Strain Compatibility Conditions Research Articles

Seismic performance and estimation of flexural bearing capacity of prefabricated engineered cementitious composite shell–reinforced concrete composite column with grouted sleeve connection

Engineered cementitious composites (ECCs) have excellent crack control capacity and pseudo-strain-hardening behaviour. They can be used to improve the damage-tolerant capacity of vulnerable parts of structures. To improve the seismic performance of a traditional prefabricated column connected with grouted sleeves (GSs), an innovative prefabricated ECC shell–reinforced concrete (RC) composite column with GSs (ECC–PRC–GSs) is proposed in this paper. Then, three-dimensional nonlinear finite element models of the ECC–PRC–GSs are established and validated with previous experimental results. Subsequently, parametric analysis of the ECC–PRC–GSs is systematically performed to assess the seismic performance of ECC–PRC–GSs. Notably, certain important design suggestions for the ECC–PRC–GSs are proposed. Finally, a theoretical model is developed to predict the flexural bearing capacity of ECC–PRC–GSs based on the strain compatibility and force equilibrium conditions. The effectiveness of the model is verified through finite element analysis and experimental results.

Study on coupled problems of thermoelasticity in Strains

In the work, within the framework of the strain compatibility conditions of Saint-Venant, two equivalent dynamic boundary value problems of thermoelasticity with respect to strains are formulated. In the case of the first boundary value problem, the dynamic equations of thermoelasticity are obtained from the compatibility conditions, in the second case, instead of the first three equations of thermoelasticity, the equations of motion expressed with respect to deformations are considered. Discrete analogues of boundary value problems are constructed using the finite-difference method in the form of explicit and implicit schemes. The solution of explicit schemes is reduced to recurrent relations with respect to deformations and temperature. Implicit schemes are solved by sequential application of the elimination method. The validity of the formulated thermoelastic boundary value problems is substantiated by comparing the numerical results of the problem of a thermoelastic parallelepiped obtained by different methods, as well as solving a similar problem in displacements.

Shear strength model for steel shape-reinforced concrete girders considering compatible steel–concrete interaction

Steel shape-reinforced concrete (SRC), also known as concrete-encased steel, offers numerous benefits compared with conventional reinforced concrete (RC) or bare steel due to its combination of RC and structural steel sections; therefore, SRC members find extensive use as transfer elements or girders in engineering applications. When evaluating SRC girders’ shear resistance, the direct strength superposition of RC and structural steel is basically used in most design codes, ignoring the steel–concrete interaction under shear, which has proven unreasonable because of the brittle shear behavior of RC. This paper suggests a novel method for assessing SRC girders’ shear resistance mechanism. The proposed method involves modeling the RC portion by a hybrid truss-arch analogy; in addition, based on the strain compatibility condition, the steel profile’s contribution is determined by applying the yielding criterion. Finally, the rationality of the proposed model is validated through a comprehensive comparison with previously published models and a set of 45 experimental test data samples available in the existing literature, and the results show that the proposed model can reasonably predict the maximum load-carrying capacity of SRC girders and is superior to existing shear strength models and code-based equations.

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

The main parameters characterizing the process of deformation of solids are displacements, strain and stress tensors. From the point of view of the strength and reliability of the structure and its elements, researchers and engineers are mainly interested in the distribution of stresses in the objects under study. Unfortunately, all boundary value problems are formulated and solved in solid mechanics mainly with respect to displacements, or an additional stress functions. And the required stresses are calculated from known displacements or stress functions. In this case, the accuracy of stress calculation is strongly affected by the error of numerical differentiation, as well as the approximation order of the boundary conditions. The formulation of boundary value problems directly with respect to stresses or strains allows to increase the accuracy of stress calculation by bypassing the process of numerical differentiation.
 Therefore, the present work is devoted to the formulation and numerical solution of boundary value problems directly with respect to stresses and strains. Using the well-known Beltrami-Miеchell equation, and considering the equilibrium equation as ah additional boundary condition, a boundary value problem(BVP) is formulated directly with respect to stresses. In a similar way, using the strain compatibility condition, the Beltrami-Mitchell type equations for strains are written.
 The finite difference equations for two-dimensional BVP are constructed and written in convenient a form for the use of iterative method. A number of problems on the equilibrium of a rectangular plate under the action of various loads applied on opposite sides are numerically solved. The reliability of the results is ensured by comparing the numerical results of the 2D elasticity problems in stresses and strains, and with the exact solution, as well as with the known solutions of the plate tension problem under parabolic and uniformly distributed loads

