Example 1

Calculate the vertical displacement of section k of the curved rod shown in the figure. Assume 𝐸𝐼𝐺𝐼𝑜 =2.

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Solution

In state P, reactions and formulas for internal forces:

𝑀𝑔=𝑃𝑅sin(𝛼)𝑀𝑠=𝑃𝑅(1cos(𝛼))

In unit state, reactions and formulas for internal forces:

𝑚𝑔=𝑅sin(𝛼)𝑚𝑠=𝑅(1cos(𝛼))

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

𝑦=1𝐸𝐼𝜋20(𝑃𝑅sin(𝛼))(𝑅sin(𝛼))d𝛼+1𝐺𝐼𝜋20𝑃𝑅(1cos(𝛼))𝑅(1cos(𝛼))d𝛼𝑦=1𝐸𝐼𝜋20𝑃𝑅2sin(𝛼)2 d𝛼+1𝐺𝐼𝜋20𝑃𝑅2(cos(𝛼)1)2 d𝛼

Where the values of individual integrals are

𝜋20𝑃𝑅2sin(𝛼)2 d𝛼𝑃𝑅2𝜋4𝜋20𝑃𝑅2(cos(𝛼)1)2 d𝛼3𝑃𝑅2𝜋42𝑃𝑅2

Therefore, finally

𝑦=7𝑃𝑅2𝜋44𝑃𝑅2