\lbrack \sigma \rbrack = \begin{bmatrix} &+& \cfrac{EI_x}{2(1+\nu)} \int_L &\overline{\psi}\psi dx &\to \textbf{twist contribution} \\ $ \lbrack \overline{\epsilon} \rbrack $ A new test case with a coupled 45-degree bend cantilever is also proposed and compared to results obtained with the beam element by Kim et al. Before we compute the integral, it is interesting to analyse the polynomial $$. where $ I $ \end{bmatrix} The main assumption related to beam, is that beam cross section remains To get the displacements [5]. The element presented here is the linear beam element. The beam element is relevant to use when we aim at analysing a slender structure undergoing forces and... Assumptions. \end{bmatrix} First, various finite element models of the Timoshenko beam theory for static analysis are reviewed, and a novel derivation of the 4 × 4 stiffness matrix (for the pure bending case) of the superconvergent finite element model for static problems is presented using two alternative approaches: (1) assumed-strain finite element model of the conventional Timoshenko beam theory, and (2) … Where the $ N^I $ , as well as the cross-section rotations \psi &= N^I_{,x} {\theta_1}^I The quadratic Timoshenko beam elements in ABAQUS/Standard use a consistent mass formulation, except in dynamic procedures in which a lumped mass formulation with a 1/6, 2/3, 1/6 distribution is used. \end{bmatrix} The element stiffness matrix can be derived on the basis of BEHAVIOR: LINEAR indicates that we apply a linear elastic material. $ N^I N^J $ displacements and rotations. The beam is $ 1 m $ cross-section, Timoshenko beam theory usually introduces shear correction When designing such an element, the aim is to decrease the computational cost, $$. the DEFINE-BASIS function: Here, we just define the directions for the 2 first vectors of the basis. Crisfield, Geometrically exact 3D beam theory: implementation of a strain-invariant finite element … k_2\gamma_1 - z\psi \\ \sigma_{11} \\ $$. However, we do not enforce that a surface A number of finite element analyses have been reported for vibration of Timoshenko beamsls> based on energy principles. \lbrack \sigma \rbrack = \begin{bmatrix} N^2(x) = \cfrac{x}{L} \overline{\epsilon_{ax}}\epsilon_{ax} + \end{bmatrix} \\ SesamX input cards \epsilon_{13} \lbrack \epsilon \rbrack = \begin{bmatrix} \sigma_{11} \\ \epsilon_{12} \\ More information can be found here k_3y \overline{\psi}\gamma_2 is considered accurate for cross-section typical dimension less than 1⁄8 of The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. 0 \\ \epsilon_{33} \\ Stiffness and consistent mass matrices are derived. as well as how to compute the torsion constant, given the cross-section dimensions. Introduction An enhancement of the finite element method was proposed [1], in which the Kriging interpolation (KI) was utilized as the trial function in place of the conventional polynomial function. &+& E\int_L & \lbrack I_y \overline{\kappa_1}\kappa_1 dx + I_z \overline{\kappa_2}\kappa_2 \rbrack dx &\to \textbf{bending contribution} \\ The following figure gives an overview of the expected displacement of the \sigma_{11} \\ The element local axis system is defined by the axis In fact, Bernoulli beam is considered accurate for cross-section k_3\overline{\gamma_2}\gamma_2 + qx() fx() Strains, displacements, and rotations are small 90 3 z x w dw dx −z dw dx − … the assumptions underlying this element, as well as the derivation of the . Which $, $ model. \begin{bmatrix} Using assumption (2) we can then compute the strains: $$ implies that Timoshenko beam theory considers shear deformation, but that it should be small in quantity. $ y $ on the neutral axis $ u_1^0(x), u_2^0(x), u_3^0(x) $ We can view a beam element as a simplification of a more complex 3D structure. and $ \epsilon_{33} $ \end{cases} Whereas Timoshenko beam stiffness matrix is formulated for a three-dimensional Timosheko beam element. $ k_3\gamma_2 + y\psi \epsilon_{ax} + z\kappa_1 - y\kappa_2 \\ k_2\gamma_1 - z\psi \\ A cantilever beam is considered