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The paper proposes an efficient sequential edge-smoothed finite element (ES-FE) analysis method enhanced with the smoothed recovery stress field to determine the collapse load limit of inelastic structures. The approach applies a modified concept of elastic compensation algorithm developed within the computationally advantageous ES-FE modelling framework and recovery stress field enhancement. It solely performs a series of standard elastic analyses to successively converge the limit load of structures at plastic collapse. Each analysis systematically adjusts the values of elastic moduli associated with some critical elements as determined by the intensity of stress resultants and hence considers inelastic stress distributions. The computing efforts are as modest as would be required to furnish standard linear elastic analysis procedures. What is important is that the proposed modulus smoothing technique enables the fast assembly of modified stiffness formulations, and the recovery stress field ensures the smoothed C0-continuous admissible stress conditions enhancing the satisfaction of yield conformity over an entire element. Various successfully solved numerical examples highlight the accuracy and efficiency of the analysis framework in overcoming the numerical challenges associated with stress singularity and volumetric locking phenomena under incompressibility conditions. They also address the determination of a lower-bound limit for sufficiently fine ES-FE models.

This study deals with calculating the resultant moments and forces around an elliptical and circular holes in a functionally graded carbon nanotube–reinforced composite (FG-CNTRC) plates. Various distributions of the reinforcement materials aligned uni-axially are investigated considering three functionally graded and uniform distributions of carbon nanotubes along thickness direction. The mechanical properties of CNTs (carbon nanotube–reinforced) are temperature- and size-dependent and are calculated by molecular dynamics simulations. Based on the gradual changes in the thickness direction and the established rule of the mixture, the effective mechanical properties of the FG-CNTRC plates are approximated. Complex variable technique is used to estimate the resultant moments and forces around the elliptical and circular holes in the FG-CNTRC plate. The impacts of CNT distribution shape, the aspect ratio of hole shape, the volume fraction of CNTs, temperature dependence, and hole orientation on the resultant moments and forces around the elliptical and circular holes are investigated in detail. The highest resultant moment and force do not necessarily reduce with increasing the volume fraction of CNT; hence, the volume fraction should increase purposefully. Aligning the CNT axis with applied loading will lead to the lowest values of the highest resultant moments and forces in the holes with less elongation in a vertical direction to the nanotube axis. According to the results, the above parameters have a significant impact on the distribution of stress resultants and moments around the cutout in CNTRC plates, and the load-bearing capacity of these plates with cutouts can be improved by careful selection of these parameters.

This work addresses the determination of yield surfaces for geometrically exact elastoplastic rods. Use is made of a formulation where the rod is subjected to an uniform strain field along its arc length, thereby reducing the elastoplastic problem of the full rod to just its cross-section. By integrating the plastic work and the stresses over the rod’s cross-section, one then obtains discrete points of the yield surface in terms of stress resultants. Eventually, Lamé curves in their most general form are fitted to the discrete points by an appropriate optimisation method. The resulting continuous yield surfaces are examined for their scalability with respect to cross-section dimensions and also compared with existing analytical forms of yield surfaces.

peer-reviewed Increasing applications of laminated composite structures necessitate the development of equivalent single layer (ESL) models that can achieve similar accuracy but are more computationally efficient than 3D or layer-wise models. Most ESL displacement-based models do not guarantee interfacial continuity of shear stresses within laminates. A possible remedy is the enforcement of interlaminar equilibrium in variational formulations, for example, in the framework of the Hellinger-Reissner variational principle, leading to a mixed force/displacement model. In this paper, the governing equations for bending and stretching of laminated beams, comprising only seven stress resultants and two displacement functionals, are obtained using global fifth-order and a local linear zigzag kinematics. As a strong-form solution technique, the differential quadrature method (DQM) is an efficient tool which can provide excellent convergence with relatively few number of grid points. However, in dealing with high-order differential equations, the conventional DQM can incur considerable errors due to the nature of numerical differentiation. Therefore, a mixed inverse differential quadrature method (iDQM) is proposed herein to solve the governing fourth-order differential equations for bending and stretching of laminated beams. This approach involves approximating the first derivatives of functional unknowns, thereby reducing the order of differentiation being performed. Using a non-uniform Chebychev-Gauss-Lobatto grid point profile, numerical results show that the accuracy of stress predictions is improved by using iDQM compared to DQM. In addition, the Cauchy’s equilibrium condition is satisfied more accurately by iDQM, especially in the vicinity of boundaries

