Compatibility Of Deformations Research Articles

3D multiscale micromechanical model of wood: From annual rings to microfibrils

A 3D micromechanical analytical-computational model of softwood, which takes into account the wood microstructures at four scale levels, from microfibrils to annual rings, is developed. For the analysis of the effect of the annual rings structure on the properties of softwood, an improved rule-of-mixture model, based on 3D orthotropic stress–strain relations and taking into account the compatibility of deformations at the interface of two phases and equilibrium of tractions at phase boundaries, is proposed. The improved rule of mixture model (IRoM) was compared with the classical rule-of-mixture (RoM) and finite element method (FEM) simulations. It was shown that IRoM gives almost as good results as FEM. The analytical model of annual rings is combined with the 3D finite element model of softwood as cellular material with multilayered, microfibril reinforced cell walls, developed by Qing and Mishnaevsky (2009a,b). Using the combined four-level model, the effect of wood density, microfibril angle (MFA) and cell shape angle (CSA) on the Young’s moduli, Poisson’s ratios and shrinkage properties of softwood has been investigated in numerical experiments. The simulations were verified by comparison with experimental data.

Analytical modeling of monolithic joints in concrete bridges

The behavior of bridge monolithic connections is modeled using a simplified mathematical model that accounts for stress equilibrium, compatibility of deformations, and the state of bond of longitudinal column bars anchored through the joint panel. In this regard, a stress gradient factor is introduced, to model the profile of bar stresses along the anchorage. To establish this factor, two independent mechanisms of stress transfer are considered: a bond mechanism between the anchored bars and the surrounding concrete and a friction mechanism between the anchored bars and the transverse bars that enclose and restrain the anchorages. The model is used for calculation of the shear stress-shear strain relationship of all tests found in the international literature on bridge monolithic connections that showed shear type of failure under simulated seismic loading. Joint strength values calculated with the proposed model are compared with the experimental results. Based on this comparison the proposed model is verified for use in interpretation of bridge monolithic connection behavior and design.

Flexural ductility of reinforced HSC beams strengthened with CFRP sheets

Externally bonding fiber reinforced polymer (FRP) sheets with an epoxy resin is an effective technique for strengthening and repairing reinforced concrete (RC) beams under flexural loads. Their resistance to electro-chemical corrosion, high strength-to-weight ratio, larger creep strain, fatigue resistance, and nonmagnetic and nonmetallic properties make carbon fiber reinforced polymer (CFRP) composites a viable alternative to bonding of steel plates in repair and rehabilitation of RC structures. The objective of this investigation is to study the effectiveness of CFRP sheets on ductility and flexural strength of reinforced high strength concrete (HSC) beams. This objective is achieved by conducting the following tasks: (1) flexural four-point testing of reinforced HSC beams strengthened with different amounts of cross-ply of CFRP sheets with different amount of tensile reinforcement up to failure; (2) calculating the effect of different layouts of CFRP sheets on the flexural strength; (3) Evaluating the failure modes; (4) developing an analytical procedure based on compatibility of deformations and equilibrium of forces to calculate the flexural strength of reinforced HSC beams strengthened with CFRP composites; and (5) comparing the analytical calculations with experimental results.

Modeling the flexural carrying capacity of corroded RC beam

Considering the change of bond strength between corroded steel and concrete, flexural carrying capacity of corroded reinforced concrete (RC) beam was calculated. On the basis of the condition of equilibrium of forces and compatibility of deformations for the whole beam, a model for the prediction of flexural carrying capacity of the corroded RC beam was proposed. Comparison of the model’s predictions with the experimental results published in the literature shows the practicality of the proposed method.