Shear behaviour of reinforced straw-bale plaster sheathed cold-formed steel-framed shear walls

The introduction of cold-formed thin-walled steel (CFS) framed systems into straw-bale construction is a new development construction. A CFS stud frame without blocking or strap-brace provides little in-plane shear stiffness, however it can provide lateral support for the thin and low-reinforcement sheathing through self-tapping screw anchorage, allowing the sheathing to subject to shear failure rather than buckling failure. Straw-bales with different thicknesses can satisfy different requirements for heat transfer and sound insulation. Since straw-bales are reliably bonded to the sheathing, the straw-bales can provide lateral support for the thin sheathing and can also effectively slow down the decline of the shear strength after the sheathing cracks. Moreover, the anchorage slip and the shear deformation of the sheathing material are the major factors affecting shear stiffness. The theory presented in this paper predicts the shear deformation response of the reinforced straw-bale plaster sheathed CFS framed shear wall specimen. The behaviour of the self-tapping screw anchorage was defined using power curve to predict the wall deformation due to anchorage slip. The deformation caused by shear in the sheathing material was estimated using refined softened truss model, which was based on the mechanical equilibrium condition, strain compatibility condition, and the constitutive relationship of softened cement mortar and mesh wire embedded in cement mortar. Comparing the theoretical results with the experimental data, the two were in good agreement. The shear strength of different configurations of reinforced straw-bale walls described in the literature were estimated, which proved the applicability of the theory. • Introduction of cold-formed thin-walled steel (CFS) framed system into straw-bale construction. • Estimation of the two major factors affecting the wall shear stiffness. • Validation of the predicted results according to experimental data. • Prediction for the shear strength of reinforced plaster straw-bale walls with different configurations.

Stress formulation of elastic wave motion.

Propagation of elastic waves in anisotropic solids is solved through a pure stress formalism. The stress solutions lead to the three wave solutions expected from the displacement formulation plus three non-propagating stresses with zero wave speed which do not satisfy the strain compatibility conditions. This work provides a different perspective to modeling elastic waves and is expected to be better suited for certain types of problems.

Efficient prediction of the effective nonlinear properties of porous material by FEM-Cluster based Analysis (FCA)

The present paper extends the FEM-Cluster based Analysis (FCA) method to the efficient prediction of effective properties of porous materials in the nonlinear range. Different from the traditional methods of homogenization or mean field theory, this new approach does not fill the pore with low modulus materials which causes numerical inaccuracy and convergence difficulties as reported in several literature. With no material filling in pores, the average strain theorem and the Hill–Mandel condition which is the basis of the Representative Volume Element (RVE) method for the prediction of effective properties of heterogeneous materials is almost impossible to implement. Furthermore, because of the lack of pore strain information, the compatibility condition of cluster strains cannot be applied in the incremental analysis of reduced order model in the nonlinear stage. In the present study we have overcome these difficulties by several approaches. First, it is found that the energy equivalence principle provides a more suitable basis for effective properties prediction of porous materials. Second, the self-equilibrium cluster stress field obtained from the interaction matrix and the cluster-based minimum complementary energy in FCA can be successfully applied to the incremental analysis of reduced order model in the nonlinear stage instead of the strain compatibility condition. However, due to the errors introduced by averaging stress and strain over each cluster, the interaction matrix becomes non-singular and the self-equilibrium properties of the cluster stress field generated from the interaction matrix become very poor. Fortunately, the most exciting discovery of the present study is that we have found a way to recover the singularity of the interaction matrix by minimizing its rank on the premise that the elastic energy norm of RVE remains unchanged within the given tolerance. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed approach.

Numerical and analytical investigations of flexural behaviours of ECC–LWC encased steel beams

By conducting a series of experimental study, the authors recently demonstrated that engineered cementitious composites (ECC) can be employed to encase the compression flanges of structural steel sections to prevent slender flanges from local buckling and lateral torsional buckling of laterally unconstrainted beams. In this context, this paper presents comprehensive numerical and analytical investigations on the flexural behaviour of ECC-lightweight concrete (LWC) encased steel beams with a wide range of steel section geometry (from compact to slender) and steel grade (from normal strength to high strength). In the numerical study, a validated three–dimensional (3D) nonlinear finite element (FE) model was adopted to carry out a parametric study on the flexural capacity of 234 encased beams with different design parameters such as steel grade, ECC and LWC strengths, compactness of steel section, beam depth to width aspect ratio and flange and web thicknesses etc. Furthermore, an analytical model was developed by using the strain compatibility and force equilibrium conditions to predict the load-deformation curves of the encased beams until the flexural failure of the beams. Finally, in order to reduce efforts needed for day-to-day design, a simplified analytical solution was also proposed, and its accuracy was validated.

