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A Beam—Just a Beam in Linear Plane Bending
Abstract
Starting from the equations of the linear, three-dimensional theory of elasticity, the displacements are expanded into power series in the width- and height-coordinates. By invoking the uniform-approximation method in combination with the pseudo-reduction technique, a hierarchy of beam theories of different orders of approximation is established. The first-order approximation coincides with the classical Euler-Bernoulli beam theory, whereas the second-order approximation delivers a Timoshenko-type of shear-deformable beam theory. Differences and implications are discussed.
... The sole assumption is the series expansion of the displacement field which is assumed to be C ∞ in transverse dimensions of the cross section (Section 4.1) or C 5 in transverse dimensions of the cross section (Sections 4.4 and 5). The aim is not here to give more simplified versions of these models which require more assumptions [19,20]. For beams with double symmetric cross sections, we show that the initial problems decouples into four subproblems and we study the coupling between these subproblems. ...
... e 3 = 0 (free traction). Now we show that with the unidimensional boundary conditions on the end cross sections of the beam (Σ 0 and Σ L ), we can obtain variational formulation of the problem (15), (17)- (20). For example suppose that the cross section Σ 0 is clamped and the cross section Σ L is subjected to an imposed torsor (R, M). ...
... For the determination of the anisotropic coupling it is necessary to express the unidimensional problem (15), (17)- (20) in terms of the displacement field. To this end, we take advantage of the smallness of the transverse dimension of the beam by taking complete series expansion in x (we assume that u is C ∞ in x ) ...
We present two new models for dynamic beams deduced from three dimensional theory of linear elasticity. The first model is deduced from virtual work considered for small beam sections. For the second model, we suppose a Taylor-Young expansion of the displacement field up to the fourth order in transverse dimensions of the beam. We consider the Fourier series expansion for considering Neumann lateral boundary conditions together with dynamical equations, we obtain a system of fifteen vector equations with the fifteen coefficients vector unknown of the displacement field. For beams with two fold symmetric cross sections commonly used (for example circular, square, rectangular, elliptical…), a unique decomposition of any three-dimensional loads is proposed and the symmetries of these loads is introduced. For these two theories, we show that the initial problem decouples into four subproblems. For an orthotropic material, these four subproblems are completely independent. For a monoclinic material, two subproblems are coupled and independent of the two other coupled subproblems. For the first model, we also give the detailed expression of these four subproblems when we consider the approximation of the displacement field used in the second model.
A novel reduced model is constructed for a linearized anisotropic rod with doubly symmetric cross-section. The derivation starts from the Taylor expansion of the displacement vector and the stress tensor. The goal is to establish rod equations for the leading order displacement and the twist angle of the mean line of the rod in an asymptotically consistent way. Fifteen vector differential equations are derived from the 3D (three-dimensional) governing system, and elaborate manipulations between these equations (including the Fourier series expansion of the lateral traction condition) lead to four scalar rod equations: two bending equations, one twisting equation, and one stretching equation. Also, recursive relations are established between the higher order coefficients and the lower order ones, which eliminate most of the unknowns. Six boundary conditions at each edge are obtained from the 3D virtual work principle, and 1D (one-dimensional) virtual work principle is also developed. The rod model has three features: it adopts no ad hoc assumptions for the displacement form and the scalings of the external loadings; it incorporates the bending, twisting, and stretching effects in one uniform framework; and it satisfies the 3D governing system in a point-wise manner.
The consistent approximation approach is a dimension reduction technique for the derivation of hierarchies of lower-dimensional analytical theories for thin structural members from the three-dimensional theory of elasticity. It is based on the uniform truncation of the potential energy after a certain power of a geometric scaling factor that describes the relative thinness of the structural member. The truncation power defines the approximation order N of the resulting theory.
