Complementary Strain Energy Research Articles

Multi-scale topology optimization of structures with multi-material microstructures using stiffness and mass design criteria

Nowadays, there is a great interest on the part of the automotive and aerospace industry to design environmentally-friendly structures. To that purpose, stiffness-oriented designs are proposed here by extending previous work on multi-scale topology optimization to the multi-material setting reformulating the problem to include appropriately mass constraints and discussing different design domain parametrizations and algorithmic strategies.On the macroscale, the problem of minimizing the compliance subject to a global mass constraint is addressed. On the microstructure scale, the multi-material design is carried out by solving the problem of minimizing the local complementary strain energy density with mass density constraint. As a result, very efficient structures composed of spatially varying porous and multi-material microstructures are obtained. The optimal design of the multi-material microstructure can be done either in a pointwise manner or in larger subdomains, to promote design uniformity. These parametrizations are here compared and discussed. Moreover, two different algorithmic strategies to solve the multi-scale problem are proposed, and their pros and cons discussed. They differ in the way the macro and micro design variables are related and updated. The macro design variables consider the mass density distribution along the structure, while the micro variables define the microstructure's topology using a multi-material SIMP interpolation scheme.The results show very efficient structures with locally optimized multi-material microstructures, which can outperform their single-material counterparts with regards to stiffness while maintaining the same mass. Additionally, the maximum stress verified on multi-material structures tends to be lower than the one obtained in the single-material counterparts.

Fracture of inhomogeneous beam with three parallel lengthwise cracks

A lengthwise fracture analysis of an inhomogeneous beam with three parallel cracks is developed. The cracks are located symmetrically with respect to the mid-span. The loading consists of two axial forces applied at the ends of the beam. A vertical notch is cut-out in the mid-span so as the upper three crack arms are free of stresses. Only half of the beam is analyzed due to the symmetry. The cracks are located arbitrary along the beam height. Therefore, the heights of the cross-sections of crack arms are different. The beam exhibits continuous material inhomogeneity in the height direction. The material has non-linear elastic behaviour. Three solutions to the strain energy release rate are derived. For this purpose, the complementary strain energy in the beam is differentiated with respect to the areas of the three cracks. The J-integral approach is applied in order to verify the solutions to the strain energy release rate. The influence of the locations of the three cracks along the beam height on the lengthwise fracture behaviour is studied in detail. The influence of the material inhomogeneity on the lengthwise fracture behaviour of the beam is also analyzed.

Optimal Plastic Reliable Design of Reinforced Concrete Beams Considering Steel Bars Volume Probability

This paper aims to investigate the plastic response of reinforced concrete tapered beams when subjected to random steel reinforcement volumes, using both deterministic and probabilistic analyses, with the complementary strain energy as a boundary in the first case, and the reliability index as a boundary in the second. The first step in this study was to use a previously studied model and perform a deterministic analysis, assuming that the complementary strain energy is a limiting factor and controller of the plastic behaviour. Next, a probabilistic analysis is applied, with the reliability index as a limitation. At the same time, the volume of the reinforcement steel used, and the complementary strain energy were treated as probabilistic variables with mean values and specific standard deviations. This novel method highlighted the plastic behaviour limiting procedure and provided results that highlighted the nature of the model’s changed behaviour when the complementary strain energy was controlled and when applying probabilistic properties with reliability index limitation.

Elasto-plastic analysis and optimal design of composite integral abutment bridge extended with limited residual plastic deformation

Due to the growing significance of structural theories concerning the composite structure analysed and designed plastically, this paper introduces a new optimisation method for controlling the plastic behaviour of a full-scale composite integral abutment bridge by employing complementary strain energy of residual forces that existed within the reinforcing rebars. Composite bridges are structures made of components such as steel and concrete, which are frequent and cost-effective building methods. Thus, various objective functions were used in this work when applying optimum elasto-plastic analysing and designing the composite integrated bridge structure that was tested experimentally in the laboratory. In contrast, the plastic deformations were constrained by restricting the complementary strain energy of the residual internal forces aiming to find the maximum applied load and the minimum number of steel bars used to reinforce the concrete column part of the structure. The numerical model employed in this paper was validated and calibrated using experimental results, which were considered inside ABAQUS to produce the validated numerical model, using concrete damage plasticity (CDP) constitutive model and concrete data from laboratory testing to solve the nonlinear programming code provided by the authors. This paper presents a novel optimization method using complementary strain energy to control the plastic behaviour of a full-scale composite integral abutment bridge, with the original contribution being the incorporation of residual forces within reinforcing rebars to limit plastic deformations. Following that, a parametric investigation of the various optimisation problems revealed how models performed variously under different complementary strain energy values, which influenced the general behaviour of the structure as it transitioned from elastic to elasto-plastic to plastic; also results showed how the complementary strain energy value is connected with the amount of damaged accrued in both concrete and steel.

