The gel space ratio of a concrete sample is given as 0.589. What is the theoretical strength (N/mm²) of that concrete sample?
Correct Answer: A. 49.04
📚 Detailed Explanation: Powers' Formula Calculation (Q20)
T.C. Powers (1958) developed an empirical formula relating the compressive strength of concrete to the gel-space ratio — the ratio of the volume of hardened cement gel to the total space available (gel + capillary pores).
Why A (49.04 N/mm²) is correct:
Powers' Formula: f'c = 240 × x³ (where x = gel-space ratio)
Given: x = 0.589
Step 1: x³ = 0.589 × 0.589 × 0.589
= 0.3469 × 0.589 = 0.20431
Step 2: f'c = 240 × 0.20431 = 49.03 ≈ 49.04 N/mm²
Powers' Formula: f'c = 240 × x³ (where x = gel-space ratio)
Given: x = 0.589
Step 1: x³ = 0.589 × 0.589 × 0.589
= 0.3469 × 0.589 = 0.20431
Step 2: f'c = 240 × 0.20431 = 49.03 ≈ 49.04 N/mm²
Checking other options: 240 × 0.65³ = 65.8 (not our x); 240 × 0.78³ = 114 (wrong); none match for x = 0.589.
Powers' Formula Reference
| Parameter | Value |
|---|---|
| Formula | f'c = 240 x³ N/mm² |
| Constant (240) | Intrinsic strength of C-S-H gel (N/mm²) |
| x (gel-space ratio) | 0.589 (given) |
| x³ | 0.589³ = 0.20431 |
| f'c | 240 × 0.20431 = 49.04 N/mm² |
Understanding the Gel-Space Ratio
x = Volume of gel / (Volume of gel + Volume of capillary pores)
As hydration proceeds, more capillary pores fill with C-S-H gel, so x increases and strength increases. A gel-space ratio of ~0.67–0.70 corresponds to typical structural concrete strength (~70 MPa range).
Key Concepts for Students
- Powers' formula: f'c = 240x³ — memorise this for numerical questions.
- The cube (x³) emphasises that strength is highly sensitive to gel-space ratio — a small increase in x gives a large strength gain.
- 240 N/mm² represents the intrinsic (maximum possible) strength of fully gel-filled cement paste.