CELE Reinforced & Prestressed Concrete — Reinforced Concrete Beams: FlexureCheat Sheet

Formula

a = (A_s × f_y) / (0.85 × f'_c × b)

Meaning

a = depth of equivalent rectangular stress block (mm); A_s = tension steel area (mm²); f_y = steel yield strength (MPa); f'_c = concrete compressive strength (MPa); b = beam width (mm)

Watch Out

Use 0.85 f'_c, NOT f'_c. This is the standard ACI/NSCP equivalent block factor. Forgetting the 0.85 multiplier will give wrong answers.

When To Use

Always the first step in beam analysis—compute the stress-block depth from equilibrium of forces.

Formula

M_n = A_s × f_y × (d − a/2)

Meaning

M_n = nominal (unfactored) moment capacity (N·mm or kN·m); d = effective depth to tension steel centroid (mm); a/2 = distance from stress block resultant to tension steel

Watch Out

Use d (effective depth), not h (total beam depth). d is measured from the top compression fiber to the centroid of the tension steel reinforcement.

When To Use

Calculate the nominal moment once 'a' is known. This is the tension force times the lever arm.

Formula

c = a / β₁

Meaning

c = neutral-axis depth (mm); β₁ = stress-block factor (0.85 for f'_c ≤ 30 MPa, linearly reduced for higher f'_c per NSCP)

Watch Out

β₁ varies with concrete strength. NSCP 2015: β₁ = 0.85 if f'_c ≤ 30 MPa; reduce by 0.05 for each 7 MPa above 30 MPa (minimum 0.65).

When To Use

Convert stress-block depth 'a' to neutral-axis depth 'c' for strain calculations and ductility checks.

Formula

ε_t = 0.003 × (d − c) / c

Meaning

ε_t = net tensile strain in the tension steel; 0.003 = crushing strain in concrete at top fiber

Watch Out

If ε_t < 0.005, the section is NOT tension-controlled; φ must be reduced below 0.90. This is the ductility gate.

When To Use

Mandatory check for every beam analysis. Determines whether section is tension-controlled (ε_t ≥ 0.005) and permits φ = 0.90.

Formula

φM_n = 0.90 × M_n (if ε_t ≥ 0.005)

Meaning

φ = resistance factor (strength-reduction factor); only 0.90 for tension-controlled flexure; lower for over-reinforced sections

Watch Out

Common exam trap: using φ = 0.90 without checking ε_t. If ε_t < 0.005, compute φ from NSCP 2015 Table (φ = 0.65 + 0.25(ε_t − 0.002)/0.003).

When To Use

Multiply nominal moment by 0.90 ONLY after confirming ε_t ≥ 0.005. Always verify ductility before using φ = 0.90.

Value

0.003

Symbol

ε_c

Quantity

Crushing strain in concrete

Value

0.005

Symbol

ε_t,min

Quantity

Minimum tensile strain for tension-control

Value

0.85 (for f'_c ≤ 30 MPa)

Symbol

α₁

Quantity

Equivalent block coefficient

Value

0.85

Symbol

β₁

Quantity

Stress-block factor (f'_c ≤ 30 MPa)

Value

0.90

Symbol

φ

Quantity

Resistance factor (tension-controlled flexure)

Term

Effective depth (d)

Example

For a 500 mm deep beam with 25 mm bars, cover = 40 mm, d ≈ 500 − 40 − 12.5 = 447.5 mm.

Definition

Vertical distance from extreme compression fiber to centroid of tension reinforcement; primary design variable.

Term

Tension-controlled section

Example

A singly reinforced beam with ρ = 0.008 typically produces ε_t ≈ 0.012, ensuring tension-control and ductility.

Definition

Beam in which ε_t ≥ 0.005 (net tensile strain); steel yields before concrete crushes; permits φ = 0.90 per NSCP 2015.

