Elasto Plastic Bending Moment Of Rectangular Beam Calculator

Elasto Plastic Bending Moment Formula:

\[ M_{ep} = \frac{\sigma_0 \cdot b \cdot (3d^2 - 4\eta^2)}{12} \]

Yield Stress (σ₀):

Breadth of Rectangular Beam (b):

Depth of Rectangular Beam (d):

Depth of Outermost Shell Yields (η):

Elasto Plastic Bending Moment (Mₑₚ):

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1. What is Elasto Plastic Bending Moment?

Elasto plastic bending moment occurs when the beam fibres beyond y=η have yielded. It represents the moment capacity of a beam in the elasto-plastic state where some fibers have yielded while others remain elastic.

2. How Does the Calculator Work?

The calculator uses the Elasto Plastic Bending Moment formula:

\[ M_{ep} = \frac{\sigma_0 \cdot b \cdot (3d^2 - 4\eta^2)}{12} \]

Where:

\( M_{ep} \) — Elasto plastic bending moment (N·m)
\( \sigma_0 \) — Yield stress of the material (Pa)
\( b \) — Breadth of rectangular beam (m)
\( d \) — Depth of rectangular beam (m)
\( \eta \) — Depth of outermost shell yields (m)

Explanation: This formula calculates the bending moment when the beam is in an elasto-plastic state, accounting for the yielded outer fibers while the inner core remains elastic.

3. Importance of Elasto Plastic Bending Moment Calculation

Details: Calculating the elasto-plastic bending moment is crucial for structural analysis and design, particularly in understanding the behavior of beams beyond the elastic limit and determining their load-carrying capacity in the plastic range.

4. Using the Calculator

Tips: Enter yield stress in Pascals, beam dimensions in meters. All values must be positive numbers. Ensure consistent units throughout the calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the elasto-plastic bending moment?
A: It helps engineers understand the transitional behavior of beams from elastic to fully plastic states, which is crucial for designing structures with adequate safety margins.

Q2: How does this differ from fully plastic bending moment?
A: Elasto-plastic moment occurs when only outer fibers have yielded, while fully plastic moment occurs when the entire cross-section has yielded.

Q3: What happens when η = 0?
A: When η = 0, the beam is in fully elastic state, and the formula reduces to the maximum elastic moment.

Q4: What happens when η = d/2?
A: When η = d/2, the entire cross-section has yielded, and the formula gives the fully plastic moment capacity.

Q5: Can this formula be used for non-rectangular sections?
A: No, this specific formula is derived for rectangular cross-sections. Other cross-sectional shapes have different formulas for elasto-plastic bending moment.