Columns are the most noticeable feature of a structure and are often used to support gravity loads transmitted from the floors or roofs of buildings. Strength, economy, adaptability, good fire resistance, and robustness are all advantages of in-situ reinforced concrete columns. During the design of columns, sound engineering judgement is often needed to balance their location, size, and shape with horizontal element spans and economy.

Circular columns are often designed for the ultimate axial load, N_{Ed}, ultimate design moment, M_{Ed}, and ultimate shear force V_{Ed}. For internal columns, moments may generally be assumed to be nominal when compared with external columns.

Due to their uniform strength in all directions, circular concrete columns are frequently employed in the design of pilings and bridge piers, and are very convenient for seismically active areas. Furthermore, it is much easier to confine the concrete using special reinforcement in circular columns than in other shapes.

**Theoretical Background**

When the neutral axis remains within a section, the basic equations for a sectionâ€™s equilibrium under combined bending and axial load are as follows:

N_{Rd} = f_{av}bx + âˆ‘f_{s}A_{s} â€”â€”- (1)

M_{Rd} = f_{av}bx(h/2 â€“ Î²x) + âˆ‘f_{s}A_{s}(h/2 â€“ d_{i}) â€”â€”- (2)

In Equation (2), moments have been taken aboutÂ the concrete sectionâ€™s centroid. The summation signs represent a summation of all layers of reinforcement in the section. Tensile stresses must be considered negative when summing them up. *d _{i}* is the distance between the sectionâ€™s compressive face and the i

^{th }layer of reinforcement.

We can substitute 0.459f_{ck} for f_{av} and 0.416 for Î²Â by assuming that the partial safety factors for the steel and concrete are 1.15 and 1.5, respectively, and that Î±_{cc} is 0.85. For cases in which the neutral axis remains within the section, the resulting equations are rigorous. More complex expressions must be resolved when the entire section is inÂ compression for the following situations;

(1) the portion of the parabolic curve cut off by the bottom of the section and

(2) the reduction in the ultimate strain at the compressive face

Additionally, for situations when the concrete strength is greater than 50 N/mm^{2}, more complicated equations are required. Because of how complex the resulting equations are, it is inappropriate to present them here. Using design charts is an easier method.

Some design charts for circular columns are given below. Since six reinforcing bars are the minimum that can be employed in a circular section, this is the assumption made while drawing the charts. It is discovered that no specific arrangement of reinforcement in relation to the axis of bending will always result in minimal strength. As a result, the charts are produced to provide a lower bound envelope to the interaction diagrams for different bar arrangements.

**Design Example of RC Circular Columns **

Verify the resistance of 6H25 bars to withstand the loads in a column of a high-rise building in accordance with EN1992-1-1 incorporating Corrigendum January 2008 and the UK national annex. The design information is as follows;

Height of column = 5m

f_{ck} = C25/30

f_{yk} = 500 MPa

Diameter of column = 400 mm

Design axial load;Â N_{Ed} = 1500.0 kN

Moment about y-axis at top; Mtopy = 66.0Â kNm

Moment about y-axis at bottom;Â Mbtmy = 32.0Â kNm

Moment about z-axis at top;Â Mtopz = 25.0Â kNm

Moment about z-axis at bottom; M_{btmz} = 5.5Â kNm

**Column geometry**

Overall diameter;Â h = 400 mm

Clear height between restraints about y-axis;Â ly = 5000Â mm

Clear height between restraints about z-axis;Â l_{z} = 5000Â mm

Stability in the z direction;Â Braced

Stability in the y direction;Â ** **Braced

**Concrete details**

Concrete strength class; C25/30

Partial safety factor for concrete (2.4.2.4(1));Â Î³_{C} = 1.50

Coefficient Î±_{cc} (3.1.6(1));Â Î±_{cc} = 0.85

Maximum aggregate size; d_{g} = 20 mm

**Reinforcement details**

Nominal cover to links;Â c_{nom} = 35 mm

Longitudinal bar diameter;Â Ï†= 25 mm

Link diameter;Â Â Ï†_{v} = 8 mm

Total number of longitudinal bars;Â N = 6

Area of longitudinal reinforcement;Â A_{s} = N Ã— (Ï€ Ã— Ï†^{2} / 4) = 2945 mm^{2}