Term

Under-reinforced beam

Example

ρ = 0.010 < ρ_b = 0.029 for f'_c = 28 MPa, f_y = 415 MPa → under-reinforced, ductile.

Definition

Steel ratio ρ < ρ_b; tension steel reaches yield before concrete reaches crushing strain; ductile failure mode.

Term

Over-reinforced beam

Example

ρ = 0.035 > ρ_b = 0.029 → over-reinforced; concrete fails first, sudden collapse risk.

Definition

Steel ratio ρ > ρ_b; concrete crushes before steel yields; brittle failure with no warning; φ < 0.90.

Term

Equivalent rectangular stress block

Example

Rectangular block depth a = 85.6 mm replaces the actual curved diagram for equilibrium calculations.

Definition

Simplified stress distribution (0.85 f'_c over depth a) replacing the actual parabolic concrete stress curve; per ACI 318 / NSCP.

Formula

ρ_b = 0.85 × β₁ × (f'_c / f_y) × [600 / (600 + f_y)]

Meaning

ρ_b = balanced steel ratio (steel yields and concrete crushes simultaneously); limit case between under- and over-reinforced.

Watch Out

NSCP 2015 does NOT permit ρ > ρ_max; ρ_max ≠ 0.75ρ_b (old rule). Use the tension-control limit: ρ_max ≈ 0.85β₁(f'_c/f_y)(0.375).

When To Use

Reference benchmark to check if design is under- or over-reinforced. ρ < ρ_b → under-reinforced (ductile); ρ > ρ_b → over-reinforced (brittle).

Formula

ρ_max = 0.85 × β₁ × (f'_c / f_y) × [0.003 / (0.003 + 0.005)]

Meaning

ρ_max = maximum steel ratio for tension-controlled section (NSCP 2015); ensures ε_t ≥ 0.005 for ductility; simplifies to ≈ 0.85β₁(f'_c/f_y)(0.375).

Watch Out

ρ_max depends on ε_t = 0.005 minimum; this is NOT the old 0.75ρ_b rule. For ε_t = 0.005, the factor is 0.375 (= 0.003 / 0.008).

When To Use

Design check: confirm ρ ≤ ρ_max to guarantee tension-control and φ = 0.90.

Formula

ρ_min = max[1.4 / f_y, √f'_c / (4 × f_y)]

Meaning

ρ_min = minimum steel ratio to prevent sudden brittle fracture upon concrete cracking; two expressions, take the larger.

Watch Out

With f'_c in MPa and f_y in MPa, the first term gives ρ_min ≈ 0.0034 for f_y = 415 MPa; the second is usually larger. Always compute both.

When To Use

Every beam design: confirm ρ ≥ ρ_min to ensure cracking is gradual, not sudden. Protects against under-reinforcement.

Formula

ρ = A_s / (b × d)

Meaning

ρ = steel ratio (reinforcement percentage); A_s in mm², b and d in mm.

Watch Out

Denominator is b × d, not b × h. Use effective depth d, not total height h.

When To Use

Convert steel area to ratio form for checking limits; compare ρ against ρ_min and ρ_max.

Value

0.0288

Symbol

ρ_b

Quantity

Typical balanced ratio (f'_c = 28 MPa, f_y = 415 MPa)

Value

0.0183 (≈ 0.375 × ρ_b)

Symbol

ρ_max

Quantity

Typical max tension-controlled ratio

Value

0.00337 (= √28 / (4 × 415))

Symbol

ρ_min

Quantity

Typical minimum ratio

Value

0.002 (f_y / E_s = 415 / 200,000)

Symbol

ε_y

Quantity

Yield strain in steel

Term

Balanced steel ratio (ρ_b)

Example

For f'_c = 28 MPa, f_y = 415 MPa: ρ_b ≈ 0.0288 (2.88%); designs with ρ < ρ_b are under-reinforced.

Definition

Steel ratio at which concrete and steel reach their limits (0.003 strain and f_y yield) simultaneously; the critical transition point.

