A beam is said to be of uniform strength when which internal response remains constant along its length under the given loading?

Introduction / Context:
The concept of a beam of uniform strength guides tapered or profiled beams where the cross-section varies so that material is used efficiently; the maximum allowable stress is reached everywhere along the beam.

Given Data / Assumptions:

• Beam may have varying cross-section.
• Allowable stress is the design limit.
• Loading and support conditions define the bending moment diagram.

Concept / Approach:
Uniform strength means the extreme-fiber bending stress σ = M / Z is constant along the member. Since M varies with x, Z must vary so that M/Z remains constant, leading to a variable section (e.g., parabolic depth variation for UDL).

Step-by-Step Solution:
Define target stress σ_allow.At any section: σ_allow = M(x) / Z(x).Thus choose Z(x) ∝ M(x) to keep σ constant.For rectangular sections, Z ∝ b h^2; shaping h(x) to match M(x) achieves uniform strength.

Verification / Alternative check:
Check that at each x, computed σ(x) equals σ_allow within tolerance; this confirms uniform strength.

Why Other Options Are Wrong:

• Bending moment same throughout: rarely true except for trivial cases; not the definition.
• Deflection same throughout: physically impossible; deflection is zero at fixed supports, etc.
• Shear stress same throughout: not a standard design goal for uniform strength.

Common Pitfalls:

• Confusing constant bending stress with constant bending moment.

Final Answer:
bending stress is same throughout the beam