Optimal Elasto-Plastic Analysis of Prestressed Concrete Beams by Applying Residual Plastic Deformation Limitations

This work introduces elasto-plastic analysis of prestressed reinforced concrete beams under different prestressing conditions by limiting the residual plastic behaviour inside the steel bars using complementary strain energy. A non-linear optimal method was used to limit residual plastic deformations in steel bars, including prestressed tendons, used to reinforce beams from two previous research investigations. This was considered by using an optimization approach with an objective function to find the maximum loading while applying constraints on the complementary strain energy of residual internal forces in steel elements to control residual plastic deformations. Thus, an elasto-plastic optimization programme was linked to models simulated by ABAQUS, as concrete was calibrated by the concrete damage plasticity (CDP) model. Some variables were considered regarding the force applied inside prestressed tendons and the number of tendons used inside the models. Thus, limits on complementary strain energy affected load values and model damage where an increase in the permissible strain energy value leads to an increase in the corresponding loading values produced; thus, this produces a higher stress intensity in steel and tension-damaged areas in concrete. Based on these data, many comparisons have been made to determine when beams behaved elastically and how they turned into plastic.

Reliability‑based probabilistic numerical plastically limited analysis of reinforced concrete haunched beams

This research proposes a novel method that considers the optimal reliability-based design of reinforced concrete haunched beams subjected to probabilistic concrete properties and complementary strain energy values. The theory is applied twice, once to evaluate a deterministic solution, where the plastic behaviour is constrained by the complementary strain energy of residual stresses induced within steel bars. Secondly, the same method is considered for a probabilistic solution where the concrete characteristics—compressive strength and modulus of elasticity—and complementary strain energy value vary randomly. The reliability index acts as a bound for the solution. It is important to mention that the model utilised in this publication is derived from recent research after being calibrated using Abaqus. This work represents an extension of that recent research that exclusively considered deterministic work. This research led deterministically to new estimates for the complementary strain energy, which will be interpreted as reflecting the behaviour of plastic thresholds and quantifying the associated loads. Afterwards, uncertainty is studied when probability interferes, showing various load values and damage in concrete and steel when the complementary strain energy and concrete properties are probabilistically evaluated, giving a less reliable solution as the load reduces. These effects are reflected in the varying plastic behaviour of the investigated haunched beams.

Limited Optimal Plastic Behavior of RC Beams Strengthened by Carbon Fiber Polymers Using Reliability-Based Design.

The plastic behavior of strengthened haunched beams utilizing carbon fiber-reinforced polymers (CFRP) was investigated using a probabilistic design that took into account random concrete properties, CFRP properties, and complementary strain energy values, with the reliability index serving as a limiting index, as the proposed method considers a novel method that deals with probabilistic parameters for models with limited plastic behavior designed based on the reliability index. The data used in this research were gathered and evaluated in a recent study on simply supported haunched beams reinforced with carbon fiber-reinforced polymers. The purpose of this research was to use the reliability limitation index for simulated strengthened haunched beams by taking into account randomness in concrete and CFRP properties and the complementary strain energy value, which is considered a plastic behavior controller that provides an illustration of the damage amount within the reinforcement steel bars. The results indicate how randomness affects the behavior of the presented models, which are chosen to have different numbers of CFRP strips. The variable randomness affects load and deflection values where the reliability index value increases as the corresponding load value decrease, reflecting the increased probability of failure in models subjected to higher loading conditions, while tension concrete damage percentages are reflected in the damage pattern presented in the results, showing that as the produced load increases, so does the damage intensity. It is also obvious that the reliability index served as a limitation index while taking concrete characteristics and complementary strain energy as random variables.