Term

Maximum tension-controlled ratio (ρ_max)

Example

If ρ_b = 0.0288, then ρ_max ≈ 0.0183 (requires ε_t ≥ 0.005 for ductility).

Definition

Largest allowed steel ratio per NSCP 2015 to ensure ε_t ≥ 0.005 and φ = 0.90; typically ≈ 0.375 × ρ_b.

Term

Minimum steel ratio (ρ_min)

Example

ρ_min = max(0.00337, 0.00131) = 0.00337 for f'_c = 28 MPa, f_y = 415 MPa.

Definition

Lower bound on reinforcement to prevent sudden flexural failure at first crack; ensures steel limits extend concrete cracking.

Note

Balanced condition is UNSTABLE in real practice; designs deviate to either under- or over-reinforced states.

Equation

At balanced state: ε_t = f_y / E_s and ε_c = 0.003 simultaneously

Conditions

Concrete crushes (ε_c = 0.003) at the same load that steel yields (ε_t = 400 MPa / 200,000 MPa = 0.002 for f_y = 415 MPa at yield); neutral axis position is fixed by strain compatibility.

Note

This is the gate: if your analysis gives ε_t < 0.005, you cannot use φ = 0.90; must reduce φ or redesign with less steel.

Equation

ε_t ≥ 0.005 ⟹ φ = 0.90 (tension-controlled)

Conditions

NSCP 2015 requirement for full φ = 0.90 in flexure; lower strains imply reduced φ per interpolation rules.

Formula

R_n = M_u / (φ × b × d²)

Meaning

R_n = nominal moment strength coefficient (MPa); M_u = factored (design) moment (N·mm); φ = 0.90 for tension-controlled.

Watch Out

Use factored moment M_u, NOT unfactored M. Units must be consistent (M_u in N·mm or kN·m with R_n in Pa or MPa).

When To Use

First step in beam design: compute the required R_n, then use it to find ρ and A_s via the quadratic formula.

Formula

ρ = (0.85 × f'_c / f_y) × [1 − √(1 − 2 × R_n / (0.85 × f'_c))]

Meaning

ρ = required steel ratio; derived from the quadratic solution of the moment equation; 0.85 × f'_c is the stress-block coefficient.

Watch Out

The square root term can only exist if R_n ≤ 0.85 f'_c × ρ_max. If R_n exceeds this, the beam is too small or over-capacity → use doubly reinforced or larger section.

When To Use

After computing R_n, use this formula (or quadratic equation) to find ρ. Then A_s = ρ × b × d.

Formula

A_s = ρ × b × d

Meaning

A_s = required area of tension steel (mm²); ρ from previous formula; b and d in mm.

Watch Out

Check that ρ_min ≤ ρ ≤ ρ_max. If ρ < ρ_min, increase A_s to satisfy minimum. If ρ > ρ_max, reject the design and increase section (b, d) or use doubly reinforced beam.

When To Use

Final step: convert ratio ρ to actual steel area. Round up to nearest standard bar size and number.

Formula

a = (A_s × f_y) / (0.85 × f'_c × b) [check]

Meaning

Reverse check: compute the resulting stress-block depth from the proposed A_s to verify the design is consistent.

Watch Out

Do NOT skip this verification step. A careless error in rounding bar sizes can invalidate the entire design.

When To Use

After selecting A_s, compute a to confirm M_n ≥ M_u / φ and ε_t ≥ 0.005.

Term

Factored moment (M_u)

Example

Dead load moment = 100 kN·m, live load moment = 50 kN·m → M_u = 1.2(100) + 1.6(50) = 200 kN·m.

Definition

Design moment including load factors (1.2D + 1.6L); input to the design equations; always ≥ unfactored moment.

Term

Moment strength coefficient (R_n)

Meaning

Normalized moment capacity per unit beam cross-section (b × d²); measure of beam efficiency.

