Eurocode 2 Beam Design Workflow
1
Material Design Strengths & Effective Depth ($d$)
Cl. 3.1.6 & Cl. 4.4.1
Calculate design concrete compressive strength $f_{cd}$ and steel yield strength $f_{yd}$ using partial safety factors $\gamma_c = 1.5$ and $\gamma_s = 1.15$:
$$f_{cd} = \frac{\alpha_{cc} \cdot f_{ck}}{\gamma_c} = \frac{0.85 \cdot f_{ck}}{1.5}, \quad f_{yd} = \frac{f_{yk}}{\gamma_s} = \frac{f_{yk}}{1.15}$$
Symbol Notation Key
$f_{cd}$= design concrete compressive strength (MPa)
$f_{ck}$= characteristic cylinder concrete strength (MPa)
$\alpha_{cc}$= long-term compressive coefficient (0.85 per UK NA)
$\gamma_c$= concrete partial safety factor ($\gamma_c = 1.5$)
$f_{yd}$= design steel yield strength (MPa)
$f_{yk}$= characteristic steel yield strength (MPa)
$\gamma_s$= steel partial safety factor ($\gamma_s = 1.15$)
Determine effective depth $d$ from overall depth $h$, nominal cover $c_{nom}$, link diameter $\phi_{link}$, and main bar radius $\phi/2$:
$$d = h - c_{nom} - \phi_{link} - \frac{\phi}{2}$$
Symbol Notation Key
$d$= effective depth of tension reinforcement (mm)
$h$= overall section depth (mm)
$c_{nom}$= nominal concrete cover to stirrups (mm)
$\phi_{link}$= shear link stirrup bar diameter (mm)
$\phi$= main tension longitudinal bar diameter (mm)
2
ULS Flexural Bending Factor ($K$) & Decision Tree
Cl. 6.1
Calculate normalized bending factor $K$ to check section moment capacity:
$$K = \frac{M_{Ed}}{b \cdot d^2 \cdot f_{ck}}$$
Symbol Notation Key
$K$= normalized flexural bending factor (dimensionless)
$M_{Ed}$= design ULS bending moment at section (kNm)
$b$= section width (mm)
$d$= effective depth of section (mm)
$f_{ck}$= characteristic concrete strength (MPa)
✓ Path A: $K \le K_{bal} = 0.167$
Singly reinforced section.
Lever arm $z = d \left[0.5 + \sqrt{0.25 - \frac{K}{1.134}}\right] \le 0.95d$.
Tension steel: $A_{s,req} = \frac{M_{Ed}}{f_{yd} \cdot z}$.
Lever arm $z = d \left[0.5 + \sqrt{0.25 - \frac{K}{1.134}}\right] \le 0.95d$.
Tension steel: $A_{s,req} = \frac{M_{Ed}}{f_{yd} \cdot z}$.
⚠ Path B: $K > 0.167$
Doubly reinforced section required.
Compression steel $A_{s2} = \frac{(K - 0.167)f_{ck}bd^2}{f_{yd}(d - d')}$.
Tension steel $A_{s1} = \frac{0.167 f_{ck}bd^2}{f_{yd}(0.82d)} + A_{s2}$.
Compression steel $A_{s2} = \frac{(K - 0.167)f_{ck}bd^2}{f_{yd}(d - d')}$.
Tension steel $A_{s1} = \frac{0.167 f_{ck}bd^2}{f_{yd}(0.82d)} + A_{s2}$.
