Empirical Formula for Secant Modulus Proposed by Hognestad in ACI Code Calculator

Secant Modulus Formula:

\[ E_c = 1,800,000 + (460 \times f'_c) \]

Secant Modulus (E_c):

1. What is the Secant Modulus Formula by Hognestad?

Definition: This empirical formula estimates the secant modulus of elasticity (E_c) for normal weight concrete based on its compressive strength.

Purpose: It provides engineers with a practical method to determine concrete's stiffness for structural analysis and design.

2. How Does the Formula Work?

The formula is expressed as:

\[ E_c = 1,800,000 + (460 \times f'_c) \]

Where:

\( E_c \) — Secant modulus of elasticity (psi)
\( f'_c \) — Compressive strength of concrete at 28 days (psi)

Explanation: The formula establishes a linear relationship between concrete's stiffness and its compressive strength.

3. Importance in Structural Engineering

Details: The secant modulus is crucial for:

Calculating deflections under service loads
Analyzing structural behavior
Designing reinforced concrete members
Performing nonlinear analysis

4. Using the Calculator

Tips:

Enter the 28-day compressive strength (f'_c) in psi
Optionally specify expected variation (default ±5%)
Results show both calculated modulus and expected range

5. Frequently Asked Questions (FAQ)

Q1: What is the typical range for f'_c in this formula?
A: The formula is valid for normal weight concrete with f'_c between 3,000 psi and 12,000 psi.

Q2: Why include a variation percentage?
A: Concrete properties naturally vary; ±5% accounts for typical material variability.

Q3: How does this compare to the ACI 318 formula?
A: Hognestad's formula often gives slightly higher values than ACI 318's \( E_c = 57,000 \sqrt{f'_c} \).

Q4: When was this formula developed?
A: Hognestad proposed this empirical relationship in the 1950s based on experimental data.

Q5: Does this apply to lightweight concrete?
A: No, this formula is specifically for normal weight concrete (density ~145 lb/ft³).