Term

Design spiral

Example

Given M_u = 300 kN·m, choose b = 350 mm, d = 550 mm, compute R_n, solve for ρ, pick bar sizes to match A_s = ρbd.

Definition

Process of selecting beam dimensions and steel to satisfy moment capacity and all code limits (ρ_min, ρ_max, ductility, etc.).

Note

The design equation avoids iteration by using the quadratic formula; both the quadratic form and ρ formula are equivalent.

Equation

M_n = A_s × f_y × (d − a/2) ≥ M_u / φ

Conditions

Equilibrium condition plus demand check; rearranged into the quadratic form to solve for ρ directly.

Note

This equilibrium condition links a (stress-block depth) and A_s (steel area); substituting into the moment equation yields the design formula.

Equation

0.85 × f'_c × a × b = A_s × f_y [force balance]

Conditions

Compression force in concrete equals tension force in steel at ultimate; the foundation of the stress-block method.

Formula

M_u = M_1 + M_2 = A_s1 × f_y × (d − a₁/2) + A's × f's × (d − d')

Meaning

M_u = total moment capacity (N·mm); M_1 = singly reinforced pair (compression block + tension steel); M_2 = compression-steel couple (A's × f's); d' = distance from compression fiber to centroid of compression steel.

Watch Out

Compression steel contributes only if it yields (ε's ≥ ε_y); always check that ε's ≥ f_y / E_s before using f's = f_y.

When To Use

When a singly reinforced beam (with ρ = ρ_max) cannot carry M_u; superpose singly-reinforced moment and compression-steel moment.

Formula

ε's = 0.003 × (c − d') / c [compression steel strain]

Meaning

ε's = compressive strain in the compression reinforcement; must be ≥ 0.002 to assume f's = f_y.

Watch Out

If ε's < 0.002, compression steel is elastic; use f's = E_s × ε's = 200,000 × ε's (do not assume f's = f_y).

When To Use

Verify that compression steel yields before calculating its contribution to moment and force balance.

Formula

a = [A_s × f_y − A's × f's] / (0.85 × f'_c × b) [modified force balance]

Meaning

a = stress-block depth when compression steel is present; net tension force (A_s − A's) balances compression block.

Watch Out

Compression steel reduces the required stress-block depth a; as A's increases, a decreases. This is the point of adding compression steel.

When To Use

Replace the singly-reinforced force-balance equation when A's ≠ 0.

Formula

A's ≥ (A_s − 0.85 × β₁ × f'_c / f_y × b × d) × [ensure compression steel carries part of moment]

Meaning

Minimum compression steel needed if singly reinforced limit is exceeded; algebraic condition.

Watch Out

This is a design guideline, not a hard code rule; NSCP may impose additional requirements on compression reinforcement ratio.

When To Use

Sizing compression steel in a doubly reinforced design.

Value

0.002 (= 400 MPa / 200,000 MPa for f_y = 415 MPa)

Symbol

ε_y

Quantity

Compression steel yield strain (typical)

Value

40 mm

Symbol

cc

Quantity

Typical cover for compression steel (top)

Term

Compression steel (A's)

Example

A 25 mm bar at 40 mm from top surface; carries compressive stress and contributes an additional moment couple.

Definition

Reinforcement in the compression zone of a beam, placed near the top fiber; intended to resist compressive moments and control deflection.

Term

Doubly reinforced beam

Example

M_u = 400 kN·m exceeds single-reinforced capacity of 300 kN·m → add A's to increase M_n via superposition.

Definition

Beam with both tension steel (A_s, bottom) and compression steel (A's, top); used when singly reinforced capacity is insufficient or to limit deflection.

Term

Compression-steel couple (M_2)

Example

A's = 400 mm², f's = 415 MPa, d − d' = 450 mm → M_2 = 400 × 415 × 450 ≈ 74.7 kN·m.

Definition

Moment contribution from the pair of compressive force in the top steel and tensile force from additional bottom steel; M_2 = A's × f's × (d − d').