In this paper a robust and effective 4-node shell element for the structural analysis of thin structures is described. A Hu–Washizu functional with independent displacements, stress resultants and shell strains is the variational basis of the theory. Based on a previous paper an additional interpolation part using quadratic shape functions is introduced for the independent shell strains. Especially for unstructured meshes this leads to an improved convergence behavior. The expanded element formulation proves to be insensitive to mesh distortion. Another well-known feature of the mixed hybrid element is the robustness in nonlinear applications with large deformations.

The structural response of a slender beam subjected to impact by several rigid particles is investigated. A higher-order shear deformation beam model and the von-Karman geometric measure are employed. The equations of motion, continuity conditions at the points of impact by particles, and geometric and natural boundary conditions for the beam are derived by means of Hamilton’s principle. The generalized differential quadrature and Newmark time integration methods are utilized to carry out the numerical solution of the system of nonlinear partial differential equations comprised of the equations of motion and stress resultants. The durations of contact, displacements, and stress resultants are obtained for beams with various types of boundary conditions.

Computational mechanics 69(1), 1-21 (2022). doi:10.1007/s00466-021-02080-8, Published by Springer, Berlin ; Heidelberg

The static limit analysis of axially symmetric masonry domes subject to pseudo-static seismic forces is addressed. The stress state in the dome is represented by the shell stress resultants (normal-force tensor, bending-moment tensor, and shear-force vector) on the dome mid-surface. The classical differential equilibrium equations of shells are resorted to for imposing the equilibrium of the dome. Heyman’s assumptions of infinite compressive and vanishing tensile strength, alongside with cohesive-frictional shear response, are adopted for imposing the admissibility of the stress state. A finite difference method is proposed for the numerical discretization of the problem, based on the use of two staggered rectangular grids in the parameter space generating the dome mid-surface. The resulting discrete static limit analysis problem results to be a second-order cone programming problem, to be effectively solved by available convex optimization softwares. In addition to a convergence analysis, numerical simulations are presented, dealing with the parametric analysis of the collapse capacity under seismic forces of spherical and ogival domes with parameterized geometry. In particular, the influence that the shear response of masonry material and the distribution of horizontal forces along the height of the dome have on the collapse capacity is explored. The obtained results, that are new in the literature, show the computational merit of the proposed method, and quantitatively shed light on the seismic resistance of masonry domes.

Applied Element Method (AEM) is an efficient method for numerical analysis of structures. It is capable of analyzing the structure up to collapse. In AEM, the structure is modelled as a group of rigid elements connected by springs. It has the benefit of lesser processing time and memory requirement when compared to Finite Element Method. It is also able to track crack propagation more realistically. To extend the advantages of AEM to the analysis of thin plate structures, thin plate element has to be developed. This is done in this paper. Being a rigid body method, AEM does not consider Poisson's ratio in its basic formulation. As Poisson's ratio plays a major role in the behavior of plate structures, the method to bring in Poisson effect in thin plate element is discussed. The procedure to determine displacements, strains, stresses and stress resultants are explained. The mass matrix for thin plate element is also found. To validate the thin plate element, both static and dynamic analyses are carried out. The analyses gave reliable results which show the efficiency of the thin plate element of AEM.

Transverse shear locking is an issue that occurs in Reissner–Mindlin plate and shell elements. It leads to an artificial stiffening of the system and to oscillations in the stress resultants for thin structures. The thinner the structure is, the more pronounced are the effects. Since transverse shear locking is caused by a mismatch in the approximation spaces of the displacements and the rotations, a field-consistent approach is proposed for an isogeometric degenerated Reissner–Mindlin shell formulation. The efficiency and accuracy of the method is investigated for benchmark plate and shell problems. A comparison to element formulations with locking alleviation methods from the literature is provided.

