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Objectives This study investigated the impact of implant support on the development of shear force and bending moment in mandibular free-end base removable partial dentures (RPDs). Material and methods Three theoretical test models of unilateral mandibular free-end base RPDs were constructed to represent the base of tooth replacement, as follows: Model 1: first and second molars (M1 and M2); Model 2: second premolar (P2), M1, and M2; and Model 3: first premolar (P1), P2, M1, and M2. The implant support located either at M1 or M2 sites. The occlusal loading was concentrated at each replacement tooth to calculate the stress resultants developed in the RPD models using the free-body diagrams of shear force and bending moment. Results There was a trend of reduction in the peak shear force and bending moment when the base was supported by implant. However, the degree of reduction varied with the location of implant support. The moment reduced by 76% in Model 1, 58% in Model 2, and 42% in Model 3, when the implant location shifted from M1 to M2 sites. Conclusions The shear forces and bending moments subjected to mandibular free-end base RPDs were found to decrease with the addition of implant support. However, the impact of implant support varied with the location of implant in this theoretical study.

Natural frequencies and mode shapes of two joined isotropic conical shells are presented in this study. The joined conical shells can be considered as the general case for joined cylindrical–conical shells, joined cylinder–plates or cone–plates, conical and cylindrical shells with stepped thicknesses and also annular plates. Governing equations are obtained using thin-walled shallow shell theory of Donnell and Hamilton׳s principle. The continuity conditions at the joining section of the cones are appropriate expressions among stress resultants and deformations. The equations are solved assuming trigonometric response in circumferential and series solution in meridional directions and all combinations of boundary conditions can be assumed in this method. The results are compared and validated with the available results in other investigations and also modal testing. The effects of semi-vertex angles and meridional lengths on the natural frequency and circumferential wave number of joined shells are investigated.

Timoshenko beam is an extension of Euler-Bernoulli beam to interpret the transverse shear impact. The more refined Timoshenko beam relaxes the normality assumption of plane section that remains plane and normal to the deformed centerline. The manuscript presents some exact concise analytic solutions on deflection and stress resultants of NET single-span Timoshenko beam with general distributed force and 6 kinds of standard boundary conditions, adopting its counterpart Euler-Bernoulli beam solutions. Engineering example shows that scale impact would not unveil itself for micro structure with micrometer μm-order length, yet will be prominent for nanostructure with nanometer nm-order length. When simply supported CNTs is undergone to a concentrative force at the median and complete bend moment, scale action is observed along the ensemble CNTs, while it unfurls itself the most at the position of the concentrated strength. When a clamped-free CNTs is exposed to a centralized force at the mesial and distributed force, there is no scale impact about the deflection at all positions on the left border of the concentrated strength position, while such operation inspires at once at all positions on the right margin of the concentrated strength position. When a clamped-clamped CNTs is lain under a concentrative strength at the middle, the deflection of NET Euler-Bernoulli CNTs reflects scale effect completely. Notable differences between the deflection of Euler-Bernoulli CNTs and that of Timoshenko CNTs are reflected at large ratio of diameter versus length. The deflection of NET clamped-free and simply supported Timoshenko beam doesn’t introduce surplus scale process in terms of its counterpart, NET Euler-Bernoulli beam. However, the deflection of NET clamped-clamped Timoshenko beam does involve additional scale impact solely including the method when the concentrated strength position is at the midway in the beam-length direction.

The analysis of reinforced concrete structures until failure requires the consideration of geometric and material nonlinearities. However, nonlinear analysis is much more complex and costly than linear analysis. In order to obtain a computationally efficient approach to nonlinear analysis of reinforced concrete structures, this work presents the formulation of a nonlinear plane frame element. Geometric nonlinearity is considered using the co-rotational approach and material nonlinearity is included using appropriate constitutive relations for concrete and steel. The integration of stress resultants and tangent constitutive matrix is carried out by the automatic subdivision of the cross-section and the application of the Gauss quadrature in each subdivision. The formulation and computational implementation are validated using experimental results available in the literature. Excellent results were obtained. A análise de estruturas de concreto armado até à ruína requer a consideração das não linearidades física e geométrica. Contudo, a análise não linear é mais complexa e possui custo computacional mais elevado que a análise linear. Com objetivo de obter uma alternativa eficiente para a análise não linear de estruturas reticuladas de concreto armado, este trabalho apresenta a formulação de um elemento finito de pórtico plano não linear. A não linearidade geométrica é tratada através do uso da formulação corrotacional e a não linearidade física é considerada através do uso de relações constitutivas apropriadas para o concreto e o aço. A integração dos esforços e da matriz constitutiva tangente é realizada pela subdivisão automática da seção transversal em faixas seguida pela uso da quadratura de Gauss em cada faixa. A formulação e implementação computacional são validadas através da comparação com resultados experimentais, tendo sido obtidos excelentes resultados.

It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.

A novel method is developed to determine the optimal stress fields for the hybrid stress element. It provides a straightforward way as to how and why the resulting element can improve its displacement counterpart. A new inner product with material weighting matrix is defined to derive this quantitative method. It reveals the relationship in quantity of exact similarity degrees between different stress modes. It is different from the methods based on the conventional energy product which can only tell whether or not the stress and strain are orthogonal to each other because they are considered as mathematical vectors without any physical meaning. The strategy including two steps is proposed to determine the desired stress field. Firstly, the basic stress modes are broken into a set of sub-modes, where the necessary and unnecessary sub-modes are independent from each other because all of them are uniaxial. Secondly, all sub-modes are compared with their basic mode. The sub-mode with largest similarity degree with the basic mode implies that it represents the most important features inside the basic mode so it is selected as the optimal assumed stress mode for hybrid element. The 2D 4-node and 3D 8-node hybrid elements are illustrated by the present approach. Numerical examples are provided to compare the performances of element derived from different assumed stress fields.

