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Chapter 2: Problem 1

The rails of a railroad track are welded together at their ends (to form continuous rails and thus eliminate the clacking sound of the wheels) when the temperature is \(10^{\circ} \mathrm{C}\) What compressive stress \(\sigma\) is produced in the rails when they are heated by the sun to \(52^{\circ} \mathrm{C}\) if the coefficient of thermal expansion \(\alpha=12 \times 10^{-6 /^{\circ} \mathrm{C}}\) and the modulus of elasticity \(E=200 \mathrm{GPa} ?\)

Short Answer

Expert verified
The compressive stress is 100.8 MPa.

Step by step solution

01

Identify the given data

We are given the following data from the problem:- Initial temperature, \( T_i = 10^{\circ} C \)- Final temperature, \( T_f = 52^{\circ} C \)- Coefficient of linear expansion, \( \alpha = 12 \times 10^{-6} \/ ^{\circ} C \)- Modulus of elasticity, \( E = 200 \, \text{GPa} = 200 \times 10^9 \, \text{Pa} \)
02

Determine the temperature change

Calculate the difference between the initial and final temperatures to find the temperature change, \( \Delta T \):\[ \Delta T = T_f - T_i = 52^{\circ} C - 10^{\circ} C = 42^{\circ} C \]
03

Calculate the thermal strain

Thermal strain caused by the temperature change is given by the formula:\[ \text{Thermal Strain} = \alpha \cdot \Delta T \]Substitute the known values:\[ \text{Thermal Strain} = 12 \times 10^{-6} \/^{\circ} C \times 42^{\circ} C = 504 \times 10^{-6} \]
04

Calculate the compressive stress

Using Hooke's Law to relate stress and strain, \( \sigma = E \cdot \text{Thermal Strain} \):\[ \sigma = 200 \times 10^9 \, \text{Pa} \times 504 \times 10^{-6} \]\[ \sigma = 100800 \times 10^3 \, \text{Pa} = 100.8 \, \text{MPa} \]
05

Verify final solution

After calculating, verify that all units are properly converted and equations have been applied correctly. The compressive stress produced due to temperature increase is \( 100.8 \, \text{MPa} \).

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Key Concepts

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

Coefficient of Thermal Expansion
The coefficient of thermal expansion is a measure of how much a material expands when its temperature changes. This property is essential when designing structures that are subject to temperature fluctuations.
Consider materials like metals, which expand as they heat up.
The coefficient of thermal expansion is often denoted by the symbol \( \alpha \), and is expressed in units of per degree Celsius (\( /^{\circ} \mathrm{C} \)).
  • Materials with a high \( \alpha \) will expand more compared to materials with a low \( \alpha \).
  • Designers must accommodate thermal expansion to prevent structural damage, hence its importance.
For example, in the given exercise, the coefficient for the railroad tracks is \( 12 \times 10^{-6} /^{\circ} \mathrm{C} \), indicating it will expand by 12 micrometers per meter for each degree Celsius increase.
Modulus of Elasticity
The modulus of elasticity, also known as Young's modulus, is a material property that measures its stiffness. It describes how much a material will deform under a given load.
Represented by the symbol \( E \), it is typically expressed in pascals (Pa) or gigapascals (GPa).
  • A higher modulus of elasticity means a material is stiffer and will deform less under the same stress compared to a material with a lower modulus.
  • Materials like steel need to have a high modulus to withstand loads without bending or stretching significantly.
In the exercise we're discussing, the modulus of elasticity for the rails is 200 GPa, meaning they are quite stiff and resistant to deformation.
Hooke's Law
Hooke's Law is a principle of physics that shows the relationship between stress and strain for elastic materials. It is represented by the formula \( \sigma = E \cdot \epsilon \), where \( \sigma \) is stress, \( E \) is the modulus of elasticity, and \( \epsilon \) is strain.
  • According to Hooke's Law, the stress experienced by an object is directly proportional to the strain applied, provided the material's elastic limit is not exceeded.
  • This law allows us to calculate how much force is needed to produce a certain amount of deformation, or vice versa.
In the context of our exercise, Hooke's Law was utilized to find the compressive stress in the railroad tracks resulting from thermal expansion.
Temperature Change in Materials
Temperature change affects materials significantly, especially metals, impacting their physical dimensions. When materials are heated, their atoms gain energy and move further apart, causing an expansion.
The degree of expansion or contraction is determined by the material's coefficient of thermal expansion.
  • In our exercise, the temperature of the railroad tracks increased from \( 10^{\circ} \mathrm{C} \) to \( 52^{\circ} \mathrm{C} \), creating a temperature change of \( 42^{\circ} \mathrm{C} \).
  • This temperature change induces thermal strain, where the material's dimensions are altered.
Understanding and predicting these changes is crucial for engineers to prevent structural failures and ensure safety under varying temperature conditions.

