Introduction

A simply supported beam is a common type of beam used in construction. It is supported at both ends and is free to rotate and deflect under load. This beam is subjected to a uniformly distributed load (UDL) and has a rectangular cross-section. This question requires the determination of the ratio between the maximum normal stress and the maximum shear stress.

Calculating Maximum Normal Stress

The maximum normal stress occurs at the point where the bending moment is the highest. This point is usually at the mid-span of the beam. The maximum bending moment can be calculated using the following formula:

Mmax = wL^2/8

where w is the UDL, L is the length of the beam, and Mmax is the maximum bending moment.


The maximum stress can be calculated using the following formula:

σmax = Mmax*y/I

where y is the distance from the neutral axis to the outermost fiber, and I is the moment of inertia of the cross-section.


For a rectangular cross-section, the moment of inertia can be calculated using the following formula:

I = (b*D^3)/12

where b is the width of the beam and D is the depth of the beam.

Calculating Maximum Shear Stress

The maximum shear stress occurs at the supports of the beam. The maximum shear force can be calculated using the following formula:

Vmax = wL/2

where Vmax is the maximum shear force.


The maximum shear stress can be calculated using the following formula:

τmax = Vmax*q/Ib

where q is the first moment of area of the cross-section above the point where the shear stress is being calculated.


For a rectangular cross-section, the first moment of area can be calculated using the following formula:

q = (b*D^2)/2

Ratio of Maximum Normal Stress to Maximum Shear Stress

The ratio between the maximum normal stress and the maximum shear stress can be calculated using the following formula:

σmax/τmax = 6/(b/D)^2


Therefore, the ratio is dependent on the aspect ratio of the cross-section, which is the ratio of the width to the depth. The higher the aspect ratio, the higher the ratio of maximum normal stress to maximum shear stress. This implies that the beam is more susceptible to failure due to bending than to shear.

Conclusion

In conclusion, the ratio between the maximum normal stress and the maximum shear stress of a simply supported beam subjected to a UDL load can be calculated using the aspect ratio of the cross-section. This ratio is an important parameter in the design of beams as it determines the mode of failure.