Large Displacement Analysis Research Articles

A flexibility approach for geometric nonlinear static analysis of guyed masts

The present study proposes a novel flexibility-based approach for the large displacement analysis of guyed masts. The guyed mast is modeled as an equivalent beam–column supported by catenary cables, and the exact force–displacement relations of sagging cables and slender beam–column are accounted for with sufficient rigor. The proposed method starts with formulating two mutually exclusive sets of second-order force–displacement equations, one for the mast and the other for the cables. Flexibility matrices are derived in an incremental form for an Euler–Bernoulli beam–column using the well-known Variational Iteration Method (VIM). Since an exact Lagrangian multiplier is used in the correction functional, the displacements obtained from this numerical method are also exact. The cable behavior is studied using Irvine’s model, which is the most accurate and realistic one since it represents the equilibrium configuration of the cable using a catenary curve. But, the model is non-algebraic and is unsuitable for complex structures. Hence, Irvine’s equations are recast into an incremental form to obtain exact incremental flexibility matrices. Then, displacement compatibility is re-established between the mast and the cables at their interfaces. This method can effectively address the eccentric compression on the mast shaft from the supporting guy cables, stay slackening, postbuckling of the mast, and trace the mast’s nonlinear response for large displacements. A multi-level guyed mast is analyzed to demonstrate the practical application of the proposed method. Classic examples of single-level guyed towers are used to illustrate the behavior in the postbuckling and post-slackening region. The sensitivity of a guyed mast to cable prestress levels, as well as mast shaft imperfections, are also discussed. The solved examples are also analyzed using commercial finite element software packages, and comparisons are drawn regarding their accuracy in capturing the load–deflection response under large displacements.

Multi-segmented fifth-order polynomial–shaped shells under hydrostatic pressure

The design and construction of submerged complex shells for new applications in offshore structures are increasingly popular. Therefore, the purpose of this study is to present the analytical model and Lagrange multipliers associated with the constraint equation for large displacement analysis of a fifth-order polynomial–shaped shell under hydrostatic pressure for the first time. The shell geometry can be computed using differential geometry with a fifth-order polynomial. The energy functional of the fifth-order polynomial–shaped shell is derived based on the principle of virtual work and written in the appropriate form. The nonlinear static responses of the fifth-order polynomial–shaped shell under hydrostatic pressure can be calculated using the nonlinear finite element method via the fifth-order polynomial shape function. This study develops the model using one-dimensional beam elements divided along the shell radius. To avoid the slope of a meridian curve at the equatorial plane approaching infinity, the shell is divided into two regions defined by different surface parameters. At the junction of two adjacent regions, the continuity requirements are established as the constraint conditions using Lagrange multipliers. The numerical results from the proposed methods are demonstrated and discussed, along with the effects of varied seawater depth, thickness, and elastic modulus on the deformed configuration and principal curvature at the deformed state. The results show that the nonlinear displacement is higher than the linear one in the case of the hydrostatic pressure, whereas the case of the internal pressure has an opposite result. For principal curvatures at the apex, the principal curvatures increase as the seawater depth increases, whereas the principal curvatures decrease when the thickness and elastic modulus increase.

Effects of shear on the lateral torsional buckling of singly-symmetric I-beams

This paper presents an investigation on the negative effects of shear on the lateral torsional buckling (LTB) of singly-symmetric I-shaped beams due to web distortion associated with shear. Results from a parametric FEA study including eigenvalue buckling analyses and large-displacement analyses were utilized considering both stiffened and unstiffened webs to consider the impact of shear on the LTB resistance. Moment gradients caused by constant shear and a shear gradient were considered. The FEA solutions demonstrate that shear can significantly reduce the lateral-torsional buckling resistance. The FEA results were used to develop design equations that account for the reduction in the LTB capacity from shear. The solutions were compared with conventional methods proposed by Winter to account for the effects of web distortion on girders with slender webs. Results from the study show that (i) the average flange area should be used to account for the effects of shear of singly-symmetric beams; (ii) post-buckling strength has limited impact on the shear effects with regards to the LTB resistance; and, (iii) compared to Winter's approach, the proposed method can capture the variation of the moment reduction factor with the shear force in the unbraced length, and is able to predict moment resistances with reasonable accuracy compared to refined FEA solutions using both eigenvalue analysis and large-displacement analysis. Design examples are provided to demonstrate the calculation procedure.