Triple longitudinal fracture of inhomogeneous beam under four-point bending an analytical study by using a non-linear elastic material model

The present paper is devoted to fracture analysis of an inhomogeneous non-linear elastic beam configuration with three parallel longitudinal cracks. The material is continuously inhomogeneous along the beam height. The beam is subjected to four-point bending. The longitudinal fracture behavior is studied by applying the J-integral approach. Solutions to the J-integral are obtained for the three cracks. For this purpose, the curvature and the coordinates of neutral axes of the crack arms are determined by using the equations for equilibrium of the elementary forces in the cross-sections of different portions of the beam. The solutions to the J-integral are valid for arbitrary locations of the cracks along the beam height. Thus, the solutions are very useful for evaluating of the effects of locations of cracks on the fracture. The longitudinal fracture behavior is studied also in terms of the strain energy release rate in order to verify the solutions to the J-integral. Solutions to the strain energy release rate are derived for the three cracks by considering the complementary strain energy in the beam. The solutions to the J-integral are applied to evaluate the effects of locations of the cracks, the material inhomogeneity, the sizes of the beam cross-section, and the coordinates of the applications points of the external forces on the longitudinal fracture behavior of the beam.

Utilization of viscoelatic models with non-linear springs and dashpots in delamination study of multilayered beams

This paper analyzes delamination of multilayered inhomogeneous beam structure by utilization a non-linear viscoelastic mechanical model. The non-linear time-dependent response is treated by a non-linear spring and a non-linear dashpot connected in series to a system of two linear springs and two linear dashpots. The model is under stress that is a linear function of time. The constitutive law of the model representing a non-linear dependence between stress, strain and time is derived. The main goal of the paper is to obtain a solution of the strain energy release rate for the delamination in the multilayered inhomogeneous beam by applying the non-linear viscoelastic model. Solutions are derived by using the complementary strain energy and by analyzing the balance of the energy with taking into account the non-linear viscoelastic behaviour. For this purpose, the constitutive law of the non-linear viscoelastic mechanical model is used. The solutions are applied to obtain results for multilayered beams with non-linear variation of material properties in longitudinal direction. The influence of different parameters on the time-dependent strain energy release rate is assessed. The study indicates the effectiveness of the viscoelastic mechanical models with non-linear springs and dashpots in time-dependent delamination analyses of multilayered inhomogeneous beam structures.

Elastic potentials as yield surfaces for isotropic materials.

This paper proposes that elastic potentials, which may be rigorously formulated using the negative Gibbs free energy or the complementary strain energy density, may be used as the yield surface of elasto-plastic constitutive models. Thus, the yield surface may be assumed in some materials as an elastic potential surface for a specific level of critical complementary strain energy density. Traditional approaches, such as the total strain energy criterion, only consider second order terms, i.e., the elastic potential is centred at the origin of the current stress state. Here, first order terms are considered, and consequently, the elastic potential may be translated, which allows to reproduce the desired level of tension-compression asymmetry. The proposed approach only adds two additional parameters, e.g., uniaxial compressive and tensile yield limits, to the elastic ones. For linear elasticity, the proposed approach provides elliptical yield surfaces and shows a correlation between the shape of the ellipse and the Poisson’s ratio, which agree with published experimental data for soils and metallic glasses. This elliptical yield surface also fits well experimental values of amorphous polymers and some rocks. Besides, the proposed approach automatically considers the influence of the intermediate stress. For non-linear elasticity, a wider range of elastic potentials, i.e., yield surfaces, are possible, such as distorted ellipsoids. For the case of incompressible non-linear materials, the yield surfaces are between von Mises and Tresca ones.

Limit design of reinforced concrete haunched beams by the control of the residual plastic deformation

In this paper, a novel computational (optimization) model is presented to control the plastic behaviour of reinforced concrete haunched beams using complementary strain energy of residual forces formed inside the steel reinforcing bars. For this purpose, a numerical model validation process was held and then two non-linear optimization problems were outlined. In these optimization problems, different objective functions were considered applying the optimal elasto-plastic analysis and design of haunched reinforced concrete beams aiming to find the maximal loading or the minimum volume of the steel used to reinforce the beams as the plastic deformations are controlled by using constraints on the complementary strain energy of the residual internal forces of the steel bars. Moreover, the effect of these constraints on different haunch angle beams is studied. The applied method is described in terms of nonlinear mathematical programming and providing solutions when the plastic reserves of the body are not fully exhausted. It is worth mentioning that in this study a concrete damage plasticity constitutive model is applied in the numerical problems and calibrated in accordance with the data gained from laboratory tests. The optimal solutions of the nonlinear mathematical problems were calculated by MATLAB programming codes written by the authors taking into consideration different objective functions and equality and inequality constraints for each case. Finally, by performing a parametric study, the different optimization problems showed how beams behaved differently under different complementary strain energy limit values shifting from elastic into elasto-plastic state and then reaches the fully plastic state where results showed different comparisons taking into consideration the effect of the different complementary strain energy limit values on the maximum applied load, geometry of the beam and steel volume used to reinforce the beams. Thus, complementary strain energy limit value is used to control the plastic deformation inside steel bars during loading progress where avoiding the formation of the plastic deformation in the steel bars would reflect on the general behaviour of the haunched reinforced concrete beams.

