Beam Deflections - Cheat Sheet
Your 30-minute rapid-fire reference for beam deflection formulas, methods, boundary conditions, and exam pitfalls. Covers double integration, area-moment, conjugate beam, and superposition—all you need to ace deflection problems on the board exam.
Fundamental Differential Equation & Boundary Conditions
Formulas
EI·y″ = M(x)
Meaning
E = modulus of elasticity (N/mm²); I = second moment of inertia (mm⁴); y = deflection (mm); M(x) = bending moment function (N·mm); y″ = d²y/dx²
Watch out
Sign convention must be consistent. Most common: downward deflection = positive y; upward = negative. Some textbooks flip this—check at the start.
When to use
Master equation—always the starting point for any deflection problem; integrate once for slope (y′), twice for deflection (y)
y′ = dy/dx = (1/EI)∫M(x)dx + C₁
Meaning
y′ is slope (radians); first integral of bending moment plus constant of integration
Watch out
Do NOT forget the constant C₁. It vanishes only if you use a definite integral with proper limits.
When to use
After first integration of the differential equation; apply boundary conditions to find C₁
y = (1/EI)∫∫M(x)dx·dx + C₁·x + C₂
Meaning
Double integral of M/EI plus linear terms from constants; gives the elastic curve
Watch out
Most students lose signs or forget that each ∫∫ step requires careful algebra. Macaulay brackets 〈x−a〉ⁿ simplify multi-segment loads drastically.
When to use
After second integration; apply two boundary conditions to solve for C₁ and C₂
Important facts
- Downward deflection is typically positive in Philippine/standard convention; verify this in your problem statement.
- At a simply supported end: y = 0 but y′ ≠ 0 (roller/pin permits rotation).
- At a fixed (cantilever) end: both y = 0 and y′ = 0 (no displacement, no slope).
- At a free end of a cantilever: both y and y′ are maximum; y″ = M/EI (not zero).
- Macaulay bracket notation 〈x − a〉ⁿ = 0 if x < a; = (x − a)ⁿ if x ≥ a. This avoids piecewise integration.
- The elastic curve must be continuous and smooth (except at point loads, where curvature can jump).
Key definitions
Elastic Curve
Example
A sagging simply supported beam forms a curve concave downward; a cantilever deflects monotonically away from the fixed end.
The deflected shape of the beam's neutral axis under load; mathematically described by y(x).
Flexural Rigidity
Example
A steel beam (E ≈ 200 GPa) is much stiffer than an aluminum beam (E ≈ 70 GPa) for equal I.
The product EI (N·mm²); resists bending. Higher EI → less deflection for the same load.
Slope
Example
At a simple support, slope is nonzero; at a fixed end, slope = 0.
The derivative dy/dx; angle of rotation (radians) of the beam's tangent at any point x.
Boundary Condition
Example
For a simply supported beam: y(0) = 0 and y(L) = 0; these two equations solve for C₁ and C₂.
Mathematical constraint applied at supports to determine constants of integration. E.g., y = 0 at a pin; y = y′ = 0 at a fixed end.
Diagrams to know
- Elastic curve shape for: simply supported beam under UDL, cantilever under point load, propped cantilever.
- Sign diagram showing curvature direction (M positive = concave up; M negative = concave down).
- Beam deflection with labels: deflection y, slope θ = y′, moment M, curvature 1/ρ = M/(EI).
Method 1: Double Integration (Direct Method)
Formulas
∫EI·y″ dx = ∫M(x) dx
Meaning
First integration of the differential equation; yields EI·y′ = ∫M(x) dx + C₁
Watch out
Integration constant C₁ is critical. Never drop it; it represents a rigid-body rotation of the entire beam.
When to use
Whenever M(x) is known as a function of x and you want slope or deflection equations
∫EI·y′ dx = ∫[∫M(x) dx + C₁] dx
Meaning
Second integration; gives EI·y = ∫∫M(x) dx·dx + C₁·x + C₂
Watch out
Algebra explosions with multi-segment loads. Use Macaulay brackets 〈x − a〉ⁿ to avoid breaking the integral into pieces.
