Problem 97 - Intermediate Algebra

Question

Solve each problem by writing a variation model. Structural Engineering. The deflection of a beam is inversely proportional to its width and the cube of its depth. If the deflection of a 4 -inch-wide by 4-inch-deep beam is 1.1 inches, find the deflection of a 2 -inch-wide by 8 -inch-deep beam positioned as in figure (a) below.

Step-by-Step Solution

Answer

The deflection of the new beam is 0.275 inches.

Step 1: Understand the Inverse Variation

When a quantity varies inversely, it can be represented as \( y = \frac{k}{x} \), where \( k \) is a constant. In this problem, the deflection \( D \) is inversely proportional to the width \( w \) and the cube of the depth \( d^3 \). This relationship can be expressed as \( D = \frac{k}{w \cdot d^3} \).

Step 2: Calculate the Constant of Variation

We know that for a beam that is 4 inches wide and 4 inches deep, the deflection is 1.1 inches. Substitute these values into the equation to find \( k \): \[ 1.1 = \frac{k}{4 \cdot 4^3} \] Calculate \( 4^3 \) to get 64, so: \[ 1.1 = \frac{k}{4 \cdot 64} \] Simplify to get: \[ 1.1 = \frac{k}{256} \] Solving for \( k \), multiply both sides by 256: \[ k = 1.1 \cdot 256 = 281.6 \]

Step 3: Set Up the Equation for the New Beam

Now that we know \( k = 281.6 \), use this constant to find the deflection \( D' \) of a beam that is 2 inches wide and 8 inches deep: \[ D' = \frac{281.6}{2 \cdot 8^3} \] Calculate \( 8^3 \) (which is 512): \[ D' = \frac{281.6}{2 \cdot 512} \]

Step 4: Solve for New Deflection

Continue simplifying the equation from Step 3: \[ D' = \frac{281.6}{1024} \] Simplify \( \frac{281.6}{1024} \) to find \( D' \): \[ D' = 0.275 \]

Step 5: Conclusion

The deflection of the 2-inch-wide by 8-inch-deep beam is 0.275 inches.

Key Concepts

Deflection of a Beam

In structural engineering, understanding how beams bend and flex under load is crucial. The deflection of a beam refers to how much it bends when subjected to a force or weight.
This bending affects stability and performance, making its measurement important.
When considering deflection, several factors play significant roles:

• **Material of the beam:** Different materials have varying levels of stiffness.
• **Width (w) and depth (d):** In this context, deflection varies inversely with width and depth cubed.
• **Load applied:** The amount of force exerted influences the degree of deflection.

In simple terms, a beam will bend more if it is narrower and shallower, assuming all else is equal.

Inverse Variation

Inverse variation describes a relationship where one value increases as another decreases. This concept is useful in understanding relationships where effects differ inversely.
A common mathematical expression for inverse variation is: \( y = \frac{k}{x} \) where \( k \) is a constant.
In the context of beam deflection, this principle takes shape as: \( D = \frac{k}{w \cdot d^3} \).
Here, \( D \) represents deflection, \( w \) is the width of the beam, and \( d \) is its depth. If width or depth increases, deflection lessens proportionally, assuming the constant \( k \) is unchanged.

Structural Engineering

Structural engineering involves designing and evaluating structures to ensure they are safe and efficient. It requires a deep understanding of how forces affect buildings and other structures.
Engineers must consider materials, forces, and design principles to achieve stability and safety.

• **Deflection Calculation:** Essential for ensuring beams and other structural elements remain within safe limits under loads.
• **Material Selection:** Choosing the right material can influence deflection and overall structural integrity.
• **Load Analysis:** Predicting and calculating potential loads help in designing structures that withstand various forces without unnecessary deflection.

Structural engineering ensures that structures not only stand firm but also perform as intended under diverse conditions.

Constant of Variation

The constant of variation, often represented as \( k \), is crucial in variation models, representing the factor that relates variables that vary directly or inversely.
For beam deflection, once established for a particular set of dimensions, \( k \) remains consistent across other similar problems.
Determining \( k \) involves substituting known values into the variation formula. As in the exercise:

• If a beam (4-inch-wide, 4-inch-deep) has a 1.1-inch deflection, \( k \) can be calculated: \[1.1 = \frac{k}{4 \cdot 64}\]
• Solving gives \( k = 281.6 \).

With this \( k \), adjusting variables allows predictions of deflections for beams of other configurations, showcasing its importance in designing structures with desired characteristics.