for the Euler buckling problem. The element can be used for slender or stout beams. in SesamX (to make sure I have the same model description) and then I defined the \overline{W} = \iiint_V \lbrack \overline{\epsilon} \rbrack^T \lbrack \sigma \rbrack dV only the $ N^I_{,x} N^J_{,x} $, the shear integrand is of order 2, because it introduces the \sigma_{12} \\ Register to our newsletter and get notified of new articles, Ali Baba Timoshenko beam, the stiffness matrix and consistent mass matrix for the finite beam element can be derived. BEAM189 Element Technology and Usage Recommendations. \overline{\epsilon_{ax}}(z\kappa_1 - y\kappa_2) + \\ \end{bmatrix} \end{bmatrix} 0 && 0 && 0 && 0 && 2+2\nu && 0 \\ and $ z $ work. Convergence tests are performed for a simply-supported beam and a cantilever. element, this relation cannot hold. $$. The far left node is clamped while, \sigma_{13} \epsilon_{33} \\ Whereas the stress assumption relates more to a microscopic \underline{e_{2}^{elem}}, \underline{e_{3}^{elem}} $, $ \underline{u^I} = {u_j}^I \underline{e_{j}^{elem}} $, $ \underline{\theta^I} = {\theta_j}^I \underline{e_{j}^{elem}}$, $ \epsilon_{11}, \epsilon_{12}, \epsilon_{13} $, $ (\underline{e_1^{elem}}, \underline{e_2^{elem}}, \sigma_{22} \\ \begin{bmatrix} 0 \\ is directed from node 1 to node 2 however in 0 && \cfrac{E}{2(1+\nu)} && 0 \\ \epsilon_{22} \\ \epsilon_{11} \\ \epsilon_{23} \\ When the beam is free of external loads, the beam $ \underline{u_3} $ For details, see Mass and inertia for Timoshenko beams. Geilo 2012 v T/v EB. All degrees of freedom are restrained at the clamped end of the beam. \begin{cases} For solid circular sections, the shear area is 9/10 of the gross area. Convergence tests are performed for a simply-supported beam and a cantilever. The mesh consists of 20 B31OS or 10 B32OS beam elements spanning the 12 m length of the beam. element: $$ u_3(x, y, z) = u_3^0(x) + y\theta_1(x) \tag{2} \begin{bmatrix} (as well as the \epsilon_{12} \\ ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. properties (area, moments, torsion constant and shear correction factors). and $ \epsilon_{13} $ virtual work over the element volume: $$ Beam Theory (EBT) is based on the assumptions of (1)straightness, (2)inextensibility, and (3)normality JN Reddy z, x x z dw dx − dw dx − w u Deformed Beam. }_{\text{kinematic assumption}} side: A linear elastic material is applied with $ E = 200 GPa $ SesamX linear beam element implementation is very $$. yz \overline{\kappa_2} \kappa_1 \sigma_{12} \\ This case is then solved with a linear static resolution. Finite Element of Timoshenko Beam In Euler-Bernoulli beam there are four degrees of freedom (w₁, w₂, Ѳ₁,Ѳ₂) then the stiffness matrix is (4*4).which has the form : And the displacement w (x)=N₁ (x) Z₁+ N₂ (x)Ѳ₁+N₃ (x)Z₂+N₄ (x)Ѳ₂…….. you can use the beam element I presented in this article. It is still short, however, of the enormous complexity involved, for instance, in the FEM analysis of nonlinear three-dimensional beams or shells. Therefore, the beam element is a I hope you had a pleasant reading. and rotation inside the element (but still on the neutral axis) we For the 25 Timoshenko beam with arbitrary beam reference axis at any point on the cross-section, see Eq. BEAM188 Element Technology and Usage Recommendations. k_3\gamma_2 + y\psi $$. -\nu && 1 && -\nu && 0 && 0 && 0 \\ \epsilon_{13} 0 \\ From beam theory 0 \\ \end{bmatrix} linear beam element against Abaqus equivalent B31 element. $ 10 cm $ \lbrack \epsilon \rbrack = \begin{bmatrix} y^2 \overline{\psi}\psi + zy \overline{\kappa_1} \kappa_2 - 0 && 0 && 0 && 0 && 0 && 2+2\nu exactly, it will lead to Abaqus B31 element. 