Design of Biaxial Reinforced Concrete Columns

A biaxial column is a column that is subjected to compressive axial force and bending moment in the two planes. They are usually found at the corners of a building or at locations where the beam spans and/or loading are not equal. In the design of biaxial reinforced concrete columns, a non-linear analysis method is required, which should take into account second-order, imperfection, and biaxial bending effects.

EN 1992-1-:2004 (Eurocode 2) did not give an express method of designing biaxial columns other than working from the first principles. This may not be very easy to achieve without the use of charts or computer programs. To develop such a program, one can divide the compression zone into strips that are parallel with the neutral axis of the section, and calculate the stress in each strip using the parabolic-rectangular diagram. The force and moment at each strip in the x and y-axis can be summed up in the ultimate limit state to find the moment and axial force developed by the concrete in compression.

There are however simplified methods of dealing with biaxial bending in reinforced concrete structures. An example is the approach given in clause 5.8.9(4) of EN 1992-1-1 for the design of biaxially bent sections in slender columns. This is based on the observation that the form of the M_x – M_y interaction diagram can conveniently be represented by a super-ellipse. A super-ellipse has an equation of the form;

x^a + y^a = k —— (1)

If a = 2, this equation becomes a circle, while if a = 1 it describes a straight line. At loads approaching the squash load, the M_x – M_y interaction diagram approaches a circle, while in the region of the balance point it is close to a straight line. Clause 5.8.9(4) adopts the equation below as a means of describing the complete interaction surface;

(M_E,dz/M_R,dz )^a + (M_Ed,y/M_Rd,y )^a ≤ 1.0 —— (2)

The proximity to the squash load can be defined using the N/N_uz ratio parameter, and EN 1992-1-1 assumes the relationship between this parameter and the exponent ‘a‘ given in the Table below. Intermediate values may be interpolated.

NEd/NRd	0.1	0.7	1.0
a	1.0	1.5	2.0

In the Table above, N_Rd is the squash load of the column, and may be calculated from

N_Rd = A_cf_cd + A_sf_yd —— (3)

The difficulty with the approach from the practical point of view is that it cannot be used as a direct design method since N_Rd can only be established once the reinforcement area has been found. It, therefore, has to be used iteratively. An initial estimate is made of N/N_uz, the section is designed, a corrected value of N/N_uz can then be estimated, and the process repeated until a correct solution is obtained.

A much simpler, but considerably more approximate method to the design of biaxial columns has been adopted in BS 8110. The design is carried out for an increased uniaxial moment, which takes account of the biaxial effects. The required uniaxial moment is obtained from whichever is appropriate of the two relationships set out below:

if M_x/h’ > M_y/b’ then M’_x = M_x + βh’M_y/b’ —— (4)
if M_x/h’ < M_y/b’ then M’_y = M_y + βb’M_x /h’ —— (5)

In the above relationships, M_x and M_y are the design moments about the x and y axes, respectively, while M’_x and M’_y are the effective uniaxial moments for which the section is actually designed. b’ and h’ are the effective depths of the column section (see image below). The factor β is defined in BS 8110 as a function of N/bhf_cu. In terms of f_ck, it can be obtained from the relationship;

β = 1 – N/bhf_ck (0.3 < β < 1.0) —— (6)

This approach has the great advantage of being very simple. It is, however, an approximate approach.

Example on the Design of Biaxial Reinforced Concrete Columns to BS 8110 and Eurocode 2

A reinforced concrete column fixed at both ends is subjected to the loading condition shown below. We are required to obtain the appropriate longitudinal reinforcement for the column using BS 8110-1:1997 and Eurocode 2.

The column is carrying longitudinal and transverse beams of depth 600mm and width 300mm. It is also supported by beams of the same dimension. The centre to centre height of the column is 3500mm. The plan view of the arrangement of the beams and column is as shown below.

DESIGN ACCORDING TO BS 8110-1:1997
N = 716.88 kN;
TOP: M_x-x = 175.87 kNm; M_y-y = 35.52 kNm
BOTTOM: M_x-x = -85.832 kNm; M_y-y = -25.269 kNm

Concrete grade f_cu = 30 N/mm²
Yield Strength of reinforcement f_y = 460 N/mm²
Concrete cover = 40mm
L_o = 3.5m, Effective length L_e = 0.75 × (3500 – 600) = 2175 mm;
Size of column = 400 x 300mm;
Slenderness = 2175/300 = 7.25 < 15. Thus column is short.

