Rotation of Orthogonal XYZ Axes

Knowledge of basic mathematics and physics is required to be able to apply this procedure to well established principles of Euler’s, Hooke’s Law and Pythagorean Theorem known to science. We are now fortunate to have electronic digital computer and Microsoft Excel to numerically confirm all our derived equations relevant to the problem at hand. There is no longer the need to use approximate methods previously invented by copying antiquated literature on approximate methods before computers. Unfortunately, the whole world is still using this old method of approximations since tons of existing literature on approximate methods can be easily referred to for expediency and compliance. The steps to follow for a new paradigm of exactitude are listed here for anyone interested in  the truth without copying erroneous information.

1. Temporarily assign 1-1, 2-2 and 3-3 to the orthogonal XYZ axes so that these axes can rotate in position while the numerical assignments remain fixed. The orthogonal XYZ axes and the numerical assignments are drawn at the center of a composite circular and rectangular section.

2. The free body diagrams for a composite circular and rectangular section can now be drawn such that a variable angle representing the movement of orthogonal axes XYZ are effected. XZ axes can be used for circular and rectangular sections.

3. The stress diagram for implementing Hooke’s Law can be plotted on the XY plane. For concrete the parabola from analytic geometry can be used to derive the forces developed in a circular and rectangular section. For steel the stress diagram is linear. Integral calculus is used within the boundaries of a short and long column definitions as seen in a balanced condition of strain. The Y axis rotates in position while the X and Z axes rotates within the central angle defined by any 2 reinforcing bars in a composite circular section and in a rectangular section varies from the horizontal to the diagonal of the rectangle.

4. Write the coordinates of reinforcing bars using basic mathematics in a composite circular or rectangular section. For a rectangular section the bars are equally spaced along the sides as