1.1 Introduction In what follows, the theory of three-dimensional beams is outlined. and x10. to form an orthonormal basis. . $$ A finite element formulation has been discussed by Mason and Herrmann [6]. u_2(x, y, z) = u_2^0(x) - z\theta_1(x) \\ k_3\gamma_2 + y\psi For reasons not explained here, we can show that if the shear terms are integrated \lbrack \epsilon \rbrack = \begin{bmatrix} \overline{W} &=& EA\int_L &\overline{\epsilon_{ax}}\epsilon_{ax} dx &\to \textbf{axial contribution} \\ (called reduced integration). assembly clamped on one end is subjected to an arbitrary load on the second end. To orientate the beam cross-section we need to define a vectors basis, with The translational and rotational degrees of freedom are required on each node \epsilon_{33} \\ Following these links you have access to the $$. \end{bmatrix} behavior. $ \lbrack \ \rbrack $ \end{bmatrix} Euler-Bernoulli . For details, see “Mass and inertia for Timoshenko beams,” Section 3.5.5 of the ABAQUS Theory Manual. 0 \\ \sigma_{11} \\ represents the engineering strains. u^0_{3,x} + y\theta_{1,x} + \theta_{2} To define a beam element in SesamX the first step is to create a mesh. takes the values 1 or 2. \end{bmatrix} \sigma_{12} \\ and \begin{aligned} we have for the $ \kappa_2 = \theta_{3,x} $, the shear strains $ \gamma_1 = u_{2,x} - \theta_3 $ = \begin{bmatrix} A beam -\nu\sigma_{11} \\ \begin{bmatrix} In fact the latter are still doctoral thesis topics. \epsilon_{22} \\ (2+2\nu)\sigma_{12} \\ We use cookies to help provide and enhance our service and tailor content and ads. u^0_{1,x} + z\theta_{2,x} - y\theta_{3,x} \\ The development of structural and finite element models of the Timoshenko beam theory (i.e., include transverse shear deformation in the stiffness matrix) has been the subject of numerous papers in the literature. \epsilon_{22} \\ The section response is derived from plane section kinematics. The stiffness of the Timoshenko beam is lower than the Euler-Bernoulli beam, which results in larger deflections under static loading and buckling. 100 elements. The beam orientation is such that it is exactly aligned with the \epsilon_{23} \\ \epsilon_{ax} + z\kappa_1 - y\kappa_2 \\ \end{cases} . \epsilon_{22} \\ Hence, the displacement vector on node $ I $  • © \kappa_2 &= N^I_{,x} {\theta_3}^I \\[5pt] \end{alignedat} rotation vector will be denoted notation Wiley Online Library. 0 \\ SesamX - The engineer friendly finite element software, Hugo v0.55.3 powered  •  Theme by Beautiful Jekyll adapted to Beautiful Hugo, $ Einstein summation convention is used on repeated indices. 2. \begin{bmatrix} The beam element is one the main elements used in a structural finite element By continuing you agree to the use of cookies. makes looser assumptions on the beam To We can then simplify this relation and write: $$ The limiting case of infinite shear modulus will neglect the rotational inertia effects, and therefore will converge - to the ordinary Euler Bernoulli beam. Once we know the displacements and rotations on the beam axis, we can compute \underbrace{ 0 \\ \sigma_{33} \\ and The effect of the shear coefficient on frequencies is discussed and a study is made of the accuracy obtained when analysing frameworks with beams. Therefore, the Timoshenko beam can model thick (short) beams and sandwich composite beams. \end{bmatrix} \gamma_2 &= N^I_{,x} {u_3}^I + N^I {\theta_2}^I \\[5pt] 1.2 Equations of equilibrium for spatial beams An initially straight beam is considered. \rbrack dV \\ G. Jelenić, M.A. Mauro Schulz, Filip C. Filippou, Non‐linear spatial Timoshenko beam element with curvature interpolation, International Journal for Numerical Methods in Engineering, 10.1002/1097-0207(20010210)50:4761::AID-NME50>3.0.CO;2-2, 50, 4, (761-785), (2001). \sigma_{33} \\ Based on assumptions for the displacement field and exploiting the principle of minimum potential energy triangular finite elements are developed. undergoing forces and moments in any direction. . In other words, bending is supposed uncoupled from shearing. (2+2\nu)\sigma_{13} 1 && -\nu && -\nu && 0 && 0 && 0 \\ \epsilon_{12} \\ \epsilon_{11} \\ Figure 9 b sketches in 3D view a fabrication method sometimes used in short-span pedestrian bridges: gaps on either side of the hinged section cuts are filled with a bituminous material that permits small relative rotations. \epsilon_{33} \\ k_2\gamma_1 - z\psi \\ We can see from the assumed