Simply supported beam

Model description

Length = 1000mm

breadth = 25mm

height = 25mm

Youngs modulus = 250 GPa

Load of 10,000N applied onto line drawn across the middle of the top plane

of the beam.

One end fixed through bottom edge.

Other end constrained by displacement only along z axis.

Mesh element size : 20 mm

Result

Maximum Deformation along y axis = -25.65 mm in the middle of the beam

Equivalent Von mises stress = 953.87 Mpa (max) 2.342 MPa (min)

link to images of analysis

https://drive.google.com/open?id=133OkRJyC_OFwpjfxp2eB6JuPjDqE4wUr

Analytical solutions

Moment of inertia = 1/12 bh^3 (for rectangular cross-section)

= (1/12)*0.025*(0.025)^3

= 3.26 * 10^(-8) m^4

Shear force on the beam = f/2 = 5000N(magnitude)

Bending Moment on the beam = f*l/4 = 2500 Nm

Maximum displacement = -f*l^3/(48EI) = -25.56 mm

Maximum Bending stress = (M*(h/2))/I = 958.59 MPa

Comparison between numerically and analytically obtained values

It is observed that the analytically calculated values for displacement are very close to those obtained numerically from Ansys.

The stress value obtained by calculation is slightly greater whereas the deformation obtained by calculation is slightly lesser compared to respective numerical values obtained from ansys. However the differences are negligibly small.

Inference

In the constraints provided in the software one of the bottom edges is fixed while the other bottom edge is constrained by displacement only along an axis parallel to the rod.

such a case is practical but very difficult to analyse mathematically

So for analytical purposes two models can be considered

Model 1

We can simply assume a beam supported freely at both ends wherein the maximum bending moment and hence the maximum stress is found to be at the central section of the beam. The results obtained for this model are extremely close to the solution generated by the software. Hence we can assume that the constraints provided in the software work very similarly to the model assumed here.

Model 2

Another approach can be as follows

At one end, we can consider the entire face to be fixed and the other end to be simply supported.

for such a case we obtain a considerable moment at the fixed end which in turn is found to be the maximum moment along the rod and consequently the bending stress is found out from its value. here the max bending stress is at the fixed end. Hence this model does not comply much to the result obtained in ansys.

Akash Mitra

17/2/18