In the contribution, we deal with the exemplary case of a slender, beam-like continuum with rectangular cross-section. We introduce an extension of the consistent approximation approach towards a simultaneous, uniform truncation of the complementary energy. We show that compatible displacement boundary conditions can be derived from the Euler-Lagrange equations of the complementary energy, whereas, field equations and stress boundary conditions follow from the Euler-Lagrange equations of the potential energy. Thus fully defined boundary value problems are obtained for all orders of approximation N.
We provide an a priori-error estimate for the systematic error of the Nth-order theory. It states that the norm difference of the solution of the so derived boundary value problem of order N to the exact solution of the three-dimensional theory of elasticity declines like the geometric scaling factor to the power of N+1. Thus the approximation character of the approach is proved.
Furthermore, we outline why the approach is to prefer over the - in engineering mostly used - approach of a fixed displacement field ansatz with unknown coefficients. We show that the later approach leads in general to theories of higher complexity without increasing the approximation accuracy compared to a consistent approximation.
By combining the uniform-approximation technique with the pseudo-reduction approach, a consistent second-order plate theory is developed with recourse to neither kinematical assumptions nor to shear-correction factors by truncation of the elastic energy. The governing partial differential equations and the expressions for the stress resultants are compared with those of other authors. Free coefficients of the resulting displacement “ansatz” are determined a posteriori, in order to satisfy three-dimensional boundary conditions and local equilibrium equations.
The uniform-approximation approach is an a-priori assumption free structured approach for the derivation of hierarchies of lower-dimensional theories for thin structures with increasing approximation accuracy. In this publication, we derive a second-order consistent plate theory for monoclinic material and investigate several theories that arise from the original theory by a pseudo-reduction approach which aims to reduce the number of PDEs that are to solve. A one-variable model that governs only the interior solution is presented and, in addition, an extended two-variable model that also covers edge effects. Since the second introduced variable is a rotation of a vector field, we have to uniquely identify the rotation dependent parts in general gradients of the vector field, which is resolved by the introduction of an orthogonal decomposition. The final two-variable model is equivalent to the Reissner–Mindlin theory for the special case of isotropic material, whereas the one-variable model is equivalent to the first Reissner PDE. In contrast to this special case, the two-variable model is a coupled system of two PDEs for general monoclinic material.
The "consistent approximation" technique is a method for the derivation of analytical theories for thin structures from the settled three-dimensional theory of elasticity. The method was successfully applied for the derivation of refined plate theories for isotropic and anisotropic plates. The approach relies on computing the Euler-Lagrange equations of a truncated series expansion of the potential energy. In this thesis we extend the approach given in Kienzler (2002) towards the simultaneous truncation of a series expansion of the dual energy. The computation of the Euler-Lagrange equations of the truncated series expansion of the dual energy ensures a rigorous derivation of compatible boundary conditions. The series expansions of both energies are gained by Taylor-series expansions of the displacement field. We show that the decaying behavior of the energy summands is initially dominated by characteristic parameters that describe the relative thinness of the structure. Consequently, the energy series are truncated with respect to the power of the characteristic parameters. For the case of a homogeneous, one-dimensional structural member with rectangular cross-section we proof an a-priori error estimate that provides the mathematical justification for this method. The estimate implies the convergence of the solution of the truncated one-dimensional problem towards the exact solution of three-dimensional elasticity as the thickness goes to zero. Furthermore, the error of the Nth-order one-dimensional theory solution decreases like the (N 1)th-power of the characteristic parameter, so that a considerable gain of accuracy could be expected for higher-order theories, if the structure under consideration is sufficiently thin. The untruncated one-dimensional problem is equivalent to the three-dimensional problem of linear elasticity. We prove that the problem decouples into four independent subproblems for isotropic material: a rod-, a shaft- and two orthogonal beam-problems. A unique decomposition of any three-dimensional load case with respect to the direction and the symmetries of the load is introduced. It allows us to identify each part of the decomposition as a driving force for one of the four (exact) one-dimensional subproblems. Furthermore, we show how the coupling behavior of the four subproblems can be derived directly from the sparsity scheme of the stiffness tensor for general anisotropic materials. Since all propositions are proved for the exact one-dimensional problem, they also hold for any approximative Nth-order theory. The approach is applied to derive a new second-order beam theory for isotropic material free of a-priori assumptions, which in particular does not require a shear-correction. The theory is in general incompatible with the Timoshenko beam theory, since it contains three in general independent load resultants, whereas Timoshenko's theory only contains one. Furthermore, Timoshenko's theory ignores any effects in width direction. However, the assumption of a simple load case allows for a vis-a-vis comparison of both differential equations and in turn, two shear-correction factors for the use in Timoshenko's theory can be derived.