3
Minimum & Maximum Reinforcement Limits
Cl. 9.2.1.1
Enforce brittle failure prevention limits for tension reinforcement:
$$A_{s,min} = 0.26 \frac{f_{ctm}}{f_{yk}} b_t d \ge 0.0013 b_t d, \quad A_{s,max} = 0.04 A_c$$
Symbol Notation Key
$A_{s,min}$= minimum tension steel area for ductility (mm²)
$f_{ctm}$= mean tensile strength of concrete ($0.30 f_{ck}^{2/3}$, MPa)
$f_{yk}$= characteristic steel yield strength (MPa)
$b_t$= mean width of tension zone (mm)
$d$= effective depth (mm)
$A_{s,max}$= maximum allowable steel area ($0.04 A_c$, mm²)
$A_c$= total concrete area ($b \cdot h$, mm²)
4
ULS Shear Verification & Strut Angle ($\theta$)
Cl. 6.2.2 & Cl. 6.2.3
Calculate concrete shear capacity $V_{Rd,c}$ without shear links:
$$V_{Rd,c} = \left[ C_{Rd,c} k (100 \rho_l f_{ck})^{1/3} + k_1 \sigma_{cp} \right] b_w d \ge v_{min} b_w d$$
Symbol Notation Key
$V_{Rd,c}$= concrete shear resistance without links (kN)
$C_{Rd,c}$= concrete shear factor ($0.18 / \gamma_c = 0.12$)
$k$= scale size effect factor ($1 + \sqrt{200/d} \le 2.0$)
$\rho_l$= longitudinal tension steel ratio ($A_{sl}/(b_w d) \le 0.02$)
$b_w$= web width of section (mm)
$v_{min}$= minimum shear strength factor ($0.035 k^{3/2} f_{ck}^{1/2}$)
✓ Path A: $V_{Ed} \le V_{Rd,c}$
Provide minimum nominal shear links:
$\frac{A_{sw}}{s} = \frac{\rho_{w,min} \cdot b_w \cdot \sin\alpha}{1}$.
$\frac{A_{sw}}{s} = \frac{\rho_{w,min} \cdot b_w \cdot \sin\alpha}{1}$.
⚠ Path B: $V_{Ed} > V_{Rd,c}$
Designed shear links required.
Variable strut inclination $21.8^\circ \le \theta \le 45^\circ$.
$\frac{A_{sw}}{s} = \frac{V_{Ed}}{z \cdot f_{ywd} \cdot \cot\theta}$.
Variable strut inclination $21.8^\circ \le \theta \le 45^\circ$.
$\frac{A_{sw}}{s} = \frac{V_{Ed}}{z \cdot f_{ywd} \cdot \cot\theta}$.
5
SLS Deflection Check (Span-to-Depth Ratio)
Cl. 7.4.2
Verify limiting span-to-effective-depth ratio $(l/d)_{lim}$:
$$\left(\frac{l}{d}\right)_{lim} = K_{def} \left[ 11 + 1.5\sqrt{f_{ck}} \frac{\rho_0}{\rho} + 3.2\sqrt{f_{ck}} \left( \frac{\rho_0}{\rho} - 1 \right)^{3/2} \right] \cdot F_1 F_2 F_3$$
Symbol Notation Key
$(l/d)_{lim}$= limiting allowable span-to-effective-depth ratio
$K_{def}$= structural system factor (1.0 simple, 1.3 continuous, 0.4 cantilever)
$\rho_0$= reference reinforcement ratio ($\sqrt{f_{ck}} \times 10^{-3}$)
$\rho$= required tension reinforcement ratio ($A_{s,req} / (b d)$)
$F_1, F_2, F_3$= flange, long span ($l > 7$m), and stress ratio correction factors
RC Beam Calculation Sandbox
Live EC2 EngineBeam Width $b$300 mm
Beam Depth $h$500 mm
Design Moment $M_{Ed}$180 kNm
Design Shear Force $V_{Ed}$120 kN
CAD Cross-Section & Reinforcement
Effective Depth $d$
440 mm
Flexural $K$ Factor
0.104
Required Steel $A_{s,req}$
1020 mm²
Concrete Shear $V_{Rd,c}$
68.4 kN
Status
✓ PASS (Singly Reinforced)
Eurocode 2 Column Design Workflow
1
Effective Height ($l_0$) & Slenderness Ratio ($\lambda$)
