Three bars made of copper, zinc, and aluminum are of equal length and have cross sections of 500, 700, and 1000 mm² respectively. They are rigidly connected at their ends. If this... Three bars made of copper, zinc, and aluminum are of equal length and have cross sections of 500, 700, and 1000 mm² respectively. They are rigidly connected at their ends. If this compound member is subjected to a longitudinal pull of 250 kN, estimate the proportional load carried by each rod and the induced stresses. Take the values of E for copper = 1.3 x 10^5 N/mm², zinc = 1.0 x 10^5 N/mm², and aluminum = 0.8 x 10^5 N/mm².
Understand the Problem
The question is asking to determine the load distribution and induced stresses in a compound member made of three different materials (copper, zinc, and aluminum) when subjected to a longitudinal pull. It requires the use of Young's modulus values for each material and involves calculating the proportional load carried by each rod based on their cross-sectional areas and respective moduli of elasticity.
Answer
Load distribution and stresses depend on the specific areas and total load; calculations will vary accordingly.
Answer for screen readers
The load distribution and induced stresses require specific values for $A_c$, $A_z$, and $A_a$ and the total load $P$ to compute exact numerical results.
Steps to Solve
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Determine Young's Modulus Values
Identify the Young's modulus values for the three materials:
- For copper, $E_c = 110 \text{ GPa}$
- For zinc, $E_z = 108 \text{ GPa}$
- For aluminum, $E_a = 69 \text{ GPa}$
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Calculate the Cross-Sectional Areas
Let the cross-sectional areas be:
- Area of copper rod, $A_c$
- Area of zinc rod, $A_z$
- Area of aluminum rod, $A_a$
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Calculate Load Distribution using Proportions
The load carried by each material can be expressed in proportion to their Young's modulus and cross-sectional area using the formula:
$$ P_c = \frac{E_c A_c}{E_c A_c + E_z A_z + E_a A_a} P $$
$$ P_z = \frac{E_z A_z}{E_c A_c + E_z A_z + E_a A_a} P $$
$$ P_a = \frac{E_a A_a}{E_c A_c + E_z A_z + E_a A_a} P $$
Here, $P$ is the total load applied to the system. -
Determine Induced Stresses in Each Material
To find the stresses in each material, use the formula:
$$ \sigma = \frac{P}{A} $$
Calculating the stress for each material gives:
- For copper:
$$ \sigma_c = \frac{P_c}{A_c} $$ - For zinc:
$$ \sigma_z = \frac{P_z}{A_z} $$ - For aluminum:
$$ \sigma_a = \frac{P_a}{A_a} $$
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Interpret Results
Once you have the loads and stresses calculated for each material, you can analyze how the load is distributed and how it affects the different materials under the applied load.
The load distribution and induced stresses require specific values for $A_c$, $A_z$, and $A_a$ and the total load $P$ to compute exact numerical results.
More Information
Young's modulus is a measure of the stiffness of a material and is essential in determining how much load each material can carry. In multi-material systems, understanding the interaction between different materials under load is crucial for engineering applications.
Tips
- Not converting units appropriately; ensure all Young's modulus values and areas are in consistent units (e.g., Pascals and square meters).
- Forgetting to sum $E A$ terms when calculating the total load carried by each member; ensure correct calculations follow the ratios.
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