91Ó°ÊÓ

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

Step by step solution

01

Understanding the Beam's Stiffness Formula

The stiffness \( S \) of a rectangular beam is given by the formula \( S = k imes w imes d^3 \), where \( w \) is the beam's width, \( d \) is the beam's depth, and \( k \) is the proportionality constant.

02

Identify Constraints from the Log's Dimensions

The beam must fit inside a cylindrical log with a diameter of 12 inches. The largest possible width and depth for the beam are determined by fitting inside a circle, which implies the cross-section of the beam is bounded by the relation: \( w^2 + d^2 = 12^2 = 144 \).

03

Express Stiffness in Terms of Single Variable

We express either \( w \) or \( d \) in terms of the other variable using the circle equation. For simplicity, let's express \( w \) in terms of \( d \): \( w = \sqrt{144 - d^2} \). Substitute into stiffness equation: \( S = k \times \sqrt{144 - d^2} \times d^3 \).

04

Maximize Stiffness Function

To maximize \( S \), we find the derivative \( \frac{dS}{dd} \) and set it to zero. Compute: \( S = k \times d^3 \times \sqrt{144 - d^2} \). Use chain rule and product rule for differentiation. Solve for \( d \) when \( \frac{dS}{dd} = 0 \).

05

Solve for Optimal Dimensions

By solving \( \frac{dS}{dd} = 0 \), determine that \( d = \sqrt{36} = 6 \) inches. Using this depth, \( w = \sqrt{144 - d^2} = \sqrt{108} = 6 \sqrt{3} \) inches.

06

Graph Stiffness as a Function of Width

Graph \( S(w) = w \times (12^2 - w^2)^{3/2} \) by substituting \( k=1 \). The graph should exhibit that the maximum stiffness occurs at the calculated width from Step 5.

07

Graph Stiffness as a Function of Depth

Graph \( S(d) = d^3 \times \sqrt{144 - d^2} \) with \( k=1 \). The graph will show a peak stiffness around the depth calculated in Step 5.

08

Compare Graph Results with Calculations

The graphs of \( S(w) \) and \( S(d) \) should show maximum stiffness at width \( 6\sqrt{3} \) and depth \( 6 \), in agreement with analytical calculations. Changing the value of \( k \) would only scale the vertical axis of the graph, not the shape or location of the maxima.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rectangular beam

A rectangular beam is a structural element used in construction that has a rectangular cross-section. The stiffness of such a beam is crucial in determining its ability to resist bending when subjected to various loads. In this context, the stiffness is given by the formula: \[ S = k \times w \times d^3 \] where \( S \) represents the stiffness, \( w \) is the width of the beam, \( d \) is the depth, and \( k \) is a proportionality constant.

• Width \( (w) \): This is one of the sides of the rectangular cross-section and directly affects the stiffness proportionally.
• Depth \( (d) \): This is the other side of the rectangle perpendicular to the width. The depth is exponentially more influential due to its cubic relationship in the formula.
• Cylindrical Constraint: When cutting a beam from a circular log, both the width and depth of the beam must fit within the diameter of the log. For a 12-inch diameter, this constraint becomes \( w^2 + d^2 \leq 144 \).

Understanding these parameters helps in designing a beam that will be efficient under load, providing maximum stiffness within the physical limitations.

proportionality constant

The proportionality constant, denoted as \( k \), plays a critical role in the stiffness formula for a beam. This constant determines the scale of stiffness according to the material properties and the geometric configuration.

• Effect on Stiffness: Since \( S = k \times w \times d^3 \), altering \( k \) scales the entire stiffness value. A larger \( k \) indicates a stiffer beam, while smaller values suggest less stiffness.
• Material and Property Dependence: The value of \( k \) often reflects the intrinsic properties of the beam's material, like the modulus of elasticity and the beam's shape factors.
• Constant Assumption for Simplicity: In the given exercise, \( k = 1 \) is assumed for graphical analysis, simplifying the relationship between width, depth, and stiffness and allowing us to focus on geometric factors without considering different materials.

By comprehending the proportionality constant's influence, we can better predict how the beam will perform under various structural scenarios.

optimization problems

Optimization problems involve finding the best solution under a given set of constraints, often maximizing or minimizing some aspect of interest. In this exercise, the objective is to maximize the stiffness of a beam cut from a cylindrical log.

• Objective Function: The function to be maximized is \( S = k \times w \times d^3 \). The goal is to find dimensions \( w \) and \( d \) that yield the highest stiffness.
• Constraints: The primary constraint is the dimensions of the beam fitting into the cylindrical log of 12-inch diameter, expressed as \( w^2 + d^2 \leq 144 \).
• Solving the Problem: To solve, transform the function in terms of a single variable, use calculus to find the derivative, and locate critical points where stiffness reaches its maximum.

Understanding optimization helps in constructing solutions that are efficient, economical, and of utmost functional effectiveness.

graphical analysis

Graphical analysis is a powerful method to visualize how changes in variables affect an equation's outcome, aiding in the understanding and verification of solutions. Here, it is applied to the beam stiffness equation.

• Graph of Width Function: By graphing \( S(w) = w \times (12^2 - w^2)^{3/2} \), the behavior of stiffness relative to width can be observed, highlighting the maximum point where stiffness is optimal.
• Graph of Depth Function: Similarly, \( S(d) = d^3 \times \sqrt{144 - d^2} \) is graphed to see stiffness variation with depth, with a peak demonstrating maximum stiffness.
• Comparative Analysis: By comparing the graphs of \( S(w) \) and \( S(d) \), one can verify if analytical solutions match graphical peaks, offering a visual check on the calculus-derived answers.
• Effect of Changing \( k \): Varying \( k \) only alters vertical scaling in the graphs, not the location of maximum points, which remains constant due to geometry.

Graphical analysis enhances understanding by providing a visual inspection of mathematical conclusions, particularly useful in identifying optimal dimensions and verifying computed solutions.