Determination of strain path during martensitic transformation in materials with two possible transformation orientation relationships from variant self-organization

Determination of strain path across martensitic transformation in materials with two transformation orientation relationships (ORs) is challenging due to the temporal and spatial limitation of the modern techniques. In this work, an analyzing strategy, i.e., the determination of transformation path via the variant organization feature of martensite, was suggested and applied for the Ni–Mn-based alloys as an example, based on the consideration that the different crystallographic symmetries of transformation systems will result in distinct organizations of martensite variants. For the selected Ni–Mn-based alloys, the orientation examination revealed that both the K−S and the Pitsch OR are respected. Further analyses in terms of the strain and stress compatibility condition and the minimum energy criteria showed that, theoretically, the K−S path produces 2 variants as a self-accommodated variant group, whereas the Pitsch path produces 4 variants as a self-accommodated variant group. Compared with the experimental results, the Pitsch path is the energetically favorable transformation path and actually occurs in stress-free austenite of the Ni−Mn based alloys. The significance of this work is multi-fold. It first resolves the long puzzling issue of transformation strain path of Ni–Mn-based alloys. Moreover, the analyzing scheme can be generalized to determine the transformation characters for martensitic transformation in other systems with multiple possible transformation orientation relationships.

Stress analysis for a thin film bonded to an orthotropic substrate under thermal loading

In this paper, the interaction between an isotropic thin film and an orthotropic substrate is considered under the plane strain state. The perfectly bonded thin film is exposed to a predefined temperature gradient. The strain compatibility condition and the plane elasticity formulation are utilized to extract the governing integral equations. Employing the Gauss- Chebyshev discretization method, the interfacial shear as well as the flexural surface in-plane stresses are determined. The influences of the material stiffness and the orthotropy parameters on both surface and subsurface stresses are investigated. The numerical results confirm that the proper adjustment of the orthotropic substrate properties can alleviated the severe stress gradients. The results indicate that the surface in-plane stress tensile peak significantly decreases if the stiffness of the orthotropic half-plane along the film interface is lower than its corresponding value through the depth. Also, the maximum von Mises stress in the substrate decreases about 50% for an orthotropic material possessing a large shear parameter.

Laminate-based modelling of single and polycrystalline ferroelectric materials – application to tetragonal barium titanate

The present contribution deals with the development of a laminate-based model designed to study the single and polycrystalline tetragonal ferroelectric material behaviour. Laminate-based models are micromechanically motivated and consider the volume fraction of the distinct ferroelectric variants directly in their formulation. At first, a single crystal laminate-based model is established by considering the average strain and polarisation compatibility conditions. A suitable thermodynamic electric Gibbs energy and a rate-dependent dissipation equation are postulated to capture the dissipative hysteretic material response. The update of the inequality constrained volume fractions is solved by adopting a Fischer–Burmeister-type algorithm in combination with a Newton–Raphson scheme. Following the single crystal formulation, a homogenisation procedure based on random orientation of the individual grains in a polycrystalline aggregate is considered. The material properties and the polarisation switching response of the randomly oriented individual grains are averaged using a finite element framework in order to study the macroscopic polycrystalline behaviour. A parameter fitting procedure based on experimental data for single crystalline response, taken from the literature, is detailed and the material model as well as the algorithmic scheme are verified by solving representative boundary value problems. Moreover, the finite element based simulation results are compared with newly generated experimental hysteresis data for a barium titanate piezoceramic.

An eigenvalue problem for tensors used in mechanics and the number of independent Saint-Venant strain compatibility conditions

A number of questions concerning the eigenvalue problem for a tensor $$\mathop A\limits_ \approx$$ ∈ ℝ4(Ω) with special symmetries are considered; here Ω is a domain of a four-dimensional (three-dimensional) Riemannian space. It is proved that a nonsingular fourth-rank tensor has no more than six (three) independent components in the case of a four-dimensional (three-dimensional) Riemannian space. It is shown that the number of independent Saint-Venant strain compatibility conditions is less than six.

