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Chapter 5: Problem 5

Each girder of the lift bridge (see figure) is \(50 \mathrm{m}\) long and simply supported at the cnds. The design load for each girder is a uniform load of intensity \(18 \mathrm{kN} / \mathrm{m}\). The girders are fabricated by welding three steel plates so as to form an I-shaped cross section (see figure) having section modulus \(S=46,000 \mathrm{cm}^{3}\) What is the maximum bending stress \(\sigma_{\max }\) in a girder due to the uniform load?

Short Answer

Expert verified
The maximum bending stress in the girder is approximately \(3057.07 \, \mathrm{kPa}\).

Step by step solution

01

Calculate the Total Load

Given the uniform load intensity of \(18 \mathrm{kN} / \mathrm{m}\), and the length of the girder is \(50 \mathrm{m}\), the total load \(w\) on the girder is calculated as:\[w = 18 \, \mathrm{kN/m} \times 50 \, \mathrm{m} = 900 \, \mathrm{kN}\]
02

Determine the Bending Moment

For a simply supported beam with a uniform load, the maximum bending moment \(M\) occurs at the midpoint of the beam. It is given by the formula:\[M = \frac{wL^2}{8}\]Substituting the values, we have:\[M = \frac{900 \, \mathrm{kN} \times (50 \, \mathrm{m})^2}{8} = 140625 \, \mathrm{kNm}\]
03

Convert the Moment to the Correct Units

Convert the bending moment from \(\mathrm{kNm}\) to \(\mathrm{Nm}\), because 1 \(\mathrm{kN} = 1000 \, \mathrm{N}\):\[M = 140625 \, \mathrm{kNm} = 140625000 \, \mathrm{Nm}\]
04

Calculate Maximum Bending Stress

The maximum bending stress \(\sigma_{\max }\) is given by:\[\sigma_{\max } = \frac{M}{S}\]where \(S = 46000 \, \mathrm{cm}^3 = 46000 \times 10^{-6} \, \mathrm{m}^3\)Substituting the values, we have:\[\sigma_{\max } = \frac{140625000 \, \mathrm{Nm}}{46000 \times 10^{-6} \, \mathrm{m}^3} = 3057.07 \, \mathrm{kPa}\]
05

Conclude the Maximum Bending Stress

Thus, the maximum bending stress \(\sigma_{\max }\) in the girder is approximately:\[3057.07 \, \mathrm{kPa}\]

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Key Concepts

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

Simply Supported Beam
A simply supported beam is one of the most common types of structural elements used in construction and engineering. This beam is supported at both ends and is free to move longitudinally at the supports. The supports typically consist of a pin support at one end, which allows rotation, and a roller support at the other end, which allows both rotation and horizontal translation. Simply supported beams are valuable because they are straightforward to analyze. They effectively handle loads distributed along their length, which makes them a reliable choice for many practical applications. In the case of a simple span, when a load is applied, the maximum bending moment occurs at the midpoint of the beam span. When dealing with simply supported beams, understanding the types of loads applied and their locations is crucial in determining how the beam will behave under different stresses, such as bending stress.
Uniform Load
A uniform load is a type of load distribution commonly found in structural analysis. It means that the load is spread evenly across the entire span of the beam. In the given exercise, the uniform load is characterized by an intensity of 18 kN/m. This intensity indicates that every meter of the beam is subjected to a force of 18 kilonewtons. Uniform loads result in predictable forces and moments along the beam, simplifying the calculation of stresses, such as bending stress. The bending moment for a simply supported beam with a uniform load reaches its maximum at the center of the beam. For engineering purposes, knowing how to handle uniform loads efficiently is critical for designing structures that remain both safe and functional under different load conditions.
Section Modulus
The section modulus is a geometric property of the cross-section of a beam. It is vital in the design and analysis of beams and other structures subject to bending. The section modulus ( S ) gives a measure of the ability of a beam to resist bending. Calculating the maximum bending stress in a beam involves the section modulus along with the bending moment. The higher the section modulus, the greater the resistance of the cross-section to bending. It is expressed in cubic centimeters (cm鲁) or cubic meters (m鲁). In this exercise, the section modulus is given as 46,000 cm鲁. The section modulus is calculated based on the shape and size of the cross-section, and it plays a critical role in determining the bending strength of beams, particularly in complex shape designs like I-sections.
I-shaped Cross Section
An I-shaped cross-section, also known as an I-beam, is a widely used beam design in construction due to its high bending strength and efficiency. The I-section consists of a vertical "web" and horizontal "flanges" at the top and bottom. These features provide superior strength and stiffness with minimal use of material. Fabricating an I-section typically involves welding or forming steel plates into shape. The girder in the exercise composite of three steel plates, which together form this I-shaped configuration. This design allows the beam to carry heavy loads with reduced weight, making it cost-effective in terms of material usage. The geometry of the I-section also influences the section modulus. The wider the flange or the thicker the web, the greater the section modulus, leading to improved performance under bending stresses. This shape is ideal for scenarios where a beam must span a large distance while supporting uniform or concentrated loads.