Damping and Viscoelastic Dynamic Response of RC Flexural Members Strengthened with Adhesively Bonded Composite Materials

A theoretical approach for the dynamic viscoelastic response of reinforced concrete (RC) beams and one-way slabs strengthened with adhesively bonded composite materials is developed. The analytical model is based on variational principles, dynamic equilibrium, and compatibility of deformations between the structural components (RC beam/slab, adhesive, composite material). The model accounts for the deformability of the adhesive layer and for its high order stress and displacement fields. The equations of motion and the boundary, continuity, and initial conditions are derived via the extended Hamilton’s principle. The Kelvin-Voigt approach is adopted for the consideration of the viscoelastic response of the adhesive material and the internal damping in the composite material and the RC member. The Rayleigh damping model is used for the external viscous damping of the RC member. The dynamic governing equations are solved using the Newmark time integration and a multiple shooting algorithm is used for the so...

Numerical analysis of multilayered CFRP retrofitted RC beams with partial interaction

This paper presents a theoretical model to simulate the behaviour of RC beams strengthened with multilayered CFRP matrix allowing for inter-layer slip. An element of the composite beam is assumed to be subjected to a system of forces that satisfy equilibrium and compatibility of deformations. The inter-layer slip is allowed for by relating the differential strain at the interfaces between the CFRP layers and the concrete to the longitudinal shear flow at the corresponding interface through the shear stiffness of the adhesive layer. The basic differential equations are derived in terms of displacement variables and solved numerically using finite differences. The results of the numerical simulation included slip values along the interfaces, maximum slip values, stresses and strains and deflections. The results compare reasonably well with experimental findings.

The exact solution for free vibration of uniform beams carrying multiple two-degree-of-freedom spring–mass systems

The literature regarding the “exact” solutions of natural frequencies and mode shapes of a uniform beam carrying multiple two-degree-of-freedom (2-dof) spring–mass systems is rare, thus, this paper aims at studying this problem using the numerical assembly method (NAM). First of all, the equivalent springs for replacing the effect of a 2-dof spring–mass system are determined. Next, the coefficient matrix for a 2-dof spring–mass system attached to the uniform beam is derived based on the compatibility of deformations and equilibrium of forces (including moments). The coefficient matrices for the left end and right end of the beam are also derived based on the various boundary conditions of the beam. Combining the coefficient matrices for all the 2-dof spring–mass systems attached to the beam and the coefficient matrices for the boundary conditions of the beam, one obtains the overall coefficient matrix of the constrained beam (i.e., the beam carrying any number of 2-dof spring–mass systems). The product of the overall coefficient matrix and the vector for all the integration constants yields a set of simultaneous equations. Let the coefficient determinant of the last simultaneous equations equal to zero, one obtains the frequency equation. The roots of the last frequency equation denote the natural frequencies of the constrained beam. Substituting the roots of the frequency equation into the set of simultaneous equations one may determine the associated mode shapes of the constrained beam. In this paper, the “exact” solution refers to the one obtained from the “continuous” model instead of the “discrete” mode, besides, the accuracy of the analytical-and-numerical combined method (ANCM) given by the existing literature is dependent on the total number of vibration modes considered, but this is not true for the accuracy of the NAM adopted here. To confirm the reliability of the presented theory, all the numerical results obtained from NAM are compared with the corresponding ones obtained from the conventional finite element method (FEM) and good agreement is achieved.

Dynamic Behavior of Reinforced Concrete Beams Strengthened with Composite Materials

The dynamic behavior of reinforced concrete (RC) beams strengthened with externally bonded composite materials is analytically investigated. The analytical model is based on dynamic equilibrium, compatibility of deformations between the structural components (RC beam, adhesive, composite material) and the concept of the high order approach. The equations of motion along with the boundary and continuity conditions are derived using Hamilton’s variational principle and the kinematic relations of small deformations. The mathematical formulation also includes the constitutive laws that are based on beam and lamination theories, and the two-dimensional elasticity representation of the adhesive layer including the closed form solution of its stress and displacement fields. The Newmark time integration method, which is directly applied to the resulting set of coupled partial differential equations, is adopted. This procedure yields a set of ordinary differential equations, which are analytically or numerically solved in every time step. The response of a strengthened beam to different dynamic loads that include impulse load, harmonic load, and seismic base excitation is numerically investigated. The numerical study highlights some of the phenomena associated with the dynamic response and explores the capabilities of the proposed model. The paper closes with a summary and conclusions.

