Proof of the Exact Method
This software was made more than 2 decades ago to make it easier for a structural engineer to analyze any reinforced concrete rectangular and circular section that has bedeviled many in their search for analytical method in reinforced concrete. This limitation forced them to copy others for expediency and compliance to the many regulations adopted by many agencies of the approximate methods invented before the advent of electronic digital computers.
When Microsoft Excel 95 became available for use, my software was programmed by William R. Mulford of Westbury, New York with me to verify the hundreds of integral calculus equations involved in the analysis for the minimum column yield capacity for a composite circular and rectangular column section in reinforced concrete.
I used the 2 pivot points (one for concrete and the other for steel) as opposed to the current one pivot point (for concrete alone) being used today. This software was advertised in an ASCE magazine and registered in the Library of Congress in 1996 but was globally ignored by the structural community. I kept a copy of this software for my personal research until 8 years later I had to publish the equations in a book entitled “Analytical Method in Reinforced Concrete†by Universal Publishers of Boca Raton, Florida in 2004. Any computer expert can use the equations in this book to create a software of the true analytical method in structural analysis. Here are the justifications for this software.
1. Euler’s Principle. Any section is subjected to an axial load and a bending moment. Euler submitted his differential equation for this truth. Experts copied this differential equation and tried to adopt this principle to their approximate methods. However, there was no way for Euler to simplify the hundreds of integral calculus equations for the forces that are developed in a composite circular and rectangular sections before the advent of electronic digital computer and Microsoft Excel. This became the beginning of deliberate ignorance of worldwide experts who were just copying others before them and the tons of literature supporting approximate methods which can be proven by Microsoft Excel as erroneous without the rotation of orthogonal XYZ axes in conjunction with Euler’s, Hooke’s law and Pythagorean Theorem.
2. Hooke’s Law. The stress is proportional to strain as seen in the equation, ? = PL/AE where ? is the elongation, P is the axial load, A is the area of the section, E is the Young’s modulus of elasticity and L is the length of the section. Strain is represented as (?/L) and stress is represented in the equation as (P/A). Compressive stresses in concrete is parabolic and universally accepted. Parabola’s equation is in analytic geometry and its properties are known and therefore we can derive its application in structural mechanics. Current method uses the rectangular stress block which is not even statically equivalent to a parabolic stress but copied worldwide. It is of course erroneous to think so, much less copy the literature supporting this concept. Forces developed in a composite circular and rectangular sections are not even integrated as it should for exactitude in the current approximate methods. For steel material, the stress is linear and tension and compression strain are equal. For a composite section there will be 2 pivot points, one for steel and the other for concrete strain. Current method uses only one pivot point and that is for concrete only. It will be unavoidable for steel to be in the plastic condition for all compressive stresses in the concrete. When concrete is cracked, stress in steel is limited to fy from the neutral axis. This is not done in the current method when compressive depth of concrete is less than balanced condition. For steel sections, a column software is not necessary since Microsoft Excel workbook is sufficient for superposition of rectangular sections to arrive at a minimum column yield capacity by rotation of orthogonal XYZ axes.
3. Pythagorean Theorem. The resultant bending moment is the square root of Mx2 and My2. This is applied with rotation of orthogonal XYZ axes to determine the minimum column yield capacity of a circular and rectangular section within the elastic limit. We know that basic mathematics and physics do not lie. Here the coordinates of reinforcing bars with respect to the center of section are mathematically written for composite circular and rectangular sections. ACI regulations for spacing and clearances of reinforcing bars are followed for accuracy. Development length of reinforcing bar is evaluated from the equation, Ld = 0.094 Asfy/(fc’)1/2 where As = area of reinforcing bar, fy = yield stress of steel reinforcing bar and fc’ = ultimate compressive stress of concrete.
4. Minimum yield capacity envelope. This is a by-product of the column capacity software. This graph is formed by the standard key points as the boundary of the elastic limit for short and long column definitions, with the horizontal axis as the bending moment and the vertical axis as the axial load. The point where the axial load is zero on the horizontal axis represents the beam condition of this column section. This graph is unvarying since it is only a function of the strength of materials used. However, external loads to which this section is subjected is variable in any location and regulated by local codes and practices. Once the external load is determined, it is plotted on this envelope and depending on its position on the graph, will determine whether the application of the section is short or long column automatically. Once the external load is inside the envelope of minimum yield column capacity, the factor of safety in design is calculated where failure is impossible to occur. This procedure is never done in practice because the current method uses the interaction formula and guesswork for factor of safety in design
Conclusion: I wish that structural experts worldwide will open their eyes to the brainwashing going on by existing literature on approximate methods and do the right thing now that we are in the age of electronic digital computer and Microsoft Excel.
Ramon V. Jarquio, P.E.