Post-buckling behavior of variable-arc-length elastica pipe conveying fluid including effects of self-weight and pressure variation

This paper investigated the post-buckling behaviors of a variable-arc-length (VAL) elastica pipe caused by internal transporting fluid motion, accounting for the effects of the self-weight and variation of internal fluid pressure. Both weak-form and strong-form formulations of the VAL elastica pipe were developed. The weak-form formulation was derived by the virtual work principle, and later solved using the nonlinear finite element method (FEM). The strong-form formulation was derived using equilibrium of the force and moment equations, the geometrical relationship of the differential pipe segment, and the energy conservation of the transported fluid. Because of the large displacement analysis, the variation in the internal pressure inside the pipe was considered, which was taken into account by energy conservation based on the Bernoulli principle. Then, the set of nonlinear governing first-order differential equations, corresponding to the two-point boundary value problem, was conveniently solved using the shooting optimization method (SOM). The numerical results obtained from the presented FEM and SOM analyses independently verified each other, with good agreement between these two methods. The parametric study considered the effects of gravity load (pipe and fluid self-weights) and internal fluid pressure on the post-buckling characteristics of the VAL pipe. The numerical results showed that without the pipe and internal fluid weights, the transporting fluid exerted an axial compressive force on the pipe, causing post-buckling phenomena. Beyond the critical state, the support rotation and the length of the VAL pipe increased as the internal fluid velocity decreased. Finally, by including the effect of self-weight, the mode exchange of the odd buckling modes (1st and 3rd modes) was found from the load-displacement path.

An enhanced corotational Virtual Element Method for large displacements in plane elasticity

An enhanced virtual element formulation for large displacement analyses is presented. Relying on the corotational approach, the nonlinear geometric effects are introduced by assuming nodal large displacements but small strains in the element. The element deformable behavior is analyzed with reference to the local system, corotating with the element during its motion. Then, the large displacement-induced nonlinearity is accounted for through the transformation matrices relating the local and global quantities. At the local level, the Virtual Element Method is adopted, proposing an enhanced procedure for strain interpolation within the element. The reliability of the proposed approach is explored through several benchmark tests by comparing the results with those evaluated by standard virtual elements, finite element formulations, and analytical solutions. The results prove that: (i) the corotational formulation can be efficiently used within the virtual element framework to account for geometric nonlinearity in the presence of large displacements and small strains; (ii) the adoption of enhanced polynomial approximation for the strain field in the virtual element avoids, in many cases, the need for ad-hoc stabilization procedures also in the nonlinear geometric framework.

A corotational isogeometric assumed natural strain shell element in updated lagrangian formulation for general geometric nonlinear analysis of thin-walled structures

A new general geometric nonlinear curved quadrilateral shell element, which can handle large displacements and large rotations, is presented in this paper. It is formulated based on the incremental virtual work equation in updated Lagrangian formulation for general large displacement analysis. The kinematics of shell element is described using non-uniform B-spline patch, which results in the element displacement field with higher order continuity and the element performance more resistive to shear locking and membrane locking. In order to perfectly avoid such membrane locking and shear locking behaviors, the assumed in-plane membrane strain field and the assumed transverse shear strain field for the incremental covariant Green-Lagrange strain are constructed as linear combinations of non-uniform B-spline basis functions with degrees lower than those adopted by the original displacement field interpolation. In addition, based on Euler parameters, the corotational procedure available for high order isogeometric shell element is developed to extract the strain-producing deformational displacements and rotations and to update the element stresses and realistic internal force vectors. The accuracy and robustness of proposed shell element are demonstrated through the solutions of various benchmark problems.

Geometrically consistent nonlinear plane strain and stress constitutive models: Application to soft-material oscillations

Position-gradient constraints are used in this study to develop new geometrically consistent plane strain and stress constitutive models for the nonlinear large-displacement analysis of soft materials. The position gradients are used to define strain and stress constraints that enter into the formulation of the deformation tensor, strain-energy density function, and new constitutive models. The new approach is applied to three nonlinear hyperelastic constitutive models for incompressible materials: Neo-Hookean, Mooney Rivlin, and Yeoh Models. The underlying geometric plane-stress and plane-strain assumptions are preserved by using the finite elements (FE) of the absolute nodal coordinate formulation (ANCF). Limitations of widely used incompressibility penalty functions that produce stresses at initial stress-free configurations are discussed. Several numerical examples developed using planar ANCF shear-deformable beam and rectangular elements are used to assess the proposed formulation and verify its results using spatial solid elements implemented in commercial FE software. It is demonstrated that the ANCF plane-stress and plane-strain formulations for the incompressible materials predict accurately soft-material oscillations, demonstrating that stiffer behavior can be attributed to material-model inconsistencies and cannot always be attributed to locking.

