Working-stress (elastic) theory for a cracked rectangular beam: If m is the modular ratio, r = As/(b d) is the steel ratio, and d is the overall effective depth, the neutral axis depth x from the top is given by which expression?
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Ax = d * [ - m r + sqrt( m^2 r^2 + 2 m r ) ]
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Bx = d * [ m r + sqrt( m^2 r^2 + 2 m r ) ]
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Cx = d / (1 + 3 m r)
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Dx = d * (2 m r)
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Ex = d / (1 + m r)
Answer
Correct Answer: x = d * [ - m r + sqrt( m^2 r^2 + 2 m r ) ]
Explanation
Introduction / Context:For a cracked elastic section (concrete in tension neglected), the neutral axis of a singly reinforced beam is obtained using the transformed-section method. The neutral axis passes through the centroid of the transformed section (compression concrete + transformed steel area).
Given Data / Assumptions:
- Modular ratio m = Es/Ec.
- Steel ratio r = As/(b d).
- Rectangular section; concrete tension ignored, linear elasticity.
Concept / Approach:
Locate the centroid (neutral axis) of the transformed section by equating moments of areas about the neutral axis: (b * x) * (x/2) = m As (d − x). This yields a quadratic in k = x/d.
Step-by-Step Solution:
Write (b/2) x^2 = m As (d − x).Divide by b d^2 and set r = As/(b d), k = x/d → (1/2) k^2 = m r (1 − k).Rearrange: k^2 + 2 m r k − 2 m r = 0.Solve for k: k = - m r + sqrt( m^2 r^2 + 2 m r ).Hence x = d * [ - m r + sqrt( m^2 r^2 + 2 m r ) ].Verification / Alternative check:
Check limiting cases: as r → 0, k → 0; as r increases, k grows sublinearly, consistent with mechanics.
Why Other Options Are Wrong:
Options (b), (c), (d), and (e) do not satisfy the derived quadratic or have incorrect functional forms.
Common Pitfalls:
Using area equilibrium instead of first moment of area; forgetting to neglect concrete in tension when cracked; algebraic sign errors.
Final Answer:
x = d * [ - m r + sqrt( m^2 r^2 + 2 m r ) ]