Deflections of Structures - Cheat Sheet
Your last-minute reference for virtual work, Castigliano's theorem, and deflection calculations. Master truss and beam deflections in 30 minutes.
Energy Methods Fundamentals
Formulas
U = ∫(M²/2EI)dx
Meaning
U = strain energy; M = bending moment; E = modulus of elasticity; I = second moment of inertia
Watch out
Must integrate over entire structure. Units: if M in N·m, EI in N·m², result is in Joules
When to use
When storing bending energy in beams/frames; foundation for Castigliano's theorem
U_axial = N²L/(2AE)
Meaning
U = axial strain energy; N = axial force; L = member length; A = cross-section area; E = modulus
Watch out
This is for ONE member. Sum over all members. N² always positive (energy is never negative)
When to use
For truss members and axially loaded columns; key for truss deflection energy
Important facts
- Virtual work is the foundation of all deflection methods for statically indeterminate structures
- A unit load (not real) must be applied in the SAME DIRECTION as the desired deflection
- Energy methods yield a SINGLE deflection at a time; to find multiple deflections, repeat the process
- Strain energy is always positive (stored in the structure); virtual work is a scalar quantity
- These methods apply uniformly to trusses, beams, frames, and cables — no special tricks needed
Key definitions
Strain Energy
Example
A bent beam stores strain energy proportional to M²/EI along its length
Internal energy stored in a deformed elastic structure; numerically equal to external work done by applied loads.
Virtual Work
Example
Apply a unit load at point C to find deflection there; multiply by real forces at each member
Work done by a virtual (imaginary) load system moving through real displacements; used to extract deflections without solving for all unknowns.
Compatibility Condition
Example
In indeterminate analysis, deflection equations from virtual work enforce that the structure remains intact
Mathematical relationship ensuring displacements are continuous and consistent with support constraints.
Diagrams to know
- Real force diagram and unit load diagram superimposed (conceptually) to show how virtual work combines them
- Sign convention: tension = positive, compression = negative in truss members
- Bending moment diagram under both real load and unit load
Virtual Work (Unit-Load Method)
Formulas
δ = Σ(nNL/AE)
Meaning
δ = deflection; n = member force from unit load; N = real member force; L = member length; AE = axial stiffness
Watch out
Signs matter: like signs (both tension or both compression) give positive nN; opposite signs give negative (error-prone). Verify geometry of unit load carefully
When to use
ALWAYS use this for truss joint deflections; straightforward, one formula per member
δ = ∫(mM/EI)dx
Meaning
δ = linear deflection; m = moment from unit load; M = real bending moment; EI = flexural stiffness
Watch out
Integration limits MUST match the structure span. If beam is L = 4 m, integrate 0 to 4, not 0 to L symbolically
When to use
For beam and frame deflections; must integrate piecewise if M or m changes expression
θ = ∫(mM/EI)dx where m is moment from unit MOMENT
Meaning
θ = angular deflection (rotation); m = moment from a unit couple (moment) applied at the point
Watch out
Apply a unit COUPLE (moment), not a force. The moment diagram m is different from the force case
When to use
When the exam asks for 'slope' or 'rotation angle' at a point
Important facts
- Virtual work is exact — no approximation; valid for ANY elastic material and load case
- The unit load is dimensionless in the calculation; its 'magnitude' of 1 simplifies the integral or sum
- For trusses with equal AE in all members, factor out AE from the sum: δ = (1/AE)Σ(nNL)
- Piecewise integration must be done carefully at discontinuities (supports, loads); easy source of errors
- The method works for ANY shape of M and m, as long as EI is constant (or split by region)
Key definitions
Unit Load (Unit Force)
Example
To find vertical deflection at C, apply 1 kN downward at C; member forces become n values
A dimensionless load of magnitude 1 (in the direction sought) applied to find a deflection; internal forces scale with it.
Unit Moment
Example
Apply 1 kN·m counterclockwise at point B to find the slope at B
A couple of magnitude 1 applied to find a rotation; used instead of a unit force.
Piecewise Integration
Example
A cantilever under UDL has one M(x) expression; if a point load is added, split at the load point
Breaking the integral into regions where M(x) and m(x) have different expressions (e.g., at supports or load points).