Characteristic yield strength; f_{yk} = 500 N/mm^{2}

Partial safety factor for reinft (2.4.2.4(1)); Î³_{S} = 1.15

Modulus of elasticity of reinft (3.2.7(4)); E_{s} = 200 kN/mm^{2}

**Fire resistance details**

Fire resistance period;Â Â R = 60 min

Exposure to fire;Â Exposed on more than one side

Ratio of fire design axial load to design resistance;Â Â m_{fi} = 0.70

**Axial load and bending moments from frame analysis**

Design axial load;Â N_{Ed} = 1500.0 kN

Moment about y axis at top; M_{topy} = 66.0Â kNm

Moment about y axis at bottom;Â M_{btmy} = 32.0Â kNm

Moment about z axis at top;Â M_{topz} = 25.0Â kNm

Moment about z axis at bottom; M_{btmz} = 5.5Â kNm

Beam/slab concrete strength class;Â **C25/30**

**Beams/slabs providing rotational restraint about y axis**

Depth on side A; h_{A1y} = 500 mm

Width on side A; b_{A1y} = 300 mm

Length on side A; l_{A1y} = 4500 mm

Depth on side B;Â h_{B1y} = 500 mm

Width on side B; b_{B1y} = 300 mm

Length on side B; l_{B1y} = 6000 mm

**Beams providing rotational restraint about z axis**

Depth on side A; h_{A1z} = 500 mm

Width on side A;Â b_{A1z} = 300 mm

Length on side A;Â l_{A1z} = 3500 mm

Depth on side B;Â Â h_{B1z} = 500 mm

Width on side B; b_{B1z} = 300 mm

Length on side B; l_{B1z} = 3500 mm

Relative flexibility end 2 for buckling about y axis;Â k_{2y} = **1000.000**

Relative flexibility end 2 for buckling about z axis; k_{2z} = **1000.000**

**Calculated column properties**

Area of concrete;Â A_{c} = Ï€ Ã— h^{2} / 4 = 125664 mm^{2}

Column characteristic comp. cylinder strength;Â f_{ck} = 25 N/mm^{2}

Column design comp. strength (3.1.6(1)); f_{cd} = a_{cc} Ã— f_{ck} / Î³_{C} = 14.2 N/mm^{2}

Column mean value of cyl. strength (Table 3.1); f_{cm} = f_{ck} + 8 MPa = 33.0 N/mm^{2}

Column secant modulus of elasticity (Table 3.1);Â E_{cm} = 22000 MPa Ã— (f_{cm} / 10 MPa)^{0.3} = 31.5 kN/mm^{2}

Beam/slab characteristic comp. cylinder strength;Â f_{ck_b} = 25 N/mm^{2}

Beam/slab mean value of cyl. strength (3.1.6(1)); f_{cm_b} = f_{ck_b} + 8 MPa = 33.0 N/mm^{2}

Beam/slab secant mod. of elasticity (Table 3.1);Â E_{cm_b} = 22000 MPa Ã— (f_{cm_b} / 10 MPa)^{0.3} = 31.5 kN/mm^{2}

**Rectangular stress block factors**

Depth factor (3.1.7(3));Â Î»_{sb} = 0.8

Stress factor (3.1.7(3)); Î· = 0.9

**Strain limits**

Compression strain limit (Table 3.1); Îµ_{cu3} = 0.00350

Pure compression strain limit (Table 3.1);Â Îµ_{c3} = 0.00175

Design yield strength (3.2.7(2)); f_{yd} = f_{yk} / Î³_{S} = 434.8 N/mm^{2}

**Check nominal cover for fire and bond requirements**

Min. cover reqd for bond (to links) (4.4.1.2(3)); c_{min,b} = max(Ï†_{v}, Ï† â€“ Ï†_{v}) = 17 mm

Min axis distance for fire (EN1992-1-2 T 5.2a);Â a_{fi} = 40 mm

Allowance for deviations from min cover (4.4.1.3);Â Dc_{dev} = 10 mm

Min allowable nominal cover;Â c_{nom_min} = max(a_{fi} â€“ Ï†/2 â€“ Ï†_{v}, c_{min,b} + Dc_{dev}) = 27.0 mm

*PASS *â€“ the nominal cover is greater than the minimum required

**Effective depth and inertia of bars for bending about y axis**

For the purposes of determining the bending capacity and interaction diagrams in this calculation, bending about the y axis is taken to be when there are two furthest equidistant bars on each side of the column centreline. Bending about the z axis is taken to be when there is one furthest bar on each side of the column centreline.

