Thursday, August 18, 2011

Column Design (Part 3)

Let's continue from the previous post ...
The criteria to select whether we should use the Euler formula or the J. B. Johnson formula to calculate for the critical load (Pcr) of the column is related to the value of the actual slenderness ratio or column constant, Cc. It is defined as
E = Modulus of elasticity of the material of the column
Sy = Yield strength of the column material

The use of the above column constant (Cc) is as follows,
  1. Determine the length and end fixity of the column
  2. Define the value of the constant (K) according to the type of end fixity
  3. Compute the effective length (Le) from Le = KL
  4. From the cross section shape and dimensions, compute the radius of gyration (r) from r = sqrt(I/A)
  5. Compute the slenderness ratio from Slenderness ratio = Le/r_min = KL/r_min
  6. From the material of the column, compute the column constant (Cc) as per the above formula
  7. Check whether KL/r > Cc?
    • If yes, the column is Long: Use the Euler formula
    • If no, the column is Short: Use the J. B. Johnson formula
In the next post, let's check the formulas of critical load (Pcr) of Euler and J. B. Johnson.

Wednesday, August 17, 2011

Column Design (Part 2)

From Column Design (Part 1), we know that a column will tend to buckle  about the axis for which the radius of gyration (r) and the moment of inertia (I) are minimum. Another important parameter for column design is the effective length (Le) of the column. The effective length is defined as

Le = KL

L = Actual length of the column between its supports
K = Constant value dependent on the end fixity of the column as following.
  • A pinned-end column is guided so that the end of the column cannot sway from side to side, but it can rotate with no resistance at the end.
  • A fixed-end column is held against rotation at the support.
The higher constant value of K as shown as the "practical values" in the above table is recommended because in reality it is particularly difficult to achieve a true fixed-end column because of lacking of rigidity of the support.

The slenderness ratio is the ratio of the effective length of the column to the least radius of gyration.

Slenderness ratio = Le/r_min = KL/r_min

The slenderness ratio will be used to select the method of performing the analysis of straight, centrally loaded columns. Two methods will be presented in the next post.
  1. The Euler formula for long, slender columns
  2. The J. B. Johnson formula for short columns