A new nonlinear cross-section deformable beam formulation based on generalized beam theory (GBT) is presented for elastic/elastoplastic analyses of thin-walled members undergoing arbitrary deformations, such as large deflections, finite rotations, distortional/local buckling, and out-of-plane warping. For rigorous numerical analyses of thin-walled structures, considering both the global and local deformation effects, shell finite elements are widely used. This paper aims at providing a more computationally efficient and structurally clarifying alternative to simulate prismatic and curved thin-walled members. Compared to the traditional beam elements and other beam formulations based on higher-order beam theories, we improved the kinematic description of member cross-section displacement field, where the kinematic parameterization is performed on two scales, i.e., global member scale and local wall scale; especially, the local wall deformations are described by means of the predetermined GBT modes which are structurally meaningful and allow for the cross-section deformations. Beam equations of equilibrium are built on the local wall scale in terms of shell-type stress resultants and stress couples; therefore, the present beam formulation owns the feature of a shell model. A Galerkin method based beam finite element is developed to solve the equilibrium equations. Finally, six illustrative examples are examined for the validity of the proposed beam formulation.

In this paper, correspondence relations between the solutions of the static bending of functionally graded material (FGM) Reddy–Bickford beams (RBBs) and those of the corresponding homogenous Euler–Bernoulli beams are presented. The effective material properties of the FGM beams are assumed to vary continuously in the thickness direction. Governing equations for the titled problem are formulated via the principle of virtual displacements based on the third-order shear deformation beam theory, in which the higher-order shear force and bending moment are included. General solutions of the displacements and the stress resultants of the FGM RBBs are derived analytically in terms of the deflection of the reference homogenous Euler–Bernoulli beam with the same geometry, loadings and end conditions, which realize a classical and homogenized expression of the bending response of the shear deformable non-homogeneous FGM beams. Particular solutions for the FGM RBBs under specified end constraints and load conditions are given to validate the theory and methodology. The key merit of this work is to be capable of obtaining the high-accuracy solutions of thick FGM beams in terms of the classical beam theory solutions without dealing with the solution of the complicated coupling differential equations with boundary conditions of the problem.

Compared to straight fiber path laminates, variable-angle tow (VAT) laminates are known to redistribute the in-plane stress resultants for improving structural buckling response. The VAT laminates ...

Mixed 4-node shell elements with the drilling rotation and Cosserat-type strain measures based on the three-field Hu–Washizu principle are proposed. In the formulation, apart from displacement and rotation fields, both strain and stress resultant fields are treated as independent. The elements are derived in the framework of a general nonlinear 6-parameter shell theory dedicated to the analysis of multifold irregular shells. The novelty of the developed elements stems from the fact that the measures of assumed strains and stress resultants are asymmetric. The original interpolation of drilling and bending components of strains and stress resultants is proposed. In the formulation of new mixed elements, two variants of the interpolation of membrane components are used and interpolation of the independent fields is defined in the natural or skew coordinates. Accuracy and efficiency of the developed elements are tested in several linear and nonlinear numerical examples. It is shown that smaller number of independent parameters in the interpolation of membrane components gives more accurate results. The proposed mixed 4-node elements enable the use of large load steps in nonlinear computations. Moreover, they require significantly less equilibrium iterations than other shell elements formulated in the 6-parameter shell theory.

The present article attempts to investigate transient thermoelastic problem of a functionally graded thin rectangular plate with thermosensitive and spatial variable dependent material properties. ...

Elevated cylindrical metal silos are often supported on a ring beam which rests on discrete column supports. The ring beam plays an important role in redistributing the majority of the discrete forces from the column supports into a more uniform stress state in the cylindrical wall, which is necessary to reduce the potential for buckling of the shell wall. Traditional design treatments assumed that the discrete support forces can be fully redistributed by the ring beam, producing circumferentially uniform axial membrane stresses in the silo shell. But it has previously been shown that only a very stiff ring beam can produce even relatively uniform axial membrane stresses in the shell. The ring beam stiffness must be evaluated relative to the stiffness of the silo shell in compatible deformation, since even a thin shell is extremely stiff in its own plane. A test for the ring beam to shell stiffness ratio was previously devised by the authors to determine the ring beam stiffness needed to achieve uniform membrane stresses. This new study assesses ring beams of practical dimensions and stiffness, to evaluate their effectiveness in smoothing the shell axial membrane stresses towards the uniform condition and then to find their effect on the buckling resistance of the shell. This study explores the stress resultants in a closed section ring beam of practical dimensions that is less stiff than the ideal stiffness using a finite element parametric study of the flexible ring beam and connected silo shell. The results show that the ring beam stress resultants, relative to those in an isolated ring beam under the same loading, can be directly related to the established shell to ring beam stiffness ratio (ψ). This treatment may be used to significantly enhance current design practice which, according to the Eurocode EN 1993-4-1 for silos, assumes that the shell applies the full uniform transverse loading onto an isolated ring beam.