In the present paper, a method for the modelling of the mechanical behaviour of composite plates, especially hybrid laminates and sandwich plates, is proposed. The chosen method is the computational homogenisation, or the so-called FE^2, for plates. The principle is to split the considered problem into two scales: on the one hand, the two-dimensional FE computation of a plate is performed on the macroscale. On the other hand, a three-dimensional FE problem is computed on the mesoscale, which enables the discretisation of the layers organisation. From each integration point of the macroscale, the deformations are projected to the Representative Volume Element (RVE) on the mesoscale, where a Dirichlet boundary value problem is solved. Finally, the homogenisation of the stresses is conducted to define the stress resultants of the macroscale. The proposed method presents the advantage to take into account any material behaviour without any transformation of the constitutive laws, even non-linear ones. Furthermore, the new projection strategy, presented in this work, considers the thickness change of the plate, which enables a resolution of the Poisson locking. The method is illustrated by the simulation of the mechanical behaviour of both sandwich plate and of hybrid laminate.

In this article, we prove that the assumption of vanishing normal stress for the generalized plane stress problem can be further weakened, so that the normal stress is harmonic. The stress state under this harmonic assumption can also be decomposed into the plane stress state and the shear stress state. Finally, a necessary and sufficient condition is presented for the existence of the biharmonic Airy stress function for the Filon averaged in-plane stresses, which is shown to be weaker than the assumption of the harmonic normal stress.

An approach to interrogate changes in flexibility regarding the severity of damage is presented. The approach is built on two results. The first shows that the influence lines of stress resultants at damage locations are a basis for the difference in the displacement field between the damaged and reference states for arbitrary loading. The second result shows that the coefficients that combine the basis are the magnitudes of discontinuities in the differential displacement field at the damage locations. The approach quantifies the damage severity by finding changes in sectional stiffness that justify, over reasonable smearing distances, the computed discontinuities. It is shown that at damage locations where the relevant stress resultant is statically determinate the solution is in closed form, while in the indeterminate case it is the fixed point of a nonlinear function.

In present work, a meshfree framework for the numerical simulation of elasto-plasticity deformation of ship structure is established. The reproducing kernel particle method (RKPM) is adopted to discretize the calculation domain, and the meshfree shell formulation is derived based on Mindlin shell theory. Then a general elasto-plasticity model is established under the meshfree shell framework, and the stress resultants are avoided since the shell formulation is based on degenerated continuum approach. By removing the normal stress from the yield function, the zero normal stress assumption is implemented. Several numerical examples are studied and the accuracy of the present elasto-plasticity model is verified. Finally, the dynamic response of the ship cabin and real ship structure under impact load are calculated, the capability of the present meshfree approach applied in complex engineering structures analysis is demonstrated.

Several numerical examples for cracked folded structures are analyzed to investigate the mixed-mode stress resultant intensity factors (SRIFs). The kinematic formulations of structures are derived by the first order shear deformation plate theory. A Galerkin meshfree six degrees of freedom (6DOFs) flat shell is employed, in which the reproducing kernel (RK) is used as the meshfree interpolant. A diffraction method, visibility criterion and enriched basis are introduced to model the through crack. J-integral is evaluated based on the stress resultants and is decomposed into the symmetric and asymmetric parts for extracting the mixed-mode SRIFs. Not only the stiffness matrix but also the J-integral are discretized by nodes. They are numerically integrated by the stabilized conforming nodal integration (SCNI) and the sub-domain stabilized conforming integration (SSCI) techniques. The effectiveness of the meshfree modeling and accuracy in the SRIFs are discussed.

This paper presents the results of an investigation concerning the influence of corners with standard radii in the local buckling behaviour of thin-walled rectangular hollow section (RHS) members subjected to combinations of axial force and biaxial bending. The calculation of the half-wave length leading to the minimum critical bifurcation load is performed by means of a Generalized Beam Theory (GBT) specialization, developed taking advantage of the assumption that the stress resultants are uniform along the member length. This assumption makes it possible to obtain semi-analytical solutions, adopting half-wave sinusoidal amplitude functions for the GBT cross-section deformation modes, and leads to a numerical implementation that (i) is able to quickly solve a large number of problems and (ii) provides physical insight into the critical buckling mode mechanics, through a shell-like stress resultant-based energy criterion, as well as the modal decomposition features of the GBT semi-analytical solutions. Members with cross-sections belonging to the EN10219-2 database are analysed and their critical buckling coefficients are compared with those provided by available analytical expressions and/or currently included in steel design codes, namely Eurocode 3 and the North American Specification for Cold-Formed Steel Members. Analytical expressions are developed to take into account the (usually) beneficial influence of the rounded corners to the RHS member critical buckling stress.

Empty open-top cylindrical steel tanks are susceptible to buckling when subjected to external pressure due to wind or partial vacuum due to blocked vents. A “wind girder” is commonly used at the top of the tank wall to increase its strength against external pressure instability. The wind pressure varies around the circumference of the tank, but is relatively constant up its height. A series of cosine functions is typically used to describe the variation of wind pressure around the circumference. Expressions for stress resultants, that form the basis of widely used design specifications such as API 650 and SH 3046, are based on simple mechanical models that adopt simplified pressure distributions and ignore interactions between the wind girder and the tank shell. Furthermore, in classical treatments a tributary height is postulated for wind loading on the girder, and this height is taken as independent of the properties of the girder and the shell. The purpose of this study is to develop a rational procedure to determine the stress resultants in a wind girder. Pursuant to this goal, Vlasov’s curved beam theory is used to derive the stress resultants and displacements for an isolated wind girder under a pressure distribution defined in terms of cosine functions. A parametric study employing finite element analysis is conducted to investigate the interaction of the wind girder with the tank shell. The stress resultants and the tributary height are found to be closely related to a shell-girder stiffness ratio that is devised in this study. This stiffness ratio was developed by considering the relative radial stiffness of the cylindrical shell and that of the wind girder. The changes in response quantities are expressed as functions of the shell-girder stiffness ratio. The developed expressions are presented in a form that is immediately useful for adoption into design standards.