Area per bar;Â A_{bar} = Ï€ Ã— Ï†^{2} / 4 = 491 mm^{2}

Radial dist from column centre to longitudinal bar;Â Â Â r_{l} = h/2 â€“ c_{nom} â€“ Ï†_{v} â€“ Ï†/2 = 144.5 mm

Subtended angle between adjacent bars; Î± = (360 deg) / N = 60.0 deg

Layer 1;Â d_{y1} = h/2 + r_{l} Ã— cos(Î±/2) = 325.1 mm

2nd moment of area of reinft about y axis; I_{y1} = 2 Ã— A_{bar} Ã— (d_{y1} â€“ h/2)^{2} = 1537 cm^{4}

Layer 2;Â d_{y2} = h / 2 + r_{l} Ã— cos[(2 â€“ 1) Ã— a + a/2] = 200.0 mm

2nd moment of area of reinft about y axis;Â I_{y2} = 2 Ã— A_{bar} Ã— (d_{y2} â€“ h/2)^{2} = 0 cm^{4}

Layer 3;Â d_{y3} = h / 2 + r_{l} Ã— cos[(3 â€“ 1) Ã— a + a/2] = 74.9 mm

2nd moment of area of reinft about y axis;Â I_{y3} = 2 Ã— A_{bar} Ã— (d_{y3} â€“ h/2)^{2} = 1537 cm^{4}

Total 2nd moment of area of reinft about y axis; I_{sy} = 3075 cm^{4}

Radius of gyration of reinft about y axis;Â i_{sy} = âˆš(I_{sy} / A_{s}) = 102 mm

Effective depth about y axis (5.8.8.3(2));Â d_{y} = h / 2 + i_{sy} = 302 mm

**Effective depth of bars for bending about z axis**

Layer 1 (tension face);Â d_{z1} = h / 2 + r_{l} = 344.5 mm

2nd moment of area of reinft about z axis; I_{z1} = A_{bar} Ã— (d_{z1} â€“ h / 2)^{2} = 1025 cm^{4}

Layer 2;Â d_{z2} = h / 2 + r_{l} Ã— cos[(2 â€“ 1) Ã— a] = 272.3 mm

2nd moment of area of reinft about z axis;Â I_{z2} = 2 Ã— A_{bar} Ã— (d_{z2} â€“ h/2)^{2} = 512 cm^{4}

Layer 3; d_{z3} = h / 2 + r_{l} Ã— cos[(3 â€“ 1) Ã— a] = 127.8 mm

2nd moment of area of reinft about z axis; I_{z3} = 2 Ã— A_{bar} Ã— (d_{z3} â€“ h/2)^{2} = 512 cm^{4}

Layer 4; d_{z4} = h / 2 + r_{l} Ã— cos[(4 â€“ 1) Ã— a] = 55.5 mm

2nd moment of area of reinft about z axis;Â I_{z4} = 1 Ã— A_{bar} Ã— (d_{z4} â€“ h/2)^{2} = 1025 cm^{4}

Total 2nd moment of area of reinft about z axis;Â I_{sz} = 3075 cm^{4}

Radius of gyration of reinforcement about z axis; i_{sz} = âˆš(I_{sz} / A_{s}) = 102 mm

Effective depth about z axis (5.8.8.3(2)); d_{z} = b / 2 + i_{sz} = 302 mm

**Relative flexibility at end 1 for buckling about y axis**

Second moment of area of column; I_{y} = Ï€ Ã— h^{4} / 64 = 125664 cm^{4}

Second moment of area of beam on side A;Â Â I_{A1y} = b_{A1y} Ã— h_{A1y}^{3} / 12 = 312500 cm^{4}

Second moment of area of beam on side B;Â I_{B1y} = b_{B1y} Ã— h_{B1y}^{3} / 12 = 312500 cm^{4}

Relative flexibility (PD6687 cl. 2.10);Â k_{1y} = max(0.1, (E_{cm} Ã— I_{y} / l_{y}) / [2 Ã— E_{cm_b} Ã— (I_{A1y}/l_{A1y} + I_{B1y}/l_{B1y})]) = 0.103

