Structural Engineering
Strength of materials, concrete design, steel structures, and structural analysis
Table of Contents
1. Stress & Strain
Types of Stress
Normal Stress (σ)
σ = P/A
Perpendicular to cross-section
- Tensile (+): Pulls apart
- Compressive (-): Pushes together
Shear Stress (τ)
τ = V/A
Parallel to cross-section
- Single shear: τ = P/A
- Double shear: τ = P/2A
Strain & Deformation
| Property | Formula | Description |
|---|---|---|
| Normal Strain | ε = ΔL/L | Change in length per unit length |
| Shear Strain | γ = τ/G | Angular distortion (radians) |
| Young's Modulus | E = σ/ε | Slope of stress-strain (elastic region) |
| Shear Modulus | G = τ/γ | G = E/[2(1+ν)] |
| Poisson's Ratio | ν = -ε_lat/ε_axial | Typically 0.25-0.35 for metals |
Axial Deformation
δ = PL/AE
For non-uniform members: δ = Σ(PᵢLᵢ/AᵢEᵢ)
Thermal Stress
Free Expansion
δ_T = αLΔT
No stress if free to expand
Constrained Member
σ_T = EαΔT
α = coefficient of thermal expansion
2. Beam Analysis
Bending Stress
σ = Mc/I = M/S
M = bending moment, c = distance from neutral axis, I = moment of inertia, S = section modulus
Shear Stress in Beams
τ = VQ/Ib
- V = shear force
- Q = first moment of area above/below point
- I = moment of inertia of entire section
- b = width at point of interest
Common Beam Formulas
| Beam Type | Max Moment | Max Deflection |
|---|---|---|
| SS - Center Point Load | PL/4 | PL³/48EI |
| SS - Uniform Load | wL²/8 | 5wL⁴/384EI |
| Cantilever - End Load | PL | PL³/3EI |
| Cantilever - Uniform Load | wL²/2 | wL⁴/8EI |
| Fixed Both Ends - Center Load | PL/8 | PL³/192EI |
| Fixed Both Ends - Uniform Load | wL²/12 | wL⁴/384EI |
Moment of Inertia (I)
Rectangle
I = bh³/12
About centroidal axis parallel to b
Circle
I = πd⁴/64 = πr⁴/4
Triangle
I = bh³/36
About centroidal axis
Parallel Axis Theorem
I = I_c + Ad²
3. Columns & Compression Members
Euler's Buckling Formula
P_cr = π²EI/(KL)²
Critical (buckling) load for long slender columns
Effective Length Factor (K)
K = 1.0
Pinned-Pinned
K = 0.5
Fixed-Fixed
K = 0.7
Fixed-Pinned
K = 2.0
Fixed-Free
Slenderness Ratio
λ = KL/r
- r = radius of gyration = √(I/A)
- Long column: Euler formula applies
- Short column: Direct compression (P/A)
- Intermediate: Use empirical formulas
4. Reinforced Concrete Design
USD (Ultimate Strength Design)
φMₙ ≥ Mᵤ
Design strength must exceed factored load
Flexural Design Formulas
| Parameter | Formula |
|---|---|
| Nominal Moment | Mₙ = Asfy(d - a/2) |
| Compression Block Depth | a = Asfy/(0.85f'c·b) |
| Neutral Axis Depth | c = a/β₁ |
| Steel Ratio | ρ = As/bd |
| Minimum Steel Ratio | ρ_min = 1.4/fy or √f'c/(4fy) |
| Maximum Steel Ratio | ρ_max = 0.75ρ_b |
Beta Factor (β₁)
- f'c ≤ 28 MPa: β₁ = 0.85
- 28 < f'c ≤ 56 MPa: β₁ = 0.85 - 0.05(f'c - 28)/7
- f'c > 56 MPa: β₁ = 0.65
Shear Design
Concrete Shear Strength
Vc = 0.17√f'c·bw·d