When to use
To find the complete deflection function y(x)
Important facts
- Double integration always works but can be tedious for complex loading.
- For piecewise loading (e.g., different load in each segment), either write separate M(x) per segment OR use Macaulay brackets.
- Constants of integration are found by applying EXACT number of boundary conditions (typically 2 for a beam: y at two points or y and y′ at one).
- Sign errors in M(x) propagate directly to y(x); draw a free body and moment diagram first.
- The method yields exact solutions for linearly elastic beams with smooth loading.
Key definitions
Macaulay (Singularity) Brackets
Example
For a simply supported beam with UDL w from x = a to x = b, write M(x) = ... − w/2·[〈x−a〉² − 〈x−b〉²] + ... and integrate twice directly.
Notation 〈x − a〉ⁿ that equals zero if x < a, and (x − a)ⁿ if x ≥ a; allows one continuous M(x) formula over entire span.
Particular vs Homogeneous Solution
Example
For cantilever with point load at free end, y_p comes from M(x) integration; C₁ = C₂ = 0 from BC y(0) = y′(0) = 0.
EI·y″ = M has a general (homogeneous) solution y_h = C₁·x + C₂ plus a particular y_p from integrating M; total y = y_p + y_h.
Diagrams to know
- Moment diagram M(x) used in the double-integration formula.
- Bending-moment equations for simple load cases (cantilever, simple beam, overhang).
Method 2: Area-Moment (Moment-Area) Theorems
Formulas
θ_B − θ_A = ∫(M/EI) dx from A to B = Area of M/EI diagram between A and B
Meaning
Theorem 1: Change in slope between two points = area under the M/EI diagram
Watch out
Area must be measured in (M/EI) space. If EI is constant, Area = (∫M dx)/EI. Watch unit consistency (moment in N·mm, EI in N·mm²).
When to use
When you know slope at one point (e.g., zero at fixed end) and want slope at another; super fast for cantilevers
t_B/A = ∫(M/EI)·x_B dx from A to B = Area(M/EI) × x̄_B
Meaning
Theorem 2: Vertical deviation of B from tangent line at A = area of M/EI × horizontal distance from B to centroid of area
Watch out
x̄_B is measured from point B. If calculating from left, measure backward to the area's centroid. Sign matters: positive area → downward deviation.
When to use
Finding deflections at specific points. For cantilever with horizontal tangent at fixed end, t_B/A = deflection at B
Important facts
- Theorem 1 is purely slope-focused; Theorem 2 gives deflection.
- Both theorems assume constant EI. If EI varies, integration must account for it.
- The M/EI diagram shape depends on loading: triangular for point loads, parabolic for UDL, piecewise linear for step loads.
- Centroid locations for common shapes: triangle = 1/3 from apex; rectangle = 1/2; parabola = 3/8 or 5/8 depending on orientation.
- Area-moment is fastest for cantilevers and single-point queries; less efficient if you need the entire y(x) function.
Key definitions
M/EI Diagram
Example
For a simply supported beam with point load P at midspan, M/EI is triangular, peaking at L·P/(4EI) at center.
A plot of bending moment divided by flexural rigidity; used as the 'load' on the conjugate beam or for area-moment calculations.
Tangential Deviation (t_B/A)
Example
For a cantilever, the tangent at the fixed end is horizontal; tangential deviation equals the actual deflection at the free end.
Vertical distance between point B and the tangent line drawn at point A; measured perpendicular to the tangent.
Centroid of M/EI Area
Example
For a triangular M/EI (cantiever), centroid is 1/3 of base from the apex; for parabolic (simply supported UDL), centroid is at 3/8 from peak.
The geometric center of the M/EI diagram; used in the second moment calculation for deflection.
Diagrams to know
- M/EI diagram for cantilever (triangular), simply supported beam under point load (double triangle), and simply supported under UDL (parabolic).
- Tangent line construction: draw tangent at a known-slope point, measure perpendicular distance to another point.