Relative flexibility end 2 for buckling about y axis;Â k_{2y} = 1000.000

**Relative flexibility at end 1 for buckling about z axis**

Second moment of area of column;Â I_{z} = Ï€ Ã— h^{4} / 64 = 125664 cm^{4}

Second moment of area of beam on side A;Â I_{A1z} = b_{A1z} Ã— h_{A1z}^{3} / 12 = 312500 cm^{4}

Second moment of area of beam on side B;Â I_{B1z} = b_{B1z} Ã— h_{B1z}^{3} / 12 = 312500 cm^{4}

Relative flexibility (PD6687 cl. 2.10);Â k_{1z} = max(0.1, (E_{cm} Ã— I_{z} / l_{z}) / [2 Ã— E_{cm_b} Ã— (I_{A1z}/l_{A1z} + I_{B1z}/l_{B1z})]) = 0.100

Relative flexibility end 2 for buckling about z axis; k_{2z} = 1000.000

Calculated effective length (cl. 5.8.3.2)

Eff. length about y axis (braced) (5.8.3.2(3));Â Â Â Â Â Â Â Â Â Â Â Â Â

l_{0y} = 0.5 Ã— l_{y} Ã— [(1 + k_{1y}/(0.45+k_{1y})) Ã— (1 + k_{2y}/(0.45+k_{2y}))]^{0.5} = **3851** mm

Eff. length about z axis (braced) (5.8.3.2(3));Â Â Â Â Â Â Â Â Â Â Â Â Â

l_{0z} = 0.5 Ã— l_{z} Ã— [(1 + k_{1z}/(0.45+k_{1z})) Ã— (1 + k_{2z}/(0.45+k_{2z}))]^{0.5} = **3843** mm

**Column slenderness about y axis**

Radius of gyration;Â i_{y} = h / 4 = 10.0 cm

Slenderness ratio (5.8.3.2(1));Â Â l_{y} = l_{0y} / i_{y} = 38.5

**Column slenderness about z axis**

Radius of gyration;Â i_{z} = h / 4 = 10.0 cm

Slenderness ratio (5.8.3.2(1)); l_{z} = l_{0z} / i_{z} = 38.4

**Design bending moments**

Frame analysis moments about y axis combined with moments due to imperfections (cl. 5.2 & 6.1(4))

Ecc. due to geometric imperfections (y axis); e_{iy} = l_{0y} /400 = 9.6 mm

Min end moment about y axis;Â M_{01y} = min(abs(M_{topy}), abs(M_{btmy})) + e_{iy} Ã— N_{Ed} = 46.4 kNm

Max end moment about y axis;Â M_{02y} = max(abs(M_{topy}), abs(M_{btmy})) + e_{iy} Ã— N_{Ed} = 80.4 kNm

**Slenderness limit for buckling about y axis (cl. 5.8.3.1)**

Factor A;Â A = 0.7

Mechanical reinforcement ratio;Â w = A_{s} Ã— f_{yd} / (A_{c} Ã— f_{cd}) = 0.719

Factor B;Â B = âˆš(1 + 2 Ã— w) = 1.562

Moment ratio; r_{my} = M_{01y} / M_{02y} = 0.577

Factor C;Â C_{y} = 1.7 â€“ r_{my} = 1.123

Relative normal force;Â n = N_{Ed} / (A_{c} Ã— f_{cd}) = 0.843

Slenderness limit; l_{limy} = 20 Ã— A Ã— B Ã— C_{y} / âˆš(n) = 26.7

l_{y} > l_{limy} â€“ Therefore, second order effects must be considered

Frame analysis moments about z axis combined with moments due to imperfections (cl. 5.2 & 6.1(4))

Ecc. due to geometric imperfections (z axis); e_{iz} = l_{0z} /400 = 9.6 mm

Min end moment about z axis;

M_{01z} = min(abs(M_{topz}), abs(M_{btmz})) + e_{iz} Ã— N_{Ed} = 19.9 kNm

Max end moment about z axis;Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

