For a simply supported beam carrying a uniformly distributed load of w per unit length over the entire span, what is the qualitative shape of the shear force diagram?
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AA single right-angled triangle (linear variation across the span)
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BTwo right-angled triangles
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COne equilateral triangle
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DTwo equilateral triangles
Answer
Correct Answer: A single right-angled triangle (linear variation across the span)
Explanation
Introduction / Context:Internal shear force and bending moment diagrams are fundamental for beam design. Recognising their shapes for common loadings lets engineers size members quickly and check results by inspection.
Given Data / Assumptions:
- Simply supported beam (pin and roller supports).
- Uniformly distributed load (UDL) = w per unit length across entire span L.
- Static, prismatic beam in the usual sign convention (upward reactions positive).
Concept / Approach:Shear force V(x) equals the algebraic sum of vertical forces to the left (or right). Under a UDL, dV/dx = −w, so V varies linearly. For symmetry, reactions are equal (wL/2 each). The diagram is a straight line decreasing from +wL/2 at the left support to −wL/2 at the right support—i.e., a right-angled triangle.
Step-by-Step Solution:
Compute reactions: R_A = R_B = wL/2.Integrate load: dV/dx = −w → V(x) = R_A − w x.Evaluate endpoints: V(0) = +wL/2; V(L) = −wL/2.Plot linear variation → triangular SFD.Verification / Alternative check:The slope of the bending moment diagram equals V; since V is linear, moment is quadratic (parabolic), consistent with standard beam theory.
Why Other Options Are Wrong:Two triangles or equilateral forms do not match the single linear ramp imposed by constant w.
Common Pitfalls:Misplacing signs at supports; forgetting that UDL produces linear shear and parabolic moment.
Final Answer:
A single right-angled triangle (linear variation across the span)