Lesson #29

Reading: 12:1-2

 

 

Beam Deflections

 

Deflections in beams can come from bending and shear effects, but the effect of shear forces in small compared to that of bending and can be ignored.

 

Earlier in the semester, we looked at a beam in pure bending and derived an equation for strain in the cross section of the beam. 

 

dx = r dq   or          dq/dx = 1/r

 

Du = -y dq             and        e = du/dx = -y dq/dx = -y/r

 

Assuming a linear elastic material so that s = E e and considering equilibrium, we derived

 

s =-M y / I

 

Applying these equations, we find

 

e = -y/r = s / E = -M y / (E I)

 

So now,

 

1 / r = M / (E I)

 

The curvature, one over the radius of curvature, can be found from calculus as

 

1 / r = ( d²v/dx²) / [ 1 + (dv/dx)²]^3/2

 

where v is the vertical deflection in the beam.  If we make a further assumption that displacements are small, we can say

 

v << 1                and        dv/dx << 1

 

Therefore, the denominator is essentially 1 and

 

1 / r = d²v/dx²

 

 

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