Stage 6 / reinforcement output

ULS reinforcement output.

The reinforcement route converts the ULS bending moment from the strip-on-foundation solve into a required area of tension reinforcement per meter width, using EC2 strain compatibility on the parabola–rectangle concrete law and an elastic–perfectly-plastic steel law. The derivation below is the screening-level theory backing that output.

Scope and intent

1. Problem class and meaning of the result

The reinforcement route is a strip ULS reinforcement screen. It takes the design bending moment per meter of strip width that the beam-on-foundation solver reports and returns the tension reinforcement area required to develop that moment, assuming a singly reinforced rectangular section and EC2 material laws.

Scope boundary. The result is a one-dimensional strip reinforcement quantity. It does not produce a two-dimensional bar layout, does not size shear links, does not verify serviceability crack widths or deflections, and does not account for punching at column heads. It is a preliminary sizing screen that the engineer then carries into a dedicated structural design tool.
Primary quantities
MEd
Design bending moment for the modelled strip width; this is kN·m/m when b = 1.0 m.
As,req
Required tension reinforcement area for the modelled strip width; this is mm²/m when b = 1.0 m.
μ, ω
Dimensionless bending moment and mechanical reinforcement ratio [-].
ξlim
Limiting neutral-axis ratio for tension-controlled (ductile) failure [-].

Sensitivity to member depth

3. Why As,req can rise with h on a Winkler strip

On a rigid-support beam M is fixed by statics, so increasing h reduces μ and reduces As,req monotonically. On a Winkler foundation that intuition fails: the moment is not given by statics. EI grows as h3, λ grows as h13/16, and the peak moment grows along with λ. Whether As,req increases or decreases with h is decided by which power of h dominates: the moment increase or the lever-arm gain.

EI ∝ h3, ks ∝ h−1/4 (Vesić soft coupling)
λ = (4 EI/(b ks))1/4 ∝ h13/16 ≈ h0.81
d ≈ h − constant ⇒ d ∝ h
μ = MEd · 106 / (bw d2 fcd), ω ≈ μ for small μ
As,req = ω bw d fcd/fyd ≈ MEd/(d fyd) for small μ

The Hetényi closed forms in the beam chapter §6 give MEd(h) for each canonical load case. Carrying those exponents through the small-μ approximation As,req ≈ MEd/(d fyd):

Direction of As,req(h) sensitivity by load case in the small-μ regime.

Load caseMEd ∝ hαAs,req ∝ hα−1Effect of growing h
Concentrated load (point or short patch)h0.81h−0.19As,req decreases slowly
Patch UDL (Lpatch < λ)h1.63h0.63As,req increases with h
Full-strip UDL on free-ended strip≈ 0≈ 0As,req hits As,min floor

Crossover at α = 1: any load case where the strip-on-foundation solver returns MEd growing faster than h1 drives As,req upward with h. The patch-UDL case crosses that boundary; the point-load case does not.

Worked illustration. Patch UDL load case, b = 1 m, fck = 30 MPa, cnom + φbar/2 = 36 mm, MEd,1 = 100 kN·m at h1 = 0.30 m. Going to h2 = 0.50 m: M2/M1 = (0.5/0.3)1.63 ≈ 2.15, MEd,2 ≈ 215 kN·m, d1 = 264 mm, d2 = 464 mm; μ1 = 0.072 → As,1 ≈ 904 mm²/m, μ2 = 0.050 → As,2 ≈ 1094 mm²/m. The +21% jump in As comes entirely from the foundation interaction; the section check is unchanged.
  • This is a property of the soil-supported analysis, not of the EC2 section check. The μ–ω–As relationship in §5 is monotonic; what changes is the MEd input that the strip solver delivers.
  • The closed-form scalings are exact in the small-μ limit (ω ≈ μ). For μ approaching the ductility bound μlim ≈ 0.372 the relationship saturates, but the qualitative direction is preserved.
  • If MEd is taken from an external structural model rather than the Stage 6 strip solver, the conventional intuition (more h → less As) does apply. The two checks are separate problems and should be kept separate in the audit trail.
  • The same logic applies inversely to ks: stiffening the soil column reduces λ and reduces MEd for the patch-UDL case, lowering As,req ∝ ks−1/2.