Inclusion of the vertical seismic acceleration in one-story buildings analysis

The principal parameters that control the elastic behavior of one-story reinforced concrete building models are analyzed in this paper. Models are subjected to seismic ground motions considering the action of the three-recorded ground accelerations, u x , Uy and u z . The horizontal components, u x and u y , are considered acting in a critical angle of incidence determined for each analyzed model. This effect has been thoroughly studied in previous works. The influence of the vertical ground acceleration on the seismic response has not been exhaustively studied. Adequate and simple methods to consider the combined effects of the three ground motion components for seismic design are proposed in this work. Three-dimensional one-story models, characterized by parameters as the lateral period of vibration, the lateral-transverse period ratio and the lateral-torsional period ratio, the vertical period, the slenderness of the comer columns, the eccentricity, the aspect ratio of the plan, the beam-column stiffness ratio, and the border conditions of the floor slab, are studied. Seismic responses are obtained using a time history response analysis method. Models are subjected to different ground motions. The compatibility of vertical deformations is enforced, and the transversal and torsional stiffnesses of the elements are considered. Maximum local and global responses are studied. The responses of vertically flexible models are especially affected by the inclusion of vertical ground acceleration. It should be considered in the seismic design. Additionally, a diaphragm behaves like a floor slab under vertical cyclic loads. It doesn't act like a rigid body in the vertical direction, particularly if its dimensions are large.

THE EXACT SOLUTIONS FOR THE NATURAL FREQUENCIES AND MODE SHAPES OF NON-UNIFORM BEAMS WITH MULTIPLE SPRING–MASS SYSTEMS

For a beam carrying n spring–mass systems, if the left side and right side of each attaching point and each end of the beam are regarded as nodes, then considering the compatibility of deformations and the equilibrium of forces between the two adjacent beam segments at each attaching point and incorporating with the equation of motion for each spring–mass system, simultaneous equations may be obtained for the ν th attaching point, where the unknowns for the simultaneous equations are composed of the integration constants for the eigenfunctions of the νth and ( ν+ 1)th beam segments and the associated modal displacements of the ν th sprung mass. It is evident that if these unknowns are considered as the nodal displacements, then the coefficient matrix of the simultaneous equations will be equivalent to the element stiffness matrix for the ν th attaching point (associated with the ν th and ( ν+ 1)th beam segments). In view of the last fact, one may use the numerical assembly method (NAM) for the conventional finite element method to obtain the overall simultaneous equations for the overall ( n) attaching points (associated with the overall ( n+ 1) beam segments) by taking into account the boundary conditions of the whole beam. The solutions for the coefficient determinant of the overall simultaneous equations to be equal to zero will give the “exact” natural frequencies of the constrained beam (carrying multiple ( n) spring–mass systems) and the substitution of each corresponding values of the integration constants into the associated eigenfunctions for each attaching point will determine the corresponding mode shapes. Since no discretization on the continuous beam was made in the present approach (NAM), the natural frequencies and the corresponding mode shapes obtained are the exact ones.

A coupled BEM‐flexibility force method for bending analysis of internally supported plates

AbstractThis paper presents a new method for the analysis of plates in bending with internal supports. The proposed method can be regarded as an extension of the well‐known force method (the flexibility matrix method) in the matrix analysis of structures. The solution is performed through two phases: the released plate phase, in which the plate is released from all internal supports and solved using the Boundary Element Method (BEM). The effect of internal supports is considered in the second phase, where a series of unit virtual loads is placed instead of the unknown redundant reactions at internal supports. The flexibility matrix is formed and compatibility of deformations at the locations of internal supports is satisfied. Hence, the corresponding system of equations is solved for the unknown redundant forces at internal supports. The final solution of the problem consists of the summation of two phases: the released plate phase and the cases of virtual unit loads phase. An efficient solution algorithm is developed to solve both phases simultaneously. The main advantages of the present formulation are: (1) the present formulation increases the versatility of the BEM as it allows the re‐usability of standard BEM codes for solution of plates in bending to be used in solving problems having internal supports, with even no modifications; and (2) the two solution phases are completely uncoupled; therefore it is easy to trace behaviour of the plate due to failure of one or more of the internal supports without re‐analysis.Several numerical examples are analysed. The results are compared to those of analytical and finite element models to demonstrate the accuracy and the validity of the present formulation. The present formulation is used also to study the differences between the finite element and boundary element modelling for building slabs. Copyright © 2002 John Wiley & Sons, Ltd.

