Reinforced Concrete Fundamentals: WSD and USD - Cheat Sheet
Your last-minute reference for the two design philosophies, strength-reduction factors, material properties, and the equivalent stress block. Master these 30 minutes before the exam.
Design Philosophies: WSD vs USD
Formulas
φ·Mn ≥ Mu
Meaning
φ = strength reduction factor; Mn = nominal strength; Mu = factored (ultimate) moment
Watch out
Do NOT use Mu with allowable stresses; do NOT forget the φ factor
When to use
USD/LRFD design check — the governing equation for all USD design
fs ≤ Fs (allowable)
Meaning
fs = actual service stress; Fs = allowable stress (e.g., 0.45f'c for concrete in WSD)
Watch out
WSD is elastic, linear, and NOW HISTORICAL; NSCP 2015 is USD; mixing them loses marks
When to use
WSD only — comparing elastic stresses to code-permitted limits
Mu = 1.2MD + 1.6ML (+ others)
Meaning
Load factor combination per NSCP 2015 for ultimate design load
Watch out
Use 1.2D + 1.6L, NOT just D + L; always check ALL NSCP load combinations for governing Mu
When to use
Whenever computing factored moment or shear for gravity loads
Important facts
- NSCP 2015 adopts USD/LRFD as the primary design method; WSD is alternate/historical.
- USD is safer (factors up loads, reduces strength) and accounts for variability.
- WSD uses linear elastic theory; USD assumes inelastic behavior at failure.
- Both methods yield approximately the same safety factor when applied correctly.
- Exam: you MUST know both methods; mixing them is a critical error.
Key definitions
Working Stress Design (WSD)
Example
Concrete stress ≤ 0.45f'c; steel stress ≤ 0.50fy — old Philippine code method, still examinable
Service-load elastic method: keep stresses below allowable fractions of material strength.
Ultimate Strength Design (USD) / LRFD
Example
Mu = 1.2D + 1.6L; then φ·Mn ≥ Mu with φ = 0.90 for flexure
Factor loads up, reduce nominal strength by φ; design so φ·Mn ≥ Mu — NSCP 2015 standard method.
Diagrams to know
- Load factor combinations (NSCP Table 202.3.1.1)
- Stress-strain curves: concrete (curved) vs steel (linear, then yield)
- Elastic vs plastic neutral axis shift
Material Properties
Formulas
Ec = 4700√f'c (MPa)
Meaning
Ec = modulus of elasticity of concrete; f'c = 28-day compressive strength (MPa)
Watch out
Formula gives Ec in MPa when f'c is in MPa; do NOT forget the square root; applies to normal-weight only
When to use
Computing Ec for normal-weight concrete; needed for modular ratio n and WSD transformed sections
n = Es / Ec
Meaning
Modular ratio: ratio of steel to concrete moduli; Es = 200,000 MPa (constant for rebar)
Watch out
n ≈ 8–10 typically; do NOT use Es = Ec; n depends on f'c, so changes with concrete strength
When to use
WSD transformed-section analysis; converts steel area to equivalent concrete area
Common values
Value
200,000 MPa
Symbol
Es
Quantity
Steel modulus Es
Value
21, 28, 35, 42 MPa
Symbol
f'c
Quantity
Typical concrete strength (PH)
Value
415 MPa
Symbol
fy
Quantity
Typical steel yield (PH Grade 415)
Value
8–10
Symbol
n
Quantity
Typical modular ratio
Important facts
- Ec formula (4700√f'c) is for normal-weight concrete only; lightweight concrete has lower Ec.
- Concrete modulus INCREASES with f'c: higher strength → stiffer material.
- Steel modulus Es is CONSTANT regardless of fy grade.
- Modular ratio n is typically 8–10; remember n ≈ 8 for quick mental checks.
- Yield strain εy = fy/Es; for 415 MPa: εy = 415/200,000 = 0.00208 ≈ 1/500
Key definitions
Concrete compressive strength f'c
Example
f'c = 21, 28, 35, 42 MPa are common in Philippine design
28-day cylinder strength (MPa); primary material property for all RC design.
