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Chapter 1: Problem 62

Perform the indicated operations. In designing a building, it was determined that the forces acting on an I beam would deflect the beam an amount (in \(\mathrm{cm}\) ), given by \(\frac{x\left(1000-20 x^{2}+x^{3}\right)}{1850},\) where \(x\) is the distance (in \(m\) ) from one end of the beam. Find the deflection for \(x=6.85 \mathrm{m}\). (The 1000 and 20 are exact.)

Short Answer

Expert verified
The deflection is approximately 1.4183 cm.

Step by step solution

01

Understand the Given Expression

The formula for deflection is given by \( f(x) = \frac{x\left(1000 - 20x^2 + x^3\right)}{1850} \). We need to plug in \( x = 6.85 \) to find the deflection at that point.
02

Substitute \( x = 6.85 \) into the Expression

Replace \( x \) with 6.85 in the expression: \( f(6.85) = \frac{6.85 \times (1000 - 20 \times (6.85)^2 + (6.85)^3)}{1850} \).
03

Calculate Each Term Inside the Parentheses

Compute \( 20 \times (6.85)^2 \) and \((6.85)^3 \).- \((6.85)^2 = 46.9225\).- \(20 \times 46.9225 = 938.45\).- \((6.85)^3 = 321.493125\).
04

Calculate the Expression Inside the Parentheses

Substitute the computed values back in:\[ 1000 - 938.45 + 321.493125 = 383.043125 \]
05

Complete the Multiplication

Now, multiply \( 6.85 \) by the result from Step 4: \( 6.85 \times 383.043125 = 2623.84340625 \).
06

Divide by the Denominator

Finally, divide the product by 1850: \( \frac{2623.84340625}{1850} \approx 1.4183 \).
07

Interpret the Result

The deflection of the beam at \( x = 6.85 \) meters is approximately 1.4183 cm.

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Key Concepts

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

Deflection
Deflection in the context of beam analysis refers to the degree to which a structural element, such as a beam, will bend under a load. When forces act upon a beam, it will naturally undergo some bending or flexing. This bending is crucial to consider in engineering, as too much deflection can compromise structural integrity.
  • In practical terms, deflection is measured in units of length, often centimeters or meters, and indicates how much a point on the beam moves from its original position.
  • Ensuring that deflection remains within safe limits is critical to the engineering process for ensuring building safety and stability.
A small deflection might be insignificant, but engineers must keep deflection within a specific range to maintain the function and safety of the structure. Calculating the deflection is a crucial step in the design and analysis of structures.
Beam Analysis
Beam analysis is a subset of structural analysis that focuses on understanding how beams behave under various forces. Beams are crucial components in construction, acting as structural elements that bear loads and help distribute them across a structure.
  • The analysis involves calculating stresses, reactions, and deflections of the beams.
  • It provides insights into how a beam will respond under load, including how much it will bend or deflect.
Beam analysis helps engineers ensure that the design will support the intended loads without failing. It is an essential part of ensuring that structures like bridges or buildings remain stable under normal use and during unexpected loads, such as earthquakes or strong winds.
I Beam
An I beam is a specific structural element often used in construction due to its superior strength-to-weight ratio. Named for its characteristic cross-sectional shape that resembles the letter "I," it offers several benefits.
  • I beams are designed to bear loads primarily in one direction and excel in supporting bending loads.
  • The design optimizes material distribution, allowing it to resist bending efficiently with less material.
The top and bottom flanges of an I beam spread pressure over a wider area, while the vertical "web" resists shear forces. This combination provides excellent load-bearing capabilities while minimizing the amount of material used, making I beams a cost-effective choice in building construction.
Mathematical Modeling
Mathematical modeling involves using mathematical expressions to represent real-world systems in a simplified form. In the case of beam deflection analysis, mathematical modeling allows engineers to predict how a beam will behave under certain conditions.
  • These models are built using known equations, like the formula for beam deflection, and take into account the material properties and geometry of the beam.
  • Models help perform complex calculations efficiently, allowing engineers to simulate different scenarios.
By using mathematical models, engineers can test hypothetical changes in design and predict how these changes would affect deflection, stress, and other factors. This approach aids in creating more efficient, safer structures without the need for extensive physical testing.

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