material science
Incorporating our prediction to the mathematical formula of metallurgy of metals per sample of failure as a proof of concept (No FEA-Only theoretical calculations).
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ASR STRESS STRAIN CURVE
Understanding the Stress-Strain Curve
The stress-strain curve is a fundamental tool in structural engineering, offering insights into a material's response to loading and aiding in optimized product design. Here's a breakdown of key concepts:
Stress: Force applied to a material's cross-sectional area.
Strain: Deformation of a material relative to its original length.
Proportional Limit: The region on the curve where stress and strain have a linear relationship (Hooke's Law applies).
Elastic Limit: The maximum stress a material can sustain before permanent deformation occurs.
Yield Strength: Stress at which a material retains 0.2% permanent deformation after the load is removed.
Ultimate Strength: The peak stress a material can withstand before failure.
Fracture Limit: The point of catastrophic material breakage.
Elastic Region: Stress range where the material returns to its original shape after the load is released.
Plastic Region: Stress range where the material experiences permanent deformation.
Strain Hardening: Strengthening of a material through plastic deformation, increasing its yield strength.
Necking: Localized reduction in a material's cross-sectional area after reaching ultimate strength.
Interpreting the Curve
Ductility: A ductile material has a long plastic region, indicating its ability to deform before failing. Ductile materials are valuable in structures where warning signs of impending collapse are desirable.
Energy Dissipation: The area under the curve represents the material's capacity to absorb energy. Plastic deformation reduces this capacity.
b8 (SS 304) and b8m (ss 316)
ASTM A193/A193M: Standard Specification for Alloy-Steel and Stainless Steel Bolting Materials for High-Temperature or High-Pressure Service and Other Special Purpose Applications
Scope:
Provides specifications for various grades of alloy and stainless steel bolting materials intended for use in pressure vessels, valves, flanges, and fittings.
Covers bolts, screws, studs, stud bolts, nuts, and other bolting components.
Specifies chemical composition, mechanical properties, heat treatment, and testing requirements.
Common Grades:
B5: Chromium-molybdenum steel (heat-treated) for moderate temperature applications.
B6: Corrosion-resistant martensitic stainless steel (heat-treated).
B7: Chromium-molybdenum steel with excellent high-temperature strength (heat-treated).
B8 and B8M: Austenitic stainless steels (classes 1 and 2) with varying corrosion resistance and mechanical properties (solution heat-treated).
B16: Chromium-molybdenum-vanadium steel, known for superior high-temperature creep resistance (heat-treated).
20 random samples from our dataset
Machine Failure Label: Indicates whether the machine has failed for any of the following five independent failure modes:
Tool Wear Failure (TWF):
Tool replacement or failure at a random time between 200 and 240 mins.
Heat Dissipation Failure (HDF):
Process failure if air-process temperature difference < 8.6 K and rotational speed < 1380 rpm.
Power Failure (PWF):
Process failure if power (torque * rotational speed) < 3500 W or > 9000 W.
Overstrain Failure (OSF):
Process failure if tool wear * torque exceeds specific thresholds based on product variants.
Random Failures (RNF):
A 0.1% chance of process failure, unrelated to process parameters, occurring for 5 data points.
b8 (SS 304) MECHANICAL PROPERTIES
MACHINE DESIGN
Temperature Difference: Temperature changes can cause materials to expand or contract. This thermal expansion or contraction can induce stress in a bolted joint. The stress due to thermal expansion (σ) can be calculated using the formula:
σ=EαΔT
where E is the modulus of elasticity of the material, α is the coefficient of thermal expansion, and ΔT is the change in temperature.
Rotational Speed: The rotational speed itself does not directly cause stress in a bolt. However, if the bolt is part of a rotating assembly (like in an engine or a turbine), the centrifugal force due to rotation can cause stress. The centrifugal force (F) on a mass (m) rotating at a speed (ω) at a radius ® from the center of rotation is given by:
F=mω2r
This force can contribute to the tensile load on the bolt.
Torque: The torque applied to a bolt causes a tension force in the bolt that can be calculated using the formula:
F=KdT
where T is the torque, K is a constant that depends on the friction conditions, and d is the nominal bolt diameter.
