Thermal Expansion of wires within a structure - see picture

[Xand888]

Homework Statement

dB = 12mm (0.012m), dC = 20mm (0.02m),
E = 140GPa and the coefficient of thermal expansion, α = 12e-6/°C.

Homework Equations

δ = (R*L)/(E*A)

The Attempt at a Solution

Please see picture for my attempt.

I don't know how the thermal expansion, which creates a thermal strain, will affect the moment balance or compatibility relationship. I know that the factor of safety will mean I'd be dividing the ultimate load by 5. Would the ultimate load be twice the static load?

Any help would be greatly appreciated!

 

[KataKoniK]

Hello, thank you for your post. Let's break down the problem and see if we can come up with a solution.

First, let's start with the given values: dB = 12mm, dC = 20mm, E = 140GPa, and α = 12e-6/°C. From these values, we can calculate the change in length (δ) of the bar using the formula δ = α*ΔT*L, where α is the coefficient of thermal expansion, ΔT is the change in temperature, and L is the original length of the bar.

In this case, we are given the change in length of the bar (dB and dC), so we can rearrange the formula to solve for the change in temperature (ΔT). This gives us ΔT = δ/(α*L).

Now, let's consider the compatibility relationship. This relationship states that the total change in length of a structure must be equal to the sum of the changes in length of its individual components. In this case, we have two components: the bar with length dB and the bar with length dC. Therefore, the compatibility relationship can be written as:

δ = δB + δC

Substituting in the values we calculated earlier, we get:

α*ΔT*L = α*ΔTB*dB + α*ΔTC*dC

Now, let's consider the moment balance. This relationship states that the sum of the moments acting on a structure must be equal to zero. In this case, we have a force acting on the bar with length dB (from the weight of the object) and a force acting on the bar with length dC (from the reaction force at the support). Therefore, the moment balance can be written as:

(Force on dC)*(Length of dC) - (Force on dB)*(Length of dB) = 0

Substituting in the values we calculated earlier, we get:

(F*dC) - (F*dB) = 0

Solving for F, we get:

F = dC/dB

Now, we can use this value of F to calculate the ultimate load (P) using the formula P = (Factor of Safety)*F. In this case, the factor of safety is given as 5, so we get:

P = 5*dC/dB

Finally, to answer your question about the ultimate

 

[Mene]

I would like to first clarify that the provided information is not sufficient to fully analyze the thermal expansion of wires within a structure. In order to accurately determine the effects of thermal expansion, we would need to know the material and dimensions of the wires, as well as the temperature change and the overall structure's geometry and boundary conditions. However, based on the given information, I can provide some general insights and suggestions for further analysis.

Firstly, the thermal expansion of wires within a structure can have significant effects on the structural integrity and stability of the overall system. When the temperature changes, the wires will expand or contract, causing changes in their length and resulting in thermal strains. These strains can lead to stresses and deformations in the wires, which can affect the overall behavior of the structure.

To analyze the effects of thermal expansion, we can use the equation provided in the homework statement, δ = (R*L)/(E*A), where δ is the thermal strain, R is the coefficient of thermal expansion, L is the change in length, E is the Young's modulus of the wire material, and A is the cross-sectional area of the wire. Using this equation, we can calculate the thermal strain for each wire and determine if it will cause any significant changes in length.

Furthermore, we can also consider the effects of thermal expansion on the overall structure's stability and compatibility. If the wires are connected to other structural elements, their expansion or contraction may cause changes in the forces and moments acting on those elements. This can affect the overall equilibrium and stability of the structure, and it is important to consider these effects in the design and analysis of the structure.

In terms of the factor of safety, it is not clear how it would be affected by thermal expansion. The ultimate load would not necessarily be twice the static load, as it depends on the specific behavior and properties of the structure and its components. It may be necessary to perform further analysis or experiments to determine the ultimate load under thermal expansion conditions.

In conclusion, thermal expansion of wires within a structure can have significant effects on its behavior and stability. It is important to consider these effects in the design and analysis of structures, and further analysis may be necessary to accurately determine the impact of thermal expansion on a specific system.