Effective depth about x-x axis
h’ = (400 – 40 – 12.5 – 10) = 337.5 mm

Effective depth about y-y axis
b’ = (300 – 40 – 12.5 – 10) = 237.5 mm

M_x/h’ = 521.109 kN; M_y/ b’ = 149.558  kN

Axial-load ratio
N_ratio = (N × 1000) / (F_cu × b × h) = (716.88 × 1000) / (30 × 400 × 300) = 0.1991

From Table 3.22 of BS 8110-1:1997, for N_ratio = 0.1991
Therefore, β = 1 – 1.1644(0.1991) = 0.7681

As M_x/h’ exceeds M_y/b’
M_x‘ = M_x + βh’/b’M_y
M_x‘ = 175.87 + [0.7681 × (0.3375)/(0.2375 ) × 35.52] = 214.64 kNm

Minimum Eccentricity in Columns
According to clause 3.8.2.4 of BS 8110-1:1997, at no section in a column should the design moment be taken as less than that produced by considering the design ultimate axial load as acting at a minimum eccentricity, e_min equal to 0.05 times the overall dimension of the column in the plane of bending considered but not more than 20 mm.

Where biaxial bending is considered, it is only necessary to ensure that the eccentricity exceeds the minimum about one axis at a time.

In the x-x direction = e_min = 0.05 × 400 = 20mm, therefore adopt 20mm, M_x = 716.88 × 0.02 = 14.3376 kNm < 214.64 kNm

In the y-y direction = e_min = 0.05 × 300 = 15mm, therefore adopt 15mm, M_y = 716.88 × 0.015 = 10.7532 kNm < 214.64 kNm

Section design ratios for chart entry
Axial load ratio N_ratio = (N × 1000)/(F_cu × b × h) = (716.88 × 1000)/(30 × 400 × 300) = 0.1991

M_ratio = M/(F_cu × b × h²) = (214.64 × 10⁶) / (30 × 300 × 400²) = 0.149

With d’/h = 337.5/400 = 0.84375

From chart, (ρ × F_y)/(F_cu) = 0.286 Therefore, ρ = (0.286 × 30) / 460 = 0.01865
A_SCreq = 0.01865 × 400 × 300 = 2238.26 mm²

Provide 6Y25mm (A_sprov = 2946 mm²)
Maximum area of reinforcement = 0.06bh = 0.06 × 400 × 300 = 7200 mm²

DESIGN ACCORDING TO EUROCODE 2
Clause 5.8.9(2) of EN 1992-1-1:2004 permits us to perform separate designs in each principal direction, disregarding biaxial bending as a first step. Imperfections need to be taken into account only in the direction where they will have the most unfavourable effect. However, in this example, we have carried out imperfection analysis in both directions.

N_Ed = 716.88 KN

Elastic Moments
Y – direction: M₀₁ = 175.87 kNm; M₀₂ = -85.832 kNm
Z – direction: M₀₁ = 35.52 kNm; M₀₂ = -25.269 kNm

Clear column height (L) = 3500 – 600 = 2900 mm

Calculation of the effective height of the column (L_o)
Let us first of all calculate the relative stiffnesses of the members in the planes of bending.

In the y-direction;
Second moment of area of beam 1 (I₁) = bh³/12 = 0.3 × 0.6³/12 = 0.0054 m⁴
Stiffness of beam 1 (since E is constant) = 4I₁/L = (4 × 0.0054) / 6 = 0.0036

Second moment of area of beam (I₂) = bh³/12 = 0.3 × 0.6³/12 = 0.0054 m⁴
Stiffness of beam 2 (since E is constant) = 4I₂/L = (4 × 0.0054) / 3.5 = 0.00617

Second moment of area of column (I_c) = bh³/12 = 0.0016 m⁴
Stiffness of column = 4I_c/L = (4 × 0.0016) / 3.5 = 0.001828

For compression members in regular braced frames, the slenderness criterion should be checked with an effective length l₀ determined in the following way:

L_o = 0.5L √[(1 + k₁)/(0.45 + k₁)) × (1 + k₂)/(0.45 + k₂))]

Where;
k₁, k₂ are the relative flexibilities of rotational restraints at ends 1 and 2 respectively
L is the clear height of the column between the end restraints

k = 0 is the theoretical limit for rigid rotational restraint, and k = ∞ represents the limit for no restraint at all. Since fully rigid restraint is rare in practise, a minimum value of 0.1 is recommended for k₁ and k₂.

In the above equations, k₁ and k₂ are the relative flexibilities of rotational restraint at nodes 1 and 2 respectively. If the stiffness of adjacent columns does not vary significantly (say, the difference not exceeding 15% of the higher value), the relative flexibility may be taken as the stiffness of the column under consideration divided by the sum of the stiffness of the beams (or, for an end column, the stiffness of the beam) attached to the column in the appropriate plane of bending.