Method 3: Conjugate-Beam Method
Formulas
Load on conjugate = M(x)/EI (from real beam)
Meaning
The real beam's M/EI diagram becomes the distributed load on a fictitious conjugate beam
Watch out
Support types transform: real simple → conjugate simple; real fixed → conjugate free (and vice versa). Real free → conjugate fixed.
When to use
When converting deflection/slope problems into static-equilibrium problems (shear/moment in conjugate)
Slope in real beam = Shear in conjugate; Deflection in real = Moment in conjugate
Meaning
θ_real = V_conjugate; y_real = M_conjugate
Watch out
Sign consistency: upward shear in conjugate = positive slope in real; clockwise moment in conjugate = downward deflection in real (check sign convention).
When to use
Reading off conjugate-beam shear/moment diagrams to find slopes and deflections
Important facts
- The conjugate beam is not real; it's a mathematical tool to convert geometry into statics.
- If EI is constant, the conjugate loading is simply the M diagram divided by EI (or left in terms of M if you account for EI later).
- For symmetric loading, the conjugate beam has symmetric loading → symmetric shear/moment diagrams.
- Conjugate-beam method fails (indeterminate) only if the conjugate itself is statically indeterminate, which is rare for standard cases.
- Excellent for cantilevers: horizontal tangent at fixed end → zero shear at conjugate free end → moment at conjugate free = deflection.
Key definitions
Conjugate Beam
Example
A real cantilever with fixed end → conjugate free end; a real simple beam → conjugate simple beam (supports unchanged for a simple span).
A fictitious beam with same geometry as real beam but loaded with M/EI diagram and supports transformed; used to compute slopes/deflections via statics.
Support Transformation Rules
Example
A propped cantilever (fixed + pin) has conjugate: fixed end becomes free, pin becomes fixed.
Real pin ↔ conjugate pin; real fixed ↔ conjugate free; real free ↔ conjugate fixed; real overhang → conjugate overhang (opposite side).
Diagrams to know
- M/EI diagram for the real beam (becomes loading diagram for conjugate).
- Shear and moment diagrams of the conjugate beam (read these as slopes and deflections of real beam).
- Support transformations table (real to conjugate).
Method 4: Superposition & Standard Formulas
Formulas
δ_total = δ_1 + δ_2 + δ_3 + ... (if all loads act on same beam structure)
Meaning
Total deflection is sum of deflections from each load component; valid for linear elastic behavior
Watch out
Superposition applies ONLY if the structure is linear and supports are unchanged. Cannot mix different support conditions.
When to use
When a beam carries multiple loads and standard formulas exist for each; much faster than full integration
δ = PL³/(48EI) [Simply supported, central point load]
Meaning
P = point load (N); L = span (mm); E = modulus (N/mm²); I = moment of inertia (mm⁴); occurs at midspan
Watch out
This is the 48 formula, not 3 (cantilever) or 384 (UDL). Verify the load is truly at center; off-center requires integration or tables.
When to use
Any simply supported beam with a single point load at center
δ = 5wL⁴/(384EI) [Simply supported, uniform distributed load]
Meaning
w = UDL (N/mm); occurs at midspan for symmetric loading
Watch out
Denominator is 384, not 360 or 360/5. This is 5/384 ≈ 0.0130, memorize separately from other formulas.
When to use
Simply supported beam under full-span UDL; most common for gravity loads on floor members
δ = PL³/(3EI) [Cantilever, point load at free end]
Meaning
Maximum deflection at the free end; P = point load; L = cantilever length
Watch out
The 3 is the key: cantilever is 3, 8 (UDL), or 2 (moment). NOT 48 or 384.
When to use
Cantilever with concentrated load at tip (e.g., bracket, overhang)
δ = wL⁴/(8EI) [Cantilever, uniform distributed load over length]
Meaning
Maximum deflection at free end; w = UDL over entire length
Watch out
Denominator is 8, not 3 or 384. Cantilever UDL: factor is 1/8 (vs. 5/384 for simply supported).
When to use
Cantilever floor slab, beam, or bracket under own weight or distributed load
δ = ML²/(2EI) [Cantilever, concentrated moment at free end]
Meaning
Deflection at free end due to applied moment M at tip; L = length
Watch out
Only L² (not L³ or L⁴); moment causes less deflection than equivalent point load.