Kinematics and constitutive laws

4. Plane sections, concrete law, and steel law

The formulation is the standard EC2 ULS bending assumption for reinforced concrete: plane sections remain plane, strain varies linearly through the depth, concrete in tension is ignored, and steel and concrete are each represented by an idealized constitutive law.

ε(y) = εc(ytop − y)/x linear strain profile
σc(εc) = fcd[1 − (1 − εc/εc2)n] for 0 ≤ εc ≤ εc2
σc(εc) = fcd for εc2 ≤ εc ≤ εcu2
n = 2, εc2 = 2.0‰, εcu2 = 3.5‰ (fck ≤ 50 MPa)
εs = εcu2(d − x)/x at the tension steel level
σs(εs) = Es εs for |εs| ≤ fyd/Es
σs(εs) = sign(εs) · fyd beyond yield (horizontal branch adopted)
EC2 general form: fcd = αcc fck/γC, fyd = fyk/γS
Current public implementation: fcd = fck/γC, fyd = fyk/γS
Notation
εc, εc2, εcu2
Concrete compressive strain, strain at peak, and ultimate strain [-]. Values for fck ≤ 50 MPa.
n
Exponent of the parabolic branch (n = 2 for normal strength).
x
Neutral-axis depth from the compressed fibre [m].
εs
Tension-steel strain obtained from the strain triangle at depth d.
Es
Steel Young modulus, 200 GPa.
αcc
Sustained-load coefficient. The current public route is consistent with αcc = 1.0.
γC, γS
Partial material factors for concrete and steel. Default ULS: γC = 1.50, γS = 1.15.

Rectangular stress block

5. Equivalent rectangular block and its λ, η factors

Integrating the parabola–rectangle law across the compression zone and equating force and moment to an equivalent uniform block of depth λx and intensity η fcd produces the EC2 rectangular block with λ = 0.8 and η = 1.0 for normal-strength concrete. The factors retain the same resultant force and lever arm as the exact integral.

Cc = ∫0x σc(ε(y)) b dy = η fcd b λ x
ȳc = ∫ σc y b dy / Cc = λx / 2 (by construction)
λ = 0.8, η = 1.0 for fck ≤ 50 MPa
λ = 0.8 − (fck − 50)/400, η = 1.0 − (fck − 50)/200 for 50 < fck ≤ 90 MPa
Notation
Cc
Resultant concrete compressive force per meter of strip [kN/m].
λ
Reduced depth factor for the equivalent stress block.
η
Reduced intensity factor (η = 1.0 for normal strength; < 1 for high strength).

Design route

6. Dimensionless moment μ and mechanical ratio ω

The closed-form route for a singly reinforced rectangular section drops out of equilibrium and the rectangular stress block. Denote the normalized moment μ = MEd/(b d² fcd) and the mechanical reinforcement ratio ω = As fyd/(b d fcd). Axial equilibrium gives ω = η λ ξ with ξ = x/d, and moment equilibrium about the tension bars yields a quadratic in ω whose tension-controlled solution is the classical closed form.

μ = MEd / (b d² fcd)
N = 0: As fyd = η fcd b λ x ⇒ ω = η λ ξ
MRd about tension steel: MRd = η fcd b λ x (d − λx/2)
General closed form: μ = ω (1 − ω/(2η)) ⇒ ω = η [1 − √(1 − 2μ/η)]
Current public route: ω = 1 − √(1 − 2μ)
As,req = ω b d fcd / fyd
Notation
h, cnom, φbar, d
Section thickness, nominal cover, tension-bar diameter, and effective depth [mm]. The current implementation does not subtract a separate link diameter.
b
Reference width used in the section check (1000 mm per metre of strip).
ξ = x/d
Relative neutral-axis depth.
MRd
Section design moment resistance [kN·m/m].

The closed-form ω = η[1 − √(1 − 2μ/η)] is the tension-controlled root: the steel yields before the concrete crushes, so σs = fyd is the right assumption. The alternative compression-controlled root would require a brittle-failure interpretation and is excluded by the ductility bound below.

Implementation note. The shipped Stage 6 screen currently uses the normal-strength shortcut ω = 1 − √(1 − 2μ) directly. It therefore does not yet carry the high-strength η reduction through the flexural closed form even though the documentation keeps the general EC2 expression visible for reference.