Remember that we will have to reduce the stiffness of the beams by half to account for cracking;

k₁ = k₂ = 0.001828 / (0.0018 + 0.003085) = 0.3743

L_o = 0.5 × 2900√[((1 + 0.3743)/(0.45 + 0.3743)) × (1 + 0.3743)/(0.45 + 0.3743)] = 2647.77 mm

Compare with BS 8110’s 0.75L = 0.75 × 2900 = 2175 mm

In the z-direction;
Second moment of area of beam 3 (I₃) = bh³/12 = 0.3 × 0.6³/12 = 0.0054 m⁴
Stiffness of beam 3 (since E is constant) = 4I₁/L = (4 × 0.0054) / 3.5 = 0.00617

Second moment of area of column (I_c) = bh³/12 = 0.0009 m⁴
Stiffness of column = 4I_c/L = (4 × 0.0009) / 3.5 = 0.00102857

k₁ = k₂ = 0.00102857/(0.003085) = 0.3334

l_o = 0.5 × 2900 √[(1 + 0.3334) / (0.45 + 0.3334) × (1 + 0.3334) / (0.45 + 0.3334)] = 2675.293 mm

Radius of gyration

i_x = h/√12 = 400/√12 = 115.47
i_z = b/√12 = 300/√12 = 86.602

Slenderness in the x-direction (λ_x) = 2647.77/115.47 = 22.930
Slenderness in the z-direction (λz) = 2675.293/86.602 = 30.892

Critical Slenderness for the y-direction
λ_lim = (20.A.B.C)/√n
A = 0.7
B = 1.1
C = 1.7 – M₀₁/M₀₂ = 1.7 – [(-85.832)/175.87] = 2.188
n = N_Ed / (A_c f_cd)
N_Ed = 716.88 × 10³ N
A_c = 400 × 300 = 120000 mm²
f_cd = (α_cc f_ck)/1.5 = (0.85 × 30)/1.5 = 17 N/mm²
n = (716.88 × 10³) / (120000 × 17) = 0.3514
λ_lim = (20 × 0.7 × 1.1 × 2.188 )/√0.3514 = 56.842

22.930 < 56.842, second order effects need not to be considered in the y-direction

Critical Slenderness for the z-direction
A = 0.7
B = 1.1
C = 1.7 – M₀₁/M₀₂ = 1.7 – [(-25.269)/35.52] = 2.411
n = N_Ed / (A_c f_cd)
N_Ed = 716.88 × 10³ N
Ac = 400 × 300 = 120000 mm²
fcd = (α_cc f_ck)/1.5 = (0.85 × 30)/1.5 = 17 N/mm²

n = (716.88 × 10³) / (120000 × 17) = 0.3514

λ_lim = (20 × 0.7 × 1.1 × 2.411 )/√0.3514 = 62.634

30.892 < 62.634, second-order effects need not be considered in the z-direction

Design Moments (y-direction)

Y – direction: M₀₁ = 175.87 kNm; M₀₂ = -85.832 kNm

e₁ is the geometric imperfection = (θ_i l₀/2) = (1/200) × (2647.77/2) = 6.619 mm

Minimum eccentricity e₀ = h/30 = 400/30 = 13.333 mm. Since this is less than 20mm, take minimum eccentricity = 20mm (clause 6.1(4) EC2).

Minimum design moment = e₀N_Ed = 20 × 10^-3 × 716.88 = 14.3376 kNm

First order end moment M₀₂ = M_Top + e_iN_Ed
e_iN_Ed = 6.619 × 10^-3 × 716.88 = 4.745 KNm
M₀₂ = M_Top + e_iN_Ed = 175.87 + 4.754 = 180.624 KNm

Longitudinal Steel Area
d₂ = C_nom + ϕ/2 + ϕ_links = 40 + 12.5 + 10 = 62.5 mm

d₂/h = 62.5/400 = 0.156
Let us read from chart; d₂/h = 0.15;

M_Ed/(f_ck bh²) = (180.624 × 10⁶)/(30 × 300 × 400² ) = 0.125
N_Ed/(f_ck bh) = (716.88 × 10³) / (30 × 300 × 400) = 0.199

From the chart, (A_s F_yk)/(bhf_ck) = 0.23

Area of longitudinal steel required A_s,req = (0.23 × 30 × 400 × 300)/460 = 1800 mm²

A_s,min = 0.10 N_Ed/f_yd = (0.1 × 716.88)/400 = 0.179 mm² < 0.002 × 400 × 300 = 240 mm²
A_s,max = 0.04bh = 4800 mm²

Provide 4Y25mm (Asprov = 1964 mm²)