Generally, the finite element method (FEM) is used in the initial design stage of large-scale structures such as ships. These FEM models tend to be made by shell element because they are usually composed by plate members. However, actually, the stress distribution at the joint of structural components is quite different from the stress distribution which can be obtained by shell element. Therefore, it is desired that 3-dimensional solid elements are used for the joint. However, it is difficult to use solid element for FEM models with fine mesh in the initial design stage because it costs a lot of time in modeling and analyzing. Then, it aims to obtain the complex stress distribution of the joint of structural components at the cost equivalent to the analysis by beam and shell elements. In this study, the new elements which enable to evaluate the stress of the joint of structural components were created with low calculation cost. Then, the good results were veried when these elements were applied to some examples.

This paper present the modal analysis for the predicating the behavior of inflatable membrane structure of general square shape with a thickness in millimeter using the various smart material which optimally within structural member subjected to edge- constrained rather than applying bending or moments. A numerical solution for membranes may also be found using the finite element method. In this paper flat thin membrane choose to analysis the behavioral effect of the multi-layered membranes using the properties of different smart material and compare their results in terms of frequency and out plane deflection with mode shape. This analysis makes more effective to select the smart materials in the space technology. Vibration analysis of arbitrary square shape membrane is also done using a finite element package, ANSYS APDL. The analysis shows good agreement in finite element solutions. Keywords: Boundary condition, finite element, material property, membrane, mode shape, natural frequency, out plane displacement. I. Introduction In the field of Engineering and Architecture, membrane structures play a vital role in many ways. Examples include textile covers and roofs, aircraft and space structures, parachutes, automobile airbags, sails, windmills, human tissues and long span structures. They are typically built with very light materials which are optimally used. These structures are characterized because they are only subjected to in-plane axial forces. Even in the field of architectures and civil engineering, both pre-stressed membranes and cable networks constitute a very remarkable group. A membrane is essentially a thin shell with no flexural stiffness. Consequently a membrane cannot resist any compression at all. However, membrane theory accounts for tension and compression stresses, and the need for a computational procedure that takes into account tension stresses only is needed. In membrane theory only the in-plane stress resultants are taken into account. A numerical solution for membranes may be found using the finite element method (1-3). The deployable space structures consist of thin polymer films that offer a wider range of packaging configurations than structures with traditional deployment mechanisms. Due to the flexibility of such deployable structure like shell or membrane shows greater importance for space application and hold great promise. The material constitutive behavior and the analytical tools to analyze them are required to make advances in building cheaper, lighter and more reliable structures. Many structures are in the developing stage and the materials that are meant to serve to make these applications possible are not yet within reach. Future missions depend much on new discoveries, mainly in material manufacturing. There are several different space applications in which the use of thin membrane structures are used or being considered. Due to their light weight, high strength-to-weight ratio and ease of stowing and deploying, membranes are especially attractive for space applications. Inflatable reflectors, space-based radar, space based communication systems such as antennae and solar power collection panels on spacecraft, etc are the examples included. (4-5). The membrane material used in the numerical analysis was assumed inextensible and its weight was neglected in the determination of the equilibrium shape. They found that the membrane's mass density is of little influence on the computed natural frequencies. Other researchers used finite elements and boundary elements to model and compute natural frequencies and mode shapes of a single-anchor inflatable dam (6). This study makes impact on finding the vibration aspect on the flat membrane using the various smart materials. The pressure in an inflatable structure can also play a critical role in the suppression of vibration (7). Literature that exists on pure structural membrane components has concentrated mostly on inflated components such as beams (8), torus (Main), and inflated lenticular concentrators (9). The dynamics of the membrane themselves are of great interest though, as it is the membrane itself that is performing the useful work, and in some applications they could be attached to more traditional aerospace structures. Therefore improving understanding the behavior of the membranes appears to be important. A membrane is essentially a thin shell with no flexural stiffness. Consequently a membrane cannot resist any Compression at all. However, membrane theory accounts for tension and compression stresses. In membrane theory only the in-plane stress resultants are taken into account (10). This paper present the modal analysis for the predicating the behavior of various inflatable membrane structure of general square shape with a thickness in millimeter using the various smart materials. A numerical solution for membranes may be found using the finite element method. Finite element analysis of membrane structures for small deformations can be found in (11) but

The present study deals with vibrational behavior of two joined cross-ply laminated conical shells. The natural frequencies and mode shapes are investigated. The joined conical shells can be considered as the general case for joined cylindrical–conical shells, joined cylinder-plates or cone-plates, cylindrical and conical shells with stepped thicknesses and also annular plates. Governing equations are obtained using thin-walled shallow shell theory of Donnell type and Hamilton’s principle. The appropriate expressions among stress resultants and deformations are extracted as continuity conditions at the joining section of the cones. The equations are solved assuming trigonometric response in circumferential and series solution in meridional directions. All combinations of boundary conditions can be assumed in this method. The results are compared and validated with the available results in other investigations. The effects of semi-vertex angles, meridional lengths and shell thicknesses on the natural frequency and circumferential wave number of joined shells are investigated.

