Cheatsheet: Structural Analysis Methods

1. Fundamental Equilibrium Methods

1.1 Static Equilibrium Equations

1.2 Determinacy and Stability

1.3 Support Reactions

2. Truss Analysis

2.1 Truss Determinacy

2.2 Method of Joints

• Apply equilibrium at each joint: ΣFx = 0, ΣFy = 0
• Begin at joint with ≤2 unknown forces
• Tension: force pulls away from joint (positive)
• Compression: force pushes toward joint (negative)
• Zero-force members: unloaded joint with 2 non-collinear members

2.3 Method of Sections

• Cut through ≤3 members to isolate section
• Apply ΣFx = 0, ΣFy = 0, ΣM = 0 to free body
• Take moments about points to eliminate unknowns
• Useful for finding forces in specific members without analyzing entire truss

3. Beam Analysis

3.1 Shear and Moment Relationships

3.2 Sign Conventions

• Positive shear: left side up, right side down
• Negative shear: left side down, right side up
• Positive moment: compression on top (sagging)
• Negative moment: tension on top (hogging)

3.3 Key Diagram Properties

• Maximum moment occurs where V = 0
• Point load causes jump in shear diagram
• Point load causes slope change in moment diagram
• Distributed load causes linear variation in shear
• Distributed load causes parabolic variation in moment
• Concentrated moment causes jump in moment diagram only

3.4 Common Beam Formulas

4. Deflection Methods

4.1 Double Integration Method

• Boundary conditions: y = 0 at supports, dy/dx = 0 at fixed ends
• Solve for constants C₁ and C₂ using boundary conditions

4.2 Moment-Area Method

4.2.1 Theorems

• θB/A = angle change from A to B
• tB/A = vertical deviation of B from tangent at A
• Use conjugate beam analogy: load = M/EI, shear = slope, moment = deflection

4.3 Conjugate Beam Method

• Real beam load → Conjugate beam support reactions
• Conjugate beam load = M/EI from real beam
• Shear in conjugate beam = slope in real beam
• Moment in conjugate beam = deflection in real beam

4.4 Virtual Work/Unit Load Method

• Apply unit load at location/direction of desired deflection
• Calculate m or f from unit load
• Calculate M or F from actual loads
• Integrate or sum products

5. Influence Lines

5.1 Definition and Use

• Shows variation of response (reaction, shear, moment) at specific point as unit load moves across structure
• Used to determine critical load positions for maximum response
• Ordinate = value of response when unit load is at that position

5.2 Müller-Breslau Principle

• Remove restraint corresponding to desired response
• Apply unit displacement in direction of response
• Deflected shape is influence line (to scale)
• Valid for statically determinate and indeterminate structures

5.3 Application for Maximum Response

• Single concentrated load: place at maximum ordinate
• Uniform distributed load: place over positive (or negative) region
• Series of concentrated loads: use absolute maximum concepts
• Maximum response = Σ(Pi × yi) where Pi = load, yi = influence ordinate

5.4 Common Influence Lines (Simply Supported Beam, Length L)

6. Indeterminate Structures - Force Method

6.1 Force Method Procedure

1. Determine degree of indeterminacy (i)
2. Select i redundant forces/moments
3. Remove redundants to create determinate primary structure
4. Apply actual loads to primary structure, find displacements at redundant locations
5. Apply unit values of each redundant separately, find flexibility coefficients
6. Solve compatibility equations: Δ = Δ₀ + f₁₁X₁ + f₁₂X₂ + ... = 0
7. Superimpose to find final forces

6.2 Flexibility Coefficients

6.3 Compatibility Equations

• For each redundant: total displacement = 0 (or known value)
• Matrix form: [f]{X} = -{Δ₀} where [f] = flexibility matrix, {X} = redundants, {Δ₀} = primary structure displacements

7. Indeterminate Structures - Displacement Method

7.1 Slope-Deflection Equations

• θA, θB = rotations at A and B
• ψ = (ΔB - ΔA)/L = chord rotation
• FMAB, FMBA = fixed-end moments from applied loads
• Apply equilibrium at each joint: ΣM = 0
• Solve simultaneous equations for unknown rotations and deflections

7.2 Fixed-End Moments (FEM)

7.3 Moment Distribution Method

7.3.1 Key Parameters

7.3.2 Procedure

1. Calculate fixed-end moments for all loaded members
2. Calculate stiffness factors K and distribution factors DF at each joint
3. Lock all joints, apply loads to get FEMs
4. Release one joint at a time, distribute unbalanced moment using DF
5. Carry over half the distributed moments to far ends (COF = 0.5)
6. Repeat until moments converge
7. Sum moments at each end of each member

8. Matrix Structural Analysis

8.1 Direct Stiffness Method

8.1.1 Local Element Stiffness Matrix (2D Truss Element)

• k' = (AE/L)[1 -1; -1 1] for axial member
• Degrees of freedom: u1, u2 (axial displacements at nodes)

8.1.2 Coordinate Transformation

8.1.3 Assembly and Solution

1. Formulate element stiffness matrices in local coordinates
2. Transform to global coordinates using transformation matrix
3. Assemble global stiffness matrix [K] by adding overlapping terms
4. Apply boundary conditions (eliminate rows/columns for restrained DOFs)
5. Solve [K]{u} = {F} for unknown displacements
6. Back-calculate element forces using {f} = [k'][T]{u}

8.2 Beam Element Stiffness (Local Coordinates)

• Stiffness matrix k' is 4×4 symmetric
• k'11 = k'22 = 12EI/L³; k'12 = k'21 = 6EI/L²
• k'33 = k'44 = 4EI/L; k'34 = k'43 = 2EI/L

9. Approximate Analysis Methods

9.1 Portal Method (Frames)

• Assumes inflection points at mid-height of columns
• Assumes inflection points at mid-span of beams
• Interior columns carry twice the shear of exterior columns
• Total horizontal force distributed: Vext = V/n, Vint = 2V/n
• Use for frames with height/width ratio > 2

9.2 Cantilever Method (Frames)

• Assumes inflection points at mid-height of columns
• Assumes inflection points at mid-span of beams
• Axial forces in columns proportional to distance from centroid
• ΣMc = 0 at each floor level to find axial forces
• Use for frames with height/width ratio <>

9.3 Trusses with Loads at Panel Points

• Assume chord members resist moments as axial force couples
• Chord force F = M/h where M = moment, h = truss depth
• Web members resist shear
• Vertical web force = V / (number of verticals)

10. Special Topics

10.1 Cables

10.1.1 Parabolic Cable (Uniform Horizontal Load w)

10.1.2 Catenary Cable (Uniform Load Along Cable)

• Shape: y = (H/w)[cosh(wx/H) - 1]
• For heavy cables, chains, or self-weight dominant

10.2 Three-Hinged Arches

• Statically determinate (3 equations + 1 condition at crown hinge)
• Horizontal thrust H reduces moment compared to beam
• At crown hinge: moment = 0 provides additional equation
• For symmetric arch with center load: H = PL/(4h)
• For uniform load: H = wL²/(8h)

10.3 Composite Structures

• Transform section using modular ratio n = E1/E2
• Transformed width = actual width × n
• Calculate properties of transformed section
• Stress in material 1: σ1 = My/I
• Stress in material 2: σ2 = nMy/I