A static limit analysis approach is proposed for assessing the collapse capacity of axisymmetric masonry domes subject to horizontal forces. The problem formulation is based on the sound theoretical framework provided by the classical statics of shells. After introducing the shell stress tensors on the dome mid-surface, integral equilibrium equations are enforced for its typical part. Heyman's assumptions of infinite compressive and vanishing tensile strengths are made, with cohesionless friction behavior governing the shear strength, to characterize the admissible stress states in the dome. An original computational strategy is developed to address the resulting static limit analysis problem, involving the introduction of a mesh on the dome mid-surface, the interpolation of the physical components of the shell stress tensors on the element boundaries, and the imposition of equilibrium and admissibility conditions respectively for the elements and at the nodes of the mesh. The descending discrete convex optimization problem is solved by standard and effective optimization tools, automatically providing collapse multiplier of horizontal forces, incipient collapse mechanism and expected crack pattern. Convergence analysis, validation with experimental results available in the literature, and parametric analyses with respect to geometric parameters and friction coefficient, are presented for spherical and ellipsoidal masonry domes, proving the reliability of the proposed approach for estimating the pseudo-static seismic resistance of masonry domes., 29 pages, 11 figures

This paper presents a three-dimensional beam element for stability analysis of elastic thin-walled open-section beams in multibody systems. The beam model is based on the generalized strain beam formulation. In this formulation, a set of independent deformation modes is defined which are related to dual stress resultants in a co-rotational frame. The deformation modes are characterized by generalized strains or deformations, expressed as analytical functions of the nodal coordinates referred to the global coordinate system. A nonlinear theory of non-uniform torsion of open-section beams is adopted for the derivation of the elastic and geometric stiffness matrices. Both torsional-related warping and Wagner’s stiffening torques are taken into account. Second order approximations for the axial elongation and bending curvatures are included by additional second order terms in the expressions for the deformations. The model allows to study the buckling and post-buckling behaviour of asymmetric thin-walled beams with open cross-section that can undergo moderately large twist rotations. The inertia properties of the beam are described using both consistent and lumped mass formulations. The latter is used to model rotary and warping inertias of the beam cross-section. Some validation examples illustrate the accuracy and computational efficiency of the new beam element in the analysis of the buckling and post-buckling behaviour of thin-walled beams under various loads and (quasi)static boundary conditions. Finally, applications to multibody problems are presented, including the stability analysis of an elementary two-flexure cross-hinge mechanism.

International audience; A stress resultant shell approach is proposed to simulate the draping of textile composite reinforcements with continuous fibers. This approach naturally makes it possible to take into account the specificity of the deformation of the fibrous reinforcements by considering independent bending and tension stiffness whereas they are linked in standard shell elements. The stress resultants and stress moments are related to membrane strains and curvatures by rate constitutive equations (hypoelastic laws). The experimental tests required to identify the material properties necessary for these constitutive laws in stress resultants are presented. This approach is implemented in the ABAQUS finite element code and can be used by all users of this software. A set of forming tests showed the effectiveness of the approach through comparisons between simulation and experimental results.

