 2012 produced extensive damage to reinforced concrete (RC) precast buildings typically adopted as industrial facilities. The considered damaged buildings are constituted by one-storey precast structures with RC columns connected to the ground by means of isolated socket foundations. The roof structural layout is composed of pre-stressed RC beams supporting pre-stressed RC floor elements, both designed as simply supported beams. The observed damage pattern, already highlighted in previous earthquakes, is mainly related to insufficient connection strength and ductility or to the absence of mechanical devices, being the connections designed neglecting seismic loads or neglecting displacement and rotation compatibility between adjacent elements. Following the vulnerabilities emerged in past seismic events, the paper investigates the seismic performance of industrial facilities typical of the Italian territory. The European building code seismic assessment methodologies are presented and discussed, as well as the retrofit interventions required to achieve an appropriate level of seismic capacity. The assessment procedure and retrofit solutions are applied to a selected case study.

Research on multiple-split load sharing of two-stage star gearing system in consideration of displacement compatibility

Further analysis was developed on multiple-split loading sharing among star gears of two-stage star gearing system in aero-engine, and a meshing error was analyzed on the eccentricity error, gear thickness error, base pitch error, assembly error, and bearing manufacturing error of gear components of the star gearing system respectively. This research, in view of the floating meshing error resulting from meshing clearance variation caused by the simultaneous floating of all gears, established a refined mathematical model of two-stage multiple-split loading sharing coefficient calculation in consideration of displacement compatibility, obtained the regular curves of the load sharing coefficient, and conducted a load sharing coefficient test experiment of star gearing system to verify the research results, which can provide scientific theory evidence for proper tolerance distribution and control in design and process.

Multibody Finite Element Method and Application in Hydraulic Structure Analysis

Multibody finite element method is proposed for analysis of contact problems in hydraulic structure. This method is based on the block theory of discontinuous deformation analysis (DDA) method and combines advantages of finite element method (FEM) and the displacement compatibility equation in classical elastic mechanics. Each single block is analyzed using FEM in corresponding local coordinate system and all contacting blocks need to satisfy the displacement compatibility requirement between any two blocks in a blocky system. It is proved that this method is very efficient and practical to overcome the limitations in DDA method when tackling contact problems, such as the overlap problem and the equal strain assumption. In this paper, detailed theoretical basis and formulations are given. Two numerical examples are performed to verify the proposed method successfully. Furthermore, this method is adopted to study the stability issues of underground houses of a large hydropower station.

Indirect Inverse Substructuring Method for Multibody Product Transport System with Rigid and Flexible Coupling

The aim of this paper is to develop a new frequency response function- (FRF-) based indirect inverse substructuring method without measuring system-level FRFs in the coupling DOFs for the analysis of the dynamic characteristics of a three-substructure coupled product transport system with rigid and flexible coupling. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed-form analytical solution to inverse substructuring analysis of multisubstructure coupled product transport system is derived based on the relationship of easy-to-monitor component-level FRFs and the system-level FRFs at the coupling coordinates. The proposed method is validated by a lumped mass-spring-damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The suggested method may help to obtain the main controlling factors and contributions from the various structure-borne paths for product transport system.

Mixed-dimensional finite element coupling for structural multi-scale simulation

Multi-scale finite element (FE) modeling and analysis of engineering structures have become very necessary in order to provide both global and local structural information for a comprehensive assessment of structural safety. The multi-scale FE modeling needs FE coupling methods to combine mixed-dimensional finite elements, such as beam-to-shell and plate-to-solid, in a single structural model. This paper presents a new mixed-dimensional FE coupling method that can achieve both displacement compatibility and stress equilibrium at the interface between the different element types. The principle of virtual work is first used to derive both linear force and displacement constraint equations for the interface. A numerical method compatible with commercial FE codes is developed to figure out the linear constraint equations which satisfy both displacement compatibility and stress equilibrium conditions at the interface. The proposed coupling method is then extended to nonlinear mixed-dimensional FE coupling problems. Finally, the proposed coupling method is applied to a number of test cases including linear beam-to-plate, beam-to-shell and beam-to-solid interface problems, a linear frame structure, and a beam-to-shell buckling problem. The obtained results are also compared with those from the existing methods. It reveals that the proposed coupling method can handle both linear and nonlinear mixed-dimensional FE coupling problems more accurately than the existing methods and that it can be applied to a structure satisfactorily for multi-scale simulation.