Design Moments (z-direction)z – direction: M₀₁ = 35.52 kNm; M₀₂ = -25.269 kNm

e₁ is the geometric imperfection = (θ_i l₀/2) = (1/200) × (2675.293/2) = 6.688 mm

Minimum eccentricity e₀ = h/30 = 400/30 = 13.333 mm. Since this is less than 20mm, take minimum eccentricity = 20mm (clause 6.1(4) EC2)

Minimum design moment = e₀N_Ed = 20 × 10^-3 × 716.88 = 14.3376 kNm

First order end moment M₀₂ = M_Top + e_iN_Ed
e_iN_Ed = 6.688 × 10^-3 × 716.88 = 4.794 KNm
M₀₂ = M_Top + e_iN_Ed = 35.52 + 4.794 = 40.314 KNm

Longitudinal Steel Area
d₂ = C_nom + ϕ/2 + ϕ_links = 40 + 12.5 + 10 = 62.5 mm
d₂/h = 62.5/400 = 0.156

Reading from chart No 1; d₂/h = 0.156;
M_Ed/(f_ck bh²) = (40.314 × 10⁶)/(30 × 300 × 400²) = 0.0279
N_Ed/(f_ck bh) = (716.88 × 10³) / (30 × 300 × 400) = 0.199

From the chart, (A_s F_yk)/(bhf_ck) = 0.00 (Nominal reinforcement required)

Biaxial Effects

Check if λ_y / λ_z ≤ 2.0 and λ_z/λ_y ≤ 2.0

17.203/18.159 = 0.9473 < 2.0, and 18.159/17.203 = 1.0556 < 2.0

Furthermore, let us also check;
(e_y/h_eq) / (e_z/b_eq) ≤ 0.2 or (e_z / b_eq) / (e_y / h_eq) ≤ 0.2

The definition of eccentricity is given in Figure 5.8 of EC2

e_y = M_Ed,y/N_Ed = (180.624 × 10⁶) / (716.88 × 10³) = 251.958 mm
e_z = M_Ed,z/N_Ed = (40.314 × 10⁶) / (716.88 × 10³) = 56.235 mm

h_eq = i_z.√12 = 300 mm
b_eq = i_y.√12 = 400 mm

(e_y/h_eq) ÷ (e_z/b_eq) = 251.958/300 ÷ 56.235/400 = 5.9739 > 0.2

Therefore we have to check for biaxial bending interaction;

(M_E,dz/M_R,dz )^a + (M_Ed,y/M_Rd,y )^a ≤ 1.0

(A_s F_yk)/(bhf_ck) = (1964 × 460) / (30 × 300 × 400) = 0.251

Therefore from the chart; M_Rd/(f_ckbh²) = 0.13

M_Rd = (0.13 × 30 × 300 × 400²) × 10^-6 = 187.2 KNm

N_Rd = A_cf_cd + A_sf_yd
N_Rd = [(300 × 400 × 17) + (1964 × 400)] × 10^-3 = 2825.6 kN
N_Ed/N_Rd = 716.88 / 2825.6 = 0.2537

To evaluate the value of a, let us look at the table below as given in Clause 5.8.9(4) of EC2

By linear interpolation, a = 1.0 + [(0.2537 – 0.1 )/(0.7 – 0.1)] × (1.5 – 1.0) = 1.128

(M_E,dz/M_R,dz )^a + (M_Ed,y/M_Rd,y )^a ≤ 1.0

(40.314/187.2)^1.128 + (180.624/187.2)^1.128 = 0.1769 + 0.9606 = 1.1375 > 1.0.
This is not ok, and this shows that 4Y25 is inadequate for the biaxial action on the column.

Let us increase the area of steel to 6Y25mm (A_sprov = 2946 mm²)

Let us check again;
(A_s F_yk)/(bhf_ck) = (2964 × 460) / (30 × 300 × 400) = 0.3787
Therefore from the chart; M_Rd/(f_ckbh²) = 0.175
M_Rd = (0.175 × 30 × 300 × 400²) × 10^-6 = 252 kNm

N_Rd = A_cf_cd + A_sf_yd
N_Rd = [(300 × 400 × 17) + (2964 × 400)] × 10^-3 = 3225.6 kN

N_Ed/N_Rd = 716.88 / 3225.6 = 0.222

By linear interpolation, a = 1.0 + [(0.222 – 0.1 )/(0.7 – 0.1)] × (1.5 – 1.0) = 1.102

(M_E,dz/M_R,dz )^a + (M_Ed,y/M_Rd,y )^a ≤ 1.0

(40.314/252)^1.102 + (180.624/252)^1.102 = 0.132 + 0.6928 = 0.825 > 1.0.

This shows that 6Y25mm is adequate for the column, and shows some agreement with result from BS 8110-1:1997.

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