The spreadsheet presented in the previous post in this series uses the basic equation given below to find the location of the neutral axis:
For a cross section with any applied axial load at eccentricity e, measured from the compression face, and depth of the neutral axis X below the compression face:
INA = QNA(X + e)
Where QNA and INA are the first and second moments of area about the neutral axis.
To illustrate this relationship the value of INA has been plotted for a range of neutral axis positions for a typical precast bridge beam, shown below, and compared with the value QNA(X + e) for 5 different values of e.
The intersection of the INA and QNA(X + e) lines gives the depth of the neutral axis for the corresponding value of load eccentricity, e. This is illustrated in the second graph, where the value of (INA – QNA(X + e)) is plotted against the depth of the neutral axis, X, for a load eccentricity of 1 metre above the compression face. The intersection of this line with the X axis gives the depth of the neutral axis for the specified load eccentricity.