M_{02z} = max(abs(M_{topz}), abs(M_{btmz})) + e_{iz} Ã— N_{Ed} = 39.4 kNm

**Slenderness limit for buckling about y axis (cl. 5.8.3.1)**

Factor A;Â A = 0.7

Mechanical reinforcement ratio;Â w = A_{s} Ã— f_{yd} / (A_{c} Ã— f_{cd}) = 0.719

Factor B;Â Â B = âˆš(1 + 2 Ã— w) = 1.562

Moment ratio; r_{mz} = M_{01z} / M_{02z} = 0.505

Factor C;Â C_{z} = 1.7 â€“ r_{mz} = 1.195

Relative normal force;Â n = N_{Ed} / (A_{c} Ã— f_{cd}) = 0.843

Slenderness limit;Â l_{limz} = 20 Ã— A Ã— B Ã— C_{z} / âˆš(n) = 28.5

l_{z} > l_{limz} â€“ Second order effects must be considered

Local second order bending moment about y axis (cl. 5.8.8.2 & 5.8.8.3)

Relative humidity of ambient environment; RH = 50 %

Column perimeter in contact with atmosphere;Â u = 1257 mm

Age of concrete at loading;Â Â t_{0} = 28 day

Parameter n_{u};Â n_{u} = 1 + w = 1.719

Approx value of n at max moment of resistance; n_{bal} = 0.4

Axial load correction factor;Â K_{r} = min(1.0 , (n_{u} â€“ n) / (n_{u} â€“ n_{bal})) = 0.665

Reinforcement design strain; e_{yd} = f_{yd} / E_{s} = 0.00217

Basic curvature; curve_{basic_y} = e_{yd} / (0.45 Ã— d_{y}) = 0.0000160 mm^{-1}

Notional size of column;Â h_{0} = 2 Ã— A_{c} / u = 200 mm

Relative humidity factor (Annex B.1(1));Â Ï†_{RH} = 1 + [(1 â€“ RH / 100%) / (0.1 mm^{-1/3} Ã— (h_{0})^{1/3})] = 1.855

Concrete strength factor (Annex B.1(1));Â Î²_{fcm} = 16.8 Ã— (1 MPa)^{1/2} / âˆš(f_{cm}) = 2.925

Concrete age factor (Annex B.1(1)); b_{t0} = 1 / (0.1 + (t_{0} / 1 day)^{0.2}) = 0.488

Notional creep coefficient (Annex B.1(1));Â Ï†_{0} = Ï†_{RH} Ã— Î²_{fcm} Ã— Î²_{t0} = 2.650

Final creep development factor; (at t = âˆž); Î²_{câˆž} = 1.0

Final creep coefficient (Annex B.1(1));Â fÏ†_{âˆž} = Ï†_{0} Ã— Î²_{câˆž} = 2.650

Ratio of SLS to ULS moments; r_{My} = 0.80

Effective creep ratio; Ï†_{efy} = Ï†_{âˆž} Ã— r_{My} = 2.120

Factor b;Â b_{y} = 0.35 + f_{c} /200 â€“ Î»_{y}/150 = 0.218

Creep factor;Â K_{fy} = max(1.0 , 1 + b_{y} Ã— Ï†_{efy}) = 1.463

Modified curvature;Â Â curve_{mod_y} = K_{r} Ã— K_{fy} Ã— curve_{basic_y} = 0.0000155 mm^{-1}

Curvature distribution factor;Â c = 10

Deflection;Â e_{2y} = curve_{mod_y} Ã— l_{0y}^{2} / c = 23.0 mm

Nominal 2^{nd} order moment;M_{2y} = N_{Ed} Ã— e_{2y} = 34.6 kNm

**Design bending moment about y axis (cl. 5.8.8.2 & 6.1(4))**

Equivalent moment from frame analysis;Â Â

M_{0ey} = max(0.6 Ã— M_{02y} + 0.4 Ã— M_{01y}, 0.4 Ã— M_{02y}) = 66.8Â kNm

**Design moment;**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

M_{Edy} = max(M_{02y}, M_{0ey} + M_{2y}, M_{01y} + 0.5Ã—M_{2y}, N_{Ed} Ã— max(h/30, 20 mm))

