Input Parameters

mm · MPa · kN·m

Member Type & Strain Distribution — k₂

k₂ = 0.5 bending-only · 1.0 pure tension · intermediate for combined M+N

Section Geometry

Width b [mm]

mm

Total Depth h [mm]

mm

Nominal Cover c_nom [mm]

mm

Eff. Depth d [mm] (auto)

mm

Reinforcement

Bar Diameter φ [mm]

mm

Bar Spacing s [mm]

mm

Materials

f_ck [MPa]

MPa

f_yk [MPa]

MPa

Loading & Conditions

Service Moment M_s [kN·m]

kN·m

Quasi-permanent (unfactored) moment for the strip width b.

Loading Duration — k_t

Exposure Class — w_lim (EC2 Table 7.1N)

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Design Background — EC2 §7.3.4

Characteristic Crack Width

EN 1992-1-1:2004 §7.3.4 defines the characteristic crack width as the product of the maximum crack spacing and the effective strain difference between steel and concrete:

w_k = s_r,max × (ε_sm − ε_cm)

Effective Strain

The mean strain difference accounts for tension stiffening (concrete between cracks still carries some tension):

ε_sm − ε_cm = [σ_s − k_t·(f_ct,eff/ρ_p,eff)·(1 + α_e·ρ_p,eff)] / E_s ≥ 0.6·σ_s / E_s

where k_t = 0.4 (long-term loading) or 0.6 (short-term), α_e = E_s/E_cm is the modular ratio, and ρ_p,eff = A_s/A_c,eff.

Maximum Crack Spacing (Eq. 7.11)

s_r,max = 3.4c + 0.425·k₁·k₂·φ / ρ_p,eff

k₁ = 0.8 for high-bond (deformed) bars. k₂ = 0.5 for bending, 1.0 for pure tension, intermediate for combined M+N (e.g. k₂ = 0.75 for walls with moderate axial tension). Walls with compression (gravity) use k₂ = 0.5. c = nominal cover.

Effective Concrete Area in Tension

h_c,ef = min(2.5(h − d), (h − x)/3, h/2) A_c,eff = b × h_c,ef

Material Properties

Property	Formula	Note
f_cm	f_ck + 8 MPa	Mean compressive strength
E_cm	22 000 (f_cm/10)^0.3 MPa	EC2 Table 3.1
f_ctm (f_ck ≤ 50)	0.30·f_ck^2/3 MPa	EC2 Table 3.1
f_ctm (f_ck > 50)	2.12·ln(1 + f_cm/10) MPa	EC2 Table 3.1
α_e	E_s / E_cm	Modular ratio (≈ 6–7)

Cracked Neutral Axis (Elastic)

x = d·[−ρ·α_e + √(ρ²·α_e² + 2ρ·α_e)] where ρ = A_s/(b·d) I_cr = b·x³/3 + α_e·A_s·(d−x)² σ_s = α_e·M_s·(d−x) / I_cr

Crack Width Limits — EC2 Table 7.1N (Reinforced Concrete)

Exposure Class	Environment	w_max (quasi-permanent)
XC1	Dry or permanently submerged concrete	0.3 mm
XC2, XC3, XC4	Wet, cyclic wet/dry, moderate humidity	0.3 mm
XD1, XD2	Moderate / wet chloride exposure	0.3 mm
XD3, XS1, XS2, XS3	Cyclic / marine chloride (severe)	0.2 mm

National Annexes may adjust these values. Some countries allow 0.4 mm for XC1 under frequent load combination. The table above represents the EN 1992-1-1 recommended values.

Minimum Reinforcement (§7.3.2)

A_s,min = k_c·k·f_ct,eff·A_ct / σ_s

k_c = 0.4 for pure bending, 1.0 for pure tension. A_ct = tension zone area before cracking (≈ b·h/2 for bending). σ_s = f_yk for crack control without further limitation.

Input Parameters

mm · MPa · kN·m

Member Type

Section Geometry

Width b [mm]

mm

Total Depth h [mm]

mm

Clear Cover c [mm]

mm

Eff. Depth d [mm] (auto)

mm

Reinforcement

Bar Diameter φ [mm]

mm

Bar Spacing s [mm]

mm

Materials (IS 456:2000)

f_ck [MPa]

MPa

f_y [MPa]

MPa

Loading & Conditions

Service Moment M_s [kN·m]

kN·m

Exposure Condition

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Design Background — IS 456:2000 Annex F

IS 456:2000 Annex F — Crack Width Formula

The design surface crack width at any point is given by:

w_cr = 3·a_cr·ε_m / [1 + 2·(a_cr − c_min)/(h − x)]

where a_cr is the distance from the point considered to the surface of the nearest longitudinal bar (mm), and c_min is the minimum cover to tension reinforcement.

