Principle Of Stationary Complementary Energy Research Articles

Shear behavior of UHPC beams with small shear span to depth ratios based on MSTM

Ultra-high performance concrete (UHPC) has been gradually used in structural engineering due to its excellent mechanical performance, however, predicting the shear behavior of the UHPC beams at service load conditions is still a big challenge, especially for the beams with small shear span to depth ratio. To address this issue, this paper devotes to develop a modified strut and tie model (MSTM) to predict the shear behavior of the UHPC beams with stirrups throughout the loading process. The principle of stationary complementary energy has been introduced into the MSTM to decouple the contribution of arch action and truss action on shear forces. Based on the MSTM, the shear force resistance mechanism has been revealed, and simplified models at each load step have been developed to estimate the shear behavior of UHPC beams. 12 UHPC beams were designed and tested to validate the proposed model. The proposed model was verified to be sufficiently accurate. Moreover, it can be noted that the current design codes, including SETRA-AFGC and SIA, give overly conservative values for UHPC beams when the shear span to depth ratio is less than 2.5.

Full-range nonlinear analysis of post-tensioned anchorage zones based on modified strut-and-tie model

The strut-and-tie model (STM) has been adopted as a preferred methodology for the design of the post tensioned concentric anchorage zones. However, the STM is just a design model at ultimate limit state developed based on the elastic stress trajectory without the consideration of stress redistribution after cracking, which can not be used to predict the serviceability behavior of anchorage zones. To address this issue, this paper is devoted to developing a modified strut-and-tie model (MSTM) to predict the behavior of concentric anchorage zones throughout loading process. The compatibility condition was introduced into MSTM based on the principle of stationary complementary energy at each load step. According to MSTM, the elastic and inelastic behavior of concentric anchorage zones, including load-strain response, load-deformation response and crack width, can be estimated from loading to failure. Moreover, simplified formulas have been proposed to calculate the first cracking load, ultimate load and the maximum crack width. Finally, the proposed MSTM was validated by the testing results.

Elastic‐to‐plastic strut‐and‐tie model for multianchorage zones

The strut‐and‐tie model (STM) has been adopted as a preferred methodology for the design of disturbed regions; however, an aspect which has not yet been sufficiently cleared up is the serviceability behavior of structural concrete, especially with regard to the cracking prediction. To address this issue, this paper is devoted to developing an advanced elastic‐to‐plastic strut‐and‐tie model (EPSTM) to capture the full elastic and inelastic response of the anchorage zones. The principle of stationary complementary energy is introduced into the STM to satisfy the compatibility and generate the adaptive EPSTM at each load step. The proposed EPSTM provides a consistent analysis of the behavior of anchorage zones, including the average strain of struts and ties, load‐deformation response and the maximum crack width, throughout the loading process, and has been verified to be sufficiently accurate by comparing with testing results.

Finite element formulation for the analysis of multilayered beams based on the principle of stationary complementary strain energy

A family of finite elements for the analysis of orthotropic multilayered beams with mono-symmetric cross-sections is developed based on the principle of stationary complementary energy. The longitudinal normal stress field is postulated as polynomial and Heaviside step function series and substituted into the infinitesimal equilibrium conditions to develop expressions for the shear and transverse stress fields. The statically admissible stress fields thus derived are then adopted within the complementary energy variational principle framework to develop a family of finite elements. The distinguishing features of the solution are: (i) it captures the nonlinear distribution of the stress fields along the section depth and steep stress gradients typically occurring near bondline ends of multilayer beams, (ii) unlike conventional solutions based on the principle of stationary potential energy which predict jumps in the shear and peeling stresses at interfaces of adjacent layers, the present solution satisfies equilibrium in an exact infinitesimal sense at layer interfaces and thus ensures continuity of the stress fields across the interface, (iii) it naturally captures the effects of transverse shear and transverse normal stresses, and (iv) it consistently converges to the displacements from above, in contrast to conventional finite element solutions where convergence is typically from below. The versatility of the solution is then illustrated in applications involving wood beams and steel beams strengthened with GFRP plates and sandwich beams with soft cores.

有限位移理论线弹性力学二类和三类混合变量的变分原理及其应用

Variational principles for dual and triple mixed variables of linear elasticity with finite displacements were proposed. Considering the variation of prescribed boundary conditions and using the reciprocal theorem of finite displacements played the key and bridging roles in derivation of the above variational principles. First, in view of the variation of the prescribed geometrical boundary conditions and based on the reciprocal theorem, the principle of minimum potential energy with dual mixed variables was derived. In a similar way, the principle of stationary complementary energy with dual mixed variables was also given. Then the relation between the strain energy density and the complementary energy density was applied to the above 2 principles, and the variational principle with triple mixed variables was deduced. In turn, the principles of virtual work and virtual complementary work with dual and triple mixed variables were directly given. Meantime, the generalized variational principles were derived with the Lagrangian multiplier method. Through an example the Lagrangian multiplier method in certain cases was proved to be ineffective. The semiinverse method for construction of the functionals for generalized variational principles was also introduced. Finally, a cantilever beam with large deflection was calculated by means of the principle of minimum potential energy for dual mixed variables.