Design Oriented Stress-Strain Models for Engineered Cementitious Composites

Engineered Cementitious Composite (ECC) is known for multiple cracking and strain hardening property when loaded in tension. To incorporate this superior property in the design of structural elements, it is mandatory to develop design oriented stress-strain models considering the tensile strain hardening property. In the present study, stress-strain models are developed for both tension and compression using the empirical values available in literature. An analytical study has been conducted on the flexural response of Fiber Reinforced Polymer (FRP) reinforced ECC beams to understand the robustness of the developed stress-strain models. For this purpose, a special purpose non-linear computer program is developed based on force equilibrium and strain compatibility conditions to predict the ultimate load and deflection. The accuracy of the nonlinear computer program is validated by comparing the analytical results with experimental results available in literature which is found to be in close agreement. Hence, it is inferred that the developed stress-strain models can be used for various design purposes.

A loaded Timoshenko beam bonded to an elastic half plane

The problem of a Timoshenko beam of finite length loaded by concentrated forces and couples and perfectly bonded to a homogeneous elastic and isotropic half plane is considered in the present work. In particular, the effects induced by shear deformation of the beam on the contact stresses arising at the interface between the beam and the underlying half plane are investigated accurately. An asymptotic analysis of the stress field at the beam ends and in the neighborhood of the loaded section of the beam allows us to characterize the singular nature of the peeling and shear stresses. The problem is formulated by imposing the strain compatibility condition between the beam and the half plane, thus leading to a system of two singular integral equations with Cauchy kernel. The unknown interfacial stresses are expanded in series of Jacobi orthogonal polynomials displaying complex singularity. This approach allows us to handle the oscillatory singularity and to reduce the integral equations to a linear algebraic system of infinite equations for the unknown coefficients of the interfacial stresses, which is solved through a method of collocation. The interfacial peeling and shear stresses and, in turn, the displacement field along the contact region have been calculated under various loading conditions applied to the beam. The internal forces and bending moments along the beam have been calculated varying the shear and flexural stiffness of the beam. The complex stress intensity factors and the strength of the stress singularities have been assessed in detail.

Evaluation of the flexural behavior of composite beam with inverted-T steel girder and steel fiber reinforced ultra high performance concrete slab

Huge efforts have been made to develop ultra high performance concrete (UHPC) and exploit its remarkable properties. Among the achievements, various solutions were proposed to achieve optimal composite beams using steel fiber reinforced UHPC. In order to increase the economy in material, a composite beam combining a slab made of UHPC and a steel girder without top flange is proposed. In this composite beam, the inverted-T steel girder requires the studs be arranged on the web for the composition with the UHPC slab. Considering the absence of studies evaluating the flexural behavior of this new type of composite beam, this study examines experimentally the flexural behavior with respect to the stud spacing and slab thickness. To that goal, eight composite beams with varying stud spacing and UHPC slab thickness were fabricated together with two additional composite beams using slabs made of normal concrete without steel fiber reinforcement for comparison. In addition, an analysis method considering the tension-softening behavior of steel fiber reinforced UHPC in the material model and the corresponding strain compatibility conditions of the beam member is proposed. The comparison of the analytic and experimental results reveals the good accuracy of the predictions indicating that the empirical tension-softening curve reflects reasonably the actual behavior of steel reinforced UHPC composite member.

Monte Carlo simulations of vector pseudospins for strains: Microstructures and martensitic conversion times

We present systematic temperature-quench Monte Carlo simulations on discrete-strain pseudospin model Hamiltonians to study microstructural evolutions in ferroelastic transitions with two-component vector order parameters (NOP=2) in 2-spatial dimensions. The zero value pseudospin is the single high-temperature phase while the low-temperature phase has Nv variants. Thus the number of nonzero values of pseudospin are triangle-to-centered rectangle (Nv=3), square-to-oblique (Nv=4) and triangle-to-oblique (Nv=6). The model Hamiltonians contain a transition-specific Landau energy term, a domain wall cost or Ginzburg term, and power-law anisotropic interaction potential, induced from a strain compatibility condition. On quenching below a transition temperature, we find behaviour similar to the previously studied square-to-rectangle transition (NOP=1,Nv=2), showing that the rich behaviour found, is generic. Thus we find for two-component order parameters that the same Hamiltonian can describe both athermal and isothermal martensite regimes for different material parameters. The athermal/isothermal/austenite parameter regimes and temperature–time–transformation diagrams are understood, as previously, through parametrization of effective-droplet energies. In the athermal regime, we find rapid conversions below a spinodal-like temperature and austenite-martensite conversion delays above it, as in the experiment. The delays show early incubation behaviour, and at the transition to austenite the delay times have Vogel–Fulcher divergences and are insensitive to Hamiltonian energy scales, suggesting that entropy barriers are dominant. Systematic temperature quench experiments can look for martensite formation and growth during conversion-incubations, divergences, and distributions close to the transition.