Mean Strain ε_m

ε_m = ε₁ − b·(h−x)³ / [3·E_s·A_s·(d−x)] ε₁ = f_s·(h−x) / [E_s·(d−x)]

ε_m ≥ 0. The stiffening term accounts for concrete in tension between cracks. f_s = steel stress at the cracked section.

Distance a_cr

a_cr = √[c_min² + (s/2)²]

This is the distance from the midpoint on the tension face (between adjacent bars) to the nearest bar surface. For a point directly below a bar, a_cr = c_min; the midpoint between bars gives the maximum crack width.

Cracked Section and Steel Stress

Modular ratio: m = E_s/E_c, E_c = 5000√f_ck (IS 456 Cl 6.2.3.1) Neutral axis: x = d·[−mρ + √(m²ρ² + 2mρ)] I_cr = bx³/3 + m·A_s·(d−x)² f_s = m·M_s·(d−x) / I_cr

Crack Width Limits — IS 456:2000

IS 456 Annex F targets a design crack width ≤ 0.3 mm under service loads for ordinary conditions. In more aggressive environments the limit reduces to 0.2 mm. These align with the general serviceability philosophy of the code.

Exposure Condition	w_lim
Mild / Moderate	0.30 mm
Severe, Very Severe, Extreme	0.20 mm

Note: IS 456 does not provide a comprehensive crack width limit table in the main body of the code; the 0.3 mm value is the accepted engineering practice consistent with Annex F and the code's durability intent.

Input Parameters

mm · MPa · kN·m

Unit System

Member Type

Section Geometry

Width b [mm]

mm

Total Depth h [mm]

mm

Cover to Bar Center d_c [mm]

mm

d_c = clear cover + φ/2

Eff. Depth d [mm] (auto)

mm

Tension Reinforcement

Bar Diameter φ [mm]

mm

Bar Spacing s [mm]

mm

Compression Steel (n_c = 0 if none)

No. of Bars n_c

bars

Bar Diameter φ' [mm]

mm

Cover to Center d' [mm]

mm

Materials (ACI 318)

f'_c [MPa]

MPa

f_y [MPa]

MPa

Loading & Exposure

Service Moment M_s [kN·m]

kN·m

Exposure — ACI 224R-01 Limit

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Design Background — ACI 224R / Frosch (1999)

Frosch (1999) Crack Width Model

ACI 224R-01 recommends the Frosch (1999) physical model to estimate maximum crack widths in flexural members. The crack width at the extreme tension fiber is:

w = 2·(f_s/E_s)·β_s·√[d_c² + (s/2)²]

where:

f_s = steel stress at service load [MPa]
E_s = 200 000 MPa (steel modulus)
β_s = (h − x̄) / (d − x̄) — strain gradient factor (always ≥ 1)
d_c = distance from extreme tension fiber to centroid of nearest bar [mm]
s = bar spacing [mm]
x̄ = cracked neutral axis depth (elastic cracked section analysis)

Physical Interpretation

The term √[d_c² + (s/2)²] is the direct distance from the extreme tension face (midway between bars) to the nearest bar centroid. Multiplying by the strain at that level and a factor of 2 (for symmetry) gives the crack opening at the surface.

ACI 224R-01 Table 4.1 — Recommended Limits

Exposure Condition	w_lim (mm)	w_lim (in)
Dry air or protective membrane	0.41	0.016
Humidity, moist air, soil	0.30	0.012
Deicing chemicals	0.18	0.007
Seawater / seawater spray, wetting & drying	0.15	0.006
Water-retaining structures (ACI 350M-06)	0.10	0.004

The ACI 318 code does not prescribe explicit crack width limits; the values above are from ACI 224R-01 and ACI 350M-06. ACI 318-25 uses the indirect bar spacing method of §24.3, which implicitly targets ≈ 0.40 mm.

Doubly Reinforced Section — Cracked NA

When compression steel is present, the cracked elastic neutral axis is found from:

(b/2)·x² + [(n−1)A'_s + nA_s]·x − [(n−1)A'_s·d' + nA_s·d] = 0 I_cr = bx³/3 + nA_s(d−x)² + (n−1)A'_s(x−d')²

(n−1) is used for compression steel because concrete at that level is already included in the bx²/2 term.

