Determination of Internal Forces under Constrained Torsion of a Cantilever Thin-Walled I-beam
Objective: Verify the correctness of the determination of internal force values under constrained torsion of a thin-walled open-section bar using the example of a cantilever beam.
Problem formulation: a cantilever beam with a span of 3.0 m is subjected to a torsional moment applied at its free end. The cross-section of the beam is a thin-walled I-section. Determine the values of warping displacements and bimoments in the sections of the beam.
Source: [1] V.Z. Vlasov, Thin-walled Elastic Rods – Fizmatgiz, Moscow, 1959. – p. 568.
Initial data file:
VlasovAnalyticalSolution_I.spr
SCAD version used:
SCAD++ 23.1.1.3
Initial data:
| Е = 210000 N/mm2 | Elastic modulus |
| v = 0.3 | Poisson’s ratio |
| Section type: | Thin-walled I-section |
| h = 400 mm | Section height (along the outer edge) |
| b = 300 mm | Flange width (along the outer edge) |
| tf = 20 mm | I-beam flange thickness |
| tw = 10 mm | I-beam web thickness |
| Mx = 1000 kN/m | Torque at the free end of the cantilever |
| l = 3.0 m | Bar length |
Finite element model: the design model of the cantilever beam is a spatial bar system. The beam’s axis aligns with the x – x axis of the global coordinate system. A rigid constraint is imposed at the left end support, restricting rotations about the x – x, y – y, z – z axes, as well as linear displacements along the x – x, y – y, z – z axes of the global coordinate system. The condition of warping continuity was applied at the intermediate nodes of the model. Rigid constraints with respect to warping were imposed at the end nodes of the beam, i.e, the warping displacements at these nodes were set to zero.
Results in SCAD++:
The resulting diagrams of warping displacements and bimoments, calculated in SCAD++, are shown in Fig. 1 and Fig. 2.
Figure 1. Warping diagram under constrained torsion of a cantilever I-beam, cm-1.
Figure 2. Bimoment diagram under constrained torsion of a cantilever I-beam, Nm2.
Comparison of solutions:
|
Parameter |
Analytical solution, p. 107 [1] |
Numerical solution SCAD Office |
Deviation, % |
|
Warping displacement at the mid-span section of the beam, cm-1 |
0,014839025 |
0,0148244 |
0,02 |
|
Bimoment at the fixed support of the beam, Nm2 |
1358729,598 |
1355982,5 |
-0,32 |
|
Bimoment at the free end of the cantilever, Nm2 |
1358729,598 |
1355982,5 |
-0,32 |
Notes: In the analytical solution, the warping displacement d and bimoment B at a section located at a distance x from the edge can be calculated using the following formulas:
\[\begin{array}{l} {d(x)=\frac{M_{x} }{GI_{t} } \left(-\frac{1-\cosh \; k}{\sinh \; k} \sinh \frac{k}{l} x+1-\cosh \frac{k}{l} x\right),} \\ {B(x)=M_{x} \frac{l}{k} \left(\frac{1-\cosh \; k}{\sinh \; k} \cosh \frac{k}{l} x+\sinh \frac{k}{l} x\right),} \end{array}\] \[k=\sqrt{\frac{GI_{t} }{EI_{\omega } } } \]