When to use
Cantilever with torque or bending moment applied at the free end
θ = PL²/(16EI) [Simply supported, central point load, slope at support]
Meaning
Slope at either end of a simple beam with central point load
Watch out
This is NOT the same as deflection formula. Check whether problem asks for slope or deflection.
When to use
When rotation at support is needed (e.g., for indeterminate analysis)
θ = wL³/(24EI) [Simply supported, UDL, slope at support]
Meaning
Slope at either end (by symmetry, equal); for simple beam under full-span UDL
Watch out
Slope at support, not deflection. Note the 24 in denominator.
When to use
Needed for settling indeterminate beams or checking member compatibility
Common values
Value
L/360 (also L/240 for live + dead in some cases; check NSCP 2015)
Symbol
δ_limit
Quantity
Typical serviceability limit (live load deflection)
Value
200 GPa = 200 000 N/mm²
Symbol
E_steel
Quantity
Modulus of elasticity (steel)
Value
70 GPa = 70 000 N/mm²
Symbol
E_aluminum
Quantity
Modulus of elasticity (aluminum)
Value
≈ 30 GPa = 30 000 N/mm² (varies with grade; see ACI 318)
Symbol
E_concrete
Quantity
Modulus of elasticity (concrete)
Important facts
- Standard formulas assume constant EI along the span.
- The 48, 384, 3, 8, 2 denominators are the most frequently tested constants; flash-card them.
- For unsymmetric loads (e.g., point load not at center), location of max deflection must be found by setting y′ = 0, then substituting back.
- Superposition works for combining loads but NOT for combining boundary conditions. A fixed-end and pinned-end support cannot be 'added.'
- Deflection formulas assume elastic linear material behavior; plastic or large-deflection theory requires different equations.
Key definitions
Superposition Principle
Example
A beam with both point load and UDL: δ_total = δ_from_P + δ_from_w.
For linear systems, total response = sum of responses to each load component applied separately; valid for small deflections and elastic material.
Maximum Deflection Location
Example
Cantilever: max deflection always at free end. Simply supported with off-center load: max is NOT at midspan; must calculate location.
Point on the beam where |y| is largest. For simple symmetric loading, it's at midspan; for unsymmetric, use calculus (set dy/dx = 0).
Diagrams to know
- Elastic curves for: simply supported under central point load (bow shape), UDL (deeper parabola), cantilever under point load (cubic curve), cantilever UDL.
- Charts of standard cases: cantilever, simple beam, overhang, propped cantilever (reference tables).
Serviceability & Code Limits (NSCP 2015 / AISC 360)
Formulas
δ_max ≤ L/360 [live load only, typical floor member]
Meaning
Maximum deflection must not exceed span divided by 360; governs comfort, cracking of finishes
Watch out
This is a LIMIT, not a calculation. Always check: actual deflection ≤ L/360. If not met, increase I or reduce load.
When to use
Checking serviceability for floor beams, joists; per NSCP 2015 Section 202.1
δ_max ≤ L/240 [live + dead load combined, some members]
Meaning
More restrictive limit for members supporting brittle finishes or plaster; NSCP 2015
Watch out
Confirm with your jurisdiction. Some codes use L/360 for live load alone, others L/240 for total.
When to use
When finishes are sensitive to deflection; check local code requirements
δ_required_by_settlement ≈ support_movement [indeterminate analysis]
Meaning
In indeterminate beams, support settlement induces reactions and internal stresses; deflection compatibility equation
Watch out
Settlement is an imposed displacement, not a load. Set δ_load + δ_temperature = 0 to find unknown reactions.
When to use
Propped cantilever, continuous beam over sinking support, etc.
Important facts
- NSCP 2015 Section 202 defines serviceability: states where structure functions adequately (no excessive deflection, vibration, cracking).
- Deflection limits vary by member type: floor beams L/360, roof beams L/240, cantilevers L/180 (stricter).