Cl. 5.8.3.2
Calculate effective column height $l_0$ considering end restraint factors $k_1, k_2$:
$$l_0 = 0.5 \cdot L \cdot \sqrt{\left(1 + \frac{k_1}{0.45 + k_1}\right)\left(1 + \frac{k_2}{0.45 + k_2}\right)}$$
Symbol Notation Key
$l_0$= effective column height (mm)
$L$= clear unbraced column height (mm)
$k_1, k_2$= relative end joint flexibility coefficients ($0.1$ fixed, $1.0$ pinned)
Radius of gyration $i$ for rectangular column section $b \times h$:
$$i = \sqrt{\frac{I}{A}} = \frac{h}{\sqrt{12}} \approx 0.289 h, \quad \lambda = \frac{l_0}{i}$$
Symbol Notation Key
$i$= radius of gyration of concrete section (mm)
$I$= second moment of area ($b h^3 / 12$, mm⁴)
$A$= cross-sectional area ($b \cdot h$, mm²)
$h$= section depth in bending direction (mm)
$\lambda$= slenderness ratio (dimensionless)
2
Limiting Slenderness ($\lambda_{lim}$) & 2nd Order Curvature
Cl. 5.8.3.1
Determine limiting slenderness criterion $\lambda_{lim}$:
$$\lambda_{lim} = \frac{20 \cdot A \cdot B \cdot C}{\sqrt{n}}, \quad \text{where } n = \frac{N_{Ed}}{A_c \cdot f_{cd}}$$
Symbol Notation Key
$\lambda_{lim}$= limiting slenderness threshold
$A$= creep factor ($1 / (1 + 0.2 \varphi_{ef}) \approx 0.7$)
$B$= reinforcement factor ($\sqrt{1 + 2\omega} \approx 1.1$)
$C$= moment ratio factor ($1.7 - r_m \approx 0.7$)
$n$= relative axial force ratio
$N_{Ed}$= design axial compression force (kN)
$A_c$= concrete cross-section area (mm²)
✓ Path A: $\lambda \le \lambda_{lim}$
Short Column.
Second-order flexural effects can be safely ignored ($M_{2} = 0$).
Second-order flexural effects can be safely ignored ($M_{2} = 0$).
⚠ Path B: $\lambda > \lambda_{lim}$
Slender Column.
Must calculate 2nd order nominal curvature moment:
$M_2 = N_{Ed} \cdot e_2 = N_{Ed} \cdot \left(\frac{1}{r} \frac{l_0^2}{c}\right)$.
Must calculate 2nd order nominal curvature moment:
$M_2 = N_{Ed} \cdot e_2 = N_{Ed} \cdot \left(\frac{1}{r} \frac{l_0^2}{c}\right)$.
3
Design Bending Moment ($M_{Ed}$) & Imperfections ($e_i$)
Cl. 5.2
Include geometric execution imperfection eccentricity $e_i$ and minimum eccentricity $e_0$:
$$e_i = \theta_i \frac{l_0}{2} = \frac{l_0}{400}, \quad e_0 = \max\left(20\text{ mm}, \frac{h}{30}\right)$$
Symbol Notation Key
$e_i$= geometric imperfection eccentricity (mm)
$\theta_i$= inclination angle deviation ($1/200$)
$e_0$= minimum design eccentricity ($\max(20\text{mm}, h/30)$)
$$M_{Ed} = \max\left( N_{Ed} \cdot e_0, \quad M_{0Ed} + N_{Ed}\cdot e_i + M_2 \right)$$
Symbol Notation Key
$M_{Ed}$= total design ULS moment including 1st & 2nd order effects (kNm)
$M_{0Ed}$= 1st order ULS moment from structural analysis (kNm)
$M_2$= 2nd order nominal curvature moment (kNm)
4
$N-M$ Interaction Diagram Capacity Envelope
Cl. 6.1
Verify that applied axial load $N_{Ed}$ and total moment $M_{Ed}$ fall inside the concrete/steel interaction capacity envelope ($N_{Rd}-M_{Rd}$).