During the past few decades, various technical solutions have been developed to improve the seismic performance of structural connections, as these determine the global response of buildings as much as single structural elements. However, in the case of heritage buildings, standard strengthening systems might have collateral effects owing to their high stiffness in comparison with historic substrata. Furthermore, little technical guidance is given to designers for dimensioning the retrofit of connections. Drawing on these observations, the authors have developed, in collaboration with Cintec Ltd, two dissipative devices. These are able to restore the connection between perpendicular masonry walls and to control relative displacements, thus reducing the load stress resultants transmitted to the substratum and the risk of pull-out or punching failures. The paper gives an overview of the experimental work conducted to identify performance parameters and exemplifies how these can be used for designing the strengthening, while satisfying code performance requirements.

In this work the third-order shear deformation theory by Reddy is applied and modified to analyze delaminated orthotropic composite plates. The delaminated plate portion is captured by Reddy's traditional theory, while a novel double-plate system is developed for the undelaminated part. It is shown that in the uncracked part four conditions are required to satisfy in symmetrically delaminated plates. The conditions involve the imposition of traction-free boundaries and the interface constraints. These four conditions enable the reduction of the parameters from nine to five in the displacement field. The governing equations show significant coupling among the stress resultants of the uncracked portion, that has to be considered in the continuity conditions between the delaminated and undelaminated parts. To demonstrate the application of the present model a simply-supported delaminated plate subjected to a concentrated force is analyzed. The distribution of the mode-II and mode-III energy release rates and their ratio are calculated using the 3-dimensional J-integral. The finite element model of the plate is also created using brick-type elements. The comparison of the analytical and finite element results shows very good agreement. It is shown that the deformations around the delamination front can be captured by the third-order plate theory with high accuracy.

This paper presents a simple and accurate sandwich plate theory accounting for the transverse normal strain and interfacial stress continuity. A refined cubic polynomial is used for the transverse shear function and a linear function is adopted for the transverse normal strain. The Heaviside step function and stress continuity coefficients are employed respectively to enforce the interfacial continuity of the in-plane displacements and transverse stresses. By the enforcement of the traction conditions on the plate surfaces, there are only five independent field variables in the present sandwich plate theory. The variational consistent equilibrium equations and boundary conditions in terms of both displacements and stress resultants are derived by utilizing the variational principle. The analytical solutions of the bending analyses of sandwich plates with different aspect ratios and stiffness ratios are solved to demonstrate the accuracy of this new sandwich plate theory. The resulting analytical solutions of deflections, normal stresses and transverse shear stresses are compared with the 3D elasticity solutions, the numerical results and the results given by other sandwich plate theories. The comparison study shows that this equivalent single layer sandwich plate theory is not only simple, but also capable of achieving the accuracy of layerwise sandwich plate theories.

For partial-interaction composite beams, two beam theories (i.e., the Euler-Bernoulli and Timoshenko beam theories) are usually used to investigate their deflections, slips, and stress resultants. However, the relationships between the solutions of partial-interaction composite beams based on the two beam theories have not been discussed while the corresponding relationships for homogeneous beams have been investigated in detail for several years. By analyzing the constitutive relationships and equations of equilibrium, the authors derived the relationships of the solutions between single-span Euler-Bernoulli and Timoshenko partial-interaction composite beams. The integral constants are also presented for various boundary conditions. Through the presented relationships, the solutions of the Timoshenko partial-interaction composite beams could be readily obtained from the solutions of the corresponding Euler-Bernoulli counterparts. As a result, the more complicated flexural-slip-shear deformation analysis based on Timoshenko beam theory may be avoided for engineering designers.

The exact static solutions of shear flexible laminated curved beams are derived using a coupled field formulation. The formulation developed in this paper can be generalized to analyze isotropic and laminated curved beams. The governing equations are first derived in terms of forces or stress resultants. The spatial distribution of radial shear, tangential force, and bending moment along the arc length of the beam are derived in closed form. The displacements are then integrated exactly using the constitutive equations. Several examples of isotropic and laminated curved beams are solved and compared with available solutions.

In this paper, stress distribution of micro cantilever beams in the presence of a dielectric-layer is studied using an analytic method. The Modified Adomian Decomposition Method (MADM) is applied to obtain a semi-analytical solution for a distributed parameter model of the micro cantilever beam. The important parameters for designing and manufacturing micro-actuators such as shear force, bending moment and stress distribution along the cantilevers are computed for different values of the dielectric-layer parameter. The results of MADM are compared with the numerical results, and they found in good agreement. It is found that increase of the dielectric-layer parameter increases the dimensionless pull-in voltage, tip deflection, internal stress and bending moment of the micro cantilever actuators at the onset of pull-in instability.

This paper presents an explicit damage evaluation methodology that utilizes the fundamental principles of mechanics to relate the changes in the physical properties of an undamaged and damaged structure to measurable response quantities. The proposed methodology is based on the principle of invariant stress resultants and the Timoshenko beam theory. First, damage in a structure is expressed in terms of local decreases in the bending stiffnesses of structural members. These stiffness decreases are in turn related to appropriate response quantities utilizing the Timoshenko beam theory. Next, a set of numerical experiments using results from two dimensional finite element models of cantilever and fixed–fixed beams are introduced to evaluate the performance of the proposed damage evaluation scheme. Results indicate that the proposed methodology can identify the location, the extent, and the severity of the local stiffness degradations in beams of arbitrary aspect ratios.

Accurately accessing the three-dimensional stress distribution through the thickness surrounding the bolt hole is necessary to identify the most effective multi-bolt single-lap joint design in large-scale thick laminate structures. To decrease computing time and resources, a two-dimensional to three-dimensional global-local finite element numerical method is presented, in which both the local zone selection strategy and boundary constraints applying method are discussed in detail. And both the contact stress and the stress components around the fastener hole are analyzed by using the proposed global-local method. Through analysis, it is found that the contact stress is concentrated in the layers that are close to the shear plane; the @s"1"1 is much higher than other stress components; most of the stress components conform cosine distribution around the fastener hole; the out-of-plane stress and interlaminate shear stress are quite high in the layers close to the shear plane; the stress distributions of different layers are depended on the ply angles of the specified layer; the effectiveness of the global-local model procedure in reducing the computation effort is obvious and this methodology is applicable to other detailed stress/strain analysis problems in large-scale structures.

