The slenderness ratio of a reinforced concrete (RC) column is the ratio between the length of the column, its lateral dimensions, and end fixity. It assesses the ability of the reinforced concrete column to resist buckling pressure. The slenderness ratio is calculated by dividing the column length by its radius of gyration.
The slenderness ratio differentiates short column from long or slender column. The design of the former is controlled by column dimension and material strength whereas the design of the latter is governed by column slenderness.
A column is said to be slender if its cross-sectional dimensions are small compared to its length. If the slenderness ratio of a column is high, it will collapse under a smaller compression load in contrast to a short column with the same cross-sectional dimensions. So, the slenderness effect should be taken into consideration during the design process.
The slenderness ratio of reinforced concrete columns can be computed according to the procedures and specifications of applicable codes such as ACI 318-19 and IS 456.
How to Calculate Slenderness Ratio Based on ACI 318-19?
The degree of slenderness is generally expressed in terms of the slenderness ratio:
lu: unsupported length of the member
r: radius of gyration of its cross section
K: constant to reflect end conditions of the column; distance between two inflection points (Alignment chart is used to calculate K).
Unsupported Length of the Member (lu)
- The unsupported length (lu) of a column is measured as the clear distance between the underside of the beam, slab, or column capital above, and the top of the beam or slab below, as shown in Figure-1.
2. The unsupported length of a column may be different in two orthogonal directions depending on the supporting elements in respective directions.
Radius of Gyration of Column Cross-section (r)
The radius of gyration introduces the effects of cross-sectional size and shape on slenderness. For the same cross-sectional area, a section with a higher moment of inertia produces a more stable column with a lower slenderness ratio. The radius of gyration r is calculated using the following formula:
Ig: moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement, mm4
Ag: Gross area of column, mm2
It is possible to use the approximations of r = 0.3h for square and rectangular sections, and r = 0.25h for circular sections.
h: overall sectional dimension in the directional stability is being considered.
Effective Length Factor (k)
The effective length factor (k) reflects the end restraint (support) and lateral bracing conditions of a column. If a column is hinged at both ends, it follows a half-sine wave when it buckles, and the value of (k) factor for such type of column is 1.0. So, the effective length of the column (klu) is equal to the unsupported length of the column (lu).
A column with fully restrained end conditions develops deflected shape as shown in Figure-5. The portion of the column between the points of contraflexure follows a half-sine wave. The effective length factor k for this case is equal to 0.5.
Columns in real structures are rarely either hinged or fixed but have ends partially restrained against rotation by abutting members. The value of k, in this case, varies between 0.5 and 1.0 for laterally braced columns, Figure-6. For unbraced columns, the value of k varies between 1.0 and ∞, Figure-7.
The majority of reinforced concrete columns are considered to be laterally braced columns. The exact value depends on the degree of end restraint, that is, on the ratio of the stiffness (EI/l) of the column to the sum of stiffnesses (EI/l) of the restraining members at both ends.
ACI Criteria for Slenderness Ratio Effect
The effect of slenderness ratio for columns braced against sideway can be neglected when:
Note: As a first approximation, k may be taken equal to 1.0 in this equation
M1 :lesser factored end moment on a compression member, N·mm
M2 :greater factored end moment on a compression member.
M1/M2 : negative if the column is bent in single curvature, and positive for double curvature. M1/M2 is negative if the column is bent in single curvature, and positive for double curvature, as illustrated in Figure-8:
The effect of slenderness ratio for columns not braced against sideway can be neglected when:
Alignment Chart for Effective Slenderness Factor
If slenderness is found to be important, refine the calculation of k based on the alignment chart as shown below:
The Ψ factor at one end of the column equals the sum of the stiffness ∑(EI/L) of the columns meeting at that joint, including the column in question, divided by the sum of all the stiffnesses of the beams meeting at the joint.
E: Modulus of Elasticity
I: Moment of Inertia
L: span length measured center to center of joints.
ΨA, ΨB : Values of (Ψ ) at each end of the column
The modulus of elasticity for normal concrete is computed as:
The moment of inertia used to compute Ψshould be based on cracked section, so ACI 318-19 section 220.127.116.11.1 provides Table-1:
Table-1: Moment of Inertia Used to Compute Ψ Factor
|Member and Condition||Moment of Inertia|
|Flate Plates and Flat Slabs||0.25Ig|
The column is considered to be slender if its length is higher compared to its cross-sectional dimension i.e. slenderness ratio is used to differentiate slender column from short column.
Slenderness ratio of reinforced concrete (RC) column is the ratio between column length, its lateral dimensions, and end fixity. It assesses the ability of reinforced concrete column to resist buckling pressure.
It is used to find out the design load as well as in classifying various columns in short/intermediate/long. The slenderness ratio of a column gives an indication of buckling failure in the column. More the slenderness ratio, the more is the tendency of the column to fail by buckling effect in that direction.
1. Calculate the effective length factor (K) that ranges between 0.5 to 1.0 for the reinforced concrete column. It can be considered to be 1.0 conservatively.
2. Compute the unsupported length of the column (lu) which is measured as the clear distance between the underside of the beam, slab, or column capital above, and the top of the beam or slab below.
3. Estimate the radius of gyration (r). It is possible to use the approximations of r = 0.3h for square and rectangular sections, and r = 0.25h for circular sections.
4. Calculate slenderness ratio which is equal to effective length factor times unsupported length divided by the radius of gyration.
Buckling of the column is a form of deformation as a result of axial compression forces that leads to a bending of the column, as a result of the instability of the column. This mode of failure is quick and undesirable.
The radius of Gyration is used to describe the distribution of cross-sectional area in an RC column around its centroidal axis.
Tips and rules for design of reinforced concrete column
Euler's theory of column buckling