Problem 35 - Thomas Calculus

Question

Stiffness of a beam The stiffness \(S\) of a rectangular beam is proportional to its width times the cube of its depth. a. Find the dimensions of the stiffest beam that can be cut from a 12 -in.- -diameter cylindrical log. b. Graph \(S\) as a function of the beam's width \(w\) , assuming the proportionality constant to be \(k=1 .\) Reconcile what you see with your answer in part (a). c. On the same screen, graph \(S\) as a function of the beam's depth \(d,\) again taking \(k=1\) . Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of \(k ?\) Try it.

Step-by-Step Solution

Answer

The dimensions of the stiffest beam are 6 inches by 6 inches.

Step 1: Understand the Problem

The problem asks us to find the dimensions of the stiffest beam (maximum stiffness) that can fit inside a 12-inch diameter cylinder. The stiffness formula given is proportional to the width, times the cube of the depth: \( S = k imes w imes d^3 \). The beam is rectangular and must fit within a circle with a radius of 6 inches.

Step 2: Formulate the Stiffness Equation

Using the given relationship of stiffness, which is proportional to \( w \times d^3 \), write the stiffness as \( S = k imes w imes d^3 \) where \( k \) is the proportionality constant. Since the beam must fit within the cylindrical log, the relationship \( w^2 + d^2 = 12^2 = 144 \) holds.

Step 3: Express Width in Terms of Depth

To simplify, express width \( w \) in terms of depth \( d \) using the equation \( w^2 + d^2 = 144 \). This gives \( w = \sqrt{144 - d^2} \). Substituting this in the stiffness equation results in \( S = k \times \sqrt{144 - d^2} \times d^3 \).

Step 4: Find the Maximum Stiffness

Differentiate \( S \) with respect to \( d \) to find the value of \( d \) that maximizes stiffness. Then solve \( \frac{dS}{dd} = 0 \) for \( d \). Simplifying, you find \( d = \sqrt{36} = 6 \). Substitute \( d = 6 \) back into the equation for \( w \) to find \( w = 6 \). Hence, the dimension of the stiffest beam is \( 6 \times 6 \).

Step 5: Graph Stiffness as a Function of Width

Using an assumed proportionality constant of \( k = 1 \), graph the expression \( S = w \times (\sqrt{144 - w^2})^3 \). You'll notice that stiffness reaches a maximum when both width and depth are 6, consistent with part (a).

Step 6: Graph Stiffness as a Function of Depth

On the same graph as Step 5, plot \( S = (\sqrt{144 - d^2}) \times d^3 \) with \( k = 1 \). Compare the functional shape of this graph to the one from width \( S(w) \). Both show the maximum stiffness occurs at \( w = d = 6 \).

Step 7: Effect of Changing Proportionality Constant

Changing the proportionality constant \( k \) does not affect the point where the stiffness is maximized because \( k \) simply scales the entire function up or down uniformly. The maxima occur at \( w = d = 6 \) irrespective of \( k \).

Key Concepts

Rectangular Beam

A rectangular beam is a solid structure with a length, width, and depth (height), commonly used in engineering and construction. The rigidity or stiffness of the beam is an important property and greatly influences how the beam behaves under loads. Here, we are considering a beam's stiffness being proportional to its width and the cube of its depth. This relationship highlights the importance of each dimension in contributing to stiffness. For example, if you increase the depth of a beam even slightly, its stiffness can increase significantly because depth is cubed in the stiffness formula:

• Stiffness Formula: \( S = k \times w \times d^3 \)
• Explaining the formula: Width \( (w) \) impacts stiffness linearly, while depth \( (d) \) influences it exponentially.

This principle guides the design of beams to ensure they are neither too flexible nor overly stiff for their intended application. A well-designed beam maximizes the use of material to achieve the desired stiffness.

Cylindrical Log

When designing a beam from a cylindrical log, it's crucial to understand the geometry involved. A cylindrical log has a circular cross-section, and any beam cut from it must fit within this boundary.

• For a 12-inch diameter log, the largest square that can fit inside has a diagonal equal to the log's diameter. Hence, the radius is 6 inches.
• Using Pythagoras, the beam's width \((w)\) and depth \((d)\) inside a circle relate by: \( w^2 + d^2 = 144 \).

This relationship is crucial when determining the dimensions of the stiffest possible beam. Properly utilizing the log's diameter ensures that the beam maximizes its volume and, hence, potential stiffness.

Proportionality Constant

The proportionality constant, \(k\), in the stiffness equation \( S = k \times w \times d^3 \) serves as a scaling factor. While the absolute value of stiffness varies with different \(k\), the relative relationships between dimensions remain unchanged. Key characteristics include:

• \(k\) cannot alter the beam's dimensions that result in maximum stiffness.
• Different values of \(k\) will uniformly scale the stiffness values, affecting neither the critical dimensions of the beam nor the point of maximum stiffness.

Changing \(k\) impacts only the magnitude of stiffness, which is essential for comparing real-world beams where material properties result in different constants.

Maximum Stiffness

Achieving maximum stiffness ensures a beam can resist bending effectively under load. The solution of this exercise involves finding dimensions that satisfy maximum stiffness when cut from a cylindrical log.

• For a given circular boundary, the formulation is: Maximize \( S = k \times w \times d^3 \).
• Calculate by expressing width in terms of depth: \( w = \sqrt{144 - d^2} \).
• Differentiate \( S \) with respect to \( d \), find and solve \( \frac{dS}{dd} = 0 \) to determine the optimal depth \( d = 6 \), and subsequently, the width \( w = 6 \).

The solution reveals a square cross-section beam provides the maximum stiffness given the log diameter. It underscores the importance of proper dimensioning in beam design, with width and depth being equal for optimal stiffness.