Ductility bound

7. Limiting neutral-axis depth and the case for compression steel

The closed form is only valid while steel yields before concrete crushes (εs ≥ fyd/Es at εc = εcu2). Linear strain compatibility translates this into a limit on ξ; above that limit the section is over-reinforced and the rectangular block overestimates capacity unless compression reinforcement is introduced.

εs,min = fyd/Es
ξlim = εcu2 / (εcu2 + εs,min)
For B500B, fyd = 435 MPa, Es = 200 GPa: εs,min ≈ 2.175‰
Pure compatibility upper bound: ξlim ≈ 0.617, ωlim = η λ ξlim ≈ 0.494
Current public warning threshold: μ ≳ 0.295 ⇒ redesign or move to a dedicated double-reinforcement check
For a doubly reinforced section, general equilibrium becomes As fyd = Cc + A′s σ′s
MRd = Cc zc + A′s σ′s(d − d′)
  • The app currently warns once μ exceeds about 0.295. That is the practical singly reinforced screen used by the public route, and it is stricter than the algebraic upper bound of the closed form.
  • Once the warning is triggered, the engineer should increase h, change the material class, or move to a full doubly reinforced section design with explicit compression steel A′s.
  • The public app does not yet size A′s automatically. It reports the flexural demand and warns that the singly reinforced shortcut is no longer the right design model.
  • Tension-controlled failure is the default EC2 design target because it gives warning by cracking and deflection before collapse.

Detailing bounds

8. Minimum and maximum reinforcement area

Strain compatibility only sets the required area for the given moment. EC2 detailing rules then impose a minimum and a maximum. The minimum protects against brittle cracking under the tensile capacity that the gross uncracked section can sustain; the maximum protects against congestion and compression-controlled behaviour.

As,min = max(0.26 fctm/fyk bt d, 0.0013 bt d)
Current public route: fctm = 0.30 fck2/3
As,max = 0.04 Ac (outside lap zones)
As,eff = max(As,req, As,min)
Notation
fctm
Mean axial tensile strength of concrete [MPa].
fcm
Mean compressive cylinder strength, fcm = fck + 8 MPa. Useful for the full EC2 tensile-strength branch, though the current public route keeps the normal-strength fctm expression.
bt
Width of the tension zone of the section (b for a rectangular strip).
Ac
Cross-sectional area of concrete (b·h per meter of strip).
  • For typical slab-on-ground strip problems with moderate moment, As,min frequently governs in the span fields and away from local peaks.
  • The app reports both As,req and the governing design area As = max(As,req, As,min).
  • The high-strength EC2 fctm branch is not yet split out separately in the current screen, so high-strength concrete should be rechecked downstream in the structural workflow.

Shear screening

9. Shear capacity without stirrups (VRd,c)

The reinforcement output is a flexural quantity, but shear almost always governs detailing for thin slab strips on elastic foundation. The EC2 empirical expression for members without shear reinforcement is included here for continuity; the engineer runs it against the VEd envelope from the beam solver.

VRd,c = [CRd,c k (100 ρl fck)1/3 + k1 σcp] bw d
VRd,c,min = (vmin + k1 σcp) bw d
k = min(1 + √(200/d[mm]), 2.0)
ρl = min(Asl/(bw d), 0.02)
CRd,c = 0.18/γC, k1 = 0.15, vmin = 0.035 k3/2 fck1/2
Notation
ρl
Longitudinal reinforcement ratio using the tension steel anchored past the section.
k
Size-effect factor capped at 2.0.
σcp
Mean compressive stress from axial load (taken positive) [MPa]. Zero for an unprestressed slab strip.
bw
Minimum web width at the shear section (equal to b for a slab strip).
  • If VEd > VRd,c, the section needs shear reinforcement; that is outside the reinforcement-screen route.
  • Size effect k is meaningful for thin slabs because d is in the denominator of the √(200/d) term.

Material and detailing assumptions

10. Material factors, structural class, and the implemented EC2 cover route

The route applies EC2 partial factors and derives the effective depth from the geometric inputs. Concrete cover is built from the exposure class, concrete class, and design working life per EC2 §4.4.1. The resulting cnom feeds directly into the effective depth, so durability assumptions influence both the stiffness-compatible reinforcement demand and the practical steel area that can be placed.