Reinforcing steel yield strength fy
Example
Grade 275: fy = 275 MPa; Grade 415: fy = 415 MPa
Stress at which steel begins to yield (plastic deformation); typical 275 or 415 MPa in PH.
Modulus of elasticity Es
Example
Same for Grade 275 and Grade 415 rebar
Steel stiffness = 200,000 MPa (constant, no variation with grade)
Diagrams to know
- Stress-strain curve for concrete: parabolic, peaks at ε ≈ 0.002, then softens
- Stress-strain curve for steel: linear (elastic) until yield, then plateau (plastic)
- Modulus effect: Ec increases with √f'c (nonlinear)
Strength-Reduction Factors φ (NSCP 2015)
Formulas
φ = 0.90
Meaning
Tension-controlled flexure (beams, slabs bending about major axis)
Watch out
ONLY for tension-controlled; do NOT use for columns, shear, or compression
When to use
Any beam or slab in bending where strain εt ≥ 0.005 (well into tension control)
φ = 0.75
Meaning
Shear and torsion; also compression-controlled SPIRAL columns
Watch out
Tied columns are φ = 0.65, not 0.75; spiral vs tied is critical difference
When to use
Shear Vu and torsion Tu checks; spiral columns with εt < εy
φ = 0.65
Meaning
Compression-controlled TIED columns; bearing on concrete
Watch out
Do NOT confuse with spiral (0.75); tied columns are lower safety
When to use
Most practical columns in buildings (square/rectangular with ties); pure compression
φ = 0.65–0.90 (linear interpolation)
Meaning
Transition zone: compression-controlled to tension-controlled
Watch out
Exam rarely asks for interpolation; most problems fall into clear tension (0.90) or compression (0.65/0.75) zones
When to use
When net tensile strain εt is between εy and 0.005
Common values
Value
0.90
Symbol
φ
Quantity
Flexure (tension-controlled)
Value
0.75
Symbol
φ
Quantity
Shear, torsion
Value
0.65
Symbol
φ
Quantity
Tied column (compression)
Value
0.75
Symbol
φ
Quantity
Spiral column (compression)
Important facts
- φ = 0.90 is LARGEST → most favorable (tension-controlled flexure).
- φ = 0.65 is SMALLEST → least favorable (tied column compression).
- Spiral columns (φ = 0.75) are stronger than tied (φ = 0.65) because of confinement.
- φ does NOT apply to nominal strength Mn; it multiplies: φ·Mn ≥ Mu.
- Exam trick: 'What is φ for shear?' Answer: 0.75 (not 0.90).
Key definitions
Tension-controlled section
Example
Most practical beams are tension-controlled
Net tensile strain εt ≥ 0.005 (0.5%); steel yields well before concrete crushes; φ = 0.90.
Compression-controlled section
Example
Heavily loaded columns, short-span deep beams
Net tensile strain εt ≤ εy = fy/Es; concrete crushes first; φ = 0.65 (tied) or 0.75 (spiral).