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Most popular questions from this chapter

A plastic bar of rectangular cross section \((b=38 \mathrm{mm}\) and \(h=75 \mathrm{mm}\) ) fits snugly between rigid supports at room temperature \(\left(20^{\circ} \mathrm{C}\right)\) but with no initial stress (see figure). When the temperature of the bar is raised to \(70^{\circ} \mathrm{C}\), the compressive stress on an inclined plane \(p q\) at midspan becomes \(8.7 \mathrm{MPa}\) (a) What is the shear stress on plane \(p q ?\) (Assume \(\alpha=95 \times 10^{-6 /^{\circ}} \mathrm{C} \text { and } E=2.4 \mathrm{GPa}\) (b) Draw a stress element oriented to plane \(p q\) and show the stresses acting on all faces of this element. (c) If the allowable normal stress is 23 MPa and the allowable shear stress is \(11.3 \mathrm{MPa}\), what is the maximum load \(P(\text {in }+x \text { direction })\) which can be added at the quarter point (in addition to thermal effects given) without exceeding allowable stress values in the bar?

A sliding collar of weight \(W=650 \mathrm{N}\) falls from a height \(h=50 \mathrm{mm}\) onto a flange at the bottom of a slender vertical rod (see figure). The rod has length \(L=1.2 \mathrm{m}\) cross-sectional area \(A=5 \mathrm{cm}^{2},\) and modulus of elasticity \(E=210 \mathrm{GPa}\) Calculate the following quantities: (a) the maximum downward displacement of the flange, (b) the maximum tensile stress in the rod, and (c) the impact factor.

A brass wire of diameter \(d=1.6 \mathrm{mm}\) is stretched between rigid supports with an initial tension \(T\) of \(200 \mathrm{N}\) (see figure). (Assume that the coefficient of thermal expansion is \(21.2 \times 10^{-6 / 0} \mathrm{C}\) and the modulus of elasticity is 110 GPa. (a) If the temperature is lowered by \(30^{\circ} \mathrm{C}\), what is the maximum shear stress \(\tau_{\max }\) in the wire? (b) If the allowable shear stress is \(70 \mathrm{MPa}\), what is the maximum permissible temperature drop? (c) At what temperature change \(\Delta T\) does the wire go slack?

Two identical bars \(A B\) and \(B C\) support a vertical load \(P\) (see figure). The bars are made of steel having a stress-strain curve that may be idealized as elastoplastic with yield stress \(\sigma_{Y}\) Each bar has cross-sectional area \(A\) Determine the yield load \(P_{Y}\) and the plastic load \(P_{P^{-}}\)

A post \(A B\) supporting equipment in a laboratory is tapered uniformly throughout its height \(H\) (see figure). The cross sections of the post are square, with dimensions \(b \times b\) at the top and \(1.5 b \times 1.5 b\) at the base. Derive a formula for the shortening \(\delta\) of the post due to the compressive load \(P\) acting at the top. (Assume that the angle of taper is small and disregard the weight of the post itself.
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