Problem 6.5. Betti’s theorem

Using Betti’s theorem derive

a) midspan deflection and

b) end rotation

of a simply supported beam subjected to uniform load, p. Length of
the beam is L, its bending stiffness, EI is constant.

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Derive midspan deflection, v.

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$v = \frac{5 p L^{4}}{384 E I}$

Determine rotation of the beam end, φ.

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$φ = \frac{p L^{3}}{24 E I}$

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Steps

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Problem a)

Step 1. Assume a unit load at midspan. Draw moment diagrams from the unit load and from the original load.

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Step 2. Apply Betti’s theorem. Determine the work done by the unit force and the moment, M_p on the displacements caused by the other load. Calculate the midspan deflection from the equality of the works.

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Eq.(6-84)

1 \times v = \frac{1}{E I} \int_{L} M_{p} M_{1} d x = \frac{2}{E I} \frac{p L^{2}}{8} \frac{L}{2} \frac{2}{3} \times \frac{5}{8} \frac{L}{4} = \frac{5}{384} \frac{p L^{4}}{E I}

where the integration was performed visually as given in Hint of Problem 6.

Problem b)

Step 1. Assume a unit moment load at the end of the beam. Draw moment diagrams from the unit moment and from the original load.

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Step 2. Apply Betti’s theorem. Determine the work done by the unit moment and the real moment, M_p on the displacements caused by the other load. Calculate the midspan deflection from the equality of the works.

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Eq.(6-84)

$1 \times φ = \frac{1}{E I} \int_{L} M_{p} M_{1} d x = \frac{1}{E I} \frac{p L^{2}}{8} L \frac{2}{3} \frac{1}{2} = \frac{1}{24} \frac{p L^{3}}{E I}$

where the integration was performed visually.

Results

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Problem a)

A unit load is assumed at midspan. Moment diagrams from the unit load and from the original load are given below.

According to Betti’s theorem the work done by the unit force and the moment, M_p on the displacements caused by the other load are equal. The midspan deflection can be determined from the equality of the works as follows:

Eq.(6-84)

$1 \times v = \frac{1}{E I} \int_{L} M_{p} M_{1} d x = \frac{2}{E I} \frac{p L^{2}}{8} \frac{L}{2} \frac{2}{3} \times \frac{5}{8} \frac{L}{4} = \frac{5}{384} \frac{p L^{4}}{E I}$

where the integration was performed visually as given in Hint of Problem 6.

Problem b)

A unit moment load is assumed at the end of the beam. Moment diagrams from the unit moment and from the original load are given in the Figure:

According to Betti’s theorem the work done by the unit moment load and the real moment, M_p on the displacements caused by the other load are equal. The end rotation can be determined from the equality of the works as follows:

$1 \times φ = \frac{1}{E I} \int_{L} M_{p} M_{1} d x = \frac{1}{E I} \frac{p L^{2}}{8} L \frac{2}{3} \frac{1}{2} = \frac{1}{24} \frac{p L^{3}}{E I}$

where the integration was performed visually.