Curved Rectangular Tube Section Stress Calculations

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Curved Rectangular Tube Section Stress Calculations

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Figure 1.0

Figure 2.0 Square / Rectangular Tube Section Beam Dimensions

Stress Distribution due to bending is given by:

Equation 1
s = M · γ / ( A · e · ( r_n - 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
r_n = 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 · h_i ) / ( A · e · r_i )

Where:

h_i = distance from the center neutral axis to the inside fiber, inches ( h_i = r_n - r_i )
r_i = radius of curvature on the inside fiber, inches

Bending Stress at the Outside Fiber is given by:

Equation 3
s = ( M · h_o ) / ( A · e · r_i )

Where:

h_o = distance from the center neutral axis to the inside fiber, inches ( h_o = r_o - r_i )
r_o = radius of curvature on the outside fiber, inches
A = ( b - t ) ( t_i + t_o ) + th

Trapezoid Section Beam Stuctural Shape

r_n = [ ( b - t ) ( t_i + t_o ) + th ] / [ b [ log^e ( ( r_i + t_i ) / r_i ) + log_e ( r_o / ( r_o - t_o) ) ] + t log_e ( ( r_o - t_o ) / ( r_i + t_i ) ]

e = R - r_n

R = r_i + [ ( 1/2) h² t + (1/2) t_i ( b-t ) + ( b-t ) ( t_o ) ( h - (1/2) t_o ] / ( ht + ( b-t) (t_i + t_o) )

Where:

b = Tube Section beam width, in
h = width, in
t_i = wall thickness, in
t_o = wall thickness, in

References:

McGraw Hill Machine Design (1968)