Simplified formula (NSCP)
Stirrup Contribution
Vs = Av·fy·d/s
For vertical stirrups
5. Steel Design
LRFD vs ASD
LRFD (Load & Resistance Factor)
φRₙ ≥ ΣγᵢQᵢ
Factor loads up, resistance down
ASD (Allowable Stress)
Rₙ/Ω ≥ ΣQᵢ
Compare to allowable stress
Tension Members
- Gross Section Yielding: Pₙ = Fy·Ag
- Net Section Fracture: Pₙ = Fu·Ae
- Effective Net Area: Ae = U·An
- U (Shear Lag Factor): Depends on connection
Compression Members
Critical Stress (Fcr)
- KL/r ≤ 4.71√(E/Fy): Inelastic buckling
- KL/r > 4.71√(E/Fy): Elastic buckling (Euler)
6. Structural Analysis Methods
Truss Analysis
Method of Joints
- Isolate each joint
- Apply equilibrium: ΣFx = 0, ΣFy = 0
- Maximum 2 unknowns per joint
- Start at joint with ≤ 2 unknowns
Method of Sections
- Cut through truss (max 3 members)
- Apply equilibrium to section
- ΣFx = 0, ΣFy = 0, ΣM = 0
- Choose moment point to eliminate unknowns
Indeterminate Structures
| Method | Application |
|---|---|
| Moment Distribution | Continuous beams and frames; iterative |
| Slope-Deflection | Relates moments to rotations; simultaneous equations |
| Three-Moment Equation | Continuous beams; relates moments at three supports |
| Conjugate Beam | Deflection and slope calculation |
Moment Distribution Concepts
- Distribution Factor (DF): K/(ΣK) where K = I/L (stiffness)
- Carry-Over Factor: 0.5 for prismatic members (far end fixed)
- Fixed End Moments: Standard formulas for various loadings
7. Load Combinations (NSCP)
Strength Design Combinations
- U = 1.4D
- U = 1.2D + 1.6L + 0.5(Lr or S or R)
- U = 1.2D + 1.6(Lr or S or R) + (L or 0.5W)
- U = 1.2D + 1.0W + L + 0.5(Lr or S or R)
- U = 1.2D + 1.0E + L + 0.2S
- U = 0.9D + 1.0W
- U = 0.9D + 1.0E
Load Notation
- D: Dead load
- L: Live load
- Lr: Roof live load
- S: Snow load
- R: Rain load
- W: Wind load
- E: Earthquake load
- F: Fluid load
Strength Reduction Factors (φ)
- Flexure (tension-controlled): φ = 0.90
- Shear and torsion: φ = 0.75
- Compression (tied columns): φ = 0.65
- Compression (spiral columns): φ = 0.75
- Bearing: φ = 0.65
8. Connections
Bolted Connections
Bolt Shear Strength
Rₙ = Fₙᵥ × Aᵦ × n
n = number of shear planes
Bearing Strength
Rₙ = 2.4dtFᵤ
Standard holes
Welded Connections
Fillet Weld Strength
Rₙ = 0.6FEXX × (0.707 × a) × L
- FEXX = electrode strength (e.g., E70 = 482 MPa)
- a = weld leg size
- L = effective length
Key Takeaways for the Board Exam
- ✓Bending stress: σ = Mc/I = M/S
- ✓Euler buckling: Pcr = π²EI/(KL)²
- ✓RC nominal moment: Mₙ = Asfy(d - a/2)
- ✓Compression block: a = Asfy/(0.85f'c·b)
- ✓K factors: P-P=1.0, F-F=0.5, F-P=0.7, F-Free=2.0
- ✓SS uniform load: M_max = wL²/8
- ✓β₁ = 0.85 for f'c ≤ 28 MPa
- ✓Shear design: φVₙ ≥ Vᵤ; Vₙ = Vc + Vs
Structural Topics
Stress & StrainBeam AnalysisColumnsReinforced ConcreteSteel DesignStructural AnalysisLoad CombinationsConnections