In this study, the buckling of two joined isotropic conical shells under axial compression and simply supported boundary conditions is investigated. The governing equations are obtained using thin-walled shallow shell theory of Donnell-type and theorem of minimum potential energy. The continuity conditions at the joining section of the cones are appropriate expressions among stress resultants and deformations. The equations are solved by assuming trigonometric response in circumferential and series solution in meridional direction. The results are validated in comparison with the available results in the literature. Effects of semi-vertex angles and meridional lengths on the buckling load and typical mode shape and circumferential wave number are investigated. In addition, the application of the results of this study on the load carrying capacity of these structures is presented.

We present in this work the theoretical aspect and numerical implementation of new enriched finite elements with smeared-embedded softening hinges for the analysis of Timoshenko and Euler-Bernoulli beams at failure. On one hand, similarly to the embedded hinge model, the softening hinge is described by a cohesive model between the stress resultants and the displacement jumps. On the other hand, the stress field in the element is evaluated from a constitutive relation at the cross-section level as in the classical smeared hinge model. The crucial point is the introduction of a consistent inelastic strain field dependent on both the interpolation scheme (not merely the characteristic length) of underlying finite elements and the hinge mode of interested. Furthermore, the traction continuity condition is enforced in strong form. Based on the irreversible thermodynamics, a multisurface model for the softening hinge and an inelastic constitutive relation for the cross-section of the beam are then established. Particularly, the displacement jumps are regarded as internal variables and can be determined at the material point level of the softening hinge. Therefore, it is not necessary to introduce the ad hoc penalty elastic stiffness, the extra degrees of freedom or the static condensation at the element level as in the embedded hinge model. Besides the above characteristics, the proposed method also defines a very convenient framework for its numerical implementation. Standard finite elements of beams/rods and well-developed algorithms for inelastic material models can be straightforwardly employed with minor modifications. Several representative numerical examples are presented, verifying that the proposed model is mesh-size objective, stress-locking free and frame invariant owing to its correct resolution of the kinematics and statics of the softening hinge.

An accurate 3d mixed beam element that is efficient especially in nonlinear analysis is presented in this paper. The mathematical theory is based on Hu-Washizu principle that uses three-fields in the variational form. The composition of the variational form ensures independent selection of displacement, stress and strain fields. Timoshenko beam theory is extended to three dimensions for deriving strains from displacement field. Numerical integration of stress–strain relations along control sections is carried out for numerical analysis. The finite element approximation for the beam uses shape functions for section forces that satisfy equilibrium and discontinuous section deformations along the control sections of the beam. This form of the element permits coupling of the stress resultants and eliminates necessity of displacement components along the beam element except at the nodes. Consequently the element is free from shear-locking. Numerical examples on uniform and tapered structural members with solid and hollow circular sections demonstrate the nonlinear interaction between axial, shear force, bending moments and torsion, where the results compare well with closed form solutions and available data in literature. Furthermore, the proposed element has superior performance in both linear and nonlinear analysis compared to a locking-free higher order displacement based 3d Timoshenko beam element.

In practice, a structure is subjected to given loads and boundary conditions, and a multitude of stress and strain states may exist in the structure; hence, optimal construction of a laminate in a structure cannot be sought by considering only a limited number of stress resultants in the existence of multiple load cases. Then, another design objective based on optimization of a laminate for the worst possible load case emerges which is formulated as a minimax problem whose solution is shown to be equivalent to singular value minimization problem. As the squares of singular values are the bounds of power, energy and power spectral density ratios between the input and output vectors, shaping the singular values of a composite material is equivalent to shaping the response of the material. As a novel approach, singular values are used for the layout optimization of laminate. In this method, the main idea is minimization of the largest singular value of the transfer function matrix between force/moment resultants and outputs stress/strain. Thus the overall optimization problem is reduced to a simple minimization problem. Numerical examples and finite element simulations are presented for several test problems. In particular, it is shown that the use of singular values and singular vectors is computationally advantageous in case of multiple load case.

An analytical study of local buckling of discrete laminated plates or panels of fiber-reinforced plastic structural shapes is presented. Two cases of composite plate analyses with two opposite edges simply supported and the other two opposite edges either both rotationally restrained or one rotationally restrained and the other free, are studied. Generic loading cases with combined linearly varying axial and in-plane shear loading are considered. A variational formulation of the Ritz method is used to establish the eigenvalue problem for the local buckling behavior, and explicit expressions for predictions of the plate buckling stress resultants, in terms of the rotationally restrained stiffness, the plate aspect ratio, and the ratios of applied stress resultants, are developed. Based on different boundary and loading conditions, simple and explicit local buckling solutions for several special cases are further reduced. Validity of the explicit solutions presented is demonstrated by a good agreeme...

The Kirchhoff and Mindlin plate theories are applied in this study to calculate the stresses and the energy release rates in delaminated orthotropic composite plates. A novel double-plate system is developed with the imposition of the kinematic continuity constraints in the interface plane. The governing equations of the system were derived in both cases. As a demonstrative example a simply-supported plate subjected to a point force was analyzed using Lévy plate formulation and the problem was solved by a state-space model. The distribution of the stress resultants and the interlaminar stresses in the uncracked part were also determined. Moreover, the distributions of the mode-II and mode-III energy release rates along the crack front were calculated by the J-integral. The 3D finite element model of the plate was created providing reference data for the analytical model. The results show that the displacement and stress fields obtained from the Kirchhoff and Mindlin theories are quite similar, but in the case of the energy release rates, transverse shear effect is necessary to consider to obtain reasonably good agreement between the analytical and numerical results.