Loading–shape interaction analysis of membrane structures subjected to deformation dependent hydrostatic loadings

Due to their flexibility, membrane structures loaded and/or supported by gas and/or fluid undergo large deformations when subjected to external loads, for example when a relatively flat membrane deflects under accumulative ponding. The most noteworthy feature during the deformation process is the loading–shape interaction since changes in the form of the structure induce changes in the loading profiles, and vice versa. Description of the hydrostatic loading is complex since the discontinuity of structural domain and fluid domain occurs with the deformation of membrane. In this paper, an interactive model dealing with accurate hydrostatic load description and nonlinear finite element analysis is employed to analysis the structural response. The hydrostatic loading is described in vectors by fluid level and loading direction in current mesh of the membrane. In case to conserve a specific volume, the free surface height is calculated with a Newton algorithm. Equivalent nodal forces evaluation on different elements are deduced especially when they are partially or fully loaded by judging the spatial relation between the element and the fluid domain, and integrating with shape functions and area coordinates in a local coordinate system. In case of multiple fluid domains, the hydrostatic load description procedure is proposed to be conducted for each fluid domain with care in identifying respective loading direction and then sum them up to get the total hydrostatic loading forces. Two numerical examples, (a) single fluid domain case: a flat membrane subjected to accumulative water loading, and (b) multiple fluid domains case: an air-inflated dam partially filled with water and loaded by lateral pressures of water, are presented to demonstrate correctness, performance and effectiveness of the proposed method in large displacement analysis of flexible membrane structures subjected to deformation dependent hydrostatic loading.

Large displacement analysis of egg-shaped containment shells using Lagrange multipliers

In this paper, a large displacement analysis of three-center-combined spherical shells, used as egg-shaped containment structures, is performed using the first and second fundamental forms of the shell surface with two different radii of curvature. The energy functional of the three-center-combined spherical shells system is derived using a variational formulation, and is written in the appropriate form to reduce the computation time in an iterative procedure. The numerical results in terms of displacement, membrane stress, and bending moment resultants are obtained by nonlinear finite element analysis with a fifth-order polynomial shape function. At the spherical shell edge, the Lagrange multipliers technique is used to enforce the discontinuity effect. The main conclusions are that the displacement responses, membrane stress, and bending moment resultants depend on the geometric parameters of the three-center-combined spherical shells. The Lagrange multipliers correspond to the internal forces due to the displacement and the slope at the shell edge in order to sustain the smooth curve for deformed configuration of the three-center-combined spherical shells under uniform pressure. The volume capacity of the three-center-combined spherical shells are more than the spherical shells with a constant radius. Finally, the results indicate that the linear assumption is good enough for all practical purposes. However, the large displacement analysis becomes significant on the tangential displacement, meridional, and circumferential moments under the large value of uniform pressure.

Large displacement analysis of two-layer beam-columns taking into account slip and uplift

PurposeThis article presents a geometrically non-linear finite element formulation for the analysis of planar two-layer beam-columns taking into account the inter-layer slip and uplift.Design/methodology/approachThe co-rotational method is adopted, in which the motion of the element is decomposed into a rigid body motion and a small deformational one. The geometrically linear formulation can be used in the local frame and automatically be transformed into a geometrically nonlinear one. In co-rotational frame, both layers are assumed to be discretely connected at the element ends. Slips and uplifts are assumed to be small. Consequently, the condition of non interpenetration between the layers can be treated using a node-to-node contact algorithm. The resolution methods such as penalty (PM) and augmented Lagrangian method (ALM) with Uzawa updating scheme can be used.FindingsThe non-penetration condition between the layers of composite beams can be formulated by using contact law. It is found that despite a low convergence rate of augmented Lagrangian method compared to penalty method, the former prevents the unrealistic penetration. Besides, it is shown that the buckling load of the composite beam-column is largely affected by the uplift stiffness of the connectors.Originality/valueThe proposed finite element model is capable of simulating accurately the geometrically non-linear behavior of planar two-layer beam-columns taking into account the inter-layer slip and uplift. Regarding uplift, the non-penetration condition is strictly enforced by considering rigorous contact conditions at the interface. The constraint problem is solved using the penalty method or the augmented Lagrangian method with the Uzawa updating scheme.

Large displacement analysis of laminated beam-type structures

The paper presents a finite element formulation for analysis of 3D framed structures with thin-walled laminated composite cross-sections, using a co-rotational approach. The formulation considers the effects of large displacements on the response of space frames subjected to conservative and static external loads. Classical lamination theory for thin fiber-reinforced laminates has been employed. Stability analysis is performed in load deflection manner using co-rotational formulation. The shear strain of the middle surface is assumed to be zero, and the cross-section is not distorted in its own plane. In order to illustrate the application of the proposed formulation, several numerical examples are presented.