S = clamp(4 + ΔSlife − ΔShs − ΔSslab − ΔSQC, 1, 6)
ΔSlife = +2 for 100-year working life, −1 for ≤ 25 years, 0 otherwise
ΔShs = 1 when fck reaches the exposure-dependent high-strength threshold
cmin,b = max(φbar, 6 mm) + 5 mm if dG > 32 mm
cmin = max(cmin,dur, cmin,b, 10 mm)
cnom,raw = max(cmin + Δcdev + Δcuneven, cfloor)
cnom,recommended = 5 · ceil(cnom,raw/5)
cnom,used = cnom,override or cnom,recommended when no override is given
Notation
cmin,b
Minimum cover for bond, equal to bar diameter.
cmin,dur
Minimum cover for durability, from EC2 Table 4.4N using the derived structural class S and the relevant exposure column.
Δcdev
Allowance for execution deviation (EC2 §4.4.1.3).
dG
Maximum aggregate size [mm]. The current route adds 5 mm to cmin,b if dG > 32 mm.
cfloor
Minimum cover floor from ground casting assumptions: 40 mm for prepared ground, 75 mm for unprepared ground.
  • Exposure fallbacks used by the current app are explicit: XF1/XF2/XF3 and XA1/XA2 map to the XC4 corrosion-cover column; XF4 and XA3 map to XD3.
  • The automatic structural-class route starts from S = 4, then adjusts for working life, high-strength concrete, slab-like geometry, and special quality control.
  • Common ULS defaults are γC = 1.50 and γS = 1.15.
  • If the engineer enters a manual cnom below the recommended value, the app keeps it but issues a warning.
  • For fire design, a separate cmin,fire from EN 1992-1-2 may govern. That route is outside the current screen.
  • Concrete composition requirements for XF/XA exposure classes remain an engineer check outside the current app.

Practical translation

11. From required steel area to bars per meter or bar spacing

The public output is an area per meter width. Structural detailing requires translating that area into a bar diameter and spacing, or into a number of bars per meter. For a chosen bar diameter φ, the single-bar area is fixed and the spacing follows directly.

Abar = π φ2 / 4
nbars/m = As,prov / Abar
s = 1000 Abar / As,prov
As,prov = 1000 Abar / s
Notation
Abar
Area of one reinforcing bar [mm²].
As,prov
Area actually provided by the chosen diameter-and-spacing arrangement for a one-meter strip [mm²/m].
s
Bar spacing [mm centre-to-centre].
  • The structural design workflow should round As,prov upward, not to the nearest nominal spacing.
  • Spacing must still satisfy EC2 detailing limits for crack control, bond, aggregate passing, and concreting tolerances.
  • Because the app currently sizes only one governing strip area, the engineer still needs to assign whether that steel belongs in the top or bottom zone of the slab strip.

Outputs and limitations

12. Reported quantities and boundaries of validity

The public output is an indicative tension reinforcement area per meter width consistent with the section’s moment capacity under the assumed plane-section and EC2 material laws. It is a structural-geotechnical screening quantity and is explicitly not a substitute for a full structural design workflow.

  • No bar schedule, no two-dimensional layout, no detailing at openings or supports.
  • The current route sizes reinforcement from |MULS,max| only; it does not yet resolve separate top and bottom reinforcement zones from the sign of the strip moment field.
  • Serviceability verifications (crack width wk, deflection, stress limitation) are outside this route.
  • Shear, punching, torsion, fatigue, and anchorage checks are outside this route.
  • Compression reinforcement is not sized; the app warns once μ exceeds about 0.295 rather than auto-sizing A′s.
  • The current public route assumes a rectangular strip section only. Flanged T-sections, ribbed slabs, and orthotropic plate reinforcement are outside this screen.
  • The underlying specification anchor is the full technical specification.

References

References

  • EN 1992-1-1:2004+A1:2014 — Eurocode 2: Design of concrete structures. Part 1-1: General rules. §3.1, §3.2, §6.1, §9.2.1.
  • NBN EN 1992-1-1 ANB:2010 — Belgian National Annex to Eurocode 2.
  • EN 1990:2002+A1:2005 — Basis of structural design (partial factors).
  • fib Model Code for Concrete Structures 2010 — background for the parabola–rectangle law and the rectangular block derivation.