Structural Analysis The Analytical Method
This book illustrates the analytical procedures for predicting the capacities
of circular and rectangular sections in concrete and steel materials. It introduces
the capacity axis in the analysis, which is a geometric property not
considered in all the current solutions in standard literature. It precludes the
use of the current standard interaction formula for biaxial bending, which
is a crude and inefficient method. More importantly, the analytical method
will prove the necessity of utilizing the capacity axis not only for determining
the minimum capacity of a section for biaxial bending but also as a reference
axis to satisfy the equilibrium of external and internal forces. Under the
current standard interaction formula for biaxial bending, the satisfaction of
equilibrium conditions is not possible. Proving the equilibrium condition is
the fundamental principle in structural mechanics that every analyst should
be able to do.
Chapter 1 covers the derivation of equations required for the prediction
of the capacity of the footing foundation subjected to a planar distribution
of stress from soil bearing pressures. The capacity of the footing is defined
by a curve wherein the vertical axis represents the scale for total vertical
load on the footing, and the horizontal axis represents the scale for maximum
moment uplift capacity. This capacity curve encompasses all states of loading
in the footing including cases when part of the footing is in tension. Hence,
it becomes an easier task for a structural engineer to determine whether a
given footing with a known allowable soil pressure is adequate to support
the external loads. There is no need to solve for biquadratic equations to
determine solutions for a footing with tension on part of its area. The Excel
spreadsheet only requires entering the variable parameters such as footing
dimensions and allowable maximum soil pressure.
The procedures in the derivation are very useful in the prediction of
capacities of steel sections that are normally subjected to linear stress conditions.
This is the subject of Chapter 2.
Chapter 1 also includes the derivation of Boussinesq’s elastic equation
for the dispersion of uniform and triangular surface loads through the soil
medium. The derived equations will be useful in the exact value of average
pressure to apply in the standard interaction formula for settlement of footing
foundations without using the current finite-element method or charts
for this problem.
Chapter 2 deals with the application of the analytical method to predict
the capacities of steel pipe in circular or rectangular sections. The steps in
Chapter 1 together with the principle of superposition will be utilized to
derive the equations for the square and rectangular tubular sections. Equations
for the outer section are derived first, followed by the inner section.
The difference between the outer and inner section will determine the yield
capacity of any steel tubing.
The equations derived for the rectangular section in conjunction with
the principle of superposition will be utilized for the steel I-sections. Here,
the position of the capacity axis is chosen at the diagonal of the outer
rectangular section. The value calculated along the capacity axis represents
the component of the resultant bending moment capacity of the section. To
obtain the resultant bending moment requires the calculation of the component