4. Moment Magnification at Ends of Compression Member

A detailed calculation for load combination 4 (gravity plus wind) is shown below to illustrate the procedure. Table 3 summarizes the magnified moment computations for the exterior columns.

    ACI 318-19 (6.6.4.6.1b)

Where:

    ACI 318-19 (6.6.4.6.2)

ACI 318-19 (6.6.4.6.2(b)) will be used for comparison purposes with results obtained from spColumn model. However, (a) and (c) can also be used to calculate the moment magnifier.

∑P_u is the summation of all the factored vertical loads in the first story, and ∑P_c is the summation of the critical buckling load for all sway-resisting columns in the first story.

    ACI 318-19 (6.6.4.4.2)

Where:

    ACI 318-19 (6.6.4.4.4)

There are three options for calculating the effective flexural stiffness of slender concrete columns (EI)_eff. The second equation provides accurate representation of the reinforcement in the section and will be used in this example and is also used by the solver in spColumn.

Further comparison of the available options is provided in “Effective Flexural Stiffness for Critical Buckling Load of Concrete Columns” technical note.

    ACI 318-19 (Table 6.6.3.1.1(a))

    ACI 318-19 (19.2.2.1.b)

β_ds is the ratio of maximum factored sustained shear within a story to the maximum factored shear in that story associated with the same load combination. The maximum factored sustained shear in this example is equal to zero leading to β_ds = 0.        ACI 318-19 (6.6.3.1.1)

For exterior columns with one beam framing into them in the direction of analysis (12 columns):

With 8-#8 reinforcement equally distributed on all sides and 22 in. x 22 in. column section à I_se = 352.6 in.⁴.

        ACI 318-19 (6.6.4.4.4(b))

k = 1.693 (calculated previously).

For exterior columns with two beams framing into them in the direction of analysis (4 columns):

        ACI 318-19 (Figure R6.2.5.1)

(Column essentially fixed at base)        ACI 318-19 (Figure R6.2.5.1)

Using Figure R6.2.5.1 from ACI 318-19 à k = 1.527 as shown in the figure below for the exterior columns with two beams framing into them in the directions of analysis.

Figure 3 - Effective Length Factor (k) Calculations for Exterior Columns with Two Beams Framing into them in the Direction of Analysis

For interior columns (8 columns):

    ACI 318-19 (Table 6.6.3.1.1(a))

    ACI 318-19 (19.2.2.1.b)

For the column below level 2:

For the column above level 2:

For beams framing into the columns:

Where:

    ACI 318-19 (19.2.2.1.b)

    ACI 318-19 (Table 6.6.3.1.1(a))

    ACI 318-19 (Figure R6.2.5.1)

(Column essentially fixed at base)        ACI 318-19 (Figure R6.2.5.1)

Using Figure R6.2.5.1 from ACI 318-19 à k = 1.614 as shown in the figure below for the interior columns.

Figure 4 - Effective Length Factor (k) Calculations for Interior Columns

With 8-#8 reinforcement equally distributed on all sides and 24 in. x 24 in. column section à I_se = 439.1 in.⁴.

        ACI 318-19 (6.6.4.4.4(b))

For load combination 4:

        ASCE 7-10 (2.3.2-3)

        ACI 318-19 (6.6.4.6.2(b))

        ACI 318-19 (6.6.4.6.1)

        ACI 318-19 (6.6.4.6.1)

P_u = 722.0 kip

A summary of the moment magnification factors and magnified moments for the exterior column for all load combinations using both equation options ACI 318-19 (6.6.4.4.4(a)) and (6.6.4.4.4(b)) to calculate (EI)_eff is provided in the table below for illustration and comparison purposes. Note: The designation of M₁ and M₂ is made based on the second-order (magnified) moments and not based on the first-order (unmagnified) moments.