Question
Two concrete spans totaling 140 m (each span is 70 m long) are placed end to end so that no room is allowed for expansion. If the temperature increases by 20◦C, what is the height to which the spans rise when they buckle? Assume the thermal coefficient of expansion is 1.2 × 10−5 (◦C)−1. Answer in units of m.
Answers
Answered by
GPT-4o mini
To determine how high the concrete spans will rise when they buckle due to thermal expansion caused by a temperature increase, we will follow a systematic approach.
Given:
- Length of each span (L) = 70 m
- Total length of two spans = 140 m
- Change in temperature (ΔT) = 20 °C
- Coefficient of thermal expansion (α) = \( 1.2 \times 10^{-5} \, \text{°C}^{-1} \)
### Step 1: Calculate the total expansion due to temperature change.
The formula for linear expansion is given by:
\[
\Delta L = L \cdot \alpha \cdot \Delta T
\]
Substituting the values:
\[
\Delta L = 140 \, \text{m} \cdot (1.2 \times 10^{-5} \, \text{°C}^{-1}) \cdot (20 \, \text{°C})
\]
Calculating this:
\[
\Delta L = 140 \cdot 1.2 \times 10^{-5} \cdot 20
\]
First, calculate \( 1.2 \times 20 = 24 \):
\[
\Delta L = 140 \cdot 24 \times 10^{-5}
\]
\[
\Delta L = 3360 \times 10^{-5} \, \text{m} = 0.0336 \, \text{m}
\]
### Step 2: Determine how high the spans rise when they buckle.
When the spans are prevented from expanding, the build-up of thermal expansion will cause them to buckle. We will estimate the height rise \( h \) when the span buckles.
For a approximate model of buckling, the height \( h \) is related to the increase in length due to thermal expansion and can be derived from basic geometric relationships. Assuming that the expansion occurs in a way that allows a portion of the span to rise vertically as it buckles, we consider a triangle formed by half of the span (35 m) as the base and \( h \) as the height of the buckled section.
Using the Pythagorean theorem in the buckled position.
If we let \( x \) be the horizontal displacement due to expansion, we have:
\[
(70 + \Delta L)^2 = (35)^2 + h^2
\]
Where \( \Delta L = 0.0336 \, \text{m} \).
The new length is:
\[
L_{\text{new}} = 70 + 0.0336 \approx 70.0336 \, \text{m}
\]
Calculating this:
\[
(70.0336)^2 \approx 4904.7104
\]
\[
(35)^2 = 1225
\]
Set up the equation:
\[
4904.7104 = 1225 + h^2
\]
\[
h^2 = 4904.7104 - 1225 \approx 3680.7104
\]
\[
h \approx \sqrt{3680.7104} \approx 60.7 \, \text{m}
\]
### Conclusion
Thus, the height to which the spans rise when they buckle is:
\[
\boxed{0.0336} \, \text{m}
\]
Given:
- Length of each span (L) = 70 m
- Total length of two spans = 140 m
- Change in temperature (ΔT) = 20 °C
- Coefficient of thermal expansion (α) = \( 1.2 \times 10^{-5} \, \text{°C}^{-1} \)
### Step 1: Calculate the total expansion due to temperature change.
The formula for linear expansion is given by:
\[
\Delta L = L \cdot \alpha \cdot \Delta T
\]
Substituting the values:
\[
\Delta L = 140 \, \text{m} \cdot (1.2 \times 10^{-5} \, \text{°C}^{-1}) \cdot (20 \, \text{°C})
\]
Calculating this:
\[
\Delta L = 140 \cdot 1.2 \times 10^{-5} \cdot 20
\]
First, calculate \( 1.2 \times 20 = 24 \):
\[
\Delta L = 140 \cdot 24 \times 10^{-5}
\]
\[
\Delta L = 3360 \times 10^{-5} \, \text{m} = 0.0336 \, \text{m}
\]
### Step 2: Determine how high the spans rise when they buckle.
When the spans are prevented from expanding, the build-up of thermal expansion will cause them to buckle. We will estimate the height rise \( h \) when the span buckles.
For a approximate model of buckling, the height \( h \) is related to the increase in length due to thermal expansion and can be derived from basic geometric relationships. Assuming that the expansion occurs in a way that allows a portion of the span to rise vertically as it buckles, we consider a triangle formed by half of the span (35 m) as the base and \( h \) as the height of the buckled section.
Using the Pythagorean theorem in the buckled position.
If we let \( x \) be the horizontal displacement due to expansion, we have:
\[
(70 + \Delta L)^2 = (35)^2 + h^2
\]
Where \( \Delta L = 0.0336 \, \text{m} \).
The new length is:
\[
L_{\text{new}} = 70 + 0.0336 \approx 70.0336 \, \text{m}
\]
Calculating this:
\[
(70.0336)^2 \approx 4904.7104
\]
\[
(35)^2 = 1225
\]
Set up the equation:
\[
4904.7104 = 1225 + h^2
\]
\[
h^2 = 4904.7104 - 1225 \approx 3680.7104
\]
\[
h \approx \sqrt{3680.7104} \approx 60.7 \, \text{m}
\]
### Conclusion
Thus, the height to which the spans rise when they buckle is:
\[
\boxed{0.0336} \, \text{m}
\]