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Section 2-15 Stresses in Thick-Walled Cylinders

A thick-walled cylinder subjected to external or internal pressure, or both, has radial and tangential stress with values which are dependent upon the radius of the element under consideration. A thick-walled cylinder may also be stressed longitudinally. In design, gun barrels, high-pressure hydraulic cylinders, and pipes carrying fluids at high pressures, must all be considered as thick-walled cylinders.

In determining the radial stress sigma_r and the tangential stress sigma_t, we make use of the assumption that the longitudinal elongation is constant around the circumference of the cylinder. In other words, a right section of the cylinder remains plane after stressing.

Referring to Figure [2-26], we designate the inside radius of the cylinder by a, the outside radius by b, the internal pressure by p_i, and the external pressure by p_o. Let us consider the equilibrium of an infinitely thin semicircular ring cut from the cylinder at radius r and having a unit length. Taking a summation of forces in the vertical direction equal to zero, we have 


     2*sigma_t*dr + 2*sigma_r*r - 2*(sigma_r+dsigma_r)
        * (r+dr) = 0                                    [a]
Simplifying and neglecting higher-order quantities gives 

     sigma_t - sigma_r - r* dsigma_r/dr = 0             [b]
Equation [b] relates the two unknowns sigma_t and sigma_r, but we must obtain a second relation in order to evaluate them. The second equation is obtained from the assumption that the longitudinal deformation is constant. Since both sigma_t and sigma_r are positive for tension, equation [2-22] can be written 

     epsilon_l = - mu * sigma_t / E - mu * sigma_r / E  [c]
where mu is Poisson's ratio and epsilon_l is the longitudinal unit deformation. Both of these are constants, and so equation [c] can be rearranged in the form 

     - E*epsilon_l/mu = sigma_t + sigma_r
                        = 2 * C1                        [d]

Fig02-26a

Fig02-26b

Next, solving equations [b] and [c] to eliminate sigma_t produces 


     r*dsigma_r/dr + 2*sigma_r = 2*C1                   [e]
Multiplying equation [e] by r gives 

     r^2*dsigma_r/dr + 2*r*sigma_r = 2*r*C1             [f]
We note that 

     d/dr*(r^2*sigma_r) = r^2*dsigma_r/dr + 
                            2*r*sigma_r                 [g]
Therefore 

     d/dr*(r^2*sigma_r) = 2*r*C1                        [h]
which, when integrated, gives 

     r^2*sigma_r = r^2*C1 + C2                          [i]
where C2 is a constant of integration. Solving for sigma_r, we obtain 

     sigma_r = C1 + C2 / r^2                            [j]
Substituting this value in equation [d], we find 

     sigma_t = C1 - C2 / r^2                            [k]

To evaluate the constants of integration, note that, at the boundaries of the cylinder, 


     sigma_r = -p_i  at r = a 
     sigma_r = -p_o  at r = b
Substituting these values in equation [j] yields 

     -p_i = C1 + C2/a^2
     -p_o = C1 + C2/b^2                                 [l]
The constants are found by solving these two equations simultaneously. This gives 

     C1 = (p_i*a^2-p_o*b^2)/(b^2-a^2)      
     C2 = a^2*b^2*(p_o-p_i)/(b^2-a^2)                   [m]
Substituting these values into equations [j] and [k] yields 

     sigma_t = (p_i*a^2 - p_o*b^2 - a^2*b^2*
                   (p_o-p_i)/r^2) / (b^2 - a^2)         [2-53]
     sigma_r = (p_i*a^2 - p_o*b^2 + a^2*b^2*
                   (p_o-p_i)/r^2) / (b^2 - a^2)         [2-54]
In the previous equations positive stresses indicate tension and negative stresses compression.

Let us now determine the stresses when the external pressure is zero. Substitution of p_o = 0 in equations [2-53] and [2-54] gives 


     sigma_t = a^2*p_i/(b^2-a^2)*(1+b^2/r^2)            [2-55]
     sigma_r = a^2*p_i/(b^2-a^2)*(1-b^2/r^2)            [2-56]
These equations are plotted in Figure [2-27] to show the distribution of stresses over the wall thickness. The maximum stresses occur at the inner surface, where r=a. There magnitudes are 

     sigma_t = p_i*(b^2+a^2)/(b^2-a^2)                  [2-57]  
     sigma_r = -p_i                                     [2-58]  




The stresses in the outer surface of a cylinder subjected 
only to external pressure are found similarly.  They are


     Sigma_t = -p_o*(b^2+a^2)/(b^2-a^2)                 [2-59]
     Sigma_r = -p_o                                     [2-60]






Figure [2-27a]


Figure [2-27b]



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Mechanical Engineering Design  
Section 2-15  Stresses in Thick-Walled Cylinders



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 6/18/10

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