1. C-Shaped Core Wall Biaxial Strength Calculations

The following three figures display the section’s strain diagram, internal forces and the corresponding moment arms in the necessary nomenclature to prepare for the strength calculations of each of the following:

• Design Axial Strength (φP_n)

Figure 3 shows the strain diagram for the reinforcement and concrete based on the neutral axis location and angle values provided. The internal forces for the reinforcement and concrete compression block are calculated based on the strain values. φ is calculated based on the strain in the extreme tension reinforcement layer.

• Design Flexural Strength (φM_nx)

The flexural strength φM_nx can be calculated using force values and moment arms from the x-axis (r_y) as shown in Figure 4.

• Design Flexural Strength (φM_ny)

The flexural strength φM_ny can be calculated using force values and moment arms from the y-axis (r_x) as shown in Figure 5.

Figure 3 - Design Axial Strength (φP_n) Calculations

Figure 4 - Design Flexural Strength (φM_nx) Calculations

Figure 5 - Design Flexural Strength (φM_ny) Calculations

1.1. Neutral Axis Location and Concrete Compression Force

The trial-and-error process for calculating the neutral axis depth and angle α is not required in this example since these values are given (c = 36.12 in. and α = 120^o)   . Where c is the distance from the fiber of maximum compressive strain to the neutral axis and α is the angle of the neutral axis.     ACI 318-19 (22.2.2.4.2)

    ACI 318-19 (Table 21.2.2)

    ACI 318-19 (22.2.2.4.1)

    ACI 318-19 (22.2.2.1)

Where:

a = Depth of equivalent rectangular stress block

    ACI 318-19 (Table 22.2.2.4.3)

   (Compression) ACI 318-19 (22.2.2.4.1)

Calculate the area of the section subject to compression and its centroid by examining the four sub segments as shown in the following figure:

Note that and are the coordinates of the centroid of the entire cross-section (uncracked core wall section).

Figure 6 - Cracked Concrete Wall Section Centroid Calculations

1.2. Determination of Reinforcement Strains and Forces

The following shows the calculations of forces in the reinforcement layers with the extreme tension (at bar 1) and extreme compression (at bar 29) strains. The calculations for the rest of layers are shown the table at the end of this section.

For extreme tension reinforcement layer (at bar 1):

(Tension)

For extreme compression reinforcement layer (at bar 29):

The area of the reinforcement in this layer is included in the area used to compute C_c (a = 30.70 in. > d₂₉ = 5.40 in.). As a result, it is necessary to subtract 0.85f_c’ from f_s29 before computing F_s29:

(Compression)

The same procedure shown above can be repeated to calculate the forces in the remaining reinforcement locations, results are summarized in the table shown in the next page:

1.3. Calculation of P_n, M_nx and M_ny