Comparison of bending in beams

BENDING IN BEAMS

Objective

To compare the stress, strain and displacement of the following beams:

1. Rectangular section
2. I section
3. C section

Introduction to Finite Element Analysis(FEA)

Engineering analysis deals with finding various engineering parameters like displacement,stress and strain. Static analysis is done when the force does not change with respect to time. The displacements are computed from the stiffness matrix and followed by the derived qualtities like stress,strain etc.

For hand claculations the Flexure formulae is used for calculating bending stresses,

σ=My/I

where M=Bending moment

y=Distance from the neutral axis

I=Moment of inertia

Steps involved

1.Creating a 3d model: The Sections were available from structural steel options from tools tab. The corresponding sketches are modified if required. The sketches were extruded 400mm and the models were obtained as shown in figure 1.1,1.2,1.3

Figure 1.1 Beam with Rectangular section

Figure 1.2 Beam with I section

Figure 1.3 Beam with C section

2.Material definition: Alloy steel was applied(Yield strength=620MPa) as shown in fig 1.4

Figure 1.4 Material Selection

3.Meshing the model using appropriate elements.
Basically the CAE softwares divide the solid model into small elements whose size can be specified by the user but for this analysis. Mesh size is selected by trial and error method until the solution converges.

4.Boundary conditions:

• Applying fixed constraint on one of the side.
  The structure must be constrained inorder for the force acting on it to make deformation in the structure otherwise there won't be any stress as there is no resistance. Fixed constraint was applied to the left side.
• Load of 1500N is applied on the smaller face which induces bending stresses.

5.Results

1.Rectangular section

Fig 1.5 shows the stress,strain,dispalcement and Factor of safety(FOS=Ultimate stress/working stress)

1. Von mises stress is minimum for Rectangular section =9.6 MPa
2. Strain is minimum compared to other two sections.
3. Displacement is less than that of C section and slightly less than I section beam.
4. FOS of 65 is attained which is the maximum and may be called overdesigned. It will increase the cost due to more material usage and added weight.

Figure 1.5 A comparison of stress,strain,displacement and FOS of rectangular section beam

2. I section beam

Fig 1.6 shows the stress,strain,dispalcement and Factor of safety

1. Von mises stress=23.4 MPa, which is in between the Rectangular section and the C section beam
2. Strain is in between compared to other two sections.
3. Displacement is less than that of C section but slightly less than I section beam.
4. FOS of 26 is attained which is also overdesigned. It will increase the cost due to more material usage and added weight.

Figure 1.6 A comparison of stress,strain,displacement and FOS of I section beam

3. C section Beam

Fig 1.7 shows the stress,strain,dispalcement and Factor of safety

1. Von mises stress=41.68 MPa, maximum of the other two sections considered.
2. Strain is in maximum of the other two sections.
3. Displacement is maximum for C section.
4. FOS of 15 is attained which is optimum for a beam and lowering cost without effecting strength.

Figure 1.7 A comparison of stress,strain,displacement and FOS of a C section beam

CONCLUSION

Section Von Mises Stress Strain FOS Displacement

Rectangular 9.6MPa 3.043xe-05 65 3.102xe-02mm

I 27.4MPa 7.655xe-05 20 8.556xe-02mm

C 41.68MPa 1.429xe-04 15 3.021xe-01mm

So considering all the three beam sections the best one would be the I section beam in case of bending even though the stresses induced are greater than the rectangular section , the second moment of intertia of I section beam is better and the symmetric geometry of I section helps it to take more load with stablility. Cost reduction is another important factor as I section gives maximum strength with less weight. So I section is used in connecting rods, rails,etc.