We prove the existence of an exact one-dimensional representation of the three-dimensional theory of linear elasticity for a structural member with constant, two-fold symmetric cross-section, without the use of any a-priori assumptions. We show that the general problem of three-dimensional elasticity decouples into four independent one-dimensional subproblems for isotropic material: a rod-, a shaft- and two orthogonal beam-problems. A unique decomposition of any three-dimensional load case with respect to the direction and the symmetries of the load is introduced. It allows us to identify each part of the decomposition as a driving force for one of the four one-dimensional subproblems. Furthermore, we show how the coupling behavior of the four subproblems can be derived directly from the sparsity scheme of the stiffness tensor for general anisotropic materials.
For homogeneous plates, the highest order term of transverse shear and normal stresses is of second order in thickness. To take this effect into account, we show that the thickness-wise expansion of the potential energy must be truncated at least from fifth order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zeroth-order displacement field. These lead to an analytical expression for two-dimensional potential energy in terms of the zeroth-order displacement field and its derivatives, which include non-standard shearing and transverse normal energies and coupled stretching-shearing, bending-shearing and stretching-transverse normal energies. As a consequence, this potential energy satisfies the stability condition of Legendre-Hadamard, which is necessary for the existence of a minimizer.
In this paper, the uniform-approximation technique in combination with the pseudo-reduction technique is applied in order to derive consistent theories for isotropic and anisotro-pic plates. The approach is used to assess and validate the plate theories already established in the literature. Further lines of research are indicated.
By Fourier‐series expansion in thickness direction of the plate with respect to a basis of scaled Legendre polynomials, several equivalent (and therefore exact) two‐dimensional formulations of the three‐dimensional boundary‐value problem of linear elasticity in weak formulation for a plate with constant thickness are derived. These formulations are sets of countably many PDEs, which are power series in the squared plate parameter. For the special case of a homogeneous monoclinic material, we obtain an approximative plate theory in finitely many PDEs and unknown variables by the truncation approach of the uniform‐approximation technique. The PDE system is reduced to a scalar PDE expressed in the mid‐plane displacement. The resulting second‐order theory, considered as a first‐order theory, is equivalent to the classical Kirchhoff theory for the special case of an isotropic material and equivalent to Huber's classical theory for an actual monoclinic material. However it remains shear‐rigid as a second‐order theory. Therefore, it is modified by an a‐priori assumption to a theory for monoclinic materials, that presumes the former equivalences, considered as a first‐order theory, but is in addition equivalent to Kienzler's theory as a second‐order theory for the special case of isotropy, which implies further equivalences to established shear‐deformable theories, especially the Reissner‐Mindlin theory and Zhilin's plate theory. The presented new second‐order plate theory for monoclinic materials is finally a system of two coupled PDEs of differentiation order six in two variables.
