Question
Two concrete spans of a 140 m long bridge 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 the height to which the concrete spans rise when they buckle due to thermal expansion, we need to compute the amount by which the spans would expand without room for expansion and then relate that to the buckling.
1. **Calculate the total length contraction due to temperature increase**:
The linear expansion of the spans due to temperature change can be calculated using the formula:
\[
\Delta L = L_0 \cdot \alpha \cdot \Delta T
\]
where:
- \( \Delta L \) is the change in length,
- \( L_0 \) is the original length of the spans,
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta T \) is the change in temperature.
Given:
- \( L_0 = 140 \, \text{m} \) (the total length of the bridge),
- \( \alpha = 1.2 \times 10^{-5} \, (\text{°C})^{-1} \),
- \( \Delta T = 20 \, \text{°C} \).
Substituting in the values, we compute:
\[
\Delta L = 140 \, \text{m} \cdot 1.2 \times 10^{-5} \, (\text{°C})^{-1} \cdot 20 \, \text{°C}
\]
\[
\Delta L = 140 \, \text{m} \cdot 1.2 \times 10^{-5} \cdot 20
\]
\[
\Delta L = 140 \cdot 1.2 \cdot 20 \times 10^{-5}
\]
\[
\Delta L = 3360 \times 10^{-5} = 0.0336 \, \text{m}
\]
2. **Determine height of buckling due to constraint**:
When the spans cannot expand linearly, the extra length must be accommodated laterally. For small deflections, the rise \( h \) of a laterally constrained beam (due to buckling) can be approximated by considering it's a circular arc, leading to the formula:
\[
h = \frac{L^2}{8R}
\]
where \( L \) is the horizontal length of the span or what would be the expanded length, and \( R \) is the radius of curvature needed to accommodate the length increase.
For buckling, we can first work with:
\[
R \approx \frac{\Delta L}{\Delta h}
\]
The horizontal span that would accommodate \( \Delta L \) can relate to the height rise \( h \) via \( \Delta L \) being equal to the horizontal movement that would create the height \( h \):
Reading this, with the assumption of moderate height (small angle approximation), we say:
\[
R = \frac{L^2}{8h}
\]
Rearranging gives a relationship to express height and surface length change:
\[
h = \frac{L^2}{8 \Delta L}
\]
With this, we can compute \( h \):
Set \( L = 140 m \) from the total bridge length and \( \Delta L = 0.0336 \, m \):
\[
h = \frac{(140)^2}{8 \cdot 0.0336} = \frac{19600}{0.2688}
\]
Calculate it:
\[
h \approx 72938.88 \approx 0.72939 \, \text{m}
\]
So, the height to which the spans rise when they buckle due to the temperature increase is approximately:
\[
\boxed{0.72939 \, \text{m}}
\]
1. **Calculate the total length contraction due to temperature increase**:
The linear expansion of the spans due to temperature change can be calculated using the formula:
\[
\Delta L = L_0 \cdot \alpha \cdot \Delta T
\]
where:
- \( \Delta L \) is the change in length,
- \( L_0 \) is the original length of the spans,
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta T \) is the change in temperature.
Given:
- \( L_0 = 140 \, \text{m} \) (the total length of the bridge),
- \( \alpha = 1.2 \times 10^{-5} \, (\text{°C})^{-1} \),
- \( \Delta T = 20 \, \text{°C} \).
Substituting in the values, we compute:
\[
\Delta L = 140 \, \text{m} \cdot 1.2 \times 10^{-5} \, (\text{°C})^{-1} \cdot 20 \, \text{°C}
\]
\[
\Delta L = 140 \, \text{m} \cdot 1.2 \times 10^{-5} \cdot 20
\]
\[
\Delta L = 140 \cdot 1.2 \cdot 20 \times 10^{-5}
\]
\[
\Delta L = 3360 \times 10^{-5} = 0.0336 \, \text{m}
\]
2. **Determine height of buckling due to constraint**:
When the spans cannot expand linearly, the extra length must be accommodated laterally. For small deflections, the rise \( h \) of a laterally constrained beam (due to buckling) can be approximated by considering it's a circular arc, leading to the formula:
\[
h = \frac{L^2}{8R}
\]
where \( L \) is the horizontal length of the span or what would be the expanded length, and \( R \) is the radius of curvature needed to accommodate the length increase.
For buckling, we can first work with:
\[
R \approx \frac{\Delta L}{\Delta h}
\]
The horizontal span that would accommodate \( \Delta L \) can relate to the height rise \( h \) via \( \Delta L \) being equal to the horizontal movement that would create the height \( h \):
Reading this, with the assumption of moderate height (small angle approximation), we say:
\[
R = \frac{L^2}{8h}
\]
Rearranging gives a relationship to express height and surface length change:
\[
h = \frac{L^2}{8 \Delta L}
\]
With this, we can compute \( h \):
Set \( L = 140 m \) from the total bridge length and \( \Delta L = 0.0336 \, m \):
\[
h = \frac{(140)^2}{8 \cdot 0.0336} = \frac{19600}{0.2688}
\]
Calculate it:
\[
h \approx 72938.88 \approx 0.72939 \, \text{m}
\]
So, the height to which the spans rise when they buckle due to the temperature increase is approximately:
\[
\boxed{0.72939 \, \text{m}}
\]