Bar Element
On the basis of the bar element, tension and compression as types of basic load cases will be described. First, the basic equations known from the strength of materials will be introduced. Subsequently the bar element will be introduced, according to the
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Bar Element
Abstract On the basis of the bar element, tension and compression as types of basic load cases will be described. First, the basic equations known from the strength of materials will be introduced. Subsequently the bar element will be introduced, according to the common definitions for load and deformation quantities, which are used in the handling of the FE method. The derivation of the stiffness matrix will be described in detail. Apart from the simple prismatic bar with constant cross-section and material properties also more general bars, where the size varies along the body axis will be analyzed in examples [1–9] and exercises.
3.1 Basic Description of the Bar Element In the simplest case, the bar element can be defined as a prismatic body with constant cross-sectional area A and constant modulus of elasticity E, which is loaded with a concentrated force F in the direction of the body axis (see Fig. 3.1). The unknown quantities are • the extension L and • the strain ε and stress σ of the bar dependent on the external load. The following three basic equations are known from the strength of materials: By ε(x) =
L du(x) = dx L
(3.1)
kinematics describes the relation between the strains ε(x) and the deformations u(x). By σ (x) = E ε(x) (3.2)
A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_3, © Springer-Verlag Berlin Heidelberg 2013
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3 Bar Element
Fig. 3.1 Tensile bar loaded by single force
the constitutive equation describes the relation between the stresses σ (x) and the strains ε(x) and the equilibrium condition results in σ (x) =
S(x) F S(x) = = . A(x) A A
(3.3)
The connection between the force F and the length variation L of the bar can easily be described with these three equations: L F = σ = Eε = E A L or with F=
EA L . L
(3.4)
(3.5)
The relation between force and length variation is described as axial stiffness. Hence, the following occurs for the bar regarding the tensile loading1 : EA F = . L L
(3.6)
For the derivation of the differential equation the force equilibrium at an infinitesimal small bar element has to be regarded (see Fig. 3.2). A continuously distributed line load q(x) acts as the load in the unit force per unit length.
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The parlance tension bar includes the load case compression.
3.1 Basic Description of the Bar Element
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Fig. 3.2 Force equilibrium at an infinitesimal small bar element
The force equilibrium in the direction of the body axis delivers: − S(x) + q(x)dx + S(x + dx) = 0.
(3.7)
After a series expansion of S(x + dx) = S(x) + dS(x) the following occurs − S(x) + q(x)dx + S(x) + dS(x) = 0
(3.8)
dS(x) = −q(x). dx
(3.9)
or in short:
Equations (3.1), (3.2) and (3.3) for kinematics, the constitutive equation and the equilibrium continue to apply. If the Eqs. (3.1) and (3.3) are inserted in (3.2), one obtains du(x) = S(x). (3.10) E A(x) dx After the differentiation and insertion of Eq. (3.9) one obtains d du(x) E A(x) + q(x) = 0 dx dx
(3.11)
as the differential equation for a bar wit
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