Effects of static indeterminacy on the lengthwise fracture in non-linear viscoelastic inhomogeneous beams loaded in torsion

Lengthwise fracture of statically undetermined beam structure that has non-linear viscoelastic behaviour is analyzed in this paper. The beam is loaded in pure torsion. The cross-section of the beam is a circle. The beam exhibits continuous material inhomogeneity along the radius of its cross-section. The mechanical behaviour of the beam is treated by a viscoelastic model structured by non-linear springs and non-linear dashpots. The static indeterminacy is resolved first and then the problem of the time-dependent strain energy release rate is solved by considering the complementary strain energy. The energy balance is analyzed for verification of the solution of the strain energy release rate. A parametric investigation of the strain energy release rate is carried-out.

Influence of Creep on Longitudinal Fracture of Inhomogeneous Rod Loaded in Torsion and Bending

The present paper analyzes the influence of creep on longitudinal fracture in continuously inhomogeneous rod of circular cross-section loaded in torsion and bending. The rod exhibits continuous material inhomogeneity in both radial and longitudinal directions. The creep is described by using non-linear time-dependent relations between the principle stresses and strains. A time-dependent solution to the strain energy release rate is derived by analyzing the complementary strain energy. The time-dependent strain energy release rate is found also by considering the energy balance for verification. The solutions are applied to perform a parametric study of the strain energy release rate under creep.

Refined orthotropic beam models based on Castigliano’s theorem and an approximate solution of the compatibility equation

This paper presents an extension of Castigliano’s second theorem for orthotropic beams. The notion “extension” in this context means to find more accurate results not only for the vertical deflection, but also for the horizontal deflection and the rotation angle. For this purpose the Airy stress function is calculated for an orthotropic rectangular strip based on the iterative approximation method from Boley-Tolins when distributed loads or shear tractions are assumed as surface loads. Hence all stress components can be derived from the stress function and may be inserted into the complementary strain energy. Computing the partial derivative of the complementary strain energy with respect to a scalar dummy parameter, the weighted displacement field components over the cross section are obtained. Finally the analytical results from the extended Castigliano theorem (ECT) are verified by two-dimensional (2D) analytical results for an orthotropic cantilever and a clamped-hinged beam and the outcome is also compared to Bernoulli–Euler and to Timoshenko results. The accuracy of the derived results is higher than that derived by the Timoshenko theory because the normal stress in thickness direction is properly taken into account. Moreover it is demonstrated that this lower order normal stress which is caused by distributed loads yields a mean horizontal elongation or shrinkage that is not considered by the Timoshenko beam theory. Hence, highly accurate formulations for laminated composite beams, involving sandwich-structures or multi-layered beams with imperfect bonding, may be derived from the present orthotropic beam model in the future.

An enriched model based on a complementary strain energy variational principle for stress analysis in FRP plate-strengthened beams

An enriched model based on a complementary strain energy variational principle for stress analysis in FRP plate-strengthened beams

Continuously inhomogeneous beam with longitudinal vertical cracks: an analytical investigation

The present paper is concerned with fracture analysis of an inhomogeneous beam with three longitudinal vertical parallel cracks. The three cracks are located symmetrically with respect to the mid-span. A notch is cut-out in the lateral surface of the beam in the mid-span. Only half of the beam is considered due to the symmetry. The material is continuously inhomogeneous in the width direction of the beam. Besides, the material exhibits nonlinear elastic mechanical behavior. The three cracks are located arbitrary in the width direction so as the cross-sections of the four crack arms have different width. The longitudinal fracture behavior is studied in terms of the strain energy release rate. Three solutions to the strain energy release rate are derived by differentiating the complementary strain energy with respect to the areas of the three cracks. The strain energy release rate is determined also by analyzing the balance of the energy for verification. Further verifications are carried-out by applying the J-integral approach. The influences of the locations of the three cracks, the material inhomogeneity and the beam geometry on the longitudinal fracture behavior are appraised. Results of analyses of a beam that is continuously inhomogeneous in both width and length directions are also presented.

