Two parallel bars (bronze and copper), equal length: Identify the correct combined relations among total load, shared loads, stress, and modulus

Introduction:
This problem concerns load sharing in a composite member made of two prismatic bars (bronze and copper) connected in parallel (equal length, loaded together). The key concepts are compatibility of deformation and equilibrium of forces, which together determine the stresses and load carried by each bar.

Given Data / Assumptions:

• Two bars: bronze (area Ab, modulus Eb) and copper (area Ac, modulus Ec).
• Equal initial length and common axial deformation (parallel arrangement).
• Total external load is P, shared as Pb (bronze) and Pc (copper).
• Linear elastic behavior and perfect bonding (same axial strain).

Concept / Approach:
Compatibility in parallel requires equal axial strain: εb = εc. Using Hooke’s law, ε = σ/E, so σb/Eb = σc/Ec. Force equilibrium requires total load P to equal the sum of the component loads: P = Pb + Pc. The load in each bar equals its area times its stress: Pb = Abσb and Pc = Acσc.

Step-by-Step Solution:
Write compatibility: σb/Eb = σc/Ec (equal strain).Use equilibrium: P = Pb + Pc.Relate stress to load in each bar: Pb = Abσb and Pc = Acσc.Combine these statements to identify the option that lists all correct relations.

Verification / Alternative check:
If Eb ≠ Ec, equal strain implies different stresses. When Ab and Ac are known, the system of equations can be solved for σb and σc for any given P, confirming internal consistency.

Why Other Options Are Wrong:
Option b: Incorrect—equal lengths in parallel enforce equal strain, not unequal.Option c: Uses σb twice; misses σc term, so it violates Pb + Pc definition.Option d: Equal areas do not force equal loads unless Eb also matches; stiffness matters.Option e: Equal stress would only occur if Eb = Ec under equal strain; not generally true.

Common Pitfalls:
Assuming equal stress instead of equal strain for members in parallel; overlooking the role of modulus in load sharing; mixing series vs parallel rules.

Final Answer:
σb/Eb = σc/Ec, P = Pb + Pc, Pb = Abσb, Pc = Acσc.