For a simply supported reinforced concrete beam where the ratio of effective span l to overall depth D is ≥ 1 and ≤ 2, what lever-arm approximation should be used for design?

Difficulty: Medium

0.2 (l + 2D)
0.3 (l + 3D)
0.2 (l + 3D)
0.3 (l + 2D)
0.25 (l + 2D)

Correct Answer: 0.2 (l + 2D)

Explanation:

Introduction / Context:
Short or deep members (including pile caps and beams with small span-to-depth ratios) are often designed using strut-and-tie or truss analogies. In such cases, simplified lever-arm expressions relate the resultant compressive strut location to the tensile tie (reinforcement), enabling rapid sizing of steel and verification of shear and anchorage, especially when beam theory assumptions are not strictly valid.

Given Data / Assumptions:

Member: simply supported beam behaving as a deep or short member.
Span-to-depth ratio: l/D is between 1 and 2 (inclusive).
Quantity: lever arm z to be used in design expressions for internal forces.

Concept / Approach:
For low l/D, the internal force path is better represented by a tied arch or truss model. Educational and handbook formulas provide lever-arm approximations as functions of l and D. For 1 ≤ l/D ≤ 2, a commonly recommended expression is z = 0.2 (l + 2D), balancing the influence of span and overall depth when the compression strut is steep and the tension tie is near the bottom reinforcement.

Step-by-Step Solution:

Recognize that standard beam theory (z ≈ 0.9 d) is less appropriate for deep members.Adopt the truss/strut-and-tie inspired approximation valid for 1 ≤ l/D ≤ 2.Select z = 0.2 (l + 2D) from the given options.

Verification / Alternative check:

Check limiting values: as l/D approaches 1, z tends to a function dominated by D, consistent with a deep tie; as l/D approaches 2, z scales with both l and D, reflecting intermediate behavior.

Why Other Options Are Wrong:

0.3(l + 3D) and 0.3(l + 2D): generally overpredict the lever arm for this range, potentially underestimating steel demand.0.2(l + 3D): misaligned with standard guidance for the specified l/D band.0.25(l + 2D): not a recognized approximation for this l/D range.

Common Pitfalls:

Applying slender-beam formulas to deep members; ignoring anchorage and shear capacity under truss action.Using an l/D-specific expression outside its validity range.

Final Answer:

0.2 (l + 2D)

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For a simply supported reinforced concrete beam where the ratio of effective span l to overall depth D is ≥ 1 and ≤ 2, what lever-arm approximation should be used for design?

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