Cross-Sectional Area According to Johnson's Parabolic Formula given Slenderness Ratio Calculator

Formula Used:

\[ A_{sectional} = \frac{P}{\sigma_c - (r \times \lambda)} \]

Critical Load On Column (P):

Compressive Yield Stress (σ_c):

Johnson's Formula Constant (r):

Slenderness Ratio (λ):

Column Cross Sectional Area (A_sectional):

1. What is Johnson's Parabolic Formula for Cross-Sectional Area?

Definition: This calculator determines the required cross-sectional area of a column using Johnson's parabolic formula based on critical load, compressive yield stress, Johnson's constant, and slenderness ratio.

Purpose: It helps engineers design columns that can withstand critical loads without buckling, considering material properties and geometric characteristics.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ A_{sectional} = \frac{P}{\sigma_c - (r \times \lambda)} \]

Where:

\( A_{sectional} \) — Column cross-sectional area (m²)
\( P \) — Critical load on column (N)
\( \sigma_c \) — Compressive yield stress (Pa)
\( r \) — Johnson's formula constant
\( \lambda \) — Slenderness ratio

Explanation: The formula calculates the minimum cross-sectional area needed to prevent buckling under the given load, considering material strength and column geometry.

3. Importance of Cross-Sectional Area Calculation

Details: Proper calculation ensures structural stability, prevents buckling failures, and optimizes material usage in column design.

4. Using the Calculator

Tips: Enter the critical load, compressive yield stress, Johnson's constant, and slenderness ratio. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is Johnson's parabolic formula used for?
A: It's used to determine the critical load or required cross-section for intermediate columns that fail by inelastic buckling.

Q2: How do I determine Johnson's constant (r)?
A: It's material-dependent and typically ranges between 1-10 MPa for common engineering materials.

Q3: What is the slenderness ratio?
A: It's the ratio of column length to the least radius of gyration of its cross-section, indicating column slenderness.

Q4: What units should I use?
A: Use consistent units: Newtons for load, Pascals for stress, and dimensionless for the constant and ratio.

Q5: When is this formula applicable?
A: For intermediate columns where the slenderness ratio is below the critical value for Euler buckling but above the short-column limit.