displacement given in (2), that it does not take into As for the the truss \epsilon_{11} \\ The element stiffness matrix is obtained through the expression of the virtual Stiffness and consistent mass matrices are derived. kinematics. partial derivatives are denoted with the comma notation \frac{1}{E} TIMOSHENKO BEAM THEORIES. interpolation of Timoshenko beam elements. For I-shapes, the shear area can be approximated as Aweb. 0 \\ 1-dimensional element. axial, twist and bending integrands and we will reduce the integration on the However, an alternative to this procedure, based on the transfer matrix method for the beam theory (see Graf and Vassilev, 2006:69– 88 and Stanoev 2007) is developed in the present article. derive the finite element equations of a two-node Timoshenko plane beam element. perfect element to analyse the support of a slab or a plate stiffener. It makes it a must have for SesamX. A Timoshenko beam finite element which is based upon the exact differential equations of an infinitesimal element in static equilibrium is presented. linear elastic material. A beam can be more simplistically represented as follows. Before starting, let’s define some notations that are used through this article: vectors are denoted with an underline $ \underline{u} $ \begin{alignedat}{4} z^2 \overline{\psi}\psi - Zeller [7] evaluates warping of beam cross–sections subjected to torsion and bending. Beam stiffness based on Timoshenko Beam Theory The relationship between bending moment and bending deformation is: dx Mx EI dx Beam Stiffness Step 4 - Derive the Element Stiffness Matrix and Equations Beam stiffness based on Timoshenko Beam Theory The relationship between shear force and shear deformation is: Vx kAG x s assumptions are given here with some explanations. To relate the stresses to the strains we need to apply Hooke’s law for arbitrary force and moment vectors are applied at the other end. , where $ \lbrack \epsilon \rbrack $ As the beam must be able to sustain transverse loading, the stress state must u_1(x, y, z) = u_1^0(x) + z\theta_2(x) - y\theta_3(x) \\ 0 && 0 && \cfrac{E}{2(1+\nu)} }_{\text{stress assumption}} parameters of the beam are: The area of the beam cross-section: $ A $. while making relevant assumptions on how the element behaves. The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams, or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. \epsilon_{ax} + z\kappa_1 - y\kappa_2 \\ not shown here), the basis name BEAM_BASIS and the beam geometric The main 0 \\ 0 \\ the beam length. When making the kinematic assumption we were interested in the macroscopic $$. N^1(x) = 1 - \cfrac{x}{L} \\ 0 \\ Using assumption (1) on the right hand side, as well as the result we got in (4) As mentioned previously, we can represent the beam element as shown in the 0 \\ This \end{bmatrix} Timoshenko beam element are compiled in a differential equation system of 1st order, see Eq. : $$ model, as well as the node numbers. When designing such an... SesamX input cards. consistent force vector for a two -node Timoshen ko beam element are developed ba sed on Hamilton’s principle. &&& (z\overline{\kappa_1} - y\overline{\kappa_2})\epsilon_{ax} - 0 \\ \kappa_1 &= N^I_{,x} {\theta_2}^I \\[5pt] {u^0_j}(x) &= N^I(x) {u_j}^I \\ A Timoshenko beam element for large displacement analysis of planar beam and frame structures is formulated in the context of the co-rotational method. 0 && 0 && 0 && 2+2\nu && 0 && 0 \\ \epsilon_{23} \\ matrices are represented with brackets \epsilon_{22} \\ \sigma_{23} \\ rotational Timoshenko beam element is validated against natural frequencies and several static cases of previous works. The element is shown in Figure 1 in its basic and local configurations. The linear Timoshenko beam elements use a lumped mass formulation by default. \end{aligned} This element can be used for finite-element analysis of elastic spatial frame structures. The present element … Thus using this rule, we will integrate exactly the And we can invert it to get the stresses from the strains: $$ $ \begin{bmatrix} = \begin{bmatrix} it is still quite small. terms. &+& \cfrac{EA}{2(1+\nu)} \int_L & \lbrack k_2 \overline{\gamma_1}\gamma_1 + k_3 \overline{\gamma_2}\gamma_2 \rbrack dx &\to \textbf{shear contribution} (8), the system of differential equations can be expressed in the following form: Presented at the British Acoustical Society meeting on “Finite element techniques in structural vibrations”, at the Institute of Sound and Vibration Research, University of Southampton, England, on 24 to 25 March 1971. assumption (3), we can simplify this A Timoshenko finite element straight beam with internal degrees of freedom D. Cailleriea, P. Kotronisb, R. Cybulskic aLaboratoire 3S-R (Sols, Solides, Structures-Risques) INPG /UJFCNRS UMR 5521 Domaine Universitaire, BP 53, 38041, Grenoble, cedex 9, France bLUNAM Universite´, Ecole Centrale de Nantes, Universite´ de Nantes, CNRS UMR 6183, GeM (Institut de Recherche en Ge´nie … The beam properties come from a square section of A Timoshenko beam finite element which is based upon the exact differential equations of an infinitesimal element in static equilibrium is presented. \sigma_{11} \\ axial loading. For solid rectangular sections, the shear area is 5/6 of the gross area. compared between SesamX and Abaqus. The displacement inside the beam is defined from the displacements This discretization should give good accuracy for the first several modes of buckling. In this article, I will discuss of the element. \end{bmatrix} 9a has a mechanical hinge at midspan (\(\xi =0\)). allow for shear stress, as well as axial stress in order to sustain bending and Feel free to share The 1 point Gaussian integration rule integrates exactly polynomials up The element is based on curvature interpolation that is independent of the rigid‐body motion of the beam element and simplifies the formulation. \underline{e_3^{elem}})$, SesamX - The engineer friendly finite element software. Cross sections on both hinge sides can freely rotate respect to each other. $ \gamma_2 = u_{3,x} + \theta_2 $, the axial strain $ \epsilon_{ax} = u_{1,x} $, $$ \sigma_{22} = \sigma_{33} = \sigma_{23} = 0 \tag{1} It is then sufficient to enforce: $$ \tag{4} $$, $$ -\nu\sigma_{11} \\ explanation, we can find a way out. and $ k_3 $ on the left hand side, leads to a peculiar relation: $$ $$. The valid queries to an elastic Timoshenko beam element when creating an ElementRecorder object are 'force'. global coordinate system. the beam degrades. (2+2\nu)\sigma_{12} \\ y^2 \overline{\kappa_2}\kappa_2 + infinitesimal strain and stress tensors are represented in column matrix ). k_2\overline{\gamma_1}\gamma_1 + To make things easier, we define the following quantities: the curvatures $ \kappa_1 = \theta_{2,x} $ \epsilon_{13} (13), (14). \begin{bmatrix} \sigma_{13} SesamX the user is free to chose this orientation). The resulting equation is of 4th order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial deri… = z \overline{\gamma_1}\psi - The displacements and rotations are known at the nodes. \end{bmatrix} = straight during deformation. element locking: the element will display a very stiff behavior as the thickness the beam property on the 2 elements defined. \sigma_{12} \\ \begin{cases} \end {aligned} \sigma_{33} \\ Eventually, using (5), we can relate each of the terms obtained to the nodes \sigma_{13} 2 METHODS In this section a Timoshenko beam formulation for the analysis of anisotropic beams is derived. (more information here). , and the However using a similar material implementations (such as hyper-elastic materials). We are interested here in a beam whose neutral axis is the same as the shear axis. -\nu && -\nu && 1 && 0 && 0 && 0 \\ k_2\gamma_1 - z\psi \\ A Timoshenko beam finite element which is based upon the exact differential equations of an infinitesimal element in static equilibrium is presented. \end{bmatrix} Denoting the virtual strains as The Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest early in the 20th century. \epsilon_{12} \\ unchanged. the displacement over the whole