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Learning Goal:

The method of virtual work can be applied to beams and frames. The displacement $$ of a specific point on a beam caused by external loading can be determined using the virtual work equation $1*={\displaystyle \underset{0}{\overset{L}{}}}\frac{mM}{EI}dx,$ where $1$ is the external virtual unit load acting on the beam in the directions of the deflection, $m$ is the internal virtual moment in the beam, expressed as a function of $x$ and caused by the external virtual unit load, $M$ is the internal moment in the beam, expressed as a function of $x$ and caused by real loads, $L$ is the member's length, $E$ is the modulus of elasticity, and I is the moment of inertia of the beam's cross$-$sectional area. The slope angle $$ can also be determined using the method of virtual work. Instead of a unit load, a unit couple moment is applie at the desired point. Once corresponding internal real moments $M$ and virtual moments $m_{}$ have been determined, as a function of $x,$ then the virtual work equation $1*={\displaystyle \underset{0}{\overset{L}{}}}\frac{m_{}M}{EI}dx$ can be used to find the tangential rotation.

To use the virtual work method, start by removing the real loads and place a unit load or unit couple moment at the desired location in the direction the displacement or rotation is to be determined. For example, to determine the downward deflection at $C$ due to the load $P$ in $($Figure $1),$ apply a unit virtual force down at $C.$

Establish $x$ coordinates for each region of the beam or frame that has no discontinuity of real or virtual load. In the example, three regions are required because there are discontinuities at $B$ and $C.$ With all the real loads temporarily removed and the virtual load or couple moment in place, calculate the internal moment $m$ or $m_{}$ as a function of each $x$ coordinate. Next, determine the internal real moments $M$ caused only by the real loads, using the same $x$ coordinates established for calculating the virtual moments.

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Part B

For the beam shown in $($Figure $2),$ determine the slope at A$.$ Use the method of virtual work and assume $A$ is pilhted, $B$ is a roller, and $E$ I is constant. Let a positive angle be measured counterclockwise from the horizontal.

Express your answer in terms of some or all of the variables $P,d,E,$ and I.

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Part C

For the frame shown in $($Figure $3),$ determine the horizontal deflection at $C$ when $a=10ft,b=14ft,$ and $w=120\frac{k}{f}t.$ Assume $A$ is pinned, $C$ is a roller, and $B$ is fixed. Take $E=2\mathrm{.}9{10}^4$ksi and $I=700$ in ${?}^4.$

Express your answer in the appropriate units to three significant figures.

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