Standard maximum deflection formulas (small deflection, prismatic beams) Which of the following expressions for maximum deflection δ_max are correct?
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ACantilever with end point load W: δ_max = W * L^3 / (3 * E * I)
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BSimply supported with midspan point load W: δ_max = W * L^3 / (48 * E * I)
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CCantilever with UDL w over full span: δ_max = w * L^4 / (8 * E * I)
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DSimply supported with UDL w over full span: δ_max = 5 * w * L^4 / (384 * E * I)
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EAll of the above
Answer
Correct Answer: All of the above
Explanation
Introduction / Context:Deflection control is essential for serviceability. Memorizing the classic small-deflection formulas for common loading cases saves time during preliminary design and checks.
Given Data / Assumptions:
- Slender, prismatic beams obeying Euler–Bernoulli theory.
- Small deflections; linear material behavior.
- Loads as stated (point load W or uniformly distributed load w).
Concept / Approach:The listed formulas are standard closed-form solutions obtained by integrating the differential equation E * I * d^2y/dx^2 = M(x) with appropriate boundary conditions for each case.
Step-by-Step Solution (outline):
For each load case, write bending moment function M(x).Integrate twice, apply boundary conditions to get y(x).Locate δ_max and evaluate the closed-form magnitude; results match the expressions in options a–d.Verification / Alternative check:Cross-reference with standard tables (Roark, structural handbooks) or quick energy methods (Castigliano’s theorem) which yield identical results.
Why Other Options Are Wrong:Only option e collects the correct set; each of a–d is independently correct, so “All of the above” is the best answer.
Common Pitfalls:Mixing the constants (3 vs 8 vs 48 vs 384); remembering that cantilevers deflect more than simply supported beams for similar loading is a useful check.
Final Answer:All of the above