In engineering applications if a thin plate is considered, most of the stress analyses will concentrate on the plane variation instead of the distribution in thickness direction, and hence the complex variable formalism which combines two variables into one becomes a useful and powerful tool. When a thin plate is subjected to inplane loads, uniform distribution across the plate thickness is usually assumed for the deformations and stresses, and hence the two-dimensional analysis can be employed. When a thin plate is under transverse loads or bending moments, linear distribution across the plate thickness is usually assumed and the analysis is generally called plate bending analysis. By taking the stress resultants and the deformations of the middle surface as the basic functions, the plate bending analysis can also be treated with two plane coordinate variables. When the thin plates are made by laying up various layers, the inplane and plate bending deformations may uncouple or couple depending on the symmetry or unsymmetry with respect to the middle surface of the plate. In this case, the coupled stretching-bending analysis should be employed. Since all these kinds of problems have been well treated by using the theory of complex variable, summarizing the formalisms introduced in the book (Hwu Anisotropic Elastic Plates, Springer, New York, 2010) several complex variable formalisms are presented in this chapter such as Lekhnitskii formalism and Stroh formalism for two-dimensional analysis and plate bending analysis, and Stroh-like formalism for the coupled stretching-bending analysis. In addition, the extended versions of Stroh formalism for thermoelastic problems and piezoelectric materials are also presented in the related sections of this chapter. Further extensions to magneto-electro-elastic and viscoelastic materials will be described in Chaps. 11 and 12. Moreover, through the use of Radon transform, the Stroh formalism can also be applied to the three-dimensional analysis, whose detailed description will be presented in Sect. 15.9.1.

In the present paper a mathematical model able to reproduce the mechanical response of a shape memory alloy (SMA) helical spring is presented. The proposed model is based on a simplified but effective theory of the helix that assumes small strains and large displacements. The kinematics of the helix and the related generalized strain measures are introduced, with stress resultants conjugated to them and the corresponding equilibrium equations of the curved beam. A specific constitutive law for this structural model is then presented for the SMA material, which accounts for the phase transformation due to the coupling of the shear and axial strains and to the effect of the temperature variation as well. Then, a numerical procedure, based on the backward Euler time integration algorithm and the prediction-correction technique for solving the nonlinear time step, is developed. Several numerical applications are presented to assess the reliability of the proposed SMA helix model and the implemented numerical procedure, and to investigate the response of the helix. Initially, the behavior of a circular SMA cross-section is studied, comparing the results obtained by the proposed modeling approach with the ones carried out by finite element analyses. Then, the helix structural element is considered and results are again compared with finite element outcomes. Finally, a sensitivity analysis is performed varying the pitch of the helix.

This chapter covers the stacking of single laminae, generally under different angles, to a so-called laminate. Based on the approach of the classical laminate theory, a simplified stress analysis and a subsequent failure analysis are derived, without the solution of the system of coupled differential equations for the unknown displacements in the three coordinate directions. This theory provides the solution of the statically indeterminate system based on generalized stress–strain relationship under consideration of the constitutive relationship and the definition of the so-called stress resultants.

Support structures of offshore wind turbines are subject to cyclic stresses generated by different time-variant random loadings such as wind, waves, and currents in combination with the excitation by the rotor. In the design phase, the cyclic demand on wind turbine support structure is calculated and forecasted with semi or fully probabilistic engineering models. In some cases, additional cyclic stresses may be induced by construction deviations, unbalanced rotor masses and structural dynamic phenomena such as, for example, the Sommerfeld effect. Both, the significant uncertainties in the design and a validation of absence of unforeseen adverse dynamic phenomena necessitate the employment of measurement systems on the support structures. The quality of the measurements of the cyclic demand on the support structures depends on (a) the precision of the measurement system consisting of sensors, amplifier and data normalization and (b) algorithms for analyzing and converting data to structural health information. This paper presents the probabilistic modelling and analysis of uncertainties in strain measurements performed for the purposes of reconstructing stress resultants in wind turbine towers. It is shown how the uncertainties in the strain measurements affect the uncertainty in the individual components of the reconstructed forces and moments. The analysis identifies the components of the vector of stress resultants that can be reconstructed with sufficient precision.