- Excess deflection causes: floor bounce, door jam, plaster crack, pipe rupture, equipment misalignment.
- Always separate live-load and dead-load deflections in design; some limits apply to live alone, others to combined.
- AISC 360 (steel) typically references L/360 for live load; ACI 318 (concrete) recommends similar limits for deflection control.
Key definitions
Serviceability Limit State
Example
A floor may have adequate strength (ULS) but fail serviceability if δ > L/360 (SLS), causing sag and cracking.
A design condition (separate from ultimate strength) based on functionality, comfort, and durability; includes deflection limits per NSCP 2015.
Live Load Deflection
Example
L/360 for live load means δ_LL ≤ L/360; δ_DL + δ_LL ≤ L/240 (or per code).
Deflection induced only by movable loads (people, furniture, temporary loads); typically more stringent limit than total deflection.
Diagrams to know
- Deflection vs. time plot (initial deflection + creep/long-term effects for concrete).
- Beam profile showing acceptable and unacceptable deflection envelopes.
Indeterminate Beams & Deflection Compatibility
Formulas
δ_load − δ_reaction = 0 [Propped cantilever or redundant support]
Meaning
Compatibility condition: deflection due to applied loads equals deflection due to unknown reaction; at the redundant support, net y = 0
Watch out
Signs must match: if load deflects downward (+), reaction must deflect upward (−) by same amount. Use superposition to split effects.
When to use
Finding reaction at a prop, settlement support, or indeterminate condition
δ_A = δ_B [Relative compatibility between two points]
Meaning
For continuous beams or tied supports, deflection at A must match B (if rigid connection); used to find internal reactions
Watch out
This applies only if A and B are rigidly connected. A pin or hinge breaks the continuity.
When to use
Continuous beams, two-span beams, or beams with internal supports
Important facts
- Indeterminate analysis is the primary application of deflection formulas beyond pure serviceability checks.
- The number of compatibility equations needed = degree of indeterminacy (# redundant reactions).
- Common method: remove the redundant support/reaction, calculate deflection due to remaining loads, then calculate deflection due to redundant alone, set sum = 0.
- For uniform redundancy (e.g., propped cantilever), one compatibility equation suffices.
- Continuous beams, fixed-end beams, and multi-span beams all rely on this principle in Structural Theory/Analysis courses.
Key definitions
Statically Indeterminate Beam
Example
Propped cantilever (cantilever + prop pin) has 3 reactions (V, M, H) but only 3 eq.; the prop reaction is found by setting δ_prop = 0.
A beam with more support reactions than independent equilibrium equations; cannot be solved by statics alone; requires deflection compatibility.
Consistent Deformation (Compatibility Equation)
Example
For a propped cantilever: ΣF, ΣM from statics + δ_prop = 0 (geometry) → solve for prop reaction R.
The requirement that deflections/slopes be geometrically compatible with the support conditions; the missing equation to solve statically indeterminate structures.
Redundant Reaction
Example
In a propped cantilever, the prop force is redundant (you could cantilever alone); found by δ_prop = 0.
An 'extra' reaction beyond the minimum needed for static equilibrium; determined by deflection compatibility.
Diagrams to know
- Free body diagram showing cantilever + prop; moment diagram and deflection curve with and without prop.
- Decomposition: original indeterminate problem = cantilever under loads (deflection down) + cantilever under prop reaction (deflection up to zero).
Reactions or equations
R_prop · δ_due_to_unit_R + δ_due_to_loads = 0
Note
Rearranging: R_prop = −δ_loads / δ_unit (flexibility method). This is the core of indeterminate beam solving.
Conditions
Applied at redundant support; R_prop is the unknown reaction; δ values calculated by any deflection method
Common Pitfalls & Quick Check Procedures
Important facts
- Unit trap: L⁴ and L³ amplify any m ↔ mm error catastrophically. Convert everything to N, mm, mm⁴ before plugging into formulas.
- Formula mix-up: Point load simple = 48; UDL simple = 5/384; Point cantilever = 3; UDL cantilever = 8; Moment cantilever = 2. Flash-card these.
