1. Bresler Reciprocal Load Method

1.1. Theory

This method approximates the ordinate 1/P_n on the surface S₂ (1/P_n, e_x, e_y) by a corresponding ordinate 1/P_n′ on the plane S′₂ (1/ P_n′, e_x, e_y), which is defined by the characteristic points A, B and C, as indicated in "Figure 7 - Reciprocal Load Method". For any particular cross-section, the value P_o (corresponding to point C) is the load strength under pure axial compression; P_ox (corresponding to point B) and P_oy (corresponding to point A) are the load strengths under uniaxial eccentricities e_y and e_x, respectively. Each point on the true surface is approximated by a different plane; therefore, the entire surface is approximated using an infinite number of planes.

Figure 7 - Reciprocal Load Method

The general expression for axial load strength for any values of e_x and e_y is as follows:

    PCA Notes (Chapter 7 Eq. 7)

Where:

P_ox = Maximum uniaxial load strength of the column with a moment of M_{nx_req} = P_{n_req} x e_y

P_oy = Maximum uniaxial load strength of the column with a moment of M_{ny_req} = P_{n_req} x e_x

P_o = Maximum axial load strength with no applied moments

This equation is simple in form and the variables are easily determined. Axial load strengths P_o, P_ox, and P_oy are determined using the method presented in “Interaction Diagram - Tied Reinforced Concrete Column Design Strength (ACI 318-19)” design example. Experimental results have shown the above equation to be reasonably accurate when flexure does not govern design. The equation should only be used when:

    PCA Notes (Chapter 7 Eq. 8)

1.2. Calculations

Check if flexure does not govern design:

    PCA Notes (Chapter 7 Eq. 8)

    O.K.

P_o, P_ox, and P_oy need to be determined to use this method:

P_ox is the uniaxial load strength when only M_{nx_req} acts on the column. As shown in "Figure 8 - Column Uniaxial Interaction Diagram (Pnox and Mnx_req)", for M_{nx_req} = 461.5 kip-ft, P_ox equals to 2241 kip.

Figure 8 - Column Uniaxial Interaction Diagram (P_nox and M_{nx_req})

Similarly, P_oy is the uniaxial load strength when only M_{ny_req} acts on the column. As shown in "Figure 9 - Column Uniaxial Interaction Diagram (Pnoy and Mny_req)", for M_{ny_req} = 192.3 kip-ft, P_oy equals to 2579.5 kip.

Figure 9 - Column Uniaxial Interaction Diagram (P_noy and M_{ny_req})

Note that both P_ox (2241 kip) and P_oy (2579.5 kip) are greater than the balanced axial force (1049.2 kip), so that the section is compression-controlled.

Using the values above, the axial load strength equals to:

    PCA Notes (Chapter 7 Eq. 7)

    (O.K.)