Note

For typical Philippine sections (d' = 40–60 mm, d = 450–600 mm), this condition is almost always satisfied in doubly reinforced beams.

Equation

If ε's ≥ f_y / E_s = 0.00207 (typically), then f's = f_y

Conditions

Compression steel is in the yield region; safe to assume full yield stress.

Note

Both couples are computed from the same neutral-axis position c (= a / β₁); they are coupled through force equilibrium.

Equation

Total M_n = [A_s × f_y × (d − a/2)] + [A's × f's × (d − d')]

Conditions

Superposition of moment from (1) the equivalent rectangular stress block couple and (2) the compression-steel couple.

Formula

a ≤ t_f? [check if stress block stays in flange]

Meaning

a = depth of equivalent stress block; t_f = thickness of the slab (flange). If a ≤ t_f, treat the section as a rectangular beam with width b_f.

Watch Out

If a > t_f, the stress block spans the flange AND part of the web; must account for the step-down in width. Incorrectly treating an overstressed T as rectangular is a common error.

When To Use

Always the FIRST check in T-beam analysis: determine whether the stress block is confined to the flange or extends into the web.

Formula

If a ≤ t_f: M_n = A_s × f_y × (d − a/2) [use rectangular formula with b = b_f]

Meaning

When the stress block is entirely in the flange, use the standard singly-reinforced formula with the flange width b_f.

Watch Out

Even though the web exists below the flange, ignore it for stress calculations if a ≤ t_f; the flange is fully effective.

When To Use

Simplest case; most T-beam problems in practice fall here.

Formula

If a > t_f: a_web = a − t_f, then M_n = C_f × (d − t_f/2) + C_w × (d − t_f − a_web/2)

Meaning

When stress block extends into web: split into (1) flange force C_f = 0.85 f'_c × b_f × t_f acting at t_f/2, and (2) web force C_w = 0.85 f'_c × b_w × a_web acting at t_f + a_web/2.

Watch Out

This formula is less common on exams but critical for large-capacity T-beams. Many students forget to split the forces; derive the moment for each part separately.

When To Use

T-beams with heavy reinforcement or shallow flanges; A_s is large enough to push the neutral axis deep into the web.

Formula

For design (given M_u, find A_s in a T-beam): assume a ≤ t_f first; solve as rectangular with b = b_f.

Meaning

Use the standard design procedure with flange width b_f. AFTER finding A_s, verify that a ≤ t_f. If not, re-do with split forces.

Watch Out

Most exam problems are designed so that a ≤ t_f; however, always verify this assumption in your final check.

When To Use

Initial T-beam design; start with the simpler assumption.

Value

100–200 mm

Symbol

t_f

Quantity

Typical flange thickness (Philippines)

Value

250–400 mm

Symbol

b_w

Quantity

Typical web width

Value

span/4 or b_w + 8t_f

Symbol

b_f

Quantity

Typical flange width (NSCP limit)

Term

T-beam (or I-beam)

Example

Monolithic slab (b_f = 4 m, t_f = 150 mm) integrated with beam web (b_w = 300 mm, h = 700 mm); slab is the effective flange.

Definition

Beam with a flange (wide slab) and web (narrow stem); flange acts as compression force, web resists shear; typical in floor systems.

Term

Flange width (b_f)

Example

Typical rule: b_f ≤ (span/4) or (b_w + 8t_f), whichever is smaller; limits the effective flange width in a floor system.

Definition

Width of the compression zone (slab) contributing to moment capacity; often governed by beam spacing (NSCP/ACI rules limit b_f based on span and geometry).

Term

Effective depth (d)

Example

Flange top at z = 0, slab thickness = 150 mm, web below, total depth h = 700 mm, d = 650 mm (typically h − cover − bar radius).

Definition

In a T-beam, distance from top of flange to centroid of tension steel; measured from the extreme compression fiber in the flange.

