Bending Stress At Outer Fibre Of Curved Beam Given Bending Moment Calculator

Formula Used:

\[ \text{Bending Stress at Outer Fibre} = \frac{\text{Bending moment in curved beam} \times \text{Distance of Outer Fibre from Neutral Axis}}{\text{Cross sectional area of curved beam} \times \text{Eccentricity Between Centroidal and Neutral Axis} \times \text{Radius of Outer Fibre}} \]

\[ \sigma_{bo} = \frac{M_b \times h_o}{A \times e \times R_o} \]

Bending Moment in Curved Beam (M_b):

N·m

Distance of Outer Fibre from Neutral Axis (h_o):

Cross Sectional Area of Curved Beam (A):

m²

Eccentricity Between Centroidal and Neutral Axis (e):

Radius of Outer Fibre (R_o):

Bending Stress at Outer Fibre (σ_bo):

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1. What is Bending Stress at Outer Fibre?

Bending Stress at Outer Fibre is the maximum stress experienced at the outermost fiber of a curved structural element when subjected to bending moments. It represents the highest tensile or compressive stress in the cross-section.

2. How Does the Calculator Work?

The calculator uses the bending stress formula for curved beams:

\[ \sigma_{bo} = \frac{M_b \times h_o}{A \times e \times R_o} \]

Where:

\( \sigma_{bo} \) — Bending stress at outer fibre (Pa)
\( M_b \) — Bending moment in curved beam (N·m)
\( h_o \) — Distance of outer fibre from neutral axis (m)
\( A \) — Cross sectional area of curved beam (m²)
\( e \) — Eccentricity between centroidal and neutral axis (m)
\( R_o \) — Radius of outer fibre (m)

Explanation: This formula accounts for the curved nature of the beam, where the neutral axis doesn't coincide with the centroidal axis, creating additional stress effects.

3. Importance of Bending Stress Calculation

Details: Accurate calculation of bending stress at outer fibre is crucial for designing curved structural elements, ensuring they can withstand applied loads without failure, and determining appropriate safety factors.

4. Using the Calculator

Tips: Enter all values in consistent SI units. Bending moment should be in Newton-meters, distances in meters, and area in square meters. All values must be positive and non-zero.

5. Frequently Asked Questions (FAQ)

Q1: Why is bending stress higher at outer fibres?
A: In bending, outer fibres experience the maximum deformation, resulting in higher stress levels compared to inner fibres.

Q2: How does curvature affect bending stress?
A: Curvature causes the neutral axis to shift toward the center of curvature, increasing stress at the outer fibre compared to straight beams.

Q3: What materials is this calculation valid for?
A: This calculation is valid for homogeneous, isotropic materials that follow Hooke's law within the elastic range.

Q4: When should this formula be used instead of straight beam formulas?
A: This formula should be used for beams with significant curvature (R/h < 10), where straight beam formulas become inaccurate.

Q5: What are typical applications of curved beam analysis?
A: Curved beams are commonly found in hooks, rings, arches, crane hooks, and various machine components with curved geometries.