Mechanical & Structural Engineering

📘 Key Concepts and Definitions

• Stress (σ): Force per unit area within a material. Units: Pascals (Pa) or psi.
• Strain (ε): The deformation or displacement of material that results from an applied stress. It is a dimensionless quantity.
• Young's Modulus (E): A measure of a material's stiffness, the ratio of stress to strain in the elastic region. Also known as the modulus of elasticity.
• Yield Strength (σ_y): The stress at which a material begins to deform plastically (permanently).
• Ultimate Tensile Strength (UTS): The maximum stress a material can withstand while being stretched or pulled before necking begins.
• Factor of Safety (FoS): The ratio of a material's ultimate strength to the actual working stress. A measure of design robustness.

🧮 Formulas and Equations

Stress-Strain Relationships

• Hooke's Law (Elastic Region): σ = E · ε
• Poisson's Ratio (ν): Relates lateral strain to axial strain. ν = -(ε_lateral)/(ε_axial)

Pressure Vessels (Thin-Walled)

• Hoop (Circumferential) Stress: σ_h = (Pr)/t
• Longitudinal (Axial) Stress: σ_l = (Pr)/(2t)
• Where: P = Internal Pressure, r = Vessel Radius, t = Wall Thickness. Note: Hoop stress is twice the longitudinal stress.

Beam Formulas (Common Cases)

• Bending Stress: σ_b = (My)/I
• Where: M = Bending Moment, y = Distance from neutral axis, I = Moment of Inertia.
• Shear Stress: τ = (VQ)/(It)
• Where: V = Shear Force, Q = First moment of area, I = Moment of Inertia, t = thickness.

🛠️ Tools & Standards

Failure Theories

• Maximum Normal Stress Theory (Rankine): Failure occurs when the maximum principal stress exceeds the material's yield strength. Good for brittle materials.
• Maximum Shear Stress Theory (Tresca): Failure occurs when the maximum shear stress exceeds the shear yield strength. τ_max ≥ (σ_y)/2. More conservative for ductile materials.
• Distortion Energy Theory (von Mises): Failure occurs when the distortion energy in a unit volume equals the distortion energy at yielding in a tension test. Excellent for ductile materials. σ' ≥ σ_y, where σ' is the von Mises stress.

Welding Symbols

Welding symbols on drawings convey complex information concisely. Key elements include the reference line, arrow, and tail, with symbols indicating weld type (fillet, groove, etc.) and dimensions.

🧭 Step-by-Step Guides: Mohr's Circle

Mohr's Circle is a graphical method to determine principal stresses, maximum shear stress, and stresses on an inclined plane.

1. Establish Axes: Draw a horizontal axis for normal stress (σ) and a vertical axis for shear stress (τ).
2. Plot Stress States: Plot two points representing the stress state on the x and y faces: A(σ_x, τ_xy) and B(σ_y, -τ_xy).
3. Draw the Circle: The line connecting A and B is the diameter of the circle. The center of the circle is at C = ((σ_x + σ_y)/2, 0).
4. Find Principal Stresses: The points where the circle intersects the horizontal axis are the principal stresses, σ₁ and σ₂.
5. Find Max Shear Stress: The top and bottom points of the circle represent the maximum shear stress, τ_max, which is equal to the circle's radius.

⌨️ Productivity Tips

• Fastener Torque: A common rule of thumb for estimating torque is T = K · D · F, where T is torque, K is the nut factor (typically ~0.2 for steel), D is the nominal bolt diameter, and F is the desired preload (often 75-90% of proof load).
• Beam Deflection Tables: For common loading cases (cantilever, simply supported), use pre-calculated formulas from engineering handbooks instead of deriving them from first principles.

📊 Tables & Visual Aids

Typical Material Properties (Common Metals)

🧪 Use Case: Pressure Vessel Calculation

Problem: A cylindrical pressure vessel with a 1m radius and 10mm wall thickness is made of steel with a yield strength of 250 MPa. What is the Maximum Allowable Working Pressure (MAWP) with a Factor of Safety of 4?

• The critical stress is the hoop stress: σ_h = (Pr)/t.
• Allowable stress = (Yield Strength)/(FoS) = (250 MPa)/4 = 62.5 MPa.
• Set allowable stress equal to hoop stress: 62.5 × 10⁶ Pa = (P · 1m)/(0.01m).
• Solve for P: P = (62.5 × 10⁶ Pa · 0.01m) / 1m = 625,000 Pa or 625 kPa.

🧹 Troubleshooting Common Issues

• Problem: Calculations show failure, but the part seems over-engineered.
  • Fix: Check your failure theory. Using the Tresca (Max Shear) theory is more conservative than von Mises for ductile materials. Switching to von Mises may provide a more realistic assessment. Also, check for stress concentrations that may not be captured in simple formulas.
• Problem: Beam deflection is higher than expected.
  • Fix: Verify the boundary conditions and moment of inertia (I). A small change in the cross-sectional geometry (especially height) can have a large impact on I. Ensure the correct formula for your specific support and loading condition is used.

📚 References and Further Reading

• Shigley's Mechanical Engineering Design by Budynas and Nisbett.
• Beer, Johnston, & DeWolf's "Mechanics of Materials".
• AISC "Steel Construction Manual".
• ASME Boiler and Pressure Vessel Code (BPVC).