M_{Edy} = 101.4 kNm

**Local second order bending moment about z axis (cl. 5.8.8.2 & 5.8.8.3)**

Basic curvature;Â curve_{basic_z} = e_{yd} / (0.45 Ã— d_{z}) = 0.0000160 mm^{-1}

Ratio of SLS to ULS moments; r_{Mz} = 0.80

Effective creep ratio (5.8.4(2));Â Ï†_{efz} = Ï†_{âˆž} Ã— r_{Mz} = 2.120

Factor b;Â b_{z} = 0.35 + f_{ck} / 200 MPa â€“ l_{z} / 150 = 0.219

Creep factor; K_{fz} = max(1.0 , 1 + b_{z} Ã— f_{efz}) = 1.464

Modified curvature; curve_{mod_z} = K_{r} Ã— K_{fz} Ã— curve_{basic_z} = 0.0000156 mm^{â€“}

Curvature distribution factor;Â c = 10

Deflection;Â e_{2z} = curve_{mod_z} Ã— l_{0z}^{2} / c = 23.0 mm

Nominal 2^{nd} order moment;Â Â M_{2z} = N_{Ed} Ã— e_{2z} = 34.5 kNm

Design bending moment about z axis (cl. 5.8.8.2 & 6.1(4))

Equivalent moment from frame analysis;Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

M_{0ez} = max(0.6 Ã— M_{02z} + 0.4 Ã— M_{01z}, 0.4 Ã— M_{02z}) = 31.6Â kNm

Design moment;

M_{Edz} = max(M_{02z}, M_{0ez} + M_{2z}, M_{01z} + 0.5Ã—M_{2z}, N_{Ed }Ã— max(b/30, 20 mm))

M_{Edz} =** **66.1 kNm

**Resultant design bending moment for a circular column**

Resultant design moment;Â M_{Ed} = âˆš(M_{Edy}^{2} + M_{Edz}^{2}) = **121.0** kNm

**Moment capacity about y axis with axial load N _{Ed}**

**Moment of resistance of concrete**

By iteration:-

Position of neutral axis; y = 289.8 mm

Depth of stress block; d_{sby} = min(l_{sb} Ã— y , h) = 231.8 mm

Area of concrete in compression; A_{sby} = 75498 mm^{2}

Concrete compression force (3.1.7(3));F_{yc} = h Ã— f_{cd} Ã— A_{sby} = 962.6 kN

Centroid of concrete compression from column cl; y_{sby} = 68.0 mm

Moment of resistance;Â M_{Rdyc} = F_{yc} Ã— y_{sby} = 65.4 kNm

**Moment of resistance of reinforcement**

Strain in layer 1;Â Îµ_{y1} = Îµ_{cu3} Ã— (1 â€“ d_{y1} / y) = -0.00043

Stress in layer 1;Â Ïƒ_{y1} = max(-1Ã—f_{yd}, E_{s} Ã— Îµ_{y1}) = -85.5 N/mm^{2}

Force in layer 1;F_{y1} = 2 Ã— A_{bar} Ã— s_{y1} = -83.9 kNÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

M_{Rdy1} = F_{y1} Ã— (h / 2 â€“ d_{y1}) = 10.5 kNm

Strain in layer 2; Îµ_{y2} = Îµ_{cu3} Ã— (1 â€“ d_{y2} / y) = 0.00108

Stress in layer 2;Â Ïƒ_{y2} = min(f_{yd}, E_{s} Ã— Îµ_{y2}) â€“ h Ã— f_{cd} = 204.1 N/mm^{2}

Force in layer 2; F_{y2} = 2 Ã— A_{bar} Ã— s_{y2} = 200.4 kN

Moment of resistance of layer 2; M_{Rdy2} = F_{y2} Ã— (h/2 â€“ d_{y2}) = 0.0 kNm

Strain in layer 3; Îµ_{y3} = Îµ_{cu3} Ã— (1 â€“ d_{y3} / y) = 0.00260

Stress in layer 3; Ïƒ_{y3} = min(f_{yd}, E_{s} Ã— Îµ_{y3}) â€“ h Ã— f_{cd} = 422.0 N/mm^{2}

Force in layer 3;Â F_{y3} = 2 Ã— A_{bar} Ã— s_{y3} = 414.3 kN

Moment of resistance of layer 3;Â Â M_{Rdy3} = F_{y3} Ã— (h/2 â€“ d_{y3}) = 51.8 kNm