Plastic hinge length of reinforced concrete slender shear walls

This paper proposes a simple model that can reasonably determine the potential plastic hinge region for seismic design of shear walls. To idealise the plastic hinge region, a parametric study is conducted on the curvature distribution over the member length using a non-linear, two-dimensional, finite-element analysis procedure. To solve the plastic hinge length equation derived from the idealised curvature distribution, the equations for yielding and ultimate moment capacities of shear walls are generalised on the basis of Bernoulli's principle, strain compatibility condition and equilibrium condition of forces. The generalised equations promise that the yielding and ultimate moment capacities of shear walls can be straightforwardly evaluated with relatively high accuracy. Ultimately, the plastic hinge length of the shear wall is then simply formulated as a function of longitudinal tensile reinforcement index in the boundary element, vertical shear reinforcement index in the web, and axial force index. The predicted plastic hinge lengths obtained from the current simple model are in better agreement with test results than the previous equations; the mean, standard deviation and coefficient of variation of the ratios between experiments and predictions are 1·019, 0·102 and 0·100, respectively.

휨 항복형 철근콘크리트 전단벽의 경계요소설계를 위한 변위연성비 모델제시

ABSTRACT This study established a displacement ductility ratio model for ductile design for the boundary element of shear walls. To determine the curvature distribution along the member length and displacement at the free end of the member, the distributio ns of strains and internal forces along the shear wall section depth were idealized based on the Bernoulli’s principle, strain compatibility condition, and equilibrium condition of forces. The confinement effect at the boundary element, provided by transverse reinforcement, was calculated using the stress-strain relationship of confined concrete proposed by Razvi and Saatcioglu. The curvatures corre sponding to the initial yielding moment and 80% of the ultimate state after the peak strength were then conversed into displacement values based on the concept of equivalent hinge length. The derived displacement ductility ratio model was simplified by the regression approach using the comprehensive analytical data obtained from the parametric study. The proposed model is in good agreement with test results, indicating that the mean and standard deviation of the ratios between predictions and experiments are 1.05 a nd 0.19, respectively. Overall, the proposed model is expected to be available for determining the transverse reinforcement ratio at the boundary element for a targeted displacement ductility ratio.

Ultimate load theorems for rigid plastic solids with internal degrees of freedom and their application in continual lattice shells

Abstract This paper studies solids with internal degrees of freedom using the method of Cartan moving hedron. Strain compatibility conditions are derived in the form of structure equations for manifolds. Constitutive relations are reviewed and ultimate load theorems are proved for rigid plastic solids with internal degrees of freedom. It is demonstrated how the above theorems can be applied in behavior analysis of rigid plastic continual shells of shape memory materials. The ultimate loads are estimated for rotating shells under external forces and in case of shape recovery from heating.

Direct strain tensor approximation for full‐field strain measurement methods

SUMMARYFull‐field strain measurement techniques are based on computing the spatial derivatives of numerical or functional approximations of the underlying displacement fields extracted from digital imaging methods. These methods implicitly assume that the medium satisfies the strain compatibility conditions, which are only true in the case of a continuum body that remains continuum throughout its deformation history. In the present work, we introduce a method that can be used to calculate the strain components directly from typical digital imaging data, without the need of the continuum hypothesis and the need for displacement field differentiation. Thus, it enables the measurement of strain fields from imaged surfaces that may or may not contain discontinuities. Numerical comparisons are performed on the basis synthetic data produced from an analytical solution for an elastically orthotropic open‐hole domain in tension. For performance comparison purposes, the mean absolute error distributions are calculated for the cases of both the traditional meshless random grid method, and the direct strain method introduced herein. It is established that the more refined representation of strain provided by our present approach is more accurate everywhere in the domain, but most importantly, near its boundaries. Published 2013. This article is a US Government work and is in the public domain in the USA.