Diagrams to know
- φ vs strain: step function or linear ramp from 0.65 to 0.90
Equivalent Stress Block & β₁ Factor
Formulas
a = β₁·c
Meaning
a = depth of equivalent rectangular stress block; c = neutral axis depth from compression fiber; β₁ = stress-block factor
Watch out
β₁ ≠ c; you MUST multiply c by β₁ to get block depth a
When to use
Moment capacity Mn calculation; replaces curved concrete stress diagram with rectangle
Stress block intensity = 0.85f'c
Meaning
Rectangular block has constant intensity 0.85f'c over depth a
Watch out
Intensity is 0.85f'c, NOT f'c; factor 0.85 is ALWAYS applied, regardless of f'c
When to use
Computing compression force C = 0.85f'c·a·b in moment equations
β₁ = 0.85 (for f'c ≤ 28 MPa)
Meaning
Rectangular block depth factor equals 0.85 for low-strength concrete
Watch out
Do NOT reduce β₁ until f'c EXCEEDS 28 MPa; at exactly 28 MPa, still 0.85
When to use
f'c = 21 or 28 MPa → always use β₁ = 0.85
β₁ = 0.85 − 0.05(f'c − 28)/7 (for 28 < f'c ≤ 55 MPa)
Meaning
β₁ decreases linearly: drop 0.05 for every 7 MPa above 28
Watch out
Numerator is (f'c − 28), NOT (55 − f'c); division is by 7, not other values; formula is LINEAR, not curved
When to use
f'c = 35, 42, 49 MPa; interpolating β₁ in mid-range strengths
β₁ = 0.65 (for f'c ≥ 55 MPa)
Meaning
Rectangular block floor: cannot go lower than 0.65
Watch out
Do NOT continue the linear formula past 55 MPa; cap at 0.65
When to use
High-strength concrete f'c = 55 MPa or more
Common values
Value
0.85
Symbol
β₁
Quantity
β₁ for f'c = 21 MPa
Value
0.85
Symbol
β₁
Quantity
β₁ for f'c = 28 MPa
Value
0.80
Symbol
β₁
Quantity
β₁ for f'c = 35 MPa
Value
0.75
Symbol
β₁
Quantity
β₁ for f'c = 42 MPa
Value
0.65
Symbol
β₁
Quantity
β₁ for f'c = 55 MPa
Value
0.85f'c
Symbol
Stress
Quantity
Stress block intensity
Important facts
- β₁ = 0.85 is the 'default' for practical concrete up to 28 MPa.
- β₁ DECREASES (block gets shallower) as f'c increases above 28 MPa.
- The formula 0.85 − 0.05(f'c − 28)/7 is LINEAR over 28–55 MPa range.
- Common values: β₁(21) = 0.85, β₁(28) = 0.85, β₁(35) = 0.80, β₁(42) = 0.75, β₁(55) = 0.65.
- Intensity 0.85f'c accounts for time-dependent creep and stress distribution under sustained load.
- The stress block is a convention; it simplifies analysis without sacrificing accuracy.
Key definitions
Whitney stress block (equivalent rectangular block)
Example
Replaces Hognestad parabola; makes hand calculations tractable
Simplified rectangular pressure distribution (intensity 0.85f'c, depth a = β₁c) replacing curved concrete stress for moment calculations.
Neutral axis depth c
Example
In a rectangular beam with reinforcement, c determines block depth a = β₁c
Distance from extreme compression fiber to neutral axis; measured perpendicular to bending axis.
β₁ stress-block factor
Example
β₁ = 0.85 at f'c = 28 MPa, drops to 0.80 at f'c = 35 MPa
Factor relating neutral axis depth c to equivalent rectangular block depth a; depends on f'c.
Diagrams to know
- Curved stress distribution (real concrete) vs rectangular block (Whitney equivalent)
- Graph of β₁ vs f'c: flat line at 0.85 up to 28, then linear decline to 0.65
Key Calculations & Quick Checks
Formulas
Mu = 1.2MD + 1.6ML
Meaning
Ultimate (factored) moment from NSCP gravity load combination
Watch out
Check ALL applicable NSCP load combinations; 1.2D + 1.6L is most common but not always governing
When to use
First step: compute Mu before designing section to resist it
εy = fy / Es
Meaning
Yield strain of steel; fy in MPa, Es = 200,000 MPa
Watch out
For 415 MPa: εy ≈ 0.00208; do NOT confuse with 0.005 (tension control threshold)
When to use
Determining if section is tension-controlled (εt ≥ 0.005 > εy) or compression-controlled (εt ≤ εy)
Ec = 4700√f'c
Meaning
Concrete modulus for normal-weight concrete
Watch out
f'c must be in MPa; answer is in MPa; lightweight concrete requires different formula
When to use
Computing modular ratio n; WSD elastic analysis; deflection checks
n = Es / Ec = 200,000 / Ec
Meaning
Modular ratio; transforms steel area to equivalent concrete
Watch out
n varies with f'c; typical range 8–10; do NOT assume n = 8 without calculating
When to use
WSD transformed-section analysis; converting As (steel) to nAs (equivalent concrete area)
Important facts
- Always compute Mu FIRST before attempting any strength calculation.