RC Column Calculation Sandbox
Live EC2 EngineColumn Width $b$350 mm
Column Depth $h$350 mm
Axial Load $N_{Ed}$1200 kN
First Order Moment $M_{Ed}$85 kNm
Column Cross-Section & Reinforcement
Slenderness $\lambda$
35.6
Limiting $\lambda_{lim}$
34.2
Main Steel $A_{s,req}$
1225 mm² (4H20)
Status
✓ PASS (Short Column)
Eurocode 2 Slab & Punching Shear Workflow
1
Punching Shear at Column Perimeter ($u_0$) & Control ($u_1$)
Cl. 6.4.2
Calculate column face perimeter $u_0$ and basic control perimeter $u_1$ located at distance $2.0d$ from column face:
$$u_0 = 2(c_x + c_y), \quad u_1 = 2(c_x + c_y) + 2\pi(2d) = u_0 + 4\pi d$$
Symbol Notation Key
$u_0$= loaded column perimeter length (mm)
$u_1$= basic control perimeter at distance $2.0d$ (mm)
$c_x, c_y$= rectangular column cross-section dimensions (mm)
$d$= mean effective slab depth, $d = (d_x + d_y)/2$ (mm)
2
Maximum Shear Stress at Column Perimeter ($v_{Ed,0}$)
Cl. 6.4.5
Check crushing limit of concrete at column face:
$$v_{Ed,0} = \frac{\beta \cdot V_{Ed}}{u_0 \cdot d} \le v_{Rd,max} = 0.5 \nu f_{cd}$$
Symbol Notation Key
$v_{Ed,0}$= maximum punching shear stress at column face ($N/mm^2$)
$\beta$= moment transfer factor ($\beta = 1.15$ internal, $1.4$ edge column)
$V_{Ed}$= design ultimate column shear force (kN)
$v_{Rd,max}$= maximum concrete crushing shear stress limit ($N/mm^2$)
$\nu$= concrete shear strength reduction factor, $\nu = 0.6(1 - f_{ck}/250)$
$f_{cd}$= design compressive concrete strength ($N/mm^2$)
3
Applied Shear Stress ($v_{Ed,1}$) vs Concrete Capacity ($v_{Rd,c}$)
Cl. 6.4.3 & Cl. 6.4.4
Calculate applied shear stress $v_{Ed,1}$ at control perimeter $u_1$:
$$v_{Ed,1} = \frac{\beta \cdot V_{Ed}}{u_1 \cdot d}$$
Symbol Notation Key
$v_{Ed,1}$= punching shear stress at basic control perimeter $u_1$ ($N/mm^2$)
$v_{Rd,c}$= design concrete punching shear resistance without links ($N/mm^2$)
$v_{Rd,cs}$= punching shear resistance with shear links ($N/mm^2$)
$A_{sw}$= cross-sectional area of one perimeter of shear links ($mm^2$)
$s_r$= radial spacing of shear link perimeters (mm)
✓ Path A: $v_{Ed,1} \le v_{Rd,c}$
No punching shear reinforcement required.
Concrete slab capacity alone is adequate.
Concrete slab capacity alone is adequate.
⚠ Path B: $v_{Ed,1} > v_{Rd,c}$
Provide vertical shear stud rails or link legs.
$v_{Rd,cs} = 0.75 v_{Rd,c} + 1.5 \left(\frac{d}{s_r}\right) A_{sw} f_{ywd,ef} \frac{1}{u_1 d} \ge v_{Ed,1}$.
$v_{Rd,cs} = 0.75 v_{Rd,c} + 1.5 \left(\frac{d}{s_r}\right) A_{sw} f_{ywd,ef} \frac{1}{u_1 d} \ge v_{Ed,1}$.
