Lesson #28

Reading: 11:1-2

 

 

Beam Design

 

When a beam is to be designed to carry a particular load, there are two stresses to consider - the maximum values of normal stress (tensile & compressive) and the maximum shear stress.  We have already discussed the fact that in a vast majority of beam problems, the normal stress from bending will be much larger that the shear stress.  Therefore, our design must begin with the normal stress.

 

To find the maximum normal stress, we must be able to draw the shear and moment diagrams to find M_max (positive and negative) and find the centroid and moment of inertia of the cross section.

 

Since the cross section size has not been determined yet, we define a new quantity called the section modulus.

 

S = I / y_max

 

Now, we can write the stress as

 

s_max = M_max / S

 

If the material is known so that we have an allowable stress that can be experienced, we can set s_max = s_allow and solve for a required minimum section modulus.

 

S_req = M_max / s_allow

 

This is not enough information to determine what the cross section dimensions are unless we are given information about the shape of the cross section.

 

For example, if the cross section is to be circular, we know

 

S = I / y_max = (p r⁴ / 4)/ r = p r³ / 4

 

If the cross section is to be square, we can say

 

S = I / y_max = (b⁴/12)/ (b/2) = b³ / 6

 

If the cross section is to be rectangular, we know

 

S = I / y_max = (b h³/12)/ (h/2) = b h² / 6

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