Fuzzy Multistage Optimization of Large-Scale Trusses

Fuzzy membership functions are implemented in the multistage optimization technique to improve its performance for the minimum weight design of truss structures of fixed topology. First, a large truss is split into several smaller substructures that are optimized separately. However, to preserve the unity of the original truss, interaction-balance equality constraints are introduced to account for the compatibility of deformations, and the equality of the action and reaction of forces at the points of the separation of the substructures. A penalty function, containing the equilibrium equations and interaction constraints as well as other constraints, is defined for each substructure. Then, a fuzzy membership function, returning a value between 0 and 1, is introduced for each of the equality constraints to provide some fuzzy evaluation of the degree of satisfaction. This measure is then used in the determination of an appropriate penalty parameter. The design procedure is iterative and continues until conv...

Nonlinear High-Order Analysis of Cracked RC Beams Strengthened with FRP Strips

A nonlinear analysis, based on the closed-form high-order approach, for the behavior of cracked reinforced concrete beams, retrofitted with externally bonded fiber-reinforced plastic (FRP) strips, is presented. The model is based on equilibrium, compatibility of deformations, and the nonlinear constitutive relations of the various materials. The nonlinear implicit governing equations are derived and are solved through an iterative linearized procedure. Verification of the nonlinear model is achieved through comparison of the load-deflection response with the experimental result. The results of the proposed analysis include global effects such as load-deflection response and the development of the tensile stresses in the FRP strip and localized effects such as the shear and peeling stresses near the edge of the strip. The study reveals that the extreme values of the stresses at the edge region and the in-plane stress resultant in the strip exhibits nonlinear dependence on the imposed load, which the linear approach is incapable of predicting. Thus, the use of linear analysis for cracked concrete beams \Iunderestimates\N the stresses in the FRP strip and in the adhesive layer. Prestressing of the FRP strip is investigated, revealing that the strip must be properly anchored to exploit its advantages. This paper ends with a summary and conclusions.

The equations of compatibility of deformations

Nine equations of compatibility of deformations are obtained in which, unlike the classical Saint-Venant compatibility equations, only first derivatives with respect to the coordinates occur. It is proved that, of these nine equations, only six are independent. It is shown that the classical compatibility equations can be obtained from these equations.

INFLUENCE OF COMPOSITION OF MORTAR MIXTURE ON STRENGTH AND DEFORMABILITY OF EXTERNAL WALLS/SKIEDINIO SUDĖTIES ĮTAKA IŠORINIŲ SIENOS SLUOKSNIŲ STIPRUMUI IR DEFORMA VIMUISI

The basic element of building, wall, must satisfty not only requirements for strength, stability, but also for cracking. Experiments and practice show that in many cases external decoration layers of external walls are cracked. It is because designers and builders do not take into consideration one of the basic conditions of the theory of layers—to satisfy compatibility of deformations of layers. Therefore it is necessary to know physical and mechanical properties of plasters such as strength limits of compressive and tensile strains and modulus of elasticity. In this paper, data and recommendations about strength and deformation properties of plaster are given, as well as regressive equations by which it is possible to choose the composition of composite cement-lime-sand mortar recording the following parameters: compressive and tensile strength, limit strain, modulus of elasticity. Analysis of experimental data shows that the compressive and tensile strength of mortar and modulus of elasticity mainly depend on the ratio cement/sand. Limit deformations depend on all 3 factors: cement, lime and sand.