Term

Web width (b_w)

Example

T-beam with flange width 2000 mm and web width 300 mm creates a 6.7:1 aspect ratio in compression.

Definition

Narrow width of the stem below the flange; supports shear and torsion; typically 200–400 mm.

Note

This is the happy case: most exam problems are designed this way. Use standard rectangular-beam formulas.

Equation

If a ≤ t_f: treat as rectangular with width b_f (ignore web below flange for bending).

Conditions

Stress block is fully within the flange; no material below t_f is in compression under the equivalent block.

Note

The moment is computed as sum of two couples: (C_f, T) from flange, and (C_w, T_extra) from web. Algebraically, it equals A_s × f_y × (d − a'/2) where a' is a weighted average depth.

Equation

If a > t_f: C = C_f + C_w = 0.85 f'_c × [b_f × t_f + b_w × (a − t_f)]

Conditions

Stress block extends into web; split the compression force into flange and web parts.

Formula

ε_t ≥ 0.005 ⟹ φ = 0.90 (tension-controlled)

Meaning

When net tensile strain ≥ 0.5%, section is tension-controlled; full φ = 0.90 permitted by NSCP 2015.

Watch Out

This is the GATE CONDITION. If ε_t < 0.005, you cannot use φ = 0.90; must use reduced φ or reject the design.

When To Use

After analyzing a section, compute ε_t to determine if φ = 0.90 is valid.

Formula

If 0.002 ≤ ε_t < 0.005: φ = 0.65 + 0.25 × (ε_t − 0.002) / 0.003 (transition zone)

Meaning

In the transition between compression-controlled (φ = 0.65) and tension-controlled (φ = 0.90), φ increases linearly.

Watch Out

This interpolation is per NSCP 2015; older codes used different rules. Always cite the correct code.

When To Use

Over-reinforced beams where ε_t is between yield (0.002) and the ductility threshold (0.005).

Formula

If ε_t < 0.002: φ = 0.65 (compression-controlled or brittle)

Meaning

When steel does not reach yield strain, concrete is the primary failure element; minimum φ = 0.65.

Watch Out

Sections with φ = 0.65 are NOT acceptable for ductility-critical applications; redesign to achieve ε_t ≥ 0.005.

When To Use

Heavily over-reinforced sections; rare in routine flexure but possible with high-strength steel or weak concrete.

Value

0.005

Symbol

ε_t,min

Quantity

Ductility threshold (tension-controlled)

Value

0.002 (= f_y / E_s ≈ 400 / 200,000)

Symbol

ε_y

Quantity

Yield strain (steel)

Value

0.003

Symbol

ε_c,max

Quantity

Crushing strain (concrete)

Value

0.90

Symbol

φ_max

Quantity

Maximum φ (flexure)

Value

0.65

Symbol

φ_min

Quantity

Minimum φ (flexure)

Term

Tension-controlled section

Example

A singly reinforced beam with ρ = 0.010 produces ε_t ≈ 0.012 >> 0.005 → tension-controlled, safe design.

Definition

Flexural section where ε_t ≥ 0.005; steel yields before concrete crushes; ductile, warning signs before failure; φ = 0.90.

Term

Compression-controlled section

Example

A heavily over-reinforced section (ρ = 0.04 >> ρ_max) produces ε_t ≈ 0.0008 << 0.002 → compression-controlled, unacceptable.

Definition

Flexural section where ε_t < 0.002; concrete crushes before steel yields; brittle, sudden failure with no warning; φ = 0.65.

Term

Transition zone

Example

ε_t = 0.0035 → φ = 0.65 + 0.25(0.0035 − 0.002)/0.003 = 0.65 + 0.083 = 0.733.

Definition

Range where 0.002 ≤ ε_t < 0.005; φ interpolates linearly from 0.65 to 0.90; neither fully ductile nor brittle.

Note

This strain calculation is the linchpin of ductility checks; always compute it.

