Column Buckling Calculator

Column material

Steel column

Material properties

Steel grade

Hollow section manufacture

I-beam manufacture

Hot-rolled section

Cross-section geometry

Principal axes of inertia (top view)

Diameter d, mm

Wall thickness t, mm

Web thickness tw, mm

Flange thickness tf, mm

Width b, mm

Height h, mm

Load and effective length

Column length L, m

Design load NEd, kN

Support conditions diagram

mm² mm⁴ cm⁶ MPa kN mm m Strength, flexural buckling and torsional buckling under axial compression have been checked. Cross-section strength and buckling have been checked, accounting for initial eccentricity and second-order effects. An 8 mm stirrup and a creep coefficient of 2 are assumed. Strength about the y-y axis Strength about the z-z axis Buckling about the y-y axis Buckling about the z-z axis Check passed Preliminary check passed Check failed Flexural-torsional mode Critical flexural-torsional load Ncr,TF The ratio of the longer to the shorter side of the reinforced concrete column exceeds 4:1. Change the cross-section width or height. The walls are too thin for this section to be calculated. Increase the wall thickness or reduce the section size. Check the entered values. All dimensions and the design load must be positive numbers. Invalid cross-section geometry. Check the dimensions and thicknesses. Calculation could not be completed. Check the cross-section dimensions, column length and material properties. For thicknesses over 250 mm, no design value of fy is specified. Select another cross-section. Invalid reinforcement layout, concrete cover, spacing or reinforcement ratio. The longitudinal reinforcement area is below the required minimum. Increase the bar diameter or number of bars. torsional mode Torsion constant J Effective length for torsion Lcr,T Critical torsional load Ncr,T Torsional buckling resistance Nb,T Cross-sectional area Radius of gyration iy Radius of gyration iz Yield strength fy Cross-section class Effective length Euler critical load Buckling reduction factor for the governing mode Modulus of elasticity E0,05 Modification factor kmod Design compressive strength Cross-section height h, mm Leg width a, mm Flange width b, mm Cross-section width b, mm Wall thickness t, mm Flange thickness h1, mm Flange thickness t, mm Calculation under axial compression without external bending moments or connection checks. Reinforcement area Longitudinal reinforcement ratio Slenderness λy Slenderness λz

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About Column Buckling Calculation

The results are approximate. Before use, verify the calculations against the applicable standards and consult a specialist. The developer is not responsible for the consequences of use without project verification.

The Column Buckling Calculator evaluates the strength, buckling resistance, and axial load capacity of steel, timber, and reinforced concrete columns under axial compression. For reinforced concrete columns, the calculation also accounts for the bending strength of the cross-section.

The calculation determines the column load capacity in kN, evaluates strength and buckling resistance, and identifies the most critical direction of instability. The method accounts for material properties, cross-sectional geometry, column length, and end support conditions.

Guidelines and Recommendations

General Principles of Column Calculation

Geometric properties. The cross-sectional dimensions are used to calculate the area A in mm² and the second moments of area I_y and I_z in mm⁴. For hollow sections, the area of the internal void is subtracted from the outer area. Rounded corners are considered for rectangular steel hollow sections, while principal axes of inertia are determined for angle sections.

The radii of gyration about the two axes are calculated from the cross-sectional area and the corresponding second moments of area:

i_y = √(I_y / A)

i_z = √(I_z / A)

Effective length. This accounts for the influence of end restraints on column buckling. It is calculated using the following formula:

L_cr = K × L

Here, L is the actual column length in metres, and K is the effective length factor. The following values are used:

Pinned at both ends: K = 1.0.
Fixed at one end and pinned at the other: K = 0.7.
Fixed at both ends: K = 0.5.
Fixed at one end and free at the other: K = 2.0.

Geometric slenderness. This represents the ratio of the effective length to the radius of gyration of the cross-section. It is calculated separately for each direction:

λ = L_cr / i

Both the effective length and the radius of gyration are expressed in millimetres when calculating slenderness. Higher slenderness increases the influence of buckling on load capacity. There is no single universal slenderness limit applicable to all materials and cross-sections.

Steel Column Calculation According to Eurocode 3

Steel columns are calculated using the provisions of EN 1993-1-1 "Eurocode 3: Design of Steel Structures - General Rules and Rules for Buildings".

Steel strength. The design resistance is determined from the yield strength f_y. For element thicknesses up to 16 mm, the values are 235 MPa for S235, 275 MPa for S275, and 355 MPa for S355. The yield strength decreases at greater thicknesses. For example, S355 steel with a thickness greater than 16 mm and up to 40 mm has a yield strength of 345 MPa.

The cross-section is first classified according to the width-to-thickness or diameter-to-thickness ratios of its elements. Calculations are performed for cross-section classes 1, 2, and 3. No result is provided for class 4 sections, which require an effective area calculation accounting for local buckling.