Calculation
# Given data for index 5399
air_temperature_K = 302.8 degrees Kelvin
process_temperature_K = 312.4 degrees Kelvin
rotational_speed_rpm = 1411 rev per minute
torque_Nm = 53.8 Newton-meter
K = 0.2 (Typical for Nut)
modulus_of_elasticity_Gpa = 193 GPa
nominal_bolt_diameter_inches = 1.5 inches
weight_lbs = 1.51 lbs
radius_inches = 6 inches
# Constants
g = 9.80665 # Acceleration due to gravity (m/s^2)
coefficient_of_thermal_expansion = 18.4 * 10**-6 # /°C (Assuming the higher range as relevant)
# Convert modulus of elasticity to Pa from GPa
modulus_of_elasticity_Pa = modulus_of_elasticity_Gpa * 10**9
# Convert temperatures to °C from K (for thermal expansion calculation)
delta_T_C = process_temperature_K - air_temperature_K
# Convert weight to kg from lbs
weight_kg = weight_lbs * 0.453592
# Convert radius to meters from inches
radius_m = radius_inches * 0.0254
# Convert nominal bolt diameter to meters from inches
nominal_bolt_diameter_m = nominal_bolt_diameter_inches * 0.0254
# Convert rotational speed to rad/s from rpm
rotational_speed_rad_s = rotational_speed_rpm * (2 * 3.14159265 / 60)
# Calculations
# Force due to torque
F_torque = K * torque_Nm / nominal_bolt_diameter_m
# Centrifugal force
F_centrifugal = weight_kg * rotational_speed_rad_s**2 * radius_m
# Total force on the bolt
F_total = F_torque + F_centrifugal
# Stress due to thermal expansion
sigma_thermal = modulus_of_elasticity_Pa * coefficient_of_thermal_expansion * delta_T_C
F_torque, F_centrifugal, F_total, sigma_thermal
(282.4146981627297, 2278.9697629707994, 2561.384461133529, 34091519.99999987)
The force due to torque is approximately 282.4 N.
The centrifugal force due to rotation is approximately 2278.97 N.
The stress due to thermal expansion (ℎσthermal) is approximately 34.09 MPa (megapascals).
The total force on the bolt (F) is the sum of these two forces, which is approximately 2561.38 N.
In this case, as calculated, the thermal expansion due to temperature difference is strongest predictor of failure as illustrated in our plots in previous sections.
In the stress-strain diagram below, we just model the behaviour of our material using Hooke's Law for the proportional deformation. In real-world testing, beyond the yield strength of our materials, the points will behave differently. Since we are not using FEA (Finite Element Analysis, we can't model the actual points of the strain hardening and necking region. For simplicity, we just model the 3rd stage of the material behaviour based on our synthetic dataset with the theoretical equations from this research paper.
THE ALPHA MANIFOLD
The temperature in the the bolt is lagging behind the temperature of the flange. The thermal effect can be damaging to the flange seals and seats due to axial expansion at different rates. If the thermal expansion is so great, based on the stress and strain curve, the bolt material might enter the plastic deformation (permanent deformation) thereby reducing the sealing pressure of the gasket. In the figure below, the difference of thermal expansion create force displacement in the form of axial material expansion.
As a form of one of the mitigation, avoid the sudden thermal gradients of the the line (which is an indicative of failure if there is). A lesson's learned during my Master's course of Mechanical Engineering (Heat Power Option).
INTERPRETATION
This visualization shows how the coefficients of different predictors change with respect to different values of a regularization parameter, alpha, in a predictive model.
X-axis (alpha): This is a log-scaled axis showing the range of alpha values. Alpha controls the strength of the regularization applied to the model. As alpha increases, the model coefficients typically shrink towards zero.
Y-axis (Two scales): On the left side, we have the absolute coefficient, suggesting it shows the magnitude of interaction, the effect size of each feature on the prediction. On the right side, there is a scale for coefficient values which ranges from about -0.6 to 0.6.
Lines (Coefficients of Features): Each line represents a feature (covariate) in the model, with the color representing a specific feature. The lines show how the coefficients of each feature change as alpha increases:
Air temperature (K): The coefficient remains relatively stable and then gradually decreases in magnitude as alpha increases.
Failure Type Encoded: This is an encoded categorical variable that represents different failure types. Its coefficient also decreases slightly with alpha but is relatively stable compared to other features.
Type L, M, H: These specific categories within the 'Failure Type' feature. Each has its own trajectory with respect to alpha values.
Type L: The coefficient decreases slightly as alpha increases.
Type M: The coefficient is stable and then slightly increases before decreasing.
Type H: Shows a significant decrease as alpha increases, suggesting it becomes less important in the presence of strong regularization.
Torque (Nm), Rotational Speed (rpm), Process Temperature (K): These features show a more pronounced decrease as alpha increases, especially Process Temperature which has the largest range in coefficient magnitude.
Thermal Expansion Interaction: The annotations along the y-axis on the left to indicate the relative 'interaction' between the features, a custom metric related to the failure prediction which was derived from our previous calculations. If the temperature will increase, say for example from 30-deg to 300-deg centigrade, we call it is "thermal transient" which can cause flange leakage if the bolt is fitted to mating flanges.
"The Alpha Manifold" is to describe how the manifold of coefficients changes across different alpha values, demonstrating the interplay between the model's complexity (as controlled by alpha) and the predictive importance of each feature.
The plot can inform us about which variables are most robust to regularization, which may shrink less in magnitude (suggesting they have strong predictive power) and which are more sensitive to regularization. The 'distance' mentioned refer to the changes in coefficients relative to the changes in alpha, showing which variables remain important across different model complexities which needs to be proven by theoretical calculations.
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