RC Slab Calculation Sandbox
Live EC2 EngineSlab Depth $h$250 mm
Punching Shear Load $V_{Ed}$380 kN
Slab Plan View: Column & 2.0d Control Perimeter
Control Perimeter $u_1$
4366 mm
Shear Stress $v_{Ed,1}$
0.46 N/mm²
Concrete Shear $v_{Rd,c}$
0.65 N/mm²
Status
✓ PASS (No Links Needed)
Eurocode 2 & 7 Pad Footing Workflow
1
Soil Bearing Pressure & Eccentricity ($e$) Check
EN 1997-1 & EC2
Calculate eccentric loading $e = M_{Ed}/N_{Ed}$ and check maximum soil pressure $q_{max}$:
$$e = \frac{M_{Ed}}{N_{Ed}}, \quad q_{max,min} = \frac{N_{Ed}}{B \cdot L} \left( 1 \pm \frac{6e}{B} \right) \le q_{allow}$$
Symbol Notation Key
$e$= load eccentricity at base, $e = M_{Ed}/N_{Ed}$ (m)
$N_{Ed}$= ultimate axial column load (kN)
$M_{Ed}$= ultimate column moment at footing top (kNm)
$q_{max}, q_{min}$= maximum & minimum soil contact pressures ($kPa$)
$B, L$= footing width & length planar dimensions (m)
$q_{allow}$= allowable net soil bearing capacity ($kPa$)
✓ Path A: $e \le B/6$
No soil uplift across footing base.
Trapezoidal or uniform pressure distribution.
Trapezoidal or uniform pressure distribution.
⚠ Path B: $e > B/6$
Partial soil uplift occurs.
Triangular bearing pressure: $q_{max} = \frac{2 N_{Ed}}{3 L (B/2 - e)} \le q_{allow}$.
Triangular bearing pressure: $q_{max} = \frac{2 N_{Ed}}{3 L (B/2 - e)} \le q_{allow}$.
2
Critical Bending Moment at Column Face
EC2 Cl. 6.1
Calculate maximum cantilever bending moment $M_{Ed,footing}$ at column face:
$$M_{Ed,footing} = q_{avg} \cdot L \cdot \frac{(B - a)^2}{8}, \quad A_{s,req} = \frac{M_{Ed,footing}}{f_{yd} \cdot z}$$
Symbol Notation Key
$M_{Ed,footing}$= design bending moment at column face (kNm)
$q_{avg}$= average soil pressure over cantilever projection ($kPa$)
$a$= column dimension parallel to width $B$ (m)
$A_{s,req}$= required bottom flexural reinforcement ($mm^2$)
$f_{yd}$= design yield strength of steel reinforcement ($N/mm^2$)
$z$= flexural internal lever arm ($z \approx 0.95d$) (mm)
Pad Footing Calculation Sandbox
Live EC2 EngineFooting Width $B$2.2 m
Column Load $N_{Ed}$850 kN
Pad Footing Elevation & Soil Reaction
Max Soil Pressure $q_{max}$
175.6 kPa
Allowable Soil Limit
200.0 kPa
Bearing Status
✓ PASS (Soil Pressure Safe)
Eurocode 2 & 7 Cantilever Retaining Wall Workflow
1
Rankine Active Pressure ($K_a$) & Earth Thrust ($P_{ae}$)
EN 1997-1 Cl. 9
Calculate Rankine active earth pressure coefficient $K_a$ for soil friction angle $\phi'$:
$$K_a = \tan^2\left(45^\circ - \frac{\phi'}{2}\right) = \frac{1 - \sin\phi'}{1 + \sin\phi'}$$
Symbol Notation Key
$K_a$= Rankine active earth pressure coefficient
$\phi'$= effective angle of internal friction of backfill soil (deg)
Horizontal active earth thrust force $P_{ae}$ acting at height $H/3$ above base:
$$P_{ae} = \frac{1}{2} K_a \gamma H^2, \quad M_{o} = P_{ae} \cdot \left(\frac{H}{3}\right)$$
Symbol Notation Key
$P_{ae}$= total horizontal active earth thrust per unit length ($kN/m$)
$\gamma$= unit weight of retained backfill soil ($kN/m^3$)
$H$= total vertical wall height from base to top (m)
$M_o$= overturning moment about toe of wall base ($kNm/m$)
2
Overturning & Sliding Stability Verification
EN 1997-1
Verify factors of safety for wall overturning and base sliding:
$$FOS_{over} = \frac{M_r}{M_o} \ge 2.0, \quad FOS_{slide} = \frac{(\sum W) \cdot \tan\delta}{P_{ae}} \ge 1.5$$
Symbol Notation Key
$FOS_{over}$= factor of safety against overturning about toe ($\ge 2.0$)
$FOS_{slide}$= factor of safety against base sliding ($\ge 1.5$)
$M_r$= stabilizing restoring moment from self-weight ($kNm/m$)
$M_o$= lateral earth thrust overturning moment ($kNm/m$)
$\sum W$= total vertical dead weight of stem, base, & backfill ($kN/m$)
$\delta$= base soil friction angle ($\delta \approx 2\phi'/3$) (deg)
✓ Path A: $FOS_{over} \ge 2.0$ & $FOS_{slide} \ge 1.5$
Global wall stability verified.