- Nominal strength Mn is computed using strain compatibility and stress block.
- Design strength φMn = φ × (stress block result); φ is a multiplier, not a component.
- φMn ≥ Mu is the GOVERNING INEQUALITY; design is complete when this is satisfied.
- WSD and USD Mn calculations differ: WSD uses linear elastic; USD uses stress block.
Key definitions
Factored load
Example
Mu = 1.2MD + 1.6ML is factored moment for gravity
Service load multiplied by factor (e.g., 1.2D, 1.6L) per NSCP load combination.
Nominal strength Mn
Example
Computed using equilibrium and strain compatibility; independent of φ
Calculated strength at section assuming inelastic material behavior and section cracking.
Design strength φMn
Example
Must satisfy φMn ≥ Mu
Meaning
Nominal strength reduced by φ; the 'safe' strength the section provides.
Diagrams to know
- Design workflow: Load combination → Mu → Section assumed → Strain compatibility → Stress block → Mn → φMn → Check φMn ≥ Mu
Common β₁ Calculation Examples
Important facts
- f'c = 21 MPa: β₁ = 0.85 (below threshold, use flat value)
- f'c = 28 MPa: β₁ = 0.85 (at threshold, still 0.85)
- f'c = 35 MPa: β₁ = 0.85 − 0.05(35 − 28)/7 = 0.85 − 0.05(1) = 0.80
- f'c = 42 MPa: β₁ = 0.85 − 0.05(42 − 28)/7 = 0.85 − 0.05(2) = 0.75
- f'c = 49 MPa: β₁ = 0.85 − 0.05(49 − 28)/7 = 0.85 − 0.05(3) = 0.70
- f'c = 55 MPa: β₁ = 0.85 − 0.05(55 − 28)/7 = 0.85 − 0.05(27/7) ≈ 0.658 → cap at 0.65
- f'c ≥ 55 MPa: β₁ = 0.65 (floor; do NOT go lower)
WSD Transformed Section (Elastic Analysis)
Formulas
As(transformed) = n × As (steel area replaced by n times that area in concrete units)
Meaning
Steel converted to equivalent concrete; n = Es/Ec
Watch out
Transformed section is ONLY for WSD; USD uses stress block (not linear/elastic)
When to use
WSD elastic bending: find NA by equating first moments, compute stress using I/y
fs = n × fc (at same distance from NA)
Meaning
Stress in steel at a given strain level is n times the concrete stress at that strain
Watch out
This is LINEAR ELASTIC relationship; does NOT apply when either material yields
When to use
WSD checking: if concrete stress fc is known, steel stress is n·fc
Important facts
- Transformed section is WSD-only; it assumes linear elastic response.
- Neutral axis (NA) position found by equating first moment of transformed areas.
- Stress in concrete: fc = M·y / Itransformed (standard bending formula).
- Stress in steel: fs = M·(n·y_steel) / Itransformed = n × fc.
- Check WSD: fs ≤ allowable (e.g., 0.50fy); fc ≤ allowable (e.g., 0.45f'c).
- WSD is historical in NSCP 2015; USD is standard; exams may still ask for WSD comparison.
Key definitions
Transformed section (WSD)
Example
For n = 8 and As = 1000 mm², equivalent concrete area = 8000 mm²
Hypothetical all-concrete section where steel area replaced by n·As; linear elastic analysis applies.
Diagrams to know
- Transformed section diagram: concrete width b, steel replaced by n·As at distance from face
Must remember
- NSCP 2015 = USD/LRFD standard. Governing equation: φ·Mn ≥ Mu. Do NOT mix WSD (elastic, allowable stresses) with USD (inelastic, strength reduction factors).
- Load factoring in USD: Mu = 1.2MD + 1.6ML (gravity); always check NSCP load combinations for the governing case.
