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### Curved Trapezoid Section Stress Formulas and Calculator

Considerable care must be taken in the arithmetic. The distance "e" from the center of gravity axis to the neutral axis is usually small. A numerical variation in the calculation of "e" can cause a large percentage change in the final results. Figure 2.0 Trapezoid Section Beam Dimensions

Stress Distribution due to bending is given by:

Equation 1
s = M · γ / ( A · e · ( rn - y ) )

Where:

s = the bending stress, psi
M = bending moment with respect to the centroical axis, in-lb
y = distance from the neutral axis to the point in question, inches (positive for distances toward the center of curvature, negative for distances away from the center of curvature)
A = the area of the section, in2
e = distance from the center of gravity axis to the neutral axis, inches
rn = radius of curvature of the neutral axis, inches
value of e used for base log = 2.7182818

Bending Stress at the Inside Fiber is given by:

Equation 2
s = ( M · hi ) / ( A · e · ri )

Where:

hi = distance from the center neutral axis to the inside fiber, inches ( hi = rn - ri )
ri = radius of curvature on the inside fiber, inches

Bending Stress at the Outside Fiber is given by:

Equation 3
s = ( M · ho ) / ( A · e · ri )

Where:

ho = distance from the center neutral axis to the inside fiber, inches ( ho = ro - ri )
ro = radius of curvature on the outside fiber, inches
A = ( 1/2 ) ( bi + bo ) h

Trapezoid Section Beam Stuctural Shape

rn = [ ( bi + bo ) / 2 h ] / [ ( ( bi ro - bo ri ) / h) loge ( ro / ri ) - ( bi - bo )]

e = R - rn

R = ri + [ h ( bi + 2 bo ) ] / [ 3 ( bi + bo ) ]

/ [ ht + ( bi - t ) ti ]

Where:

bi = T Section beam width, in
bo = T Section beam width, in
h = width, in

Related:

References:

• McGraw Hill Machine Design (1968)