One way to develop theories for the elastic deformation of two- or one-dimensional structures (like, e.g., shells and beams)
under a given load is the uniform-approximation technique (see [2] for an introduction). This technique derives lower-dimensional
theories from the general three-dimensional boundary value problem of linear elasticity by the use of series-expansions. It
leads to a set of power series in one or two characteristic parameters, which are truncated after a given power, defining
the order of the approximating theory. Finally, a so-called pseudo reduction of the resulting PDE system in the unknown displacement
coefficients is performed, as the last step of the derivation of a consistent theory. The aim is to find a main differential
equation system (at best a single PDE) in a few main variables (at best only one) and a set of reduction differential equations,
which express all other unknown variables in terms of the variables of the main differential equation system, so that the
original PDE system is identically solved by inserting the reduction equations, if the main variables are a solution of the
main differential equation system. To find a valid pseudo reduction by inserting the PDEs of the original system into each
other is a complicated and very time-consuming task for higher-order theories. Therefore, an structured algorithm seeking
all possibilities of valid pseudo reductions (to a given number of PDEs in a given number of variables) is presented. The
key idea is to reduce the problem to finding a solution of a linear equation system, by treating each product of different
powers of characteristic parameters with the same variable as formally independent variables. To this end, all necessary equations,
which can be build from the original PDE system, have to be identified and added to the system a-priori.
The paper shows that the application of the asymptotic expansion method to the variational formulation of the three-dimensional linear elasticity model yields the usual plate models and gives the standard a priori assumptions at the same time. This approach is particularly amenable to the error analysis between the three-dimensional and two-dimensional solutions.
Models for plates and shells derived from three-dimensional linear elasticity, based on a thickness-wise expansion of the strain energy of a thin body, are described. These involve the small thickness explicitly and accommodate combined bending and stretching in a single framework. Physically motivated local constraints on the through-thickness variation of the displacement field, required for consistency with the exact theory, are introduced. When incorporated into the energy functional, these yield an expression for the two-dimensional strain energy density that includes non-standard two-dimensional strain gradient effects.
Applying the uniform-approximation technique, consistent plate theories of different orders are derived from the basic equations
of the three-dimensional linear theory of elasticity. The zeroth-order approximation allows only for rigid-body motions of
the plate. The first-order approximation is identical to the classical Poisson-Kirchhoff plate theory, whereas the second-order
approximation leads to a Reissner-type theory. The proposed analysis does not require any a priori assumptions regarding the
distribution of either displacements or stresses in thickness direction.
Ausgehend von den Grundgleichungen der linearen dreidimensionalen Elastizittstheorie werden Verformungen und Verzerrungen in eine Potenzreihe bezglich der Dickenkoordinate entwickelt. Whrend die Kinematik und das Elastizittsgesetz lokal erfllt werden, werden die Gleichgewichtsbedingungen durch Anwendung eines Integrationsmechanismus global erfllt. Es wird gezeigt, da eine konsequente quadratische Approximation zu konsistenten Schalengleichungen fhrt. Zustzliche Annahmen oder Vernachlssigungen sind nicht erforderlich. Als Beispiel wird die Kreiszylinderschale behandelt.Starting from the basic equations of the linear three-dimensional elasticity theory, displacements and strains are expanded in power series in terms of the thickness coordinate. While the kinematic equations and Hooke's law are satisfied locally, the equilibrium equations are satisfied globally by application of an integration procedure. It can be shown, that a consequent quadratic approximation leads to consistent shell equations without requiring any additional assumption. The circular cylindrical shell is treated as a typical example.
In his pioneering work, Kirchhoff [4] established the so-called classical plate theory. With a set of (partly self-contradictory) a priori assumptions and the concepts of ersatz-shear forces as well as corner forces, Kirchhoffs theory delivers accurate results for thin plates undergoing small deformations. In a series of papers, Reissner [ 10-12] extended the theory in considering effects of shear deformations and stresses acting transversely to the plate mid-surface. The refinement of plate theories in various directions is a subject of still ongoing research. Some of the latest achievements are collected in a special issue of ZAMP [ 14] dedicated to Eric Reissner.
The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a Γ-limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U → Rn, U ⊂ Rn. We show that the L2-distance of ∇v from a single rotation matrix is bounded by a multiple of the L2-distance from the group SO(n) of all rotations.
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