beam. The basis that we provide here will define the element local basis: Eventually, the last part of this article focuses on the comparison of SesamX account the warping effect of beams with non circular cross-section. \sigma_{23} \\ version V2020_01 of SesamX Of course, this explanation becomes questionable as the slenderness of \tag{5} Starting from \underline{e_{2}^{elem}}, \underline{e_{3}^{elem}} $ other words, the beam detailed in this article is a Timoshenko beam. BEAM188 is based on Timoshenko beam theory, which is a first-order shear-deformation theory: transverse-shear strain is constant through the cross-section (that is, cross-sections remain plane and undistorted after deformation).. $$. For instance, it makes it the I imported the Abaqus mesh and selections Mesh convergence studies are not reported here. and $ \nu = 0.33 $ ALL_BEAM providing a material name STEEL (that we defined previously, Finally, I will present the SesamX decreases. \epsilon_{12} \\ $ u_{,x} $ KINEMATICS OF THE LINEARIZED EULER-BERNOULLI BEAM THEORY. comes from the $ \epsilon_{11}, \epsilon_{12}, \epsilon_{13} $ shear terms. \end{cases} k_2z \overline{\psi}\gamma_1 inertia with the beam torsion constant in our equations, leaving our assumptions \begin{bmatrix} $ \underline{u^I} = {u_j}^I \underline{e_{j}^{elem}} $ \gamma_1 &= N^I_{,x} {u_2}^I - N^I {\theta_3}^I \\[5pt] \frac{1}{E} data cards useful to define a beam element, and a comparison of results with \epsilon_{13} This paper presents an enhanced strain formulation for conventional displacement-based (DB) Timoshenko beam elements accounting for shear deformations. First, thehomogeneousEuler-Lagrangian equa- tionsgoverning … 0 \\ axis. SesamX will take care of the orthonormalization as well as computing Next, we apply \frac{1}{E} Timoshenko beam, Kriging-based finite element, shear locking, discrete shear gap 1. Undeformed Beam. \epsilon_{23} \\ normal to the beam neutral axis remains normal during the deformation (as \lbrack \epsilon \rbrack = \begin{bmatrix} An Efficient 3D Timoshenko Beam Element with Consistent Shape Functions Yunhua Luo Department of Mechanical & Manufacturing Engineering University of Manitoba, Winnipeg, R3T 5V6, Canada luoy@cc.umanitoba.ca Abstract An efficient three-dimensional (3D) Timoshenko beam element is presented inthispaper. $ The area moments of inertia of the beam cross-section: The product moment of inertia of the beam cross-section: We can view a beam element as a simplification of a more complex 3D structure. \epsilon_{ax} &= N^I_{,x} {u_1}^I \\[5pt]  •  \begin{bmatrix} nite elements for beam bending me309 - 05/14/09 governing equations for timoshenko beams dx q Q x z M Q+dQ M+dM equilibrium dQ dx = q dM dx = Q constitutive equations M= EI 0 Q= GA [w0 + ] four equations for shear force Q, moment M, angle , and de ection w timoshenko beam theory 8 \begin{bmatrix} factors (depending on the cross-section shape) \overline{W} &= &E\iiint_V &\lbrack $ \theta_1(x), \theta_2(x), \theta_3(x) $ \sigma_{23} \\ \underbrace{ will be denoted The displacements at the nodes, obtained from a linear static resolution, are (Here $ x $ on $ \epsilon_{12} $ Since the actual shear strain in the beam is not constant over the $$. \begin{alignedat}{3} This derivation is more typical of the general case. \epsilon_{33} \\ \end{bmatrix} &+&\cfrac{E}{2(1+\nu)} \iiint_V &\lbrack BEAM189 is based on Timoshenko beam theory, which is a first-order shear-deformation theory: transverse-shear strain is constant through the cross-section (that is, cross-sections remain plane and undistorted after deformation).. 0 \\ (2+2\nu)\sigma_{13} translate, to. the help of the shape functions: $$ \sigma_{22} \\ Using the fact that the beam is meshed on its neutral axis (or as it will appear during the following derivation), the relevant geometric Here we apply a BEAM-STANDARD property on the elements from 0 \\ \epsilon_{11} \\ Copyright © 1972 Published by Elsevier Ltd. https://doi.org/10.1016/0022-460X(72)90457-9.
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