Residual Stress In Beam In Fully Plastic State Calculator

Formula Used:

\[ \sigma_{Res\_plastic} = -\left(\sigma_0 + \frac{M_{rec\_plastic} \cdot y}{\frac{b \cdot d^3}{12}}\right) \]

Yield Stress (σ₀):

Fully Plastic Recovery Bending Moment (M_rec_plastic):

N·m

Depth Yielded Plastically (y):

Breadth of Rectangular Beam (b):

Depth of Rectangular Beam (d):

Residual Stress (σ_Res_plastic):

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1. What is Residual Stress in Beam in Fully Plastic State?

Residual stress in beam in fully plastic state refers to the internal stress that remains in a structural beam after it has been subjected to plastic deformation and then unloaded. These stresses exist without any external loads and result from mechanical processes that cause permanent deformation.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \sigma_{Res\_plastic} = -\left(\sigma_0 + \frac{M_{rec\_plastic} \cdot y}{\frac{b \cdot d^3}{12}}\right) \]

Where:

\( \sigma_{Res\_plastic} \) — Residual stress in beam in plastic state (Pa)
\( \sigma_0 \) — Yield stress of the material (Pa)
\( M_{rec\_plastic} \) — Fully plastic recovery bending moment (N·m)
\( y \) — Depth yielded plastically from outermost fiber (m)
\( b \) — Breadth of rectangular beam (m)
\( d \) — Depth of rectangular beam (m)

Explanation: The formula calculates the residual stress by considering the yield stress, recovery bending moment, and the geometric properties of the beam.

3. Importance of Residual Stress Calculation

Details: Calculating residual stresses is crucial for understanding the structural integrity of beams after plastic deformation, predicting fatigue life, and ensuring safety in engineering applications where components undergo significant loading and unloading cycles.

4. Using the Calculator

Tips: Enter all values in the specified units. Yield stress and geometric dimensions must be positive values. The bending moment can be negative if it represents a recovery moment in the opposite direction.

5. Frequently Asked Questions (FAQ)

Q1: What causes residual stresses in beams?
A: Residual stresses are caused by non-uniform plastic deformation during manufacturing processes, thermal treatments, or mechanical loading that exceeds the yield point of the material.

Q2: How do residual stresses affect beam performance?
A: Residual stresses can affect fatigue life, dimensional stability, and may lead to stress corrosion cracking or premature failure under subsequent loading.

Q3: Can residual stresses be measured directly?
A: Yes, through various methods including X-ray diffraction, hole-drilling, and ultrasonic techniques, though calculation provides a theoretical estimate.

Q4: Are residual stresses always undesirable?
A: Not always. In some cases, compressive residual stresses can be beneficial as they improve fatigue resistance by counteracting tensile stresses from applied loads.

Q5: How can residual stresses be relieved?
A: Through stress relief annealing, vibration stress relief, or mechanical methods like shot peening which introduces beneficial compressive stresses.