Nonlinear stability of double-deck reticulated circular shallow spherical shell

Based on the variational equation of the nonlinear bending theory of double-deck reticulated shallow shells, equations of large deflection and boundary conditions for a double-deck reticulated circular shallow spherical shell under a uniformly distributed pressure are derived by using coordinate transformation means and the principle of stationary complementary energy. The characteristic relationship and critical buckling pressure for the shell with two types of boundary conditions are obtained by taking the modified iteration method. Effects of geometrical parameters on the buckling behavior are also discussed.

Complementary Energy Based Formulation for Torsional Buckling of Columns

A unique formulation for the elastic torsional buckling analysis of columns is developed in this paper based on the principle of stationary complementary energy. It is well known that in displacement based numerical formulations, discretization errors lead to stiffer behavior; hence convergence from above. On the other hand, discretization errors in complementary energy based numerical formulations lead to softer behavior in linear elasticity problems, which is a desired feature from the engineering view point. However, complementary energy based formulations can only overpredict the buckling loads for the flexural buckling problems of columns unless the physical conditions are compromised. In this study a formulation based on the principle of stationary complementary energy is considered for the elastic torsional buckling analysis of columns. The complementary energy expression is obtained from the well known total potential energy functional by using Frederichs’ transformation. In contrast to flexural b...

Buckling analysis of thin-walled open members—A complementary energy variational principle

This first of two companion papers develops a new variational principle for the buckling analysis of thin-walled members based on the principle of stationary complementary energy. Some of the aspects of the Vlasov thin-walled beam theory (the rigid cross section assumption, and the stress expressions) are postulated to describe the behavior of members while other aspects of the theory (i.e., the zero shear strain assumption at mid-surface) are discarded. Koiter's formulation based on polar decomposition theory in finite elasticity is adopted to formulate expressions for statically admissible stress resultant fields. The stationarity conditions of the complementary energy expression are then evoked to yield the conditions of neutral stability and associated boundary conditions in which the rotation fields appear explicitly. The formulation seamlessly incorporates shear deformation effects and load position effects. Also, the Wagner effect and the mono-symmetry property which arise in displacement based formulations arise in the present formulation in a natural way.

Buckling analysis of thin-walled open members—A finite element formulation

In a companion paper, a variational principle based on the principle of stationary complementary energy was developed for the buckling analysis of thin-walled members with open cross-sections. In this paper, the variational principle is adopted to formulate a finite element buckling solution. The formulation successfully incorporates shear deformation effects, a feature that is neglected in most available buckling solutions. By adopting a non-orthogonal coordinate system, the solution successfully captures the transverse load position effect relative to the shear center. A series of examples demonstrate the validity of the finite elements formulated and their applicability to a wide variety of buckling problems. Examples include column flexural and torsional buckling, lateral torsional buckling of beams with a variety of end conditions and subjected to a variety of moment gradients. The formulation is shown to be applicable to beams with mono-symmetric sections. In all cases, the validity of the new solution is assessed and established through comparisons to well-established closed-form and/or numerical solutions.

Torsion analysis of thin-walled beams including shear deformation effects

The first part of the paper develops a theory for the torsional analysis for open thin-walled beams of general cross-sections which accounts for shear deformation effects. Statically admissible stress fields are postulated in agreement with those resulting from the Vlasov thin-walled beam theory. The principle of stationary complementary energy is then adopted to formulate the governing field compatibility condition under the stress fields postulated. The naturally arising boundary terms are found to relate the warping deformations to the internal force fields. A torsion beam example is solved using the new theory in order to illustrate its applicability to practical problems. The second part of the paper implements the solution numerically in a force-based finite element context. Two finite elements are developed by assuming linear and hyperbolic bimoment fields. The FEA solutions are shown to provide lower bound representations of the stiffness when compared to those based on conventional beam theories founded on postulated kinematic assumptions.

Reinforced Concrete Frame Element with Bond Interfaces. I: Displacement-Based, Force-Based, and Mixed Formulations

This is the first of two papers that presents the theory and applications of three different formulations of reinforced concrete frame elements with bond slip in the reinforcing bars. This paper presents the governing differential equations of the problem (strong form) and the three different element formulations (weak forms). The first is the displacement-based formulation, which is derived from the principle of stationary potential energy. The second is the force-based formulation, which is derived from the principle of stationary complementary energy. The third is the two-field mixed formulation, which is derived from the principle of stationary Hellinger-Reissner potential. The selection of the displacement and force shape functions for the different formulations is discussed. Tonti’s diagrams are used to conveniently represent the equations that govern both the strong and the weak forms of the problem. This paper derives the general matrix equations of the three formulations. Implementation of the formulations in a general-purpose nonlinear structural analysis software and a set of applications are discussed in the companion paper.

