The total active pressure after the development of tension cracks is equal to
- A½ γH²Ka − 2c′H√Ka
- B½ γH²Ka + 2c′H√Ka
- C½ γH²Ka − 2c′H√Ka − 2(c′)²/γ
- D½ γH²Ka − 2c′H√Ka + 2(c′)²/γCorrect
Solution
Concept
Active pressure with tension crack
Formula
Pa = ½γH²Ka − 2cH√Ka + 2c²/γ
Step-by-step solution
1. Crack depth zc = 2c/(γ√Ka). 2. Pa = ½ (H − zc)(γH Ka − 2c√Ka). 3. Expanding: ½γH²Ka − 2cH√Ka + 2c²/γ.
Explanation
The tensile zone is neglected, which adds the term 2c²/γ.
Why the other options are wrong
The other options give the pressure without cracks or with a wrong sign.
Shortcut
Pa = ½γH²Ka − 2cH√Ka + 2c²/γ.
Correct answer: D) ½ γH²Ka − 2c′H√Ka + 2(c′)²/γ