SUMMARY In this paper a coupled two-scale shell model is presented. A variational formulation and associated linearization for the coupled global–local boundary value problem is derived. For small strain problems, various numerical solutions are computed within the so-called FE 2 method. The discretization of the shell is performed with quadrilaterals, whereas the local boundary value problems at the integration points of the shell are discretized using 8-noded or 27-noded brick elements or so-called solid shell elements. At the bottom and top surface of the representative volume element stress boundary conditions are applied, whereas at the lateral surfaces the in-plane displacements are prescribed. For the out-of-plane displacements link conditions are applied. The coupled nonlinear boundary value problems are simultaneously solved within a Newton iteration scheme. With an important test, the correct material matrix for the stress resultants assuming linear elasticity and a homogeneous continuum is verified.Copyright © 2013 John Wiley & Sons, Ltd.

The buckling of a bar is studied analytically on the basis of a simple linear theory of gradient elasticity in the frame of the method of initial values. The method of initial values provides the values of the displacements and stress resultants throughout the bar once the initial displacements and initial stress resultants are known. We use probably for the first time the method of initial values to get critical loads of a strain gradient beam under completely different boundary conditions at the two end faces of the beam. Exact carryover matrix is presented for the classical beam and gradient beam analytically. The first mode shapes of classical beam and gradient beam are plotted. The method of initial values is also applied to the beams with variable cross-section. The priorities of the method of initial values are depicted. The variational approach gives a sixth-order ordinary differential equation for a beam in buckling. The additional boundary conditions are used to obtain critical loads. It is observed that critical loads increase dramatically for increasing values of the gradient coefficient.

SUMMARY It has been shown in a previous paper that the normal stress resultants for an arbitrary polygonal section endowed with a uniaxial elastoplastic law can be computed exactly by means of a general approach; it has been denominated fiber free as no fiber discretization of the section is required. The aim of this paper was to extend the fiber-free approach to the evaluation of the tangent stiffness matrix to allow for the analytical integration of all the basic quantities required by the section state determination occurring in nonlinear frame analysis. The numerical examples showed that the general properties of the fiber-free approach, already discussed with reference to the stress resultants, carry over to the tangent stiffness matrix as well. Namely, independent from the complexity of the constitutive law, the accuracy in the evaluation of the tangent stiffness matrix allowed by the number of fibers adopted in commercial software for nonlinear frame analysis is even worse than that associated with the computation of the stress resultants. Copyright © 2013 John Wiley & Sons, Ltd.

An elastoplastic implicit integration algorithm applicable to both plane stress and three-dimensional stress states is developed for a general class of combined nonlinear kinematic-isotropic hardening models. The algorithm is first built for three-dimensional stress states in a general manner using the return mapping procedure and the Newton-Raphson method. The plane stress constraint is then incorporated into the Newton-Raphson iteration loop derived for three-dimensional stress states. The resulting algorithm has a mode patch that makes the algorithm applicable to both plane stress and three-dimensional stress states. The algorithm is specified by assuming an advanced evolution model of multiple back stresses, and is verified by performing numerical tests using plane stress, shell, and brick elements. The numerical tests are finite element analyses of homogeneously deformed plates and a cyclically loaded single-hole plate. It is demonstrated that the developed algorithm provides the quadratic convergence of iterations for implicit stress integration in plane stress, shell, and brick elements. It is also demonstrated that the algorithm is stable even in large incremental steps.

In this paper both the static and dynamic analyses of the geometrically linear or nonlinear, elastic or elastoplastic nonuniform torsion problems of bars of constant or variable arbitrary cross section are presented together with the corresponding research efforts and the conclusions drawn from examined cases with great practical interest. In the presented analyses, the bar is subjected to arbitrarily distributed or concentrated twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. For the dynamic problems, a distributed mass model system is employed taking into account the warping inertia. The analysis of the aforementioned problems is complete by presenting the evaluation of the torsion and warping constants of the bar, its displacement field, its stress resultants together with the torsional shear stresses and the warping normal and shear stresses at any internal point of the bar. Moreover, the construction of the stiffness matrix and the corresponding nodal load vector of a bar of arbitrary cross section taking into account warping effects are presented for the development of a beam element for static and dynamic analyses. Having in mind the disadvantages of the 3D FEM solutions, the importance of the presented beamlike analyses becomes more evident.

This work develops a kinematically linear shell model departing from a consistent nonlinear theory. The initial geometry is mapped from a flat reference configuration by a stress-free finite deformation, after which, the actual shell motion takes place. The model maintains the features of a complete stress-resultant theory with Reissner-Mindlin kinematics based on an inextensible director. A hybrid displacement variational formulation is presented, where the domain displacements and kinematic boundary reactions are independently approximated. The resort to a flat reference configuration allows the discretization using 2-D Multiple Fixed Least-Squares (MFLS) on the domain. The consistent definition of stress resultants and consequent plane stress assumption led to a neat formulation for the analysis of shells. The consistent linear approximation, combined with MFLS, made possible efficient computations with a desired continuity degree, leading to smooth results for the displacement, strain and stress fields, as shown by several numerical examples.

The second-order shear deformation theory is used in this study to calculate the stresses and the energy release rates in orthotropic composite plates. A novel double-plate system is utilized with the imposition of the proper kinematic constraints in the interface plane of a double-plate system. The governing equations of the system were derived and as a demonstrative example a simply-supported plate subjected to a point force was analyzed. Using Lévy plate formulation, the plate problem was solved by a state-space model, incorporating four different regions. The distribution of the stress resultants and the interlaminar stresses in the uncracked part were also determined. Moreover, the distributions of the mode-II and mode-III energy release rates along the crack front were calculated by the J-integral. The 3D finite element model of the plate was created providing reference data for the analytical model. The results show reasonably good agreement between the analytical and numerical results. Also, the present model eliminates the physical inconsistency of previous models and reveals that under mixed-mode II/III condition, the energy release rate is not contributed by the bending, twisting moments and shear forces at all.