INFLUENCE OF COMPOSITION OF MORTAR MIXTURE ON STRENGTH AND DEFORMABILITY OF EXTERNAL WALLS

Summary The basic element of building, wall, must satisfty not only requirements for strength, stability, but also for cracking. Experiments and practice show that in many cases external decoration layers of external walls are cracked. It is because designers and builders do not take into consideration one of the basic conditions of the theory of layers—to satisfy compatibility of deformations of layers. Therefore it is necessary to know physical and mechanical properties of plasters such as strength limits of compressive and tensile strains and modulus of elasticity. In this paper, data and recommendations about strength and deformation properties of plaster are given, as well as regressive equations by which it is possible to choose the composition of composite cement-lime-sand mortar recording the following parameters: compressive and tensile strength, limit strain, modulus of elasticity. Analysis of experimental data shows that the compressive and tensile strength of mortar and modulus of elasticity m...

Compatibility of large deformations in nonlinear shell theory

In this paper, we propose to establish compatibility conditions satisfied by the membrane strain tensor Δ and the curvature strain tensor K of a shell midsurface. To this end, we use the polar decomposition RU of the linear tangent map of a mapping function φ. First, we characterize the tangential part of the derivative of R by a new variable A. Then the compatibility equations are decomposed in a natural way into a first-order differential system. The use of A greatly simplifies the differential calculus and gives an intrinsic and explicit formulation for compatibility conditions in nonlinear shell theory.

Strengthening of reinforced concrete beams with externally bonded fiber-reinforced-plastic plates: design guidelines for shear and flexure

This paper presents a comprehensive design approach for reinforced concrete flexural beams and unidirectional slabs strengthened with externally bonded fiber-reinforced-plastic (FRP) plates. The approach complies with the Canadian Concrete Standard. The paper is divided into two parts, namely flexural strengthening and shear strengthening. In the first part, analytical models are presented for two families of failure modes: classical modes such as crushing of concrete in compression and tensile failure of the laminate, and premature modes such as debonding of the plate and ripping off of the concrete cover. These models are based on the common principles of compatibility of deformations and equilibrium of forces. They can be used to predict the ultimate strength in flexure that can be achieved by such elements, given the FRP cross-sectional area, or conversely, the required FRP cross-sectional area to achieve a targeted resisting moment for rehabilitated flexural elements. In the second part, design equations are derived to enable calculation of the required cross-sectional area of shear lateral FRP plates or strips for four plating patterns: vertical strips, inclined strips, wings, and U sheet jackets. A step by step procedure is also presented along with an easy to use flow chart. Finally, a numerical design example is provided to demonstrate the applicability of the approach. This study is believed to be timely and very useful for the practicing engineer.Key words: reinforced concrete, beams, slabs, strengthening, FRP, plates, design, flexure, shear, mode of failure, adhesive.

Time dependent service load behaviour of prestressed composite tee beams

This paper is concerned with the time dependent service load behaviour of prestressed composite tee beams. The effects of creep and shrinkage of the concrete slab are modelled using the age adjusted effective modulus method and a relaxation approach. The tendon strain is determined considering compatibility of deformations and equilibrium of forces between the tendon and the composite tee beam. A parametric study is undertaken to study the influence of various aspects on the stress, strain and deformations of the concrete slab, steel beam and prestressing tendon. The effect of loading type and tendon relaxation has also been considered for various types of prestressing tendon materials. Recommendations are then made in relation to adequate span to depth ratios for varying levels of prestressing force.

Strengthening of concrete beams using fiber-reinforced platics

One application of composite materials in civil engineering is examined: the strengthening of a reinforced concrete beamin situ by externally-bonded fiber reinforced plastic (FRP). Studies of the mechanical properties of the interface and the rheological behaviour of composite materials are very important to design. For the experimental determination of the mechanical properties of the concrete/glue/plate interface, a new test is suggested. An iterative analytical model capable of simulating the bond-slop and the material non-linearity, based on the compatibility of deformations and the equilibrium of forces, is developed in order to predict the ultimate forces and deflections. A new equation is proposed to anticipate the maximal shear and normal stresses at the interface goal to anticipate the failure mode due to the debonding of the plate. Finally, a series of large-scale beams strengthened with fiber reinforced plastic is tested up to failure; load-deflection curves are measured and compared with the predicted values to study the efficiency of the externally-bonded plate and to verify the theoretical method.