Input Parameters

mm · MPa · kN·m

Member Type & Strain Distribution — k₂

k₂ = 0.5 bending-only · 1.0 pure tension · intermediate for combined M+N

Section Geometry

Width b [mm]

mm

Total Depth h [mm]

mm

Cover c [mm]

mm

Eff. Depth d [mm] (auto)

mm

Reinforcement

Bar Diameter φ [mm]

mm

Bar Spacing s [mm]

mm

Materials (TS 500)

f_ck [MPa]

MPa

f_yk [MPa]

MPa

Loading & Conditions

Service Moment M_s [kN·m]

kN·m

Loading Type — β₂

Exposure Class — w_lim (TS 500 Table 11.2)

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Design Background — TS 500:2000 §11.5

TS 500:2000 §11.5 — Crack Width

The TS 500:2000 crack width formula is based on the CEB-FIP 1990 Model Code:

w_k = β · s_m · ε_sm

β = 1.7 for members dominated by bending. The formula is the product of the mean crack spacing and the mean steel strain.

Mean Crack Spacing

s_m = 50 + 0.25·k₁·k₂·φ / ρ_r,eff [mm] k₁ = 0.8 (deformed bars), k₂ = 0.5 (bending) / 1.0 (predominantly tension)

Mean Steel Strain

ε_sm = (σ_s/E_s)·[1 − β₁·β₂·(σ_sr/σ_s)²] ≥ 0.4·σ_s/E_s β₁ = 1.0 (deformed bars), β₂ = 0.5 (sustained) / 1.0 (short-term)

σ_sr = f_ctm·(1 + α_e·ρ_eff) / ρ_eff — steel stress at cracking.

TS 500:2000 — Crack Width Limits

Exposure Class	w_lim
Dry interior	0.40 mm
Normal interior/exterior	0.30 mm
Humid / aggressive or water contact	0.20 mm

β = 1.7 is used for flexural members. The cracking stress σ_sr is derived from the gross cross-section at the crack load.

Crack Width Control

Input Parameters

Design Background — EC2 §7.3.4

Characteristic Crack Width

Effective Strain

Maximum Crack Spacing (Eq. 7.11)

Effective Concrete Area in Tension

Material Properties

Cracked Neutral Axis (Elastic)

Crack Width Limits — EC2 Table 7.1N (Reinforced Concrete)

Minimum Reinforcement (§7.3.2)

Input Parameters

Design Background — IS 456:2000 Annex F

IS 456:2000 Annex F — Crack Width Formula

Mean Strain ε_m

Distance a_cr

Cracked Section and Steel Stress

Crack Width Limits — IS 456:2000

Input Parameters

Design Background — ACI 224R / Frosch (1999)

Frosch (1999) Crack Width Model

Physical Interpretation

ACI 318-25 §24.3 — Indirect Crack Control

ACI 224R-01 Table 4.1 — Recommended Limits

Doubly Reinforced Section — Cracked NA

Input Parameters

Design Background — TS 500:2000 §11.5

TS 500:2000 §11.5 — Crack Width

Mean Crack Spacing

Mean Steel Strain

TS 500:2000 — Crack Width Limits

Crack Width Control

Input Parameters

Design Background — EC2 §7.3.4

Characteristic Crack Width

Effective Strain

Maximum Crack Spacing (Eq. 7.11)

Effective Concrete Area in Tension

Material Properties

Cracked Neutral Axis (Elastic)

Crack Width Limits — EC2 Table 7.1N (Reinforced Concrete)

Minimum Reinforcement (§7.3.2)

Input Parameters

Design Background — IS 456:2000 Annex F

IS 456:2000 Annex F — Crack Width Formula

Mean Strain εm

Distance acr

Cracked Section and Steel Stress

Crack Width Limits — IS 456:2000

Input Parameters

Design Background — ACI 224R / Frosch (1999)

Frosch (1999) Crack Width Model

Physical Interpretation

ACI 318-25 §24.3 — Indirect Crack Control

ACI 224R-01 Table 4.1 — Recommended Limits

Doubly Reinforced Section — Cracked NA

Input Parameters

Design Background — TS 500:2000 §11.5

TS 500:2000 §11.5 — Crack Width

Mean Crack Spacing

Mean Steel Strain

TS 500:2000 — Crack Width Limits

Related Calculators & Guides

Mean Strain ε_m

Distance a_cr