This paper extends an analytical method for static analysis of general cross-ply piezolaminated rectangular plates with any combination of clamped/simply-supported boundary conditions under uncoupled thermo-electro-mechanical loadings. This method is based on the novel superposition method and the first-order shear deformation theory (FSDT). The FSDT enables this expanded method to consider the effect of shear deformation of the plate. The process of applying electrical and thermal resultants causes some advantages due to its simplicity and less computational process. In this analysis displacement components are written in terms of unknown force and moment resultants. Using Fourier series for displacement components, mechanical, thermal, and/or electrical stress resultants, the complex governing differential equations of the plate are reduced to a set of linear algebraic equations with non-trivial solution. The obtained equations may be solved analytically to determine the unknown stress resultants. Several examples are proposed, and their obtained numerical results are compared with those available in the literature to verify the convergence, high accuracy, and the capability of the present method to analyze the static behavior of piezolaminated plates. It is found that there is high agreement between the present results with those obtained by other investigators.

The paper deals with geometrically nonlinear finite element analysis of folded-plate and shell structures. A Koiter asymptotic approach is proposed, based on the reuse of a linear element in the nonlinear context through a corotational formulation.The corotational approach represents a simple and effective way to satisfy the basic requirement of Koiter analysis, i.e. full objectivity in the finite element modeling. In fact, starting simply from a suitable linear finite element and implementing the corotational algebra proposed in Garcea et al. (2009), Zagari (2009) lead to objective explicit expressions for the first four variations of the strain energy which are needed by asymptotic analysis.The shell element used here is the flat shell quadrangular element with 4 nodes and 6 dofs per node proposed in Madeo et al. (2012) and called MISS-4: a mixed element, based on the Reissner–Mindlin plate theory, with an Allman-like quadratic interpolation for displacements and an equilibrated isostatic interpolation for the stress resultants. The element is free from locking and spurious zero-energy modes, so it appears a suitable candidate for nonlinear corotational analysis.The results of the numerical validation show the effectiveness and accuracy of the proposed approach, and its excellent overall robustness for both mono- and multi-modal buckling problems, also in the presence of strong nonlinear pre-critical behavior.

The buckling and postbuckling of beams is revisited taking into account both the influence of axial compressibility and shear deformation. A theory based on Reissner’s geometrically exact relations for the plane deformation of beams is adopted, in which the stress resultants depend linearly on the generalized strain measures. The equilibrium equation is derived in a general form that holds for the statically determinate and indeterminate combinations of boundary conditions representing the four fundamental buckling cases. The eigenvalue problem is recovered by consistent linearization of the governing equations, the critical loads at which the trivial solution bifurcates are determined, and the influence of shear on the buckling behavior is investigated. By a series of transformations, the equilibrium equation is rearranged such that it allows a representation of the solution in terms of elliptic integrals. Additionally, closed-form relations are provided for the displacement of the axis, from which buckled shapes are eventually obtained. Even for slender beams, for which shear deformation can usually be neglected, both the buckling and the postbuckling behavior turn out to be affected by shear not only quantitatively, but also qualitatively.

In box girder bridges, the quantity and distribution of reinforcement to be put in concrete elements of sections can be evaluated only by considering the deformation of the cross section in addition to the longitudinal analysis of the static scheme, establishing the entire state of stress of box sections. This leads to a need to evaluate the interaction between internal forces obtained by the global analysis and the ones obtained by the local analysis of the cross sections. The frame effect implies the elastic deformation of slabs and webs, whereas eccentrically applied loads lead to cross-section distortion with the loss of the box shape. Hence, the reinforcement is strongly influenced by a significant interaction between longitudinal shear and transverse bending moments, because the same reinforcement bars, although acting on different planes, face these two stress resultants. Moreover, the longitudinal reinforcement in the webs plays an important role in the failure model. In this paper, an analytical model of interaction between longitudinal shear and transverse bending is presented, based on the stress field theory and on the application of the static theorem of plasticity. The proposed model allows engineers to establish the effective quantity of transverse and longitudinal reinforcements of box girder elements, both for ordinary and prestressed concrete girders. In the model, the variation of stress field inclination between the limit state of first cracking (serviceability) and the ultimate limit state is considered, and the related consequences are discussed. Interaction domains of longitudinal shear and transverse bending are sketched and supplied through dimensionless graphs to be used by designers for actual cases of engineering practice. A case study of a prestressed girder bridge is presented to show the reliability of the proposed approach. The analytical model is validated through comparisons with experimental data; finally, the influences of prestressing, warping phenomena, and cross-section distortion are discussed.

The purpose of this study is to analytically investigate the out-of-plane static behavior of curved nanobeams with non-uniform distributed loads. Nonlocal constitutive equations are implemented into the governing equilibrium equations in cylindrical coordinates. Use of these equations enables revealing the scale effect on static response of curved nanobeams. Shear deformation is considered in the formulation of out-of-plane static behavior of nanobeams. The exact analytical solution to the theoretical model is derived by using initial value method. The fundamental matrix required for the initial value method is obtained analytically. Then, the displacement, slopes and stress resultants are found analytically using the fundamental matrix. Illustrative examples for such nano structures with non-uniform distributed loads are given in subsequent sections. Results showed that the scale effect plays crucial role at nano scale which agree with the results given in the literature. The effects of small scale parameter, slenderness ratio and opening angle on the displacements ratio, rotation angles ratio and stress resultants ratio are also investigated in details. Considering scale effect and shear deformation in the exact solutions of the governing equilibrium equations presented here, the results can be considered as a reference for nonlocal theories of curved nanobeams. (C) 2017 Elsevier Ltd. All rights reserved.