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Most popular questions from this chapter

A so-called "trapeze bar" in a hospital room provides a means for paticnts to cxercise while in bed (sec figure). The bar is \(2.1 \mathrm{m}\) long and has a cross section in the shape of a regular octagon. The design load is \(1.2 \mathrm{kN}\) applied at the midpoint of the bar, and the allowable bending stress is \(200 \mathrm{MPa}\) Determine the minimum height \(h\) of the bar. (Assume that the ends of the bar are simply supported and that the weight of the bar is negligible.)

A wood beam \(A B C\) with simple supports at \(A\) and \(B\) and an overhang \(B C\) has height \(h=300 \mathrm{mm}\) (see figure). The length of the main span of the beam is \(L=3.6 \mathrm{m}\) and the length of the overhang is \(L / 3=1.2 \mathrm{m}\). The beam supports a concentrated load \(3 P=18 \mathrm{kN}\) at the midpoint of the main span and a moment \(P L / 2=10.8 \mathrm{kN} \cdot \mathrm{m}\) at the free end of the overhang. The wood has weight density \(\gamma=5.5 \mathrm{kN} / \mathrm{m}^{3}\) (a) Determine the required width \(b\) of the beam based upon an allowable bending stress of \(8.2 \mathrm{MPa}\) (b) Determine the required width based upon an allowable shear stress of \(0.7 \mathrm{MPa}\)

A cantilever beam \(A B\) having rectangular cross sections with varying width \(b_{x}\) and varying height \(h_{x}\) is subjected to a uniform load of intensity \(q\) (see figure). If the width varies linearly with \(x\) according to the equation \(b_{x}=b_{B} x / L,\) how should the height \(h_{x}\) vary as a function of \(x\) in order to have a fully stressed beam? (Express \(h_{x}\) in terms of the height \(h_{B}\) at the fixed end of the beam.)

A cantilever beam \(A B\) with a rectangular cross section has a longitudinal hole drilled throughout its length (see figure). The beam supports a load \(P=600 \mathrm{N}\). The cross section is \(25 \mathrm{mm}\) wide and \(50 \mathrm{mm}\) high, and the hole has a diameter of \(10 \mathrm{mm}\) Find the bending stresses at the top of the beam, at the top of the hole, and at the bottom of the beam.

A cylindrical brick chimney of height \(H\) weighs \(w=12 \mathrm{kN} / \mathrm{m}\) of height (see figure). The inner and outer diameters are \(d_{1}=0.9 \mathrm{m}\) and \(d_{2}=1.2 \mathrm{m},\) respectively. The wind pressure against the side of the chimney is \(p=480 \mathrm{N} / \mathrm{m}^{2}\) of projected area. Determine the maximum height \(H\) if there is to be no tension in the brickwork.
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