- Boundary condition blunder: Fixed end requires BOTH y = 0 AND y′ = 0. Missing one leaves a free constant.
- Sign flip: Confirm your sign convention (downward = ±). Bending moment positive = concave up. If M diagram inverted, y flips sign.
- Off-center load: Max deflection is NOT at midspan if load is off-center. Must find location by setting dy/dx = 0 first.
- EI variation: If EI changes along beam (e.g., stepped section), cannot use simple formulas; must integrate piecewise or use conjugate with variable 'loading.'
- Superposition trap: Cannot add different boundary conditions. E.g., cannot superpose 'fixed at one end' + 'fixed at other end' to get 'both ends fixed.' The structure is different.
- Serviceability vs. strength: A beam can pass strength (ULS) but fail deflection (SLS). Always check both.
- Cantilever vs. simply supported: Cantilever has 1/48 · (point load factor) ≈ 6× more deflection than simple span for same load/span/EI. Know this ratio.
- Creep in concrete: ACI 318 requires long-term deflection checks for concrete beams; immediate deflection ≠ final deflection (add ~1.5× for creep).
Must remember
- The master equation EI·y″ = M(x); integrate twice, apply 2 boundary conditions to solve for y(x).
- Standard formulas: 48 (point simple), 384/5 (UDL simple), 3 (point cantilever), 8 (UDL cantilever), 2 (moment cantilever). Memorize these 5 denominators.
- Area-moment Theorem 1: slope change = area of M/EI. Theorem 2: deflection = (area of M/EI) × (distance to centroid). Perfect for cantilevers.
- Conjugate-beam method: real beam's M/EI = conjugate's load. Conjugate's shear = real slope; conjugate's moment = real deflection. Reverse support types (fixed ↔ free, pin stays pin).
- Superposition: δ_total = δ₁ + δ₂ + ... Works only if structure is linear and supports unchanged. Cannot mix different boundary conditions.
- Sign convention trap: Downward deflection positive or negative? Confirm at the start. Bending moment positive = concave up. Check all sign algebra carefully.
- Unit explosion: L⁴ and L³ are huge. Convert to consistent units (N, mm) before substituting into formulas. A 1 mm error in span ×4 power = massive error.
- Boundary conditions at supports: Pin/roller → y = 0 only. Fixed end → y = 0 AND y′ = 0. Free end → no constraints. Forget any one = wrong answer.
- Serviceability limit L/360 (live) must be checked separately from strength ULS. Beam can be strong but too springy; check both in design.
- For indeterminate beams (propped cantilever, etc.), use compatibility: δ_loads + δ_reaction = 0 at redundant support. This finds the unknown reaction.
Last minute tips
- Check units relentlessly. BEFORE you calculate, convert L to mm, loads to N, E to N/mm², I to mm⁴. Then substitute. A single m/mm flip in L⁴ term kills the answer.
- If a problem gives you multiple loads, use superposition. Find max deflection for each, add them. Faster and less error-prone than writing one mega M(x) equation.
- Cantilever is your friend: area-moment method is lightning fast. Just calculate area of M/EI triangle and centroid location; one multiplication = slope or deflection.
- Always draw the moment diagram first. It shows you the shape of the elastic curve (concave up ↔ M positive). Sanity-check: does your calculated deflection curve make sense with M?
- For the board exam: have a formula card with the 5 standard cases and a table of centroid locations (triangle 1/3, rectangle 1/2, parabola 3/8 or 5/8). Saves 10 minutes and eliminates recalculation errors.