Resultant concrete/steel force; F_{y} = 1493.3 kN

PASS â€“ This is within half of one percent of the applied axial load

**Combined moment of resistance**

Moment of resistance about y axis;Â M_{Rdy} = 127.8 kNm

**Moment capacity about z axis with axial load N _{Ed}**

Moment of resistance of concrete

By iteration:-

Position of neutral axis;Â z = 287.0 mm

Depth of stress block;Â d

_{sbz}= min(l

_{sb}Ã— z , h) = 229.6 mm

Area of concrete in compression;Â A

_{sbz}= 74628 mm

^{2}

Concrete compression force (3.1.7(3));Â F

_{zc}= h Ã— f

_{cd}Ã— A

_{sbz}= 951.5 kN

Centroid of concrete compression from column cl;Â Â Â y

_{sbz}= 69.1 mm

Moment of resistance; M

_{Rdzc}= F

_{zc}Ã— y

_{sbz}= 65.8 kNm

**Moment of resistance of reinforcement**

Strain in layer 1;Â Îµ_{z1} = Îµ_{cu3} Ã— (1 â€“ d_{z1} / z) = -0.00070

Stress in layer 1;Â Ïƒ_{z1} = max(-1Ã—f_{yd}, E_{s} Ã— Îµ_{z1}) = -140.2 N/mm^{2}

Force in layer 1;Â F_{z1} = 1 Ã— A_{bar} Ã— s_{z1} = -68.8 kN

Moment of resistance of layer 1; M_{Rdz1} = F_{z1} Ã— (h / 2 â€“ d_{z1}) = 9.9 kNm

Strain in layer 2;Â Îµ_{z2} = Îµ_{cu3} Ã— (1 â€“ d_{z2} / z) = 0.00018

Stress in layer 2;Â Ïƒ_{z2} = min(f_{yd}, E_{s} Ã— Îµ_{z2}) = 36.0 N/mm^{2}

Force in layer 2;Â F_{z2} = 2 Ã— A_{bar} Ã— s_{z2} = 35.3 kN

Moment of resistance of layer 2;Â M_{Rdz2} = F_{z2} Ã— (h / 2 â€“ d_{z2}) = -2.6 kNm

Strain in layer 3;Â Îµ_{z3} = Îµ_{cu3} Ã— (1 â€“ d_{z3} / z) = 0.00194

Stress in layer 3;Â Ïƒ_{z3} = min(f_{yd}, E_{s} Ã— Îµ_{z3}) â€“ h Ã— f_{cd} = 375.7 N/mm^{2}

Force in layer 3; F_{z3} = 2 Ã— A_{bar} Ã— s_{z3} = 368.8 kN

Moment of resistance of layer 3; M_{Rdz3} = F_{z3} Ã— (h/2 â€“ d_{z3}) = 26.6 kNm

Strain in layer 4;Â Îµ_{z4} = Îµ_{cu3} Ã— (1 â€“ d_{z4}/z) = 0.00282

Stress in layer 4;Â Ïƒ_{z4} = min(f_{yd}, E_{s} Ã— Îµ_{z4}) â€“ h Ã— f_{cd} = 422.0 N/mm^{2}

Force in layer 4; F_{z4} = 1 Ã— A_{bar} Ã— s_{z4} = 207.2 kN

Moment of resistance of layer 4; M_{Rdz4} = F_{z4} Ã— (h/2 â€“ d_{z4}) = 29.9 kNm

Resultant concrete/steel force;Â F_{z} = 1494.0 kN

PASS â€“ This is within half of one percent of the applied axial load

**Combined moment of resistance**

Moment of resistance about z axis; M_{Rdz} = **129.8** kNm

**Minimum moment capacity with axial load N _{Ed}**

Minimum moment capacity; M

_{Rd}= min(M

_{Rdz}, M

_{Rdy}) =

**127.8**kNm

** PASS** â€“ The moment capacity exceeds the resultant design bending moment

**Summary**

Description | Unit | Provided | Required | Utilisation | Result |

Moment capacity (y) | kNm | 127.79 | 101.41 | 0.79 | PASS |

Moment capacity (z) | kNm | 129.76 | 66.06 | 0.51 | PASS |

Combined capacity | kNm | 127.79 | 121.03 | 0.95 | PASS |