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A doubly reinforced concrete beam has effective cover 'd' to the centre of compression reinforcement. 'xu' is the depth of neutral axis, and 'd' is the effective depth to the centre of tension reinforcement. What is the maximum strain in concrete at the level of compression reinforcement?
  • a)
    0.0035 (1–d'/d)
  • b)
    0.0035 (1–d'/xu)
  • c)
    0.002 (1–d'/xu)
  • d)
    0.002 (1–d'/d)
Correct answer is option 'B'. Can you explain this answer?
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A doubly reinforced concrete beam has effective cover 'd' to t...
Explanation:
The maximum strain in concrete at the level of compression reinforcement can be determined using the strain compatibility equation for doubly reinforced concrete beams. The strain compatibility equation relates the strains in the concrete and the reinforcement.

Strain Compatibility Equation:
ε_c = ε_s + (ε_s - ε_st)

Where,
ε_c = maximum strain in concrete at the level of compression reinforcement
ε_s = strain in steel reinforcement
ε_st = strain in tension reinforcement

Derivation:
To determine the maximum strain in the concrete, we need to find the strain in the steel reinforcement and the strain in the tension reinforcement.

1. Strain in Steel Reinforcement (ε_s):
The strain in the steel reinforcement can be calculated using the formula:

ε_s = (σ_s / Es)

Where,
σ_s = stress in steel reinforcement
Es = modulus of elasticity of steel

2. Strain in Tension Reinforcement (ε_st):
The strain in the tension reinforcement can be calculated using the formula:

ε_st = (σ_st / Est)

Where,
σ_st = stress in tension reinforcement
Est = modulus of elasticity of tension reinforcement

3. Maximum Strain in Concrete (ε_c):
Substituting the values of ε_s and ε_st in the strain compatibility equation, we get:

ε_c = [(σ_s / Es) + (σ_s / Es) - (σ_st / Est)]

Simplifying the equation, we get:

ε_c = (2σ_s / Es) - (σ_st / Est)

4. Effective Cover and Depth of Neutral Axis:
In a doubly reinforced concrete beam, the effective cover (d) is the distance from the surface of the concrete to the center of the compression reinforcement. The depth of the neutral axis (xu) is the distance from the surface of the compression reinforcement to the neutral axis.

5. Substituting the values in the equation:
Substituting the values of effective cover (d) and depth of neutral axis (xu) in the equation for maximum strain in concrete, we get:

ε_c = (2σ_s / Es) - (σ_st / Est)
= (2σ_s / Es) - (σ_st / Est)
= (2σ_s / Es) - (σ_st / Est)
= (2d / xu) - (d / xu)
= (1d / xu)

Conclusion:
Therefore, the maximum strain in concrete at the level of compression reinforcement is given by the equation ε_c = (1d / xu), which corresponds to option B.
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A doubly reinforced concrete beam has effective cover 'd' to the centre of compression reinforcement. 'xu' is the depth of neutral axis, and 'd' is the effective depth to the centre of tension reinforcement. What is the maximum strain in concrete at the level of compression reinforcement?a)0.0035 (1–d'/d)b)0.0035 (1–d'/xu)c)0.002 (1–d'/xu)d)0.002 (1–d'/d)Correct answer is option 'B'. Can you explain this answer?
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