To determine the vertical displacement of joint A, we can use the formula for thermal expansion:
ΔL = αLΔT
where ΔL is the change in length of the bar, α is the coefficient of thermal expansion, L is the original length of the bar, and ΔT is the change in temperature.
Assuming that the bars are initially at room temperature (T1=50∘F), we can calculate the original length of each bar using the formula:
L = √(h^2 + w^2)
where h and w are the height and width of the truss, respectively. From the diagram, we can see that h = 20 ft and w = 30 ft, so:
L = √(20^2 + 30^2) = 36.06 ft
Next, we can calculate the change in temperature as:
ΔT = T2 - T1 = 150 - 50 = 100 °F
The coefficient of thermal expansion for steel is α = 6.5 × 10^-6 in/in/°F. We need to convert the length of each bar from feet to inches, so:
L = 36.06 ft × 12 in/ft = 432.72 in
Using the formula for ΔL, we can calculate the change in length of each bar:
ΔL = αLΔT = (6.5 × 10^-6 in/in/°F) × (432.72 in) × (100 °F) = 0.2814 in
Since joint A is the midpoint of the bottom chord, it is supported by two bars. The total change in length of the bars supporting joint A is therefore:
ΔLtotal = 2 × ΔL = 2 × 0.2814 in = 0.5628 in
Since the bars are in tension, their elongation will cause joint A to move downward by an equal amount. Therefore, the vertical displacement of joint A is:
Δy = -ΔLtotal = -0.5628 in
So the answer is that joint A will move downward by 0.5628 inches when the temperature changes from 50°F to 150°F.