Proceed to stem base bending moment $M_{Ed,stem}$ check.
Proceed to stem base bending moment $M_{Ed,stem}$ check.
⚠ Path B: Stability Failed
Increase base slab width $B$ or add a base shear key under stem.
3
Stem Base Flexural Bending & Reinforcement
EC2 Cl. 6.1
Calculate design moment $M_{Ed,stem}$ and main vertical rear face reinforcement $A_{s,stem}$:
$$M_{Ed,stem} = 1.35 \cdot \left[ \frac{1}{6} K_a \gamma h_{stem}^3 \right], \quad A_{s,req} = \frac{M_{Ed,stem}}{f_{yd} \cdot z}$$
Symbol Notation Key
$M_{Ed,stem}$= design flexural moment at junction of stem and base ($kNm/m$)
$h_{stem}$= clear vertical height of stem wall (m)
$A_{s,req}$= required main tension vertical reinforcement in rear face ($mm^2/m$)
$f_{yd}$= design yield strength of steel reinforcement ($N/mm^2$)
$z$= internal flexural lever arm ($z \approx 0.95 d_{stem}$) (mm)
4
Heel & Toe Base Slab Bending Design
EC2 Cl. 6.1
Calculate cantilever bending moment on heel slab due to soil load:
$$M_{Ed,heel} = 1.35 \cdot \frac{\gamma_{soil} h_{stem} b_{heel}^2}{2}, \quad A_{s,heel} = \frac{M_{Ed,heel}}{f_{yd} \cdot z}$$
Symbol Notation Key
$M_{Ed,heel}$= ultimate design moment at top face of heel slab ($kNm/m$)
$b_{heel}$= rear cantilever projection width of heel slab (m)
$A_{s,heel}$= required top tension reinforcement in heel slab ($mm^2/m$)
5
Stem Base Ultimate Shear Verification
EC2 Cl. 6.2
Verify concrete shear capacity $v_{Rd,c}$ at base of stem without links:
$$V_{Ed,stem} = 1.35 \cdot \left[ \frac{1}{2} K_a \gamma h_{stem}^2 \right], \quad v_{Ed,stem} = \frac{V_{Ed,stem}}{b \cdot d_{stem}} \le v_{Rd,c}$$
Symbol Notation Key
$V_{Ed,stem}$= ultimate shear force at stem base ($kN/m$)
$v_{Ed,stem}$= design shear stress ($N/mm^2$)
$d_{stem}$= effective depth of vertical stem wall at base (mm)
$v_{Rd,c}$= concrete design shear resistance without shear links ($N/mm^2$)
Retaining Wall Calculation Sandbox
Live EC2 EngineWall Height $H$4.5 m
Soil Friction Angle $\phi'$30°
Cantilever Wall Section & Earth Pressure
Active Pressure Coeff $K_a$
0.333
Stem Base Moment $M_{Ed}$
91.1 kNm/m
Overturning FOS
2.45
Stability Status
✓ PASS (Stable)