In the present paper, the problem of finite dimensional rectangular parallelepiped in isotropic thermoelastic medium with convective type heating is considered. The heat conduction equation (HCE) of the region is described by time HC of fractional order with Caputo derivative form. The non-linear form of heat conduction equation is converted to linear form with Kirchhoff’s transformation. Integral transform technique is used to deal with the spatial variables and Laplace transform technique is used to deal with Caputo type time fractional derivative. Inverse Laplace transform and inverse finite Fourier transform are employed to expose the solution in the transformed domain. Numerical results are obtained for temperature distribution, deflection, stress resultants and thermal stress distribution for different values of time fractional order parameter. These results are presented graphically and discussed for various values of time fractional parameters. The obtained results show significant influence of the time fractional order derivative on the temperature as well as stress distribution. Thermosensitivity plays a vital role in the analysis of any real thermoelastic problems and one should consider their effect while dealing with materials in high temperature environment.

This study commences the application of the modified couple stress theory in the analysis of spatial arbitrarily curved microbeams. The kinematic assumptions of the Timoshenko beam theory are employed. The moving trihedron of the beam axis are used to form a local Cartesian coordinate system. Displacements of the beam axis and cross-sectional rotations in the local coordinate system are considered as unknowns in the kinematic descriptions. The principle of virtual work is used to derive the equations of motion. The outputs, including the deformation measures, constitutive relations, cross-sectional stress resultants, equations of motion, and boundary conditions, are explicitly expressed in terms of the kinematic unknowns. The established general formulation is further specialized to the case of planar arbitrarily curved microbeams which cannot be found in the literature. The accuracy of the derived equations of motion is tested by a comparison with existing studies of circular microbeams. Since this is the first study for the use of the modified couple stress theory in the setting of spatial arbitrarily curved microbeams, the outputs of the present study enable further in-deep analysis to comprehend these structures.

This paper presents the semi-layerwise analysis of structural sandwich plates with through-width delamination. The mechanical model of rectangular plates is based on the method of four equivalent single layers and the system of exact kinematic conditions. An important improvement compared to a previous formulation is the consideration of linear and quadratic stretching term in the transverse displacement component. Three different delamination scenarios are investigated: core-core failure, face-core delamination and the face-face failure. By applying the first- and second-order laminated plate theories and the principle of virtual work the governing equations are derived. The equilibrium equations are solved under Levy type boundary conditions using the state-space approach. Solutions for the mechanical fields are provided and compared to 3D finite element results. The energy release rate distributions along the delamination front are also determined using the J-integral. Although the stress resultants by transverse stretching do not influence directly the J-integral, the results indicate that this effect improves the accuracy of the model in general, and substantially influences the results of the first-order plate theory in the case of the face-face delamination.

An efficient semi-analytical simulation framework is presented, that allows a highly systematic analysis of the mechanical behaviour of laminated prismatic thin-walled beams. It is excellently suitable for extensive case studies and can serve as a key ingredient for the optimization of shape, geometry and stacking sequences of laminated beams. The proposed method can be coded as a stand-alone solution and easily and directly be embedded in existing numerical optimization tools, without the overhead of common simulation tools (like FE-codes). Thus, long-winded and time consuming data transfers are avoided. Additionally, as a consequence of the structural clearness of the proposed method, the simulation framework can also be used as an environment for the development and testing of new beam models. By analogy with existing models for thin-walled beams, the kinematic of the cross-section is described by a proper set of deformation modes. There are no restrictions regarding these modes, i.e. rigid body modes (axial extension, mayor- and minor-axis flexure, and torsion) as well as higher-order modes (involving warping, distortion and transverse bending of the cross-section walls) can be considered. It is shown that for each given set of modes, the resulting system of equations can be derived - both analytically and systematically - in a straight-forward manner. In contrast to common beam formulations, the introduction of both stress resultants and cross-section values is not required.

The non-linear response of a sandwich panel with the core that consists of a functionally graded material (FGM) undergoing in-plane loading is investigated. The panel consists of two face sheets, metallic or composite laminated, and an FGM core that is a medium whose mechanical properties change through the depth facilitating a desirable response of the structure. The effect of the FGM core is introduced through the constitutive relations that affect conventional and high-order stress resultants and stress couples in the panel. The formulation employs the Extended High-Order Sandwich Panel Theory (EHSAPT) to assess the effect of the FGM material of the core. A variational approach is adopted to derive the linear and non-linear governing equations with a prescribed FGM distribution through the depth of the core. The wrinkling study of FGM panels includes two loading scenarios where in-plane loads are applied through a rigid edge beam connected to the core only and where the loads are applied through a rigid edge structure attached to both the face sheets and core causing uniform end shortening. The post-wrinkling behavior is also considered to prove that the initial pattern of wrinkles is not affected.