Diagrams to know
- Two side-by-side FBD sketches: (1) real load case, (2) unit load case; label all reactions
- Bending moment diagrams for both real and unit cases; show key ordinates (maximum, zero-crossings)
- For trusses: member force table with columns N, n, nN, L, nNL
Castigliano's Theorems
Formulas
δ_i = ∂U/∂P_i
Meaning
δ_i = deflection at point i; U = total strain energy; P_i = load at point i; ∂ = partial derivative
Watch out
MUST take partial derivative with respect to the LOAD, not position. Easy to confuse with dU/dx
When to use
When you want to avoid setting up an integral explicitly; especially useful if U is already known
θ_i = ∂U/∂M_i
Meaning
θ_i = rotation at point i; M_i = applied moment (couple) at point i
Watch out
If no moment is applied at the target point, introduce a dummy moment Q, differentiate, then set Q = 0
When to use
For rotations; algebraically identical to the force case, just with moments instead
δ = ∫(M/EI)(∂M/∂P)dx
Meaning
Expanded Castigliano form; ∂M/∂P = how the moment changes when load P increases by dP
Watch out
This is NOT a coincidence — virtual work and Castigliano are equivalent. Choose whichever feels easier
When to use
Algebraically identical to unit-load method; ∂M/∂P is exactly the moment m from the unit load
Important facts
- Castigliano's second theorem is EQUIVALENT to the unit-load method — same answer, different derivation
- The dummy-load trick is essential when no real load exists at the deflection point
- Strain energy must include ALL sources: axial, bending, shear (though shear is usually neglected for slender beams)
- Partial derivatives are with respect to loads only; position x is NOT a variable being differentiated
- Result is a single deflection; for multiple deflections, repeat for each load (or use superposition)
Key definitions
Castigliano's First Theorem
Example
∂U/∂δ = P (not commonly tested in PRC exams)
The partial derivative of total strain energy with respect to a deflection equals the conjugate force (rarely used in practice).
Castigliano's Second Theorem
Example
δ_C = ∂U/∂P_C if load P_C acts at C
The deflection at a point equals the partial derivative of strain energy with respect to the load at that point; standard tool for finding deflections.
Dummy Load Method
Example
To find slope at a point with no applied moment, add Q (unit couple), find ∂U/∂Q, set Q = 0
If no real load acts at the target point, introduce a fictitious load Q, compute ∂U/∂Q, then set Q = 0 at the end.
Diagrams to know
- Energy curve U(P) for a simple cantilever; show that slope dU/dP at load P gives deflection δ
- Free-body diagram with dummy load Q labeled; must be dimensionally compatible (force for deflection, moment for rotation)
Truss Deflections Step-by-Step
Formulas
δ_joint = Σ(nNL/AE) for all members
Meaning
Sum over all members in the truss; n and N are axial forces (tension positive), L is member length, AE is stiffness
Watch out
Sign errors are the #1 killer. Use a consistent sign convention (tension = +, compression = –). Verify n and N have correct signs before multiplying
When to use
EVERY truss deflection problem; straightforward algorithm
Common values
Value
E = 200 GPa = 200,000 MPa = 2 × 10⁵ kN/m²
Symbol
E
Quantity
Typical Young's modulus for structural steel
Value
A = 50–500 mm² (0.005–0.05 m²)
Symbol
A
Quantity
Typical cross-section area (small truss member)
Important facts
- Truss deflection depends ONLY on axial strains; bending of members is neglected (assumed rigid)
- Member length L is the distance between joints; use geometry or Pythagoras to find it
- If all members have the same AE, factor it out: δ = (1/AE)Σ(nNL); often simplifies arithmetic
- Horizontal and vertical deflections are independent; apply unit loads in the respective direction to find each
- A single joint can have multiple deflection components (horizontal, vertical); solve separately
Key definitions
Truss Member Force (Real)
Example
In a roof truss under gravity, lower chords are typically in tension; upper chords in compression
Axial force N in a member under real loads; found by method of joints or sections. Positive = tension, negative = compression.
Unit Load Member Force
Example
If a real load of 20 kN produces N = 10 kN, a unit load produces n = 0.5 kN in the same member
Axial force n in a member when a unit load (1 kN) is applied at the joint of interest. Scale of real forces.
