How to figure out if my column has minimum eccentricity?

A Concrete Column Failing in Seismic
Concrete Column Failure via Seismic Forces

A small eccentricity in a column is useful since it assumes that the moment acting on a column doesn't have a significant impact on the design and therefore can be ignored. The limits for the eccentricity are no longer part of the ACI 318 [1] but approximations have been provided below.

e = {M_u\over{P_uh}}

where:

e = the eccentricity in the column
Mu = the maximum moment
Pu = Factored axial load on the column
h = the dimension of the column perpendicular to the axis of bending

When the eccentricity (e) in the column is less than approximately 0.1 for tied columns or 0.05 for spiral columns then flexure in the column can be assumed negligible. Otherwise the column will have to be designed via the Concrete Column Interaction Diagram tables.

Column design for small eccentricity:

In order to design for a standard column with a small amount of eccentricity (detailed above) you must insure that the factored nominal force (capacity) is greater than the ultimate load (demand). Below we show the equation and steps necessary to either determine the strength of a member (given the dimensions and amount of steel), or design a beam so that it is capable of supporting a desired load.

α P_o \ge P_u

where:

= Concrete Strength Reduction Factors
α = Strength Factor (α = 0.80 for tied columns and 0.85 for spiral columns)
  • note: no symbol is given to the above factor in the ACI 318-05. To simplify the problem visually, α was added as a variable.
Po = Maximum nominal axial capacity of the column. See below to calculate.
Pu = Factored axial load on the column

and subsequently:

P_o = 0.85f_c'\left(A_g-A_{st}\right) + A_{st}f_y

where:

f'c = compressive strength of the concrete
Ag = Gross Area of the column
Ast = Area of the longitudinal steel in the column (do not take ties or spirals into account)
fy = Yield strength of the steel (typ. 40-60 ksi)

Combining the two you can come up with the following equations:

{P_u\over {α}} = 0.85f_c'\left(A_g-A_{st}\right) + A_{st}f_y

or:

{P_u\over {α}} = 0.85f_c'\left(1-ρ_g\right) + ρ_gf_y

where:

ρg = A_{st}\over {A_g}

Note: A good starting value for ρg is between 0.02 and 0.025. Remember, ρg cannot exceed 0.08.

References

  1. American Concrete Institute, "Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (ACI 318R-05)", 2005

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