Torsion Bar
The basic load type torsion for a prismatic bar is described with the help of a torsion bar. First, the basic equations known from the strength of materials will be introduced. Subsequently, the torsion bar will be introduced, according to the common defi
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Torsion Bar
Abstract The basic load type torsion for a prismatic bar is described with the help of a torsion bar. First, the basic equations known from the strength of materials will be introduced. Subsequently, the torsion bar will be introduced, according to the common definitions for the torque and angle variables, which are used in the handling of the FE method. The explanations are limited to torsion bars with circular cross-section. The stiffness matrix will be derived according to the procedure for the tension bar [1–6].
4.1 Basic Description of the Torsion Bar In the simplest case, the torsion bar can be defined as a prismatic body with constant circular cross-section (outside radius R) and constant shear modulus G, which is loaded with a torsional moment M in the direction of the body axis. Figure 4.1a illustrates the torsion bar with applied load and Fig. 4.1b shows the free body diagram.
(a)
(b)
Fig. 4.1 Torsion bar a with applied load and b free body diagram
A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_4, © Springer-Verlag Berlin Heidelberg 2013
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4 Torsion Bar
The unknown quantities are • the rotation ϕ of the end cross-sections, • the rotation ϕ(x), the shear strain γ (x) and the shear stress τ (x) on a crosssection on the inside of the bar in dependence of the external load. The following three basic equations are known from strength of materials. The interrelationships of the kinematic state variables are shown in Fig. 4.2 under consideration of a cylindrical coordinate system (x, r , ϕ).1 Fig. 4.2 Torsion bar with state variables
Kinematics describes the relation between the shear strain and the change of angle: γ (x) =
du ϕ dϕ(x) =r . dx dx
(4.1)
The constitutive equation describes the relation between the shear stress and the shear strain with τ (x) = Gγ (x) . (4.2) The internal moment M(x) is calculated through M(x) =
r τ (x)d A ,
(4.3)
A
and with the kinematic relation from Eq. (4.1) and the constitutive equation from Eq. (4.2) the following results dϕ M(x) = G dx
r 2 d A = G Ip
dϕ . dx
(4.4)
A
Hereby the elastic behavior regarding the torsion can be described through Besides the shear strain γxϕ (r, x) and the deformation u ϕ (x, r ) no further deformation parameters occur during the torsion of circular cross-sections. For clarity reasons the indexing for clear dimensions is omitted.
1
4.1 Basic Description of the Torsion Bar
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M(x) dϕ(x) = . dx G Ip
(4.5)
On the basis of this equation the interrelation between the rotation ϕ of the two end cross-sections and the torsional moment M can be described easily: ϕ =
M L. G Ip
(4.6)
The expression GI p is called the torsional stiffness. The stiffness for the torsion bar results from the relation between the moment and the rotation of the end cross-section: G Ip M = . ϕ L
(4.7)
For the derivation of the differential equation the equilibrium at the infinitesimal small torsion bar element has to be regarded (see Fig. 4.3). A continuously distributed load m t (x
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