A horizontal beam of negligible weight and of length \(2 L\) is attached to a hinge at \(x=0\) and is supported at \(x=L .\) A downward force \(F_{1}\) is applied at \(x=L / 2\), causing a small upward deflection \(D_{2}\) at the end, \(x=2 L\). The force \(F_{1}\) is removed and a downward force \(F_{2}^{\prime}\) is applied at the end, \(x=2 L\), causing a small upward deflection \(D_{1}^{\prime}\) at \(x=L / 2\). Show that †The notation \(V_{1}\) for potential energy density must not be confused with \(V\) for shearing force. \(F_{1} / D_{2}=F_{2}^{\prime} / D_{1}^{\prime}\). Can you generalize this special case into a theorem, known as the reciprocity theorem, which applies to all linear elastic systems characterized by small strains?

Step by step solution

01

Finding deflection due to force F1

To find the deflection \(D_2\) caused by force \(F_1\), we will first find the bending moment along the beam and then use the Euler-Bernoulli equation to determine the deflection. The bending moment equation for a beam with a hinge at \(x=0\) and a downward force \(F_1\) at \(x=L/2\) is given by: $M(x) = \begin{cases} F_1 x, & 0\leq x \leq \frac{L}{2}\\ F_1 (L-x), & \frac{L}{2}\leq x\leq L \end{cases}.$ Now we will use the Euler-Bernoulli equation: \(EI \frac{d^4w(x)}{dx^4} = M(x)\) Here, \(E\) is the Young's modulus, \(I\) is the area moment of inertia and \(w(x)\) is the deflection. Integrate the fourth order ODE four times to find \(w(x)\) and apply boundary conditions to find the constants of integration. Finally, find the deflection \(D_2 = w(2L)\).

02

Finding deflection due to force F2'

Now, we will find the deflection \(D_1'\) caused by force \(F_2'\) applied at \(x=2L\). We will follow the same procedure as in Step 1. The bending moment equation for this case is: $M(x) = \begin{cases} F_2' x, & 0\leq x \leq L\\ F_2' (2L-x), & L\leq x\leq 2L \end{cases}.$ Using the Euler-Bernoulli equation, integrate, and apply boundary conditions to find the constants of integration. Finally, find the deflection \(D_1' = w(L/2)\).

03

Show the required ratio

Now that we have found the deflections \(D_2\) and \(D_1'\) for both force configurations, we can find the ratio and compare them: \(F_1 / D_2 = F_2' / D_1'\) By solving for the deflections in Steps 1 and 2, we should find that the ratio holds.

04

Generalize to the reciprocity theorem

The reciprocity theorem states that for linear elastic systems characterized by small strains, the ratio of the input force to the resulting deflection is the same for any two input forces and corresponding deflections, provided that the deflections are caused by the forces acting individually. The problem demonstrates this theorem for a particular case involving a beam with forces at different locations. By understanding the underlying principles, one can apply this theorem to other linear elastic systems with small strains.

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