One-dimensional consolidation: If C_v is the coefficient of consolidation, t is the elapsed time, and d is the drainage path length, the dimensionless time factor T_v is given by:
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AT_v = (C_v * t) / d^2
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BT_v = (d^2) / (C_v * t)
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CT_v = (4 * C_v * t) / d^2
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DT_v = (C_v * t) / (4 * d^2)
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ET_v = √(C_v * t / d)
Answer
Correct Answer: T_v = (C_v * t) / d^2
Explanation
Introduction / Context:Terzaghi’s one-dimensional consolidation theory uses two dimensionless groups: the time factor T_v and the average degree of consolidation U. Correctly forming T_v is the key to relating lab or field drainage times to soil layer thickness and drainage conditions (single or double drainage).
Given Data / Assumptions:
- C_v has units of length^2/time.
- t is time since loading.
- d is the maximum drainage path length (half thickness for double drainage, full thickness for single drainage).
Concept / Approach:
Dimensional analysis and the consolidation solution define T_v = C_v * t / d^2. This non-dimensional time collapses different layer thicknesses and drainage conditions into a common framework to read U–T_v relationships.
Step-by-Step Solution:
State definition: T_v = C_v * t / d^2.Check units: (L^2/T * T) / L^2 → dimensionless.Apply d correctly depending on single vs. double drainage.Verification / Alternative check:
Standard charts (e.g., U vs. T_v) rely on this definition; laboratory oedometer test interpretations use the same form.
Why Other Options Are Wrong:
(b) is the reciprocal; (c) and (d) include an unjustified factor 4; (e) has incorrect dimensions.
Common Pitfalls:
Using layer thickness instead of drainage path; forgetting that for double drainage d = H/2.
Final Answer:
T_v = (C_v * t) / d^2