## What is a long column? Figure 1: Bridge Piers can often be long columns (Photo: source)

A long concrete column basically means that the column is subject to P-Delta forces (second-order analysis). In short columns the deformations of the column are significantly small enough that they can be ignored. In Figure 1 the concrete columns for the bridge shown could be 75 feet long before they reach solid ground (in that case a 10'-0" sq. column might still be considered a long column depending on the supports).

One differentiation that must be determined when dealing with long columns is whether or not the column is braced (braced vs. unbraced columns). The idea is that a braced column will fail without joint translation whereupon an unbraced column will fail with joint translation (unbraced columns have less capacity since the joint translation increases the P-Delta forces acting on the column).

Note: Diagonals are not the only determinant on whether or not a column is braced. A sufficiently stiff wall or floor system is adequate to resist joint translation.

A concrete column will be considered long when:

If the structure is part of a braced system then:

{kl_u\over{r}}\ge min[34-12{M_1\over{M_2}}, 40]

If the structure is part of a unbraced system then:

{kl_u\over{r}}\ge 22

where:

k = the effective length factor (can be conservatively taken as 1.0 )
lu = clear height of the column
M1 = the smaller absolute value of the two end moments acting on the column
M2 = the larger absolute value of the two end moments acting on the column
r = radius of gyration

## Find the allowable axial load of a long column (Braced Frame):

Solving for the allowable capacity is a long process, first you need to find the buckling load (Pc).

1) Find Bd:

B_d = \frac{P_{DL}}{P_{TOT}}

where:

Bd = Stiffness reduction factor for long term effects (unitless)
PDL = The factored dead load acting on the column
PTOT = The factored total load acting on the column

Note: If lateral loads are used (wind, earthquake, etc.) then Bd = 0.

2) Solve for the flexural stiffness (EI) of the system:
This term is included to take into account any reductions from cracking, non-linear effects in the stress-strain curve, and creep.

The following two equations can be used to determine the flexural stiffness. I usually use the later one since it doesn't require me to know the steel reinforcement (and solving for the moment of inertia of the steel bars is a pain).

\begin{eqnarray} && EI = \frac{0.2E_cI_g + E_sI_{se}}{I+B_d} \\ && = \frac{0.4E_cI_g}{I+B_d} \end{eqnarray}

where:

Ec = Modulus of Elasticity of Concrete = 57,000\sqrt{f^'_c}
I = The Moment of Inertia of the Column
Es = Modulus of Elasticity of Steel Rebar
Ise = Moment of Inertia of the Steel Rebar
Bd = Found in Section 1 above

3) Solve for the effective length (klu):
Before you start crunching numbers it is important to know that since second-order moments are present in this analysis. For this portion of the analysis your Moment of Inertia must be reduced to account for cracking, non-linear effects in the stress-strain curve, and creep. Your new Moments of Inertia are:

 Moments of Inertia of columns 0.70Ig Moments of Inertia of beams 0.35Ig Moments of Inertia of plates and slabs 0.25Ig

Note: This altered moment of inertia is only used to find the effective length (klu)

Now using your new Moment of Inertia found above you can find the relative stiffness perameter Ψ, by calculating the following:

\Psi =\frac{\sum _{columns}\frac{EI}{l_c}}{\sum _{beams}\frac{EI}{l_b}}

where:

Ψ = Relative stiffness parameter to be used in the "Jackson and Moreland Alignment Chart"
E = Modulus of Elasticity of Concrete = 57,000\sqrt{f^'_c}
I = The Moment of Inertia of the Column/Beam (using the lowered moments of inertia detailed above)
lc = The height of the column (centerline to centerline)
lb = The length of the beam (centerline to centerline)

Note: Ψ will include columns from the floor above and below. Also the beams from in front and in back should be included.

4) Find the Buckling Load, Pc:
The Euler formula (remember this from steel design?) will give the buckling load:

P_c = \frac{\pi^2EI}{(kl_u)^2}

where:

EI = From Section 2 above
klu = From Section 3 above

## References

1. American Concrete Institute, "Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (ACI 318R-05)", 2005
1. From Section 10.12.2
2. American Concrete Institute, "Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (ACI 318R-05)", 2005
1. From Figure R10.12.1

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