2. Bresler Load Contour Method

2.1. Theory

In this method, the surface S₃ (P_n, M_nx, M_ny) is approximated by a family of curves corresponding to constant values of P_n. These curves, as illustrated in "Figure 10 - Bresler Load Contours for Constant Pn on Failure Surface S3", may be regarded as “load contours”.

Figure 10 - Bresler Load Contours for Constant P_n on Failure Surface S₃

The general expression for these curves can be approximated by a nondimensional interaction equation of the form:

    PCA Notes (Chapter 7 Eq. 9)

Where:

M_nx = The nominal biaxial moment strength in the direction of x axis

M_ny = The nominal biaxial moment strength in the direction of y axis

M_nox = The nominal uniaxial moment strength about the x-axis

M_noy = The nominal uniaxial moment strength about the y-axis

Note that (M_nx and M_ny) are the vectorial equivalent of the nominal uniaxial moment M_n. The values of the exponents α and β are a function of the amount, distribution and location of reinforcement, the dimensions of the column, and the strength and elastic properties of the steel and concrete. Bresler indicates that it is reasonably accurate to assume that α = β; therefore, the previous equation becomes (shown graphically in "Figure 11 - Interaction Curves for Bresler Load Contour Method"):

    PCA Notes (Chapter 7 Eq. 10)

Figure 11 - Interaction Curves for Bresler Load Contour Method

When using the previous equation or figure, it is still necessary to determine the α value for the cross-section being designed. Bresler indicated that, typically, α varied from 1.15 to 1.55, with a value of 1.5 being reasonably accurate for most square and rectangular sections having uniformly distributed reinforcement.

With α set at unity, the interaction equation becomes linear:

    PCA Notes (Chapter 7 Eq. 11)

The previous equation would always yield conservative results since it underestimates the column capacity, especially for high axial loads or low percentages of reinforcement. It should only be used when:

    PCA Notes (Chapter 7 Eq. 12)

2.2. Calculations

The reference conservatively selected α = 1.0 due to a lack of available data. Check if flexure does not govern design:

    PCA Notes (Chapter 7 Eq. 12)

    (No Good)

Although , the reference decided to carry out the necessary calculations for illustration purposes.

Since the section is symmetrical:

("Figure 6 - Column Section Capacity Interaction Diagram (spColumn)")

Using the interaction equation becomes linear:

    PCA Notes (Chapter 7 Eq. 11)

    (O.K.)