Cross-sectional compression resistance. This is determined by multiplying the cross-sectional area by the steel yield strength:

N_c,Rd = A × f_y

The area A is expressed in mm², and the yield strength is expressed in MPa. The resulting force is in newtons and is converted to kN by dividing by 1000. The partial safety factors used in the resistance calculation are γ_M0 = 1.0 and γ_M1 = 1.0.

Euler critical buckling load. For each principal axis, the load at which an ideal straight column may buckle is calculated:

N_cr = π² × E × I / L_cr²

The modulus of elasticity of steel is taken as E = 210000 MPa. A smaller second moment of area or a greater effective length reduces the critical buckling load.

Reduction of load capacity. Buckling curves a, b, c, and d are used to account for initial imperfections and the characteristics of steel sections. Their imperfection factors α are 0.21, 0.34, 0.49, and 0.76, respectively.

The buckling curve is selected according to the cross-sectional shape, buckling direction, and manufacturing method. Curve a is used for hot-finished hollow sections according to EN 10210, while curve c is used for cold-formed hollow sections according to EN 10219. For I-sections, the selection also depends on the height-to-width ratio, flange thickness, and manufacturing method.

The non-dimensional slenderness and the buckling reduction factor χ are calculated in sequence:

λ_rel = √(A × f_y / N_cr)

Φ = 0.5 × [1 + α(λ_rel - 0.2) + λ_rel²]

χ = 1 / [Φ + √(Φ² - λ_rel²)]

The factor χ is limited to a maximum of 1.0. When the non-dimensional slenderness does not exceed 0.2, χ = 1.0 is used. The buckling resistance is then calculated as:

N_b,Rd = χ × A × f_y

Torsional buckling resistance. For open sections, including I-sections, channels, and angles, the possibility of buckling accompanied by cross-sectional rotation is also evaluated. The calculation uses the torsion constant J, the warping constant I_w, and the shear modulus G = E / 2.6, approximately 80769 MPa. The effective length for torsion is taken as no less than the actual column length.

The final buckling load capacity of a steel column is the lowest resistance obtained from flexural buckling about the principal axes and the corresponding torsional or flexural-torsional buckling modes.

Timber Column Calculation According to Eurocode 5

Timber columns are calculated using the provisions of EN 1995-1-1 "Eurocode 5: Design of Timber Structures - General Rules and Rules for Buildings". The properties of structural timber are based on the strength classes defined in EN 338.

Timber compression strength. For strength classes C16, C18, C24, and C30, the characteristic compressive strength parallel to the grain f_c,0,k is 17, 18, 21, and 23 MPa, respectively. The fifth-percentile modulus of elasticity E_0,05 is taken as 5400, 6000, 7400, and 8000 MPa.

Influence of service conditions. The design strength of timber is adjusted using the modification factor k_mod, which accounts for moisture conditions and load duration. The following values are used for permanent, long-term, medium-term, short-term, and instantaneous loads:

Service classes 1 and 2: 0.60, 0.70, 0.80, 0.90, and 1.10.
Service class 3: 0.50, 0.55, 0.65, 0.70, and 0.90.

The design compressive strength is calculated using the material partial safety factor γ_M = 1.3:

f_c,0,d = k_mod × f_c,0,k / 1.3

The cross-sectional compression resistance is the product of the area and the design compressive strength of timber:

N_c,Rd = A × f_c,0,d

Timber column buckling. The non-dimensional slenderness is calculated for each principal axis using the effective length, radius of gyration, and timber properties:

λ_rel = L_cr / (π × i) × √(f_c,0,k / E_0,05)

The reduction in resistance is accounted for by the buckling factor k_c. An initial imperfection factor β_c = 0.2 is used to determine this factor. When the non-dimensional slenderness does not exceed 0.3, k_c = 1.0 is used. At greater slenderness, the following relationships apply:

k = 0.5 × [1 + 0.2(λ_rel - 0.3) + λ_rel²]

k_c = 1 / [k + √(k² - λ_rel²)]

The buckling resistance is calculated as k_c × A × f_c,0,d. The calculation is performed for both axes, and the lower value is selected.

Reinforced Concrete Column Calculation According to Eurocode 2

Reinforced concrete columns are evaluated using a calculation model based on EN 1992-1-1 "Eurocode 2: Design of Concrete Structures - General Rules and Rules for Buildings". The verification accounts for concrete and reinforcement resistance, initial axial load eccentricity, and second-order effects.