Large Displacement Analysis With Material Nonlinear of Shell Structure Using New Flat Shell Elements

In this study, we propose a numerical algorithm for a nonlinear large displacement analysis method using flat shell elements in the hybrid-type penalty method (HPM). First, we describe the flat plate elements of HPM using the local coordinate system. Even with a flat plate, the adjacent elements have an angle at which a large displacement occurs. Next, we propose the relative displacement in this case. The “step-by-step method” is used as the algorithm for material nonlinear analysis and large displacement analysis. Finally, the accuracy of the solution of the proposed method is verified using a simple numerical example.

Determination of bend angle and shape imperfection effect on collapse load of pipe bends under in-plane opening moment

This study uses a three-dimensional nonlinear finite-element method to investigate the effects of bend angle ([Formula: see text]) and shape imperfection on the collapse load of pipe bends subjected to in-plane opening bending moment using large displacement analysis. ABAQUS was used to create a pipe bend model, whereas by virtue of symmetry about the longitudinal axis, half of the pipe bend model was built. The bend angle was varied from 30° to 180° with an interval of 30°, and shape imperfections, namely ovality ( Co) and thinning ( Ct), were varied from 0% to 20% each at 5% intervals. Elastic-perfectly plastic is thought to be the material of choice for the pipe bend. The results show that the bend angle has a significant impact on the collapse load of pipe bends between 30° and 120°, after which a significant decline was observed. The ovality impact increases with an increase in bend angle for pipe bends with a large thickness, whereas the opposite effect was found for smaller thicknesses with a small bend radius. The effect of bend radius and thickness was significant for large bend angles. Thinning ( Ct) and thickening ( Cth) have no effect on collapse load. A new mathematical equation has been proposed to predict the integrated effect of bend angle and shape imperfection.

Locking-free 6-noded triangular shell elements based on hierarchic optimisation

Conforming lower-order shell elements based on Reissner-Mindlin plate theory generally exhibit an over-stiff response under loading, typically manifested through various forms of locking. A recently developed hierarchic optimisation approach addresses locking by enriching the conforming strains with hierarchic strain terms towards an objective ‘smoother’ strain distribution afforded by the element, which has proven to be effective in relieving shear, membrane and distortion locking in 9-noded quadrilateral shell elements. Nevertheless, in some practical structural problems that involve complex geometry, triangular shell elements are required to avoid a highly distorted mesh of quadrilateral elements. This paper presents a family of 6-noded Reissner-Mindlin triangular shell elements based on the hierarchic optimisation approach. The proposed curved triangular shell elements not only effectively alleviate inaccuracies arising from locking, but also embrace the desirable characteristics of spatial isotropy and insensitivity to element distortion. The family of 6-noded triangular elements have been incorporated within a co-rotational framework to allow large displacement analysis of thin to moderately thick plates and shells. Several numerical examples are finally presented to demonstrate the effectiveness and accuracy of the proposed 6-noded shell element formulation as well as its superior locking-free performance compared to existing shell elements. • Novel family of 6-noded locking-free curved shell elements. • Mitigation of shear, membrane and distortion locking via hierarchic optimisation. • Objective and corrective assumed-strain variants including spatial isotropy. • Incorporation within co-rotational framework for large displacement analysis. • Element variant H3O6 offers superior performance compared to MITC6 element.

Finite element formulation of Timoshenko tapered beam-column element for large displacement analysis based on the exact shape functions

ABSTRACT A numerical formulation was carried out in this paper to produce the tangent stiffness matrix for two-nodal tapered Timoshenko beam-column elements for geometrically nonlinear analysis. The proposed solution is based on the exact shape functions and their derivatives describing the non-uniformity of the element properties. The section properties were presented as exponential functions with tapering indices to illustrate the variations in section properties along the tapered element length. The model is applicable for elements with different solid and hollow cross-sections. The proposed formulation is embedded into a Visual Basic code to carry out the analysis accompanied by many examples for validating its accuracy and efficiency. The model results are compared with those of commercial software and cited references that showed high accurate results with a small number of elements.

