Example 1

Calculate the vertical displacement of section k of the curved rod shown in the figure. Assume 𝐸⁢𝐼𝐺⁡𝐼𝑜 =2.

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Solution

In state P, reactions and formulas for internal forces:

𝑀𝑔=−𝑃⋅𝑅⋅sin⁡(𝛼)𝑀𝑠=𝑃⋅𝑅⋅(1−cos⁡(𝛼))

In unit state, reactions and formulas for internal forces:

𝑚𝑔=−𝑅⋅sin⁡(𝛼)𝑚𝑠=𝑅⋅(1−cos⁡(𝛼))

I calculate the displacement of point K using the Maxwell Mohr formula:

𝑦=1𝐸⁢𝐼⁢∫𝜋20(−𝑃⋅𝑅⋅sin⁡(𝛼))⋅(−𝑅⋅sin⁡(𝛼))d𝛼+1𝐺⁡𝐼⁢∫𝜋20𝑃⋅𝑅⋅(1−cos⁡(𝛼))⋅𝑅⋅(1−cos⁡(𝛼))d𝛼𝑦=1𝐸⁢𝐼⁢∫𝜋20𝑃⋅𝑅2⋅sin⁡(𝛼)2⁢ d⁢𝛼+1𝐺⁡𝐼⁢∫𝜋20𝑃⋅𝑅2⋅(cos⁡(𝛼)−1)2⁢ d⁢𝛼

Where the values of individual integrals are

∫𝜋20𝑃⋅𝑅2⋅sin⁡(𝛼)2⁢ d⁢𝛼→𝑃⋅𝑅2⋅𝜋4𝜋2∫0𝑃⋅𝑅2⋅(cos⁡(𝛼)−1)2⁢ d⁢𝛼→3⋅𝑃⋅𝑅2⋅𝜋4−2⋅𝑃⋅𝑅2

Therefore, finally

𝑦=7⋅𝑃⋅𝑅2⋅𝜋4−4⋅𝑃⋅𝑅2