Thickness of Tapered Bar using Temperature Stress Calculator

Thickness Formula:

\[ t = \frac{\sigma}{E \times \alpha \times \Delta T \times \frac{D2 - D1}{\ln\left(\frac{D2}{D1}\right)}} \]

Thermal Stress (σ):

Young's Modulus (E):

Coefficient of Linear Thermal Expansion (α):

1/K

Change in Temperature (ΔT):

Depth of Point 2 (D2):

Depth of Point 1 (D1):

Tolerance:

Section Thickness (t):

1. What is Thickness of Tapered Bar using Temperature Stress?

Definition: This calculator determines the required thickness of a tapered bar to withstand thermal stress caused by temperature changes.

Purpose: It helps engineers design tapered bars that can accommodate thermal expansion without failure.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ t = \frac{\sigma}{E \times \alpha \times \Delta T \times \frac{D2 - D1}{\ln\left(\frac{D2}{D1}\right)}} \]

Where:

\( t \) — Section thickness (meters)
\( \sigma \) — Thermal stress (Pascals)
\( E \) — Young's modulus (Pascals)
\( \alpha \) — Coefficient of linear thermal expansion (1/Kelvin)
\( \Delta T \) — Change in temperature (Kelvin)
\( D2 \) — Depth of point 2 (meters)
\( D1 \) — Depth of point 1 (meters)

Explanation: The formula accounts for the geometric taper of the bar and the material's response to thermal changes.

3. Importance of Thickness Calculation

Details: Proper thickness calculation prevents thermal stress failure, ensures structural integrity, and optimizes material usage.

4. Using the Calculator

Tips: Enter all required parameters including thermal stress, material properties, temperature change, and depth measurements. The tolerance field (default ±5%) allows for manufacturing variations.

5. Frequently Asked Questions (FAQ)

Q1: What is a typical value for coefficient of linear thermal expansion?
A: For steel it's about 12×10⁻⁶/K, aluminum about 23×10⁻⁶/K, but varies by material.

Q2: Why is the natural logarithm used in the formula?
A: The ln(D2/D1) term accounts for the tapered geometry's non-linear stress distribution.

Q3: How does temperature change affect the required thickness?
A: Greater temperature changes require thicker sections to accommodate the resulting thermal stress.

Q4: What happens if D1 equals D2?
A: The formula becomes undefined as it would represent a non-tapered bar (uniform thickness).

Q5: How should I set the tolerance percentage?
A: 5% is common, but adjust based on manufacturing capabilities and safety requirements.