Large displacement analysis of stiffened plates with parallel ribs under lateral pressure using FE modeling with shell elements

This study provides the load–displacement behavior of stiffened plates with parallel ribs when modeled using shell elements with and without considering the large-displacement effects in comparison with the conventional equivalent beam analysis (EBA) in which such stiffened plates are simplified into an imaginary beam with the geometrical properties of an equivalent built-up cross-section. This study highlights the advantages of FE modeling with shell elements and large-displacement analysis (LDA) over the existing EBA method and then provides a path for using this method in the ultimate limit state (ULS) design of stiffened plates. The stress distributions in the panel plate part of the stiffened plates are presented to demonstrate the significance of considering the large-displacement effects in the analysis. The numerical investigation indicated that the linear beam theory does not correctly presume the true behavior of the stiffened plates and that the corresponding load–displacement estimation does not align with its behavior predicted by the LDA. The results also demonstrate that when shell element modeling is utilized, the internal forces and stresses in the panel plate are calculated correctly, only if the large-displacement effects are considered in the analysis. Finally, a method for estimating the ultimate load capacity of stiffened plates with parallel ribs is provided, based on the large-displacement FE analysis with shell elements, which can be used to establish empirical design equations.

Deduction of ultimate equilibrium limit states for concrete gravity dams keyed into rock mass foundations based on large displacement analysis

Concrete gravity dams are mass concrete structures, often built on rock mass foundations, conceived to rely upon their weight for stability. To prevent sliding, these structures are usually keyed/embedded into the foundation, a good construction practice particularly relevant in medium to high intensity seismic zones. In stability analysis, the extra strength obtained by keying the dam into the foundation is usually either neglected or taken as a passive resistance, which, such as explored in this paper, do not reflect the real structural response in pre-collapse situations. Limit state philosophy requires the ultimate equilibrium conditions to be expressed as accurately as possible. In this paper, the rigid-body equilibrium of a wedgy model representing the dam and a downstream rock wedge is analyzed according to the large displacement regime. Failure mechanisms were identified, analytically described and numerically validated. Application to two Portuguese large concrete gravity dams led to safety factors considerably larger than those computed assuming the usual practice. The proposed approach is intended to support probabilistic and/or semi-probabilistic methodologies for safety assessment of concrete gravity dams, in the design and feasibility phases, in which the limit state approach is inherently followed.

Analisis Perpindahan Besar Akibat Gaya Non-Konservatif

Structural applications based on large displacement analysis are widespread in various fields, such as the application of aircraft structures, membranes, cables, pipelines and risers. In this study, the author reviews the structural responses based on large displacement theory and non-conservative load models, assuming linear sections and materials. The parameters reviewed are the variation of the load factor and the variation of the slope angle of the load P at the end of the span. Solving non-linear differential equations with non-conservative force, using numerical integration method. The simulation results with the variation of the load factor with the angle of inclination, vertical and horizontal displacements are obtained, where the final orientation of the acting force corresponds to the deformation of the structure, and the maximum deformation occurs at a load angle of 90o, the deformation decreases for the angle orientation is getting smaller.

A 2D field-consistent beam element for large displacement analysis using a rational Bézier representation with varying weights

This study develops a new 2D field-consistent Euler-Bernoulli beam element for the analysis of beam structures undergoing large displacements but small strains. The total Lagrangian description is considered in the derivation of finite element equations. Relationships between the cross-sectional rotation and displacements of the beam axis are derived, and consequently, the kinematics of the element are entirely described by the displacements of the beam axis. These relationships underlie the construction of field-consistent interpolations. Regarding geometric descriptions, the deformed configuration of an element is represented by a rational cubic Bézier curve. For better geometric representations of complex deformed configurations, some of the weights are treated as discrete unknowns. Several benchmark numerical examples are used to demonstrate the accuracy and robustness of the proposed beam element. The merits of considering weights as discrete unknowns are verified through comparisons between elements having varying weights and stationary weights. Furthermore, the proposed beam element is found to be naturally free from membrane locking, and thus, no additional efforts are required to handle the locking issues.

Large displacement analysis of dry-joint masonry arches subject to inclined support movements

In this paper, the structural response of a segmental dry-joint masonry arch subjected to increasing support displacements was investigated. The arch behaviour was analysed in the framework of large displacements using finite element (FE) and rigid block (RB) models, both based on a micro-modelling strategy. Once the two modelling approaches were validated against experimental results from literature, nonlinear analyses were carried out considering different configurations of support displacements (i.e. vertical, horizontal and inclined). The effect of the displacement direction on the arch response was evaluated in terms of initial hinge position, collapse mechanism, support reaction-displacements curves as well as ultimate displacement capacity. The sensitivity of the results to the number of voussoirs was also investigated. The good agreement between the results from the two models allowed to draw new conclusions about the response of segmental masonry arches to inclined support displacements in the large displacement regime. A limit displacement domain computed as a function of the direction of the support displacements is also proposed.