Diagrams to know
- Truss geometry diagram with coordinates; calculate member lengths using distance formula
- Member force table: columns for Member, N (kN), n (dimensionless), L (m), nNL
- Final sum: Σ(nNL) with units verified (e.g., kN × m = kN·m, then divide by AE in kN to get m)
Beam and Frame Deflections Step-by-Step
Formulas
δ = ∫₀ᴸ (mM/EI)dx
Meaning
Integrate over entire span L; m = moment from unit load, M = real bending moment, EI = flexural stiffness
Watch out
Integration limits must match beam span. If using symbolic integration, verify numerical result is sensible (magnitude, sign, units)
When to use
ANY beam or frame deflection; piecewise if M or m change expression
∫ₐᵇ x dx = [x²/2]ₐᵇ = (b² − a²)/2
Meaning
Common integral for linear bending moment; a, b are bounds
Watch out
Sign of limits matters; upper − lower. Common mistake: flipping the subtraction
When to use
Cantilever or simple beam with linear M(x) expressions
∫ₐᵇ x² dx = [x³/3]ₐᵇ = (b³ − a³)/3
Meaning
Common integral for parabolic moment (UDL produces M ∝ x²)
Watch out
Ensure UDL region matches integration bounds; don't integrate beyond the load
When to use
Cantilever under UDL; simple beam under UDL (in region where M is quadratic)
Common values
Value
E = 2 × 10⁵ kN/m² (or 200 GPa)
Symbol
E
Quantity
Young's modulus (structural steel)
Value
E = 2.5 × 10⁴ kN/m² (or 25 GPa)
Symbol
E
Quantity
Young's modulus (reinforced concrete, avg)
Value
I = bd³/12
Symbol
I
Quantity
Second moment of inertia (rectangular section, b × d)
Important facts
- Always draw both real M(x) and unit m(x) diagrams; verify shapes and key ordinates before multiplying
- Product mM is always positive where both are on the same side; negative if they are opposite (unusual)
- Piecewise integration requires breaking the span at every discontinuity (support, load, load-type change)
- EI is usually constant for a homogeneous beam; if it varies, split the integral by region
- For a cantilever, integrating from the FREE end simplifies M(x); avoid integrating from the fixed end (messier algebra)
Key definitions
Piecewise Segment
Example
A cantilever with point load at tip and UDL on the left half has 2 segments: each has its own M(x)
Region of a beam where bending moment M(x) has a single continuous expression; bounded by supports, loads, or discontinuities.
Flexural Stiffness
Example
EI = 200,000 kN/m² × 0.001 m⁴ = 200 kN·m² (or 2 × 10⁹ N·mm²)
Product EI; measure of a beam's resistance to bending. Higher EI → smaller deflection.
Diagrams to know
- Real bending moment diagram (to scale or with key values)
- Unit load (or unit moment) diagram for the same beam with support reactions
- Side-by-side diagrams to visualize the product mM along the span
- Table of integration results: region, M(x) expression, m(x) expression, product, integral limits, ∫mM dx
Standard Formulas (Quick Reference)
Formulas
δ_tip = PL³/(3EI)
Meaning
Deflection at free end of cantilever under end load P; L = length, EI = flexural stiffness
Watch out
This is CUBIC in L; doubling length → 8× deflection. Be careful with units (L in m, result in m)
When to use
Exam favorite; memorize. Appears in many structures
δ_mid = PL³/(48EI)
Meaning
Midspan deflection of simply supported beam under central point load P
Watch out
Coefficient 48 is easy to confuse with other standard formulas (e.g., UDL under SS beam uses different constant)
When to use
Another exam favorite; ratio is 48, not 3 (cantilever is 3EI)
δ_mid = 5wL⁴/(384EI)
Meaning
Midspan deflection of simply supported beam under uniform distributed load w (force per unit length)
Watch out
Coefficient 384 is specific to UDL on SS beam. Different boundary conditions → different constant
When to use
Common in building floors; w in kN/m, L in m
δ_cantilever_UDL = wL⁴/(8EI)
Meaning
Free-end deflection of cantilever under UDL w; L = length
Watch out
Constant is 8, not 3. UDL → higher deflection than point load at same total force
When to use
Cantilevered overhangs, shelves, etc.