This paper presents a two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates. In recent studies, a simple first-order shear deformation theory (S-FSDT) was developed and extended. It has only two variables by separating the deflection into bending and shear parts while the conventional first-order shear deformation theory (FSDT) has three variables. However, the S-FSDT provides incorrect predictions for the transverse shear forces on the insides and the twisting moments at the boundaries except simply supported plates since it does not consider in-plane rotation. The present theory also has two variables but considers in-plane rotation such that it is able to correctly predict the responses of plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with two opposite edges that are simply supported, with the other edges having arbitrary boundary conditions. Numerical results of deflections, stress resultants, buckling loads and natural frequencies are presented with the FSDT, the S-FSDT and the present theory. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the bending, buckling and free vibration responses of isotropic plates.

Earthquakes and aircraft impacts induce vibrations that propagate throughout the entire building and they need to be considered in designing SSCs (Structures, Systems and Components). Mainly linear calculation methods have been in use in design practice and the codes and standards consider damping ratios only for linear structural analyses. Induced vibrations, especially in damaged concrete structures, have not been studied extensively enough for optimization of structural frameworks and/or qualified systems and components. Experimental data on damping properties of damaged reinforced concrete are needed also for benchmarking analysis programs and methods.Recently, within IMPACT project, a new type of test series considering vibration propagation has been carried out at VTT. The test target is a reinforced concrete structure with two parallel walls connected to a floor slab. The front wall is additionally supported by triangular shaped side walls which are connected to the floor slab too. The test structure is supported on elastomeric bearing pads, with back pipes effective mainly in compression and with bars effective in tension. In order to obtain information on vibration propagation in damaged concrete structure at different levels of damage grades the same structure was tested six times. At each time the mass of the deformable stainless steel missile was 50 kg. The hit point located in the middle of the front wall. The impact velocity was about 110 m/s in the first four tests (V1A-D) and about 60 m/s in the remaining two tests (V1E and F). In this paper, numerical results on tests V1A and V1F are compared with the corresponding experimental ones.The calculated results, such as accelerations, displacements, their response spectra and strains, are compared with experimental measurements. Five finite element (FE) programs are used in computations: Abaqus, Europlexus, LS-DYNA, SOFiSTiK and an in-house code (IHC).Most of the FE-codes in the present study use shell elements. In Abaqus and SOFiSTiK non-linear behaviour of shell section is modelled by dividing the cross section into layers. Reinforcements are also modelled as layers. In Europlexus and IHC, an alternative approach is adopted in which the non-linear behaviour of concrete and reinforcement is homogenized beforehand in the shell thickness direction obtaining relations between stress resultants and generalized strains valid for the shell section. In LS-DYNA, 3D solid elements for modelling concrete and beam elements for modelling reinforcements are used.Equations of motion are integrated with explicit central difference time integration method, except in SOFiSTiK implicit integration method is used. Modelling and computations with the mentioned FE-programs are made independently of each other. Computations with LS-DYNA are carried out as blind exercises.Consideration of the results from benchmarking point of view is still on-going. However it is evident that analysed results follow reasonable well test results in main design parameter level such as maximum displacements, accelerations and strains. Also frequency spectra are estimated reasonably well.

In this paper, using an analytic technique based on complex variable method, the stress resultants and moments in infinite functionally graded plates with rectangular cutout under uniaxial tensile load are investigated. The mechanical properties of the functionally graded plate are assumed to change exponentially. The effect of different parameters such as cutout orientation, aspect ratio of the cutout, and cutout corner curvature on the distribution of stress resultants and moments is investigated. To verify the accuracy of the presented analytical method, its results are compared with the numerical results obtained from finite element modeling in ABAQUS. The comparisons demonstrate a good level of agreement between the obtained numerical and analytical results. According to the results, the above parameters have a significant impact on the distribution of stress resultants and moments around the cutout, and the load-bearing capacity of the functionally graded plates with cutouts can be improved by careful selection of these parameters.