Inverse Hooke's law and complementary strain energy in coupled strain gradient elasticity

AbstractThe inverse Hooke's law and complementary strain energy density has been examined in the context of the theory of coupled gradient elasticity for second gradient materials. To this end, it was assumed that the potential energy density is a quadratic form of the strain and of the second gradient of displacement. Existence of the coupling term significantly complicates the problem. To avoid this complication the equation for the potential energy density was transformed in order to present it as an uncoupled quadratic form of a modified strain and the second gradient of displacement or of the strain and a modified second gradient of displacement. These transformations, which is in essence a block matrix diagonalization, lead to a decoupling of strains and strain gradient in the potential energy density and makes it possible to determine tensorial relations for the compliance tensors of fourth‐, fifth‐ , and sixth‐rank. Both modifications result in the same compliance tensors and are valid for an arbitrary material symmetry class. In the case of hemitropic materials, the compliance tensors have the same symmetry and the same form as the stiffness tensors and are characterized by eight independent constants, namely the two classical isotropic constants, five constants in the strain gradient part and one constant in the coupling term. Explicit expressions for these eight parameters are obtained from the tensorial relations for the compliance tensors and are compared with the direct solution of a linear system for the compliance's. All three solutions are identical, what we consider as a verification of the presented results.

On the objectivity of the nonlinear along-fibre shear stress\u2013strain relationship for unidirectionally fibre-reinforced composites

The nonlinear along-fibre shear stress–strain relationship for unidirectionally fibre-reinforced composites has been investigated in this paper aiming at its applications in general 3D stress conditions in a consistent manner. So far, such relationship has only been addressed in plane stress conditions. In this paper, it has been shown that its straightforward generalisation to 3D stress states lacks objectivity, which is a basic requirement for all theoretical studies of physical problems. A new formulation is proposed based on the stress invariants and the complementary strain energy guided by the rational theoretical framework of nonlinear elasticity. A consistent and objective stress–strain relationship has been obtained and verified through an example of application to a torsion problem.

An Energy-Based Framework for Nonlinear Kinetostatic Modeling of Compliant Mechanisms Utilizing Beam Flexures

Abstract Although energy-based methods have advantages over the Newtonian methods for kinetostatic modeling, the geometric nonlinearities inherent in deflections of compliant mechanisms preclude most of the energy-based theorems. Castigliano's first theorem and the Crotti-Engesser theorem, which don't require the problem being solved to be linear, are selected to construct the energy-based kinetostatic modeling framework for compliant mechanisms in this work. Utilization of these two theorems requires explicitly formulating the strain energy in terms of deflections and the complementary strain energy in terms of loads, which are derived based on the beam constraint model. The kinetostatic modeling of two compliant mechanisms are provided to demonstrate the effectiveness of the explicit formulations in this framework derived from Castigliano's first theorem and the Crotti-Engesser theorem.

Optimal Plastic Analysis and Design of Pile Foundations Under Reliable Conditions

In this research, in order to evaluate the plastic limit load and also plastic design parameters of the long pile foundations subjected to horizontal loads, shakedown method is applied. In carrying out shakedown analysis and design methods, large plastic deformations and residual displacements could develop in the pile foundation which might lead to the failure of the structure. For this reason, complementary strain energy of residual forces proposed as a limit condition to control the plastic deformation of the pile structure. Furthermore, considering the uncertainties (strength, manufacturing, geometry) the limit conditions on the complementary strain energy of residual forces are assumed randomly and the reliability condition was formed by the use of the strict reliability index. The influence of the limit conditions on the plastic limit load and design parameters of the long pile in cohesionless soil subjected to lateral load were investigated and limit curves for shakedown load factors are presented. The numerical results show that the probabilistic given limit conditions on the complementary strain energy of residual forces have significant influence on the load bearing limit and the design parameters of pile foundations. The formulations of the reliability based problems lead to mathematical programming which were carried out by the use of non-linear algorithm.