- β₁ = 0.85 for f'c ≤ 28 MPa (constant). For 28 < f'c ≤ 55: β₁ = 0.85 − 0.05(f'c − 28)/7 (linear drop). For f'c ≥ 55: β₁ = 0.65 (floor).
- Strength-reduction factors: φ = 0.90 (flexure/tension-controlled), φ = 0.75 (shear, torsion, spiral columns), φ = 0.65 (tied columns, bearing). Memorize these; they appear in every problem.
- Equivalent stress block = 0.85f'c intensity over depth a = β₁·c. Replace curved concrete stress with this rectangle; simplifies hand calculations without losing accuracy.
- Material properties: Ec = 4700√f'c (normal-weight), Es = 200,000 MPa (constant), n = Es/Ec ≈ 8–10 typically. Modular ratio varies with f'c.
- Yield strain εy = fy/Es (e.g., 415/200,000 ≈ 0.00208). Tension-controlled when εt ≥ 0.005 (much larger than yield strain). This determines which φ to use.
- WSD = elastic, linear, transformed sections (As → n·As). USD = inelastic, stress block, strain compatibility. Exams test both; confusing them is fatal.
- Common concrete strengths (PH): 21, 28, 35, 42 MPa. Know β₁ for each: 0.85, 0.85, 0.80, 0.75 respectively. These appear constantly.
- Design sequence: (1) Compute factored Mu from load combination. (2) Assume section dimensions/rebar. (3) Calculate nominal Mn using strain compatibility & stress block. (4) Check φ·Mn ≥ Mu. (5) If not satisfied, revise section and repeat.
Last minute tips
- φ for SHEAR is 0.75, NOT 0.90. This is a classic trap: students confuse flexure (0.90) with shear. Write it on your page: 'Shear φ = 0.75' at the start of the exam.
- Check β₁ EARLY. Compute it first before any Mn calculation. f'c = 35 MPa → β₁ = 0.80 is the most common exam case; pre-compute: 0.85 − 0.05 = 0.80.
- NSCP load combination 1.2D + 1.6L always give Mu; then ask 'which φ applies?' If flexure (tension-controlled), φ = 0.90. If shear check, φ = 0.75. The same Mu, different φ.
- Stress-block intensity is ALWAYS 0.85f'c, NEVER f'c alone. This is the 0.85 factor; it is NOT part of β₁. Compression force C = 0.85f'c × a × b (constant).
- If a problem says 'WSD' or 'allowable stress,' switch to elastic/transformed mindset. If 'USD' or 'NSCP 2015,' use stress block. Mixing them loses the entire problem.
Comparison tables
WSD vs USD: Side-by-Side Comparison
| Aspect | WSD (Elastic, Alternate) | USD (Inelastic, NSCP 2015 Standard) |
|---|---|---|
| Load handling | Service loads (no factors) | Factored loads (1.2D, 1.6L, etc.) |
| Material behavior | Linear elastic (Hooke's law) | Inelastic at failure; stress block replaces curve |
| Section analysis | Transformed section (steel → n·As in concrete units) | Strain compatibility + stress block (0.85f'c rectangular) |
| Stress limits | fs ≤ Fs (allowable); fc ≤ Fc (allowable) | φ·Mn ≥ Mu (design strength ≥ factored demand) |
| Safety factor | Built into allowable stresses (implicit) | In load factors (up) and φ (down) — explicit |
| Governing equation | Stress check: fs ≤ 0.50fy; fc ≤ 0.45f'c | Strength check: φ·Mn ≥ Mu |
| Design philosophy | Keep stresses low under service load | Allow inelastic behavior up to ultimate failure |
| Typical φ value | N/A (implicit in allowables) | 0.90 (flexure), 0.75 (shear), 0.65 (tied column) |
| Exam relevance (PRC) | Historical, alternate; still examinable | PRIMARY method per NSCP 2015 |
Strength-Reduction Factor φ at a Glance
| Action / Element Type | φ value | Section Classification | Key Note |
|---|---|---|---|
| Flexure (beams, slabs in bending) | 0.90 | Tension-controlled | Largest φ; steel yields before concrete crushes |