Equation

ε_t = 0.003 × (d − c) / c

Conditions

Linear strain distribution from 0.003 (top) to ε_t (steel level); c = neutral-axis depth from stress block.

Note

Use 0.90 ONLY if ε_t ≥ 0.005; otherwise interpolate or use 0.65.

Equation

φ values: 0.90 (tension-control), 0.65–0.90 (transition), 0.65 (compression-control)

Conditions

NSCP 2015 flexure; ε_t determines which regime the section falls into.

Values

• Bottom only (A_s)
• Bottom (A_s) + Top (A's)
• Bottom of web (A_s)

Property

Tension steel location

Values

• Concrete block + slab (if present)
• Concrete block + compression steel
• Wide flange above web

Property

Compression zone

Values

• M_n = A_s f_y (d − a/2)
• M_n = A_s f_y (d − a/2) + A's f's (d − d')
• M_n = A_s f_y (d − a/2) if a ≤ t_f; else split forces

Property

Moment capacity equation

Values

• Most beams; simple, economical
• When M_u exceeds single capacity; deflection control
• Floor systems (monolithic slab-beam)

Property

Typical use case

Values

• Straightforward quadratic or formula
• Assume max ρ for bottom, add couple from top steel
• Check a ≤ t_f first; may need force split

Property

Design complexity

Values

• Compute ε_t from c and d; ensure ε_t ≥ 0.005
• Compute ε_t (bottom) and ε's (top); both strained
• Compute ε_t from neutral axis in web (if a > t_f)

Property

Ductility check

Values

• 0.90 if tension-controlled (typical)
• May be reduced if ε_t < 0.005 (rare for doubly reinf.)
• 0.90 if a ≤ t_f and tension-controlled

Property

φ value

Values

• Baseline (least steel)
• Higher (compression steel adds cost & congestion)
• Varies; flange efficiency can reduce total steel

Property

Steel cost impact

Values

• 0.85 β₁ (f'_c / f_y) × [600 / (600 + f_y)]
• 0.0288
• Concrete crushes and steel yields simultaneously; boundary between ductile and brittle

Property

Balanced ratio (ρ_b)

Values

• 0.85 β₁ (f'_c / f_y) × [0.003 / 0.008] ≈ 0.375 ρ_b
• 0.0183
• NSCP limit; ensures ε_t ≥ 0.005 for ductility and φ = 0.90

Property

Maximum tension-controlled (ρ_max)

Values

• max[1.4 / f_y, √f'_c / (4 f_y)]
• 0.00337
• Minimum steel to prevent brittle cracking failure; code-mandated lower bound

Property

Minimum ratio (ρ_min)

Values

• ρ_min ≤ ρ ≤ ρ_max
• 0.00337 ≤ ρ ≤ 0.0183
• Any ratio in this range is ductile, code-compliant, and will use φ = 0.90

Property

Safe design band

Values

• Steel yields first; concrete crushes after; ductile
• 0.90
• Tension-controlled (NSCP preferred)
• ✓ Acceptable; full capacity

Property

ε_t ≥ 0.005

Values

• Transition; both close to limit; mixed mode
• 0.65 + 0.25(ε_t − 0.002)/0.003
• Transition zone (reduced φ)
• ✓ Acceptable but conservative; reduced capacity

Property

0.002 ≤ ε_t < 0.005

Values

• Concrete crushes first; steel elastic; brittle
• 0.65
• Compression-controlled (discouraged)
• ✗ Not recommended; brittle failure

Property

ε_t < 0.002

Values

• M_n = A_s f_y (d − a/2) [rectangular formula]
• Entire block within flange; no web involvement
• Full flange width b_f
• Simple; treat as rectangular beam with width b_f

Property

If a ≤ t_f (stress block in flange)

Values

• Split into two couples: flange + web contributions
• Block spans flange (thickness t_f) and extends into web
• Flange full width b_f, web at b_w below
• Complex; must account for step-down in width; rare on exam

Property

If a > t_f (stress block into web)