We recast the displacement-traction problem of the Kirchhoff–Love theory of linearly elastic plates as a boundary value problem with the bending moments and stress resultants inside the middle section of the plate as the sole unknowns, instead of the displacement field in the classical formulation. To this end, we show in particular how to recast the Dirichlet boundary conditions satisfied by the displacement field of the middle surface of a plate as boundary conditions satisfied by the bending moments and stress resultants.

In this work, the problem of first order shear deformable solid circular plate under transverse load was solved mathematically. The problem considered was assumed axisymmetric. The plate and loading were considered axisymmetric. The problem was defined as a boundary value problem of a system of differential equations of equilibrium in terms of the stress resultants and the stress – resultants – displacement relations. The set of equations were considered simultaneously to express them in variable separable form. The mathematical technique of separation of variables was then used to obtain solutions for the unknown generalised displacements. Specific problems of clamped edge plates and simply supported edge plates under uniformly distributed load and point load at the centre were considered and solved using the same technique of separation of variables. The mathematical expressions obtained showed that in all cases, the deflection was expressible in terms of flexural and shear components. The maximum deflection was found to occur at the plate centre as is expected from the symmetrical nature of the problem. The shear component of the transverse deflection was found to significantly increase with significant increase in the ratio of the plate thickness to the radius (h/r0).

The dynamic behavior of a very large rectangular floating structure is considered. The structure is modelled as a plate with free edges. Two different thicknesses are considered for the model. The Mindlin plate theory is used to formulate the structure behavior. Natural frequencies, mode shapes, and stress resultants of the structure are predicted by using finite element method. For this purpose, a MATLAB code is written. The same analysis is performed by using the ANSYS software. The results of these two analysis are compared with each other and with the available results in the literature, where close agreement is observed. Therefore, the written finite element code is found to be acceptable for prediction of the dynamic behavior of very large rectangular floating structures in early stages of design.

The current work focuses on extending the analytical solution given by Greszczuk to determine the stress distribution in multilayered composite plates subjected to arbitrary in-plane loadings. This is achieved by introducing an arbitrary oriented uniaxial, biaxial, and shear loading conditions into Greszczuk solution. The present solution has exactly reproduced most of the results of the analytical solution existing in the literature. The varying parameters, such as fiber orientation angle, type and load orientation, stacking sequence, and anisotropic modulus ratio $$E_{1}/E_{2}$$ , are examined. In the second part of this study, digital image correlation (DIC) technique is used to obtain full-field surface strain measurements in unidirectional E-glass/epoxy specimens with different fiber orientation angles, in order to show its effect on stress concentration and failure modes. The analytical results obtained by the present solution compare favorably with those obtained using DIC technique. Compared to the experimental data, the accuracy of the present approach is within 10%. The simplicity of the present solution gives the design engineer a useful tool for optimizing laminate plates in the presence of a circular cutout.

Non-local viscoelasticity is a subject of great interest in the context of non-local theories. In a recent study, the authors have proposed a non-local fractional beam model where non-local effects are represented as viscoelastic long-range volume forces and moments, exchanged by non-adjacent beam segments depending on their relative motion, while local effects are modelled by elastic classical stress resultants. Long-range interactions have been given a fractional constitutive law, involving the Caputo's fractional derivative. This paper introduces a comprehensive numerical approach to calculate the stochastic response of the non-local fractional beam model under Gaussian white noise. The approach combines a finite-element discretization with a fractional-order state-variable expansion and a complex modal transformation to decouple the discretized equations of motion. While closed-form expressions are derived for the finite-element matrices associated with elastic and fractional terms, fractional calculus is used to solve the decoupled equations of motion, in both time and frequency domain. Remarkably, closed-form expressions are obtained for the power spectral density, cross power spectral density, variance and covariance of the beam response along the whole axis. Time-domain solutions are obtained by time-step numerical integration methods involving analytical expressions of impulse response functions. Numerical examples show versatility of the non-local fractional beam model as well as computational advantages of the proposed solution procedure.