In the preceding chapters it has been seen that the PVP yields the governing equations which are valid regardless of the constitutive material of the plate, but these are not sufficient in number to solve the problem. To obtain as many field equations as there are unknowns, it is essential to add the constitutive law - that is, the relation between the components of stress and components of strain - which takes into account the nature and the response of the material.

In this work, we present an exact 3D plate solution in the conventional form of 2D plate theories without invoking any of the assumptions inherent to 2D plate formulations. We start by formulating a rectangular plate problem by employing Saint Venant’s principle so that edge effects do not appear in the plate. Then the exact general 3D elasticity solution to the formulated interior problem is examined. By expressing the solution in terms of mid-surface variables, exact 2D equations are obtained for the rectangular interior plate. It is found that the 2D presentation includes the Kirchhoff, Mindlin and Levinson plate theories and their general solutions as special cases. The key feature of the formulated interior plate problem is that the interior stresses of the plate act as surface tractions on the lateral plate edges and contribute to the total potential energy of the plate. We carry out a variational interior formulation of the Levinson plate theory and take into account, as a novel contribution, the virtual work due to the interior stresses along the plate edges. Remarkably, this way the resulting equilibrium equations become the same as in the case of a vectorial formulation. A gap in the conventional energy-based derivations of 2D engineering plate theories founded on interior kinematics is that the edge work due to the interior stresses is not properly accounted for. This leads to artificial edge effects through higher-order stress resultants. Finally, a variety of numerical examples are presented using the 3D elasticity solution.

Current consensus suggests a trade-off for predicting accurate 3D stress fields in multilayered plates. Relatively expensive layerwise models facilitate accurate stress predictions from underlying model assumptions, whereas more efficient equivalent single-layer theories often rely on variationally inconsistent transverse stresses from recovery steps. In contrast, we show that variationally consistent 3D stress fields are predicted from an equivalent single-layer model using a contracted form of the Hellinger–Reissner functional. Notably, this procedure facilitates computationally efficient analysis of 3D heterogeneous plates, i.e. laminates comprising layers with material properties that differ by multiple orders of magnitude and that can also vary continuously in-plane. The model includes effects of higher-order transverse shearing and zig-zag deformations by expressing the in-plane stress field as a Taylor series expansion of global and local higher-order stress resultants. Equilibrated transverse stress assumptions are derived from Cauchy’s 3D equilibrium equations, which identically satisfy the interfacial and surface traction equilibrium conditions when applied within a variational statement that enforces the 3D equilibrium equations as constraints, hence the Hellinger–Reissner mixed-variational statement. By using inherently equilibrated stress fields as model assumptions, it is shown that only the classical membrane and bending equilibrium equations have to be enforced explicitly. As a result, a contracted Hellinger–Reissner functional emerges that requires fewer displacement Lagrange multipliers than when independent stress fields are assumed. The governing equations are derived in a generalised framework such that the order of the model, and hence its accuracy, increases automatically when implemented in a computer code. By refining the order of the stress assumptions, the model is adaptable to plate-like structures ranging from thin engineering laminates to highly heterogeneous, thick laminates comprising straight-fibre or tow-steered variable-stiffness laminates, foam, honeycomb or other compliant layers.

A higher-order beam model for analyzing the flexural response of curved multilayered beams with constant curvature and arbitrary constant thickness is developed. The new model is derived from the Hellinger–Reissner mixed variational statement and predicts inherently equilibrated 3D stresses from an equivalent single-layer model. As a starting assumption, the hoop stress is formulated as a series of higher-order stress resultants multiplied by Legendre polynomials. The governing equations are derived in a generalized manner such that the modeling order can be adjusted and is not defined a priori. Hence, the highest order Legendre polynomial determines the modeling order. The through-thickness shear and normal stresses are derived by integrating the generalized hoop stress in Cauchy’s polar equilibrium equations. As a result, all stress fields are based on the same set of variables, thereby considerably reducing the computational effort. The three stress fields, and two displacements in the radial and hoop directions are used in the Hellinger–Reissner functional to derive a new set of stress-displacement relations. The enforcement of the classical membrane and bending equilibrium equations of curved beams in the Hellinger–Reissner functional guarantees that all interlaminar and surface traction equilibrium conditions are satisfied exactly. A validation study of a composite laminate using a high-fidelity 3D finite element model shows that the stresses are captured very accurately by the present model, but with much less computational effort than the finite element model. As a result, the developed model can provide rapid and accurate insights into the expected damage onset behavior of curved laminates.

Hydroelastic interactions of a deformable floating body with random waves are investigated in time domain. Both hydroelastic motion and structural dynamics are solved by expansion of elastic modes and Fourier transform for the random waves. A direct and efficient structural analysis in time domain is developed. In particular, an efficient way of obtaining distributive loads for the hydrodynamic integral terms including convolution integral by using Fubini theory is explained. After confirming correctness of respective loading components, calculations of full distributions of loads in random waves are expedited by reformulating all the body loading terms into distributed forms. The method is validated by extensive convergence tests and comparisons against the counterparts of the frequency-domain analysis. Characteristics of motion/deformation responses and stress resultants are investigated through a parametric study with varying bending rigidity and types of random waves. Relative contributions of componential loads are identified. The consequence of elastic-mode resonance is underscored.

Eringen’s nonlocal elasticity theory is incorporated into classical beam model considering the effects of axial extension and the shear deformation to capture unique static behavior of the nanobeams under continuum mechanics theory. The governing differential equations are obtained for curved beams and solved exactly by using the initial value method. Circular uniform beam with concentrated loads are considered. The effects of shear deformation, axial extension, geometric parameters and small scale parameter on the displacements and stress resultants are investigated.