Design material properties. For concrete strength classes from C20/25 to C90/105, the design compressive strength is calculated using:

f_cd = 0.85 × f_ck / 1.5

Here, f_ck is the characteristic cylinder compressive strength of concrete in MPa. For B500 longitudinal reinforcement with a yield strength of 500 MPa and a partial safety factor of 1.15, the design yield strength is:

f_yd = 500 / 1.15 = 434.78 MPa

Reinforcement area. This is calculated from the number and diameter of the longitudinal bars. In rectangular sections, the bars are distributed around the perimeter, while in circular sections, they are distributed around the circumference. The positions of their centres are determined considering the concrete cover and 8 mm transverse reinforcement.

The minimum longitudinal reinforcement area is taken as at least 0.2% of the concrete cross-sectional area and at least 10% of the design axial force divided by the design yield strength of the reinforcement. The maximum reinforcement ratio is limited to 4%. Bar spacing and the feasibility of arranging the bars within the specified cross-section are also checked.

Cross-sectional resistance. For concrete up to C50/60, a parabolic-rectangular stress-strain diagram is used, with a strain of 2.0‰ at maximum stress and an ultimate strain of 3.5‰. For higher concrete strength classes, the diagram parameters vary according to concrete strength. The modulus of elasticity of reinforcement is taken as 200000 MPa.

The forces in the concrete and each reinforcing bar are calculated from the strain distribution. These forces are used to determine pairs of axial force N and bending resistance M for each principal axis. This produces an axial force-bending moment interaction diagram, which is used to establish the maximum allowable load.

Initial eccentricity. Even under axial compression, a minimum offset of the load line from the centre of the cross-section is considered:

e_min = max(20 mm, h / 30)

Here, h represents the cross-sectional dimension in the direction being considered. For the buckling verification, an additional initial imperfection of L_cr / 400 is considered. The greater of these values is used as the initial eccentricity.

Second-order effects. The increase in bending moment caused by deformation of the compressed column is evaluated using the nominal curvature method. A concrete creep coefficient φ = 2.0 is assumed. The limiting slenderness, above which additional bending is considered, depends on the axial compression level, reinforcement amount, and creep.

Where necessary, the additional eccentricity is calculated from the square of the effective length and the curvature:

e₂ = L_cr² / 10 × f_yd / (200000 × 0.45d_eff) × k_r × k_φ

Here, d_eff is the effective depth of the reinforcement, k_r accounts for the axial load level, and k_φ accounts for creep. The resulting eccentricity increases the design bending moment.

Determination of load capacity. For different axial force values, the calculation checks whether the cross-sectional bending resistance is at least equal to the moment produced by the force and its corresponding eccentricity:

M_Rd(N) ≥ N × e(N)

The maximum allowable axial force is determined separately for the two principal directions. Initial eccentricity is considered in the strength verification, while the buckling verification additionally accounts for second-order effects. The lowest of the calculated values is taken as the governing resistance.

Interpreting the Calculation Results

Capacity utilisation. This indicates the percentage of the design resistance used by the applied axial load N_Ed. It is calculated separately for strength and buckling resistance:

η = N_Ed / N_Rd × 100%

A value below 100% means that the corresponding verification is satisfied. At 100%, the design load equals the calculated resistance. A value above 100% indicates insufficient load capacity under the condition being checked.

Governing condition. Both verifications must be satisfied for the overall assessment to pass. If the cross-sectional strength is sufficient but buckling resistance is inadequate, the column fails the calculation. For cross-sections with different properties about their principal axes, the direction with the lower load capacity governs the result.

FAQs

Why does a column pass the strength check but fail the buckling check?

Strength describes the ability of the material and cross-section to resist a compressive load. Buckling resistance also depends on column length, end restraints, and cross-sectional stiffness. As a result, a long and relatively slender column may buckle at a load considerably below the material strength limit.

Why do columns with the same cross-sectional area have different load capacities?

Equal cross-sectional area does not mean equal stiffness. Material distributed farther from the centre of the cross-section can provide a greater second moment of area and better resistance to flexural buckling. Therefore, the section shape and the distribution of material significantly affect the calculated column load capacity.

How much do end support conditions affect column buckling resistance?

End support conditions determine the effective length, which appears squared in the denominator of Euler's critical buckling load formula. For example, with the same actual length, increasing the factor K from 1.0 to 2.0 reduces the ideal critical load by a factor of 4. The actual design buckling resistance also depends on material properties and the applicable buckling curve.

Why cannot column slenderness be assessed using a single limit?

The same geometric slenderness does not imply the same buckling safety margin for steel, timber, and reinforced concrete. Each material has different deformation properties and a specific calculation model. Therefore, the calculator uses slenderness as an intermediate characteristic, while the final assessment is based on the ratio of the applied load to the calculated resistance.

Why is bending considered for reinforced concrete columns under axial loading?

Even under nominally concentric compression, small geometric imperfections and deviations in the load position are taken into account. These produce a bending moment that may increase as the column deforms. Therefore, reinforced concrete column buckling is evaluated considering eccentricity and second-order effects rather than concrete and reinforcement areas alone.