| Shear (Vu, transverse) | 0.75 | N/A | Same as torsion; more conservative than flexure |
| Torsion (twisting) | 0.75 | N/A | Same as shear |
| Columns: TIED (square, rectangular) | 0.65 | Compression-controlled | Most common; LOWEST φ in practice; tied confinement is weak |
| Columns: SPIRAL (circular, spirally reinforced) | 0.75 | Compression-controlled | Better than tied due to confinement; still lower than flexure |
| Bearing on concrete | 0.65 | Compression | Same as tied columns |
| Plain (unreinforced) concrete | 0.60 | Brittle | Lowest; no ductility; rarely used structurally |
β₁ Stress-Block Factor Quick Lookup
| f'c (MPa) | β₁ Value | Calculation / Note |
|---|---|---|
| ≤ 28 | 0.85 | Constant; below threshold |
| 21 | 0.85 | Typical low-strength |
| 28 | 0.85 | Boundary value; still 0.85 |
| 35 | 0.80 | 0.85 − 0.05(35−28)/7 = 0.85 − 0.05 |
| 42 | 0.75 | 0.85 − 0.05(42−28)/7 = 0.85 − 0.10 |
| 49 | 0.70 | 0.85 − 0.05(49−28)/7 = 0.85 − 0.15 |
| 55 | 0.65 | 0.85 − 0.05(55−28)/7 ≈ 0.658 → floor at 0.65 |
| ≥ 55 | 0.65 | Floor (minimum); do NOT reduce further |
Material Properties & Quick Formulas
| Property | Formula / Value | Units | When Needed |
|---|---|---|---|
| Concrete modulus Ec | 4700√f'c | MPa (when f'c in MPa) | Modular ratio, WSD deflection |
| Modular ratio n | Es / Ec = 200,000 / Ec | Dimensionless | WSD transformed section |
| Steel modulus Es | 200,000 (constant) | MPa | All sections (WSD, USD) |
| Yield strain εy | fy / Es (e.g., 415/200,000) | mm/mm (or just #) | Determining section control (tension vs compression) |
| Tension-control threshold | εt = 0.005 (minimum) | mm/mm | Distinguishing φ = 0.90 from lower values |
Common NSCP Load Combinations (Gravity)
| Combination | Factored Load Expression | When to Use |
|---|---|---|
| 1 (gravity only) | 1.4D | Dead load only; no live load |
| 2 (gravity primary) | 1.2D + 1.6L | MOST COMMON; dead + live (typical building) |
| 3 (roof live load) | 1.2D + 1.6Lr + 0.8W (or similar) | Roof structures with sloped/flat; Lr = roof live load |
| 4 (wind primary) | 1.2D + 1.3W + 0.5L | When wind governs over gravity |
| 5 (earthquake primary) | 1.2D + E (or 0.9D + E) | When seismic governs |
Diagrams
Reinforced Concrete Design Fundamentals – Concept Map
Hierarchical breakdown of RC fundamentals showing how material properties, design philosophies, reduction factors, and the stress block come together. Use this to quickly see the big picture and where each concept fits.
USD Design Verification Flow (φ·Mn ≥ Mu Check)
Complete flowchart for USD design verification. Shows the decision tree for selecting φ, then the analysis steps to compute nominal strength Mn, and the final check φ·Mn ≥ Mu. This is the exam-critical workflow.
Stress-Block Factor β₁ State Transitions (f'c dependency)
State diagram showing how β₁ transitions as concrete strength increases. The factor is constant until 28 MPa, then decreases linearly, then caps at 0.65. Useful for visualizing the piecewise behavior.
Design Method Evolution & NSCP 2015 Standard
Timeline showing the shift from WSD (elastic, historical) to USD/LRFD (inelastic, modern) in Philippine codes. Helps context: WSD is still examinable but USD is the exam standard per NSCP 2015.
Strength-Reduction Factor φ Usage Frequency in Exams
Pie chart showing the relative frequency of φ factors tested in Philippine licensing exams. Flexure and shear dominate; columns are less frequent. This helps prioritize study effort.