θ_slope = PL²/(2EI) at fixed end (or other standard slopes)
Meaning
Rotation angle at a point; use standard formulas if available
Watch out
Slopes often have different coefficients than deflections; memorize separately or derive via unit-moment method
When to use
When exam asks for 'slope' or 'angle of rotation'
Important facts
- Standard formulas are time-savers in exams; derive them once, then use them freely
- All standard formulas assume constant EI, elastic material, small deflections (linear theory)
- Deflection ∝ L⁴ for cantilevers and simply supported beams (very sensitive to span length)
- Deflection ∝ 1/EI; stiffer beams (larger I or E) deflect less
- Serviceability limits (NSCP 2015) often cap deflections at L/240 to L/360; use these formulas to check compliance
Diagrams to know
- Shape of deflected curve for cantilever (concave, maximum at tip)
- Shape of deflected curve for simply supported beam (symmetric, maximum at midspan)
NSCP 2015 & ACI 318 Serviceability Limits
Common values
Value
L/360
Symbol
δ_max
Quantity
Typical deflection limit for residential floor
Value
L/240
Symbol
δ_max
Quantity
Typical deflection limit for floor with plaster ceiling
Value
L/180
Symbol
δ_max
Quantity
Typical deflection limit for cantilever
Important facts
- NSCP 2015 Section 402.4 (ACI 318): immediate deflection Δ_i is calculated from elastic deflection formulas
- Long-term deflection Δ_long = Δ_i + Δ_creep; creep factor λ accounts for sustained load and reinforcement ratio
- Typical limits: L/360 for floors (comfort criterion), L/240 for beams with plastered ceilings, L/180 for cantilevered members
- Steel structures (AISC 360): typical limit H/240 for vertical deflections of primary members; more lenient than concrete
- Always CHECK deflection adequacy; just meeting strength is NOT enough for design (serviceability governs in many cases)
Key definitions
Allowable Deflection
Example
A 6 m beam with L/240 limit allows δ_max = 6000/240 = 25 mm
Maximum permissible deflection per NSCP 2015; typically L/240 to L/360 depending on structural member type and use.
Sign Conventions & Common Errors
Important facts
- Signs of n and N must match for the product nN to be positive (both tension or both compression)
- Mismatched signs → negative product → erroneous negative deflection (student thinks structure is stiff, actually made a sign error)
- For bending, the product mM is always positive if both are on the same side of the neutral axis (typical case)
- Consistent sign convention THROUGHOUT the problem is essential; mixing conventions causes cascading errors
- Unit load ALWAYS acts in the direction of desired deflection; never apply a unit load opposite to the sought direction
Key definitions
Positive Deflection
Example
If a beam bends downward under downward load, δ is positive
Deflection in the same direction as the applied unit load (usually downward for gravity loads).
Tension Member
Example
Lower chord of a roof truss under gravity is typically in tension
Truss member pulling apart (N > 0, stretches under load); contributes positively to joint deflection.
Compression Member
Example
Upper chord of a roof truss under gravity is typically in compression
Truss member pushing inward (N < 0, shortens under load); also contributes positively if n and N both negative.
Diagrams to know
- Sign diagram for truss members: + for tension (pulling), − for compression (pushing)
- Bending moment sign convention: sagging = positive, hogging = negative (or vice versa, as long as consistent)
Must remember
- Virtual work formula: δ = Σ(nNL/AE) for trusses OR δ = ∫(mM/EI)dx for beams/frames — these are THE formulas; everything else is bookkeeping
- Unit load MUST be applied in the direction of the desired deflection; if you want vertical δ, apply vertical unit load (not horizontal, not diagonal)
- Castigliano's second theorem δ = ∂U/∂P is IDENTICAL to virtual work; ∂M/∂P is exactly the moment from the unit load — same answer, different path
- Sign convention: tension = positive, compression = negative in truss members; product nN is positive if both have same sign (both tension or both compression)
- Piecewise integration is MANDATORY when bending moment M(x) or unit moment m(x) change expression; split at every support, load, or discontinuity
- Dummy load trick: if no real load acts at the target point, introduce fictitious load Q, compute ∂U/∂Q with Q in the expressions, then substitute Q = 0 at the end
- Standard formulas are time-savers: cantilever end load = PL³/(3EI), simply supported central load = PL³/(48EI), cantilever UDL = wL⁴/(8EI) — memorize for speed
