Problem 13: Hollow Circular Beam under a Twist Load

Model Description

Model location: <altair>\utility\mbd\nlfe\validationmanual\model1.mdl

This model consists of a straight cantilever beam of a hollow circular cross-section. The beam is subjected to a torsion load at its endpoint as shown in Figure 1 below.

Figure 1. A Beam of Hollow Circular Cross Section Fixed at One End and Subject to Torsion at the Other

The twist angle and the shear strain can be obtained from simple beam theory as:

Twist angle: $θ = \frac{T l}{G J}$
Shear strain: $γ = \frac{θ r}{l}$

Here,

$l$: Beam length
$r$: Beam radius.
$J$: Polar moment of inertia about the beam axis.
$G = \frac{E}{2 (1 + ν)}$: Shear modulus
$E$: Young's modulus
$ν$: Poisson ratio

Multibody Model

The cantilever beam is modeled using 10 NLFE beam elements in MotionSolve. NLFE beam elements with a hollow tube section are used by specifying an inner and outer radius. The cantilever beam is fixed to the ground by a fixed joint. Gravity is turned off for this model.

A torque is applied at the end of the cantilever beam using a TorqueOnly Force_Vector_OneBody element. A torque of 3x107 Nmm is applied gradually, via a STEP function as:(1)

θ= \frac{T1}{GJ} = \frac{({3×10}^{7} Nmm) × (1000 mm)}{(8 {.076 × 10}^{4} N/mms) × (1 .392 × 107 mm4)}

θ = 0.026686rad

The rotation of the marker associated with the applied torque will be used to determine the twist angle. This can be found in the ABF output file model_1.abf.

Numerical Results

The multibody model described in the previous section is run for an end time of 5.0 seconds. The output request is plotted below.

Figure 4. Twist Angle of the Marker on NLFE Associated with Torque

The twist angle for the numerical result is found as:(2)

θ = 0.026690 r a d

Conclusion

The NLFE model with 10 NLFE beam elements shows close agreement to the theoretical results for this case.

	Beam Theory	Numerical	% error
Twist Angle	0.026686 rad	0.026690 rad	0.02 %