Complementary energy theorem in geometrically non-linear structural problems

The principle of stationary complementary energy for geometrically non-linear problems of beams and trusses is considered. The functional of complementary energy is formulated in terms of force quantities. The stationary conditions describe the conditions of compatibility of the structure. The applications include the problem of elastica, a truss, and a bending problem by use of beam elements.

On the principle of stationary complementary energy in finite elastostatics

In this work, we explore conditions sufficient to guarantee the existence of a principle of stationary complementary energy in finite elastostatics. We are then able to derive a principle of complementary energy. The results also apply to a large class of incompressible and almost incompressible materials.

A finite element model for plane elasticity problems using the complementary energy theorem

AbstractA finite element stress analysis capability for plane elasticity problems, employing the principle of stationary complementary energy, is developed. Two models are investigated. The first is a 24 d.o.f. rectangular finite element. The second model consists of an 18 d.o.f. triangular element. In order to allow for self‐equilibrating stresses which are continuous within the element, the well‐known Airy stress function ø is used. The function ø is represented by means of quintic Hermitian polynomials within the finite element. The values of the ø function and its derivatives up to order two are used as nodal parameters. For matching the stress function with the prescribed boundary tractions, additional equations are developed considering the force and moment equilibrium equations on the boundary consistent with the assumed stress function. These additional boundary equations are incorporated into the system equations using the Lagrangian multiplier technique. Excellent results are obtained for linear elastic problems even with coarse finite element discretization. Some examples of plane elasticity problems are solved and results compared.

A note on variational theorems in non-linear elastostatics

In a recent paper Koiter (5) discussed a principle of stationary complementary energy for the finite deformation of elastic materials. The complementary energy functional he uses depends only on the components of the nominal stress, and not on the displacement field. He, incorrectly, attributes the principle to Zubov (14).

On the Principle of Stationary Complementary Energy in the Nonlinear Theory of Elasticity

Until recently the famous principle of minimum complementary energy of the classical linear theory of elasticity has defied all attempts at an effective generalization to the nonlinear case where the strain-displacement relations are no longer linear. L. M. Zubov [4] has now taken a significant step towards the solution of this problem in the form of his principle of stationary complementary energy, expressed in terms of the Piola stress tensor. Zubov’s argument, however, is open to criticism at several points. The present paper aims at the removal of these weak points and at a simpler derivation of Zubov’s theorem. The theory is illustrated by some simple examples.

Bending of circular sandwich plates due to asymmetric temperature distribution

I. Introduction T HERMOELASTIC behavior of sandwich plates has been considered by many investigators. However, most of these investigations pertain to the analysis of rectangular plates. Thermal deflection of circular sandwich plates was investigated by Bruun 1 and Huang and Ebcioglu,2 but with the restriction of rotational symmetry. Nonrotationally symmetrical bending of circular sandwich plates was considered by the author3; however, no thermal effect was included in the analysis. In this paper the thermoelastic equations for the bending of circular sandwich plates subjected to arbitrary temperature distribution are derived by means of the principle of stationary complementary energy. To illustrate the use of these equations an example, a clamped circular sandwich plate subjected to arbitrary temperature distribution which is expressible in the form of a double infinite series, is presented. The assumptions made in the presented analysis are as follows. 1) Two faces are isotropic but may be different in material, thickness, and temperature. The Poisson's ratios of the faces are the same. Bending rigidities of the individual faces may be neglected. 2) The core is considered to be isotropic and is assumed to take only the transverse shear stress. 3) Small deflection theory is considered.

Note on the principle of stationary complementary energy applied to free vibration of an elastic body

A free vibration problem of an elastic body is treated in the present paper from the viewpoint of variational formulations. The principle of stationary potential energy is written for the problem. Then, it is generalized through the use of Lagrange's undetermined multipliers and the principle of stationary complementary energy is shown to be derivable from the generalized principle.

Complementary Energy Method for Finite Deformations

A principle of stationary complementary energy is developed for statically loaded structures with finite displacements, rotations, and strains, made of material with linear or nonlinear elasticity. The principle is based on a vectorial definition of complementary work analogous to the vectorial definition of ordinary work. It is used to solve two illustrative truss problems, including one in which even the infinitesimal deformations cannot be determined by the conventional complementary energy method. The principle is shown to include the conventional complementary energy theorem for infinitesimal deformations as a special case.