Comparison tables
Standard Deflection Formulas at a Glance
| Beam Type | Load Type | Location | Max Deflection Formula | Key Denominator | |
|---|---|---|---|---|---|
| Simply Supported | Simply Supported | Point load P at center | Midspan | PL³/(48EI) | 48 |
| Simply Supported | Simply Supported | UDL w over full span | Midspan | 5wL⁴/(384EI) | 384 (coeff 5/384) |
| Cantilever | Cantilever | Point load P at free end | Free end | PL³/(3EI) | 3 |
| Cantilever | Cantilever | UDL w over full length | Free end | wL⁴/(8EI) | 8 |
| Cantilever | Cantilever | Moment M at free end | Free end | ML²/(2EI) | 2 |
| Simply Supported | Simply Supported | Point load P at center (slope at support) | Support | θ = PL²/(16EI) | 16 |
| Simply Supported | Simply Supported | UDL w (slope at support) | Support | θ = wL³/(24EI) | 24 |
Double Integration vs. Area-Moment vs. Conjugate Beam
| Method | Best For | Output | Effort | Common Pitfall | |
|---|---|---|---|---|---|
| Double Integration | Double Integration | Complex loads, full elastic curve y(x), slopes everywhere | Complete y(x) function | High (algebra intense, especially piecewise) | Forgetting constants C₁, C₂; sign errors in integration |
| Area-Moment | Area-Moment | Single-point slopes/deflections, cantilevers, geometric shapes | θ and y at specific points | Low (geometric only, no integration) | Wrong centroid location, misidentifying area boundaries, sign confusion |
| Conjugate Beam | Conjugate Beam | Slopes/deflections via statics, symmetric beams, multiple points | θ and y from shear/moment in conjugate | Low–Medium (setup + static analysis) | Support transformation errors, wrong loading direction, confusion of V ↔ θ and M ↔ y |
| Standard Formulas | Standard Formulas | Common loads (point, UDL), quick checks, board exam | Max deflection only (or midspan for symmetric) | Minimal (memorize denominators) | Formula mix-up (48 vs. 3), unit errors, unsymmetric loads (formula doesn't apply) |
Boundary Conditions & Their Meanings
| Support Type | Real Beam BCs | Conjugate Beam Support | Physical Significance | |
|---|---|---|---|---|
| Pin/Roller | Pin/Roller | y = 0; y′ ≠ 0 (can rotate) | Pin/Roller (unchanged) | No vertical displacement; beam can rotate freely |
| Fixed (Built-in) | Fixed | y = 0; y′ = 0 (both zero) | Free end (opposite end type) | No displacement AND no rotation; rigid connection to wall |
| Free End | Free End | y ≠ 0; y′ ≠ 0 (arbitrary) | Fixed end (anchored) | Can deflect and rotate; cantilever tip or overhang |
| Continuity (internal hinge) | Continuity | y continuous; y′ has jump (slope discontinuity) | Hinge in conjugate | Deflection smooth but slope rotates; internal pin |
| Settlement/imposed displacement | Settlement | y = Δ (known value) | Conjugate: apply corresponding 'load' or reaction | Support moves down by Δ; indeterminate if redundant |
Serviceability Deflection Limits (NSCP 2015 & Common Codes)
| Member Type | Load Case | Deflection Limit | Reason / Standard | |
|---|---|---|---|---|
| Floor beams & joists | Floor beams & joists | Live load only | L/360 | Comfort, cracking prevention (NSCP 202) |
| Roof members | Roof members | Live load only | L/240 | Water ponding, equipment clearance (NSCP 202) |
| Cantilevers | Cantilevers | Live load only | L/180 | More stringent (overhang visibility, comfort) |
| Members with brittle finishes | Members with brittle finishes | Live + dead combined | L/240 or L/360 (check code) | Plaster, tile, rigid cladding crack risk |
| Concrete beams (deflection control) | Concrete beams | Immediate + long-term (creep) | L/360 (live) or per ACI 318 Table 24.2.1 | ACI 318-19 Section 24.2; long-term = 1.5–3× immediate |
Diagrams
Beam Deflection Methods & Formulas Hierarchy
Hierarchical overview of all deflection methods, formulas, and their application areas. Shows how methods relate and when each is most efficient.
Beam Deflection Problem-Solving Decision Tree
Decision tree to select the most efficient deflection method based on problem type, loading, and beam configuration. Guides exam taker to fastest solution path.
Boundary Condition Types & Real-to-Conjugate Transformation
Transformation map from real beam boundary conditions to conjugate beam conditions. Essential for conjugate-beam analysis; highlights that fixed ↔ free transformation reverses.