Indirect Inverse Substructuring Theory for Coupling Dynamic Stiffness Identification of Complex Interface Between Packaged Product and Vehicle Transport System

Inverse substructuring method has been recently proposed and applied for inverse analysis of the dynamical response of product transport system. The component-level frequency response functions (FRFs) and the coupling dynamic stiffness for facilitating the cushioning packaging design are all predicted from only the system-level FRFs. However, the system-level FRFs from coupling degree of freedoms may not be measured accurately because of the difficulties of vibration excitation and response measurement for the coupled interface between packaged product and vehicle within the limited accessible space. The aim of this paper is to develop a new FRF-based indirect inverse substructuring method for the analysis of the dynamic characteristics of a three-substructure coupled product transport system without measuring system-level FRFs at the coupling degree of freedoms. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed-form analytical solution to inverse sub-structuring analysis of multi-substructure coupled product transport system is derived based on the relationship of easy-to-monitor component-level FRFs and the system-level FRFs at the coupling coordinates.. The proposed method is validated by a lumped mass-spring-damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. Then, the FRF tests of a physical prototype of multi-substructure coupled product transport system are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The proposed method may help to obtain the main controlling factors and contributions from the various structure-borne paths for product transport system. Copyright © 2014 John Wiley & Sons, Ltd.

Warping torsion in commercial vehicle frames, taking into consideration flexible joints

The torsional behaviour of commercial vehicle frames is dominated by warping torsion, because warping is inhibited in the joints where the cross–members are attached to the side–members. This paper presents a hybrid method of analysis, which combines finite element idealization of the joint areas with analytically derived beam elements for the cross–members and side–member sections. The beam element includes warping torsion force–displacement relationships. Thus, the flexibility of the joints is included together with the compatibility of warping displacements. The method gives close agreement with experimental results.

Simulation of the interfaced structural component based on a multiple-time-scale algorithm

A multiple-time-scale algorithm is developed to numerically simulate certain structural components in civil structures where local defects inevitably exist. Spatially, the size of local defects is relatively small compared to the structural scale. Different length scales should be adopted considering the efficiency and computational cost. In the principle of physics, different length scales are stipulated to correspond to different time scales. This concept lays the foundation of the framework for this multiple-time-scale algorithm. A multiple-time-scale algorithm, which involves different time steps for different regions, while enforcing the compatibility of displacement, force and stress fields across the interface, is proposed. Furthermore, a defected beam component is studied as a numerical sample. The structural component is divided into two regions: a coarse one and a fine one; a micro-defect exists in the fine region and the finite element sizes of the two regions are diametrically different. Correspondingly, two different time steps are adopted. With dynamic load applied to the beam, stress and displacement distribution of the defected beam is investigated from the global and local perspectives. The numerical sample reflects that the proposed algorithm is physically rational and computationally efficient in the potential damage simulation of civil structures.

Force equilibrium-based finite displacement analyses for reinforced slopes: Formulation and verification

Formulation and verification for a force equilibrium-based finite displacement method (FFDM) using test results of reinforced model slopes subjected to increasing pseudo-static seismic forces are reported. The FFDM requires, in addition to force equilibrium for a sliced potential failure mass, a hyperbolic shear stress–displacement constitutive law for the backfill soils, a hyperbolic pull-out force–displacement constitutive law for the reinforcement, and a displacement compatibility requirement for adjacent soil slices. As a result, the mobilized reinforcement force is an analytical output, rather than an empiricism-based input as required in conventional limit equilibrium analyses. Analytical results from the FFDM also indicated that a brittle failure is associated with the lightly reinforced failure surface; a ductile failure is associated with the heavily reinforced failure surface, regardless of the extensibility of reinforcement investigated in the present study. Good agreements between the measured and the computed slope displacements and reinforcement forces in response to increases in pseudo-static seismic forces suggest that the FFDM can be used as an analytical tool for evaluating displacements of reinforced slopes subjected to pseudo-static seismic loads.

Seismic performance of ductile connections between precast beams and roof elements

The seismic vulnerability of precast reinforced concrete buildings is often governed by the performance of mechanical connections between precast elements. This aspect was highlighted by recent seismic events in Italy, where several collapses were registered among industrial buildings typical of Italian practice. The building damage was related to failure of connections between beams and columns and between beams and roof elements, which led to the loss of support of the structural elements. Starting from the results of an experimental campaign, the present work investigates the use of ductile connections between precast beams and roof elements suitable for both new structures and as a retrofit measure of existing ones. These connections are able to transfer the horizontal inertial loads and to accommodate deformations arising from seismic displacement compatibility. The relative rotation between the end of the roof elements and the beam, owing to seismic displacement demand, could lead to their contact. This leads to a load increase in rigid connections, which can cause their premature failure, or to horizontal relative displacements as in the ductile connections considered herein. Moreover, the connections investigated could be used to dissipate seismic energy.