Problem 23 - Calculus Of Variations, With Applications To Physics And Engineering

Question

(a) Circular plate \((r \leqq a)\) clamped. ANswER: \(\phi=0,(\partial \phi / \partial r)=0\) at \(r=a\). (b) Rectangular plate \((0 \leqq x \leqq a ; 0 \leqq y \leqq b)\) simply supported. ANgwER: \(\phi-0,\left(\partial^{2} \phi / \partial x^{2}\right)-0\) on \(x-0\) and \(x-a ; \phi-0,\left(\partial^{2} \phi / \partial y^{2}\right)=0\) on \(y-0\) and \(y=b .\) Note also that \(\left(\partial^{2} \phi / \partial y^{2}\right)=0\) on \(x=0, x=a ;\) and \(\left(\partial^{2} \phi / \partial x^{2}\right)=0\) on \(y=0, y=b\). Why? (c) Semicircular plate (r \(\leqq a ; 0 \leqq \theta \leqq \pi)\) simply supported. ANswER: Since \(\begin{aligned} K=(1 / a) \text { on } r=a & \\\ \phi & \phi 0, \quad \frac{\sigma}{a} \frac{\partial \phi}{\partial r}+\frac{\partial^{2} \phi}{\partial r^{2}}=0 \text { on } r=a(0 \leqq \theta \leqq \pi) \\ & \phi=0, \quad \frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}=0 \text { on } \theta=0, \theta=\pi . \end{aligned}\) (d) Circular plate \((r \leq a)\) free. ANswER: $$ \left.\begin{array}{r} \frac{\partial}{\partial r}\left(\nabla^{2} \phi\right)+(1-\sigma) \frac{1}{a^{2}} \frac{\partial}{\partial \theta}\left(\frac{\partial^{2} \phi}{\partial \theta \partial r}-\frac{1}{a} \frac{\partial \phi}{\partial \theta}\right)=0 \\ (1-\sigma)\left(\frac{1}{a^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}+\frac{1}{a} \frac{\partial \phi}{\partial r}\right)-\nabla^{2} \phi=0 \end{array}\right\\} \text { on } r=a $$

Step-by-Step Solution

Answer

Every specific case represents different geometries and constraints. (a) is a clamped circular plate where neither displacement nor radial change occurs at the edge. (b) depicts a rectangular plate simply supported at all its corners, and (c) characterizes a semi-circular plate also simply supported at its edges. Lastly, (d) presents an unrestricted circular plate where there is no stress or strain at the edge.

Step 1: Understanding Case (a) - Circular Plate

In case (a), we have a circular plate of radius 'a' which is clamped. The mathematical conditions representing this situation are: \( \phi=0,(\partial \phi /\partial r)=0\) at \(r=a\). The outer boundary is fixed and does not allow for any displacement, represented by \( \phi = 0\). The criteria \( (\partial \phi / \partial r) = 0 \) when \( r = a \) tells us that the rate of change of displacement concerning the radius is zero at \( r = a \), indicating that there is no radial displacement at the plate's edge.

Step 2: Understanding Case (b) - Rectangular Plate

For case (b), we have a rectangular plate defined in the domain \(0 \leqq x \leqq a ; 0 \leqq y \leqq b\). The conditions for this case are given by: \( \phi -0,(\partial^{2} \phi / \partial x^{2})-0\) at \(x=0\) and \(x=a ; \phi-0,(\partial^{2} \phi / \partial y^{2})=0\) at \(y=0\) and \(y=b\). These boundary conditions denote the scenario where the plate is simply supported, meaning there is zero bending moment and any displacement at all four edges is zero.

Step 3: Understanding Case (c) - Semicircular Plate

Case (c) presents a semicircular plate defined in the domain \(r \leqq a ; 0 \leqq \theta \leqq \pi\), and the conditions are given by: Since \( K=(1 / a)\) on \( r=a \) , \( \phi =0\), \( \partial \phi / \partial r+\partial^{2} \phi / \partial r^{2}=0\) on \( r=a(0 \leqq \theta \leqq \pi)\) and \( \phi=0,\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}=0\) on \( \theta=0,\theta=\pi\). These expressions are derived from the general equations of a semi-circular plate and denote that the edges of the semicircular plate are simply supported.

Step 4: Understanding Case (d) - Circular Plate

In case (d), we are looking at a circular plate again, but this time the plate is free, i.e., not undergone stress or fixed like in the previous cases. A pair of equations represent the conditions as follows: \(\frac{\partial}{\partial r}\left(\nabla^{2} \phi\right)+(1-\sigma)\frac{1}{a^{2}} \frac{\partial}{\partial \theta}\left(\frac{\partial^{2}\phi}{\partial \theta \partial r}-\frac{1}{a} \frac{\partial \phi}{\partial\theta}\right)=0\) and \((1-\sigma)\left(\frac{1}{a^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}+\frac{1}{a} \frac{\partial \phi}{\partial r}\right)-\nabla^{2}\phi=0 \) on \( r=a \). These equations represent the condition that stresses and strains in any direction are null at the edge of the free circular plate.

Key Concepts

Circular Plate

A circular plate is a common structural element found in engineering and physics. It is defined by its circular shape with radius 'a'. When it comes to boundary conditions for circular plates, they determine how the plate is constrained or allowed to behave at its edges.

In scenario (a) where the circular plate is clamped, both the displacement \( \phi = 0 \) and its derivative \( \frac{\partial \phi}{\partial r} = 0 \) are zero at the boundary where \( r = a \). This means the edge of the plate is fixed, and there is no displacement or movement along the radial direction. Simply put, the plate is stiffened around its circumference, and such conditions are critical when calculating stresses and deflections in the plate.

Rectangular Plate

A rectangular plate is a flat structure with length and width, typically defined within a domain \(0 \leqq x \leqq a ; 0 \leqq y \leqq b\). Rectangular plates are often used in engineering as they provide a simple geometry for analyzing stress and deflection.

The boundary conditions of a simply supported rectangular plate imply that at all four edges, the plate can rotate freely but cannot experience any displacement. Consequently, the deflection \( \phi = 0 \) at \( x = 0, x = a, y = 0, \) and \( y = b \) while the bending moment \( \frac{\partial^{2} \phi}{\partial x^{2}} = 0 \) at the sides \( x = 0 \) and \( x = a \), and similarly for the y-direction, ensuring that no bending moment is transferred at the edges.

Simply Supported Plate

Simply supported plates are structural elements which can rotate at their supports but do not displace. They include circular, semi-circular, and rectangular plates among others. This support type is crucial in structural engineering and mechanics to design stable and efficient structures.

In practice, using simply supported boundary conditions simplifies the mathematical solution of plates but also provides realistic approximations for scenarios like bridges or floor slabs. For example, the circular or semicircular plates discussed before showcase simply supported edges, which don't experience bending moments, allowing the focus to shift primarily to shear forces and deflections.

• Supports only allow reaction forces not moments.
• Span stability along with structural integrity is maintained.

Clamped Plate

A clamped or fixed plate is rigidly held along its edges, prohibiting any movement. These plates are quite rigid, hence ideal when no deformation is desired at edges.

Consider scenario (a) – a clamped circular plate. Here, both the function \( \phi = 0 \) and its radial derivative remain zero at the boundary \( r = a \). A clamped condition ensures that both displacement and slope remain zero, resulting in no rotational or translational movement at the edges. It leads to more complex stress distributions within the plate compared to simply supported ones.

These boundary conditions are integral for scenarios demanding high stability and minimal deflection, as in certain bridge decks or electronic device substrates that need to maintain precise alignment and minimal vibrational response.