Description
A cantilever is fully fixed onthe left end and loaded by a bending moment according to the following sketch. The problem is described by the following set of parameters.
| Material | Elastic-Plastic | Modulus of Elasticity | E | 210000.000 | MPa |
| Poisson's Ratio | ν | 0.000 | - | ||
| Shear Modulus | G | 105000.000 | MPa | ||
| Plastic Strength | fy | 240.000 | MPa | ||
| Geometry | Cantilever | Length | L | 2.000 | m |
| Width | w | 0.005 | m | ||
| Thickness | t | 0.005 | m | ||
| Load | Bending Moment | M | 6.000 | Nm | |
Small deformations are considered and the self-weight is neglected in this example. Determine the maximum deflection uz,max.
Analytical Solution
The cantilever is loaded by the bending moment M. The quantities of this load are discussed at first. The moment Me when the first yield occurs and the ultimate moment Mp when the structure becomes plastic hinge are calculated as follows:
The bending moment M causes the elastic-plastic state. The cross-section in the elastic-plastic state is divided into the elastic core and the plastic surface, which is described by the parameter zp according to the following diagram.
The elastic-plastic moment Mep in the cross-section has to equal to the bending moment M. The curvature κ results from this equality.
The total deflection of the structure uz,max is calculated using the Mohr's integral.
RFEM Settings
- Modeled in RFEM 5.16 and RRFEM 6.01
- The element size is lFE= 0.020 m
- Geometrically linear analysis is considered
- The number of increments is 5
- Shear stiffness of the members is neglected
Results
| Material Model | Analytical Solution | RFEM 5 | RFEM 6 | ||
| uz,max [m] | uz,max [m] | Ratio [-] | uz,max [m] | Ratio [-] | |
| Orthotropic Plastic 2D | 1.180 | 1.190 | 1.008 | 1.190 | 1.008 |
| Isotropic Plastic 2D/3D, Plate | 1.173 | 0.994 | 1.173 | 0.994 | |
| Isotropic Plastic 1D | 1.180 | 1.000 | 1.180 | 1.000 | |
| Isotropic Nonlinear Elastic 2D/3D, Plate, Mises | 1.190 | 1.008 | 1.190 | 1.008 | |
| Isotropic Nonlinear Elastic 2D/3D, Plate, Tresca | 1.190 | 1.008 | 1.190 | 1.008 | |
| Isotropic Plastic 1D | 1.180 | 1.000 | 1.180 | 1.000 | |
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