- NSCP 2015 / ACI 318 deflection limits: L/360 for floors (no plaster), L/240 with plaster, L/180 for cantilevers; always check serviceability, not just strength
- Integration limits must match the beam span; if span is L = 6 m, integrate from 0 to 6, not from 0 to L symbolically — verify limits after substituting values
- Strain energy is always positive (U ≥ 0); energy methods yield scalar results (single deflection at a time); use superposition or repeat process for multiple points
Last minute tips
- Draw BOTH diagrams (real and unit load) side by side before integrating or summing; visual check catches sign errors and integration-limit mistakes
- For truss problems: create a table with columns [Member | N (real) | n (unit) | L | nNL]; sort by magnitude to spot arithmetic errors before the final sum
- Forget memorizing 'which coefficient goes where'? Derive the cantilever formula in 30 seconds: M(x) = −Px, m(x) = −x, product = Px², integrate ∫₀ᴸ Px²/(EI)dx = PL³/(3EI) — done
- If integration feels stuck: use dimensional analysis to check your setup. ∫(mM/EI)dx should have units [moment²/(stiffness·length)] = [length], confirming the integrand is correct before grinding through algebra
- Exam trick: many deflection problems appear solvable by standard formula BUT with a twist (asymmetric loading, non-standard support). Default to virtual work / Castigliano to be safe; standard formulas are shortcuts only if the setup matches perfectly
Comparison tables
Virtual Work vs. Castigliano's Theorem Comparison
| Aspect | Virtual Work (Unit Load) | Castigliano's Second Theorem |
|---|---|---|
| Fundamental Concept | Apply unit load → compute internal forces → multiply by real forces | Differentiate strain energy U with respect to load P |
| Formula for Deflection | δ = Σ(nNL/AE) or ∫(mM/EI)dx | δ = ∂U/∂P = ∫(M/EI)(∂M/∂P)dx |
| When to Use | Any deflection, but especially straightforward for single unknowns | When U is already expressed; cleaner for multiple loads |
| Dummy Load Trick | Not needed; apply unit load directly at point of interest | Required if no real load acts; add Q, compute ∂U/∂Q, then Q = 0 |
| Algebraic Equivalence | ∂M/∂P in Castigliano IS the unit moment m | Exactly the same as virtual work; just different mental path |
| Typical Exam Use | Primary method taught; most students prefer it | Alternative derivation; shows deeper understanding |
Truss vs. Beam Deflection Methods Comparison
| Feature | Truss Joint Deflection | Beam / Frame Deflection |
|---|---|---|
| Deformation Mode | Axial strain in each member (ΔL = NL/AE) | Bending strain; curvature κ = M/EI |
| Key Formula | δ = Σ(nNL/AE) | δ = ∫(mM/EI)dx |
| Integration / Summation | Discrete sum over members; finite calculation | Continuous integral over span; may be piecewise |
| Typical Effort | Member force analysis + arithmetic; fast if <10 members | Reaction + moment diagram + integration; slower if many segments |
| Common Errors | Sign of nN (compression/tension confusion); wrong L (geometry) | Integration limits; sign of mM; piecewise segment boundaries |
| Unit Load Application | Unit force at the joint (horizontal or vertical) | Unit force or unit moment at the point (same structure, different load case) |
Standard Deflection Formula Reference
| Beam Type & Load | Deflection Formula | Coefficient | Notes |
|---|---|---|---|
| Cantilever, end point load P | δ = PL³/(3EI) | 1/3 | L³ term; memorize coefficient 3 |
| Cantilever, UDL w | δ = wL⁴/(8EI) | 1/8 | L⁴ term; coefficient 8 (not 3) |
| Simply supported, central load P | δ = PL³/(48EI) | 1/48 | Coefficient 48 (not 3); symmetric case |
| Simply supported, UDL w | δ = 5wL⁴/(384EI) | 5/384 | L⁴ term; coefficient 384 (memorize for exam) |
| Cantilever, slope at fixed end (UDL) | θ = wL³/(6EI) | 1/6 | Rotation angle, not deflection; L³, not L⁴ |
Diagrams
Deflection Methods Hierarchy
Comprehensive mind map showing the hierarchy of deflection methods, from energy-based approaches (virtual work and Castigliano) down to standard formulas and critical checks. Use this to navigate the chapter conceptually.
Deflection Problem-Solving Algorithm
Step-by-step flowchart guiding students through any deflection problem. Start with structure type, apply appropriate method, perform calculations, and validate results against serviceability limits.
Common Deflection Calculation Errors Distribution
Distribution of typical mistakes students make on deflection exams. Most errors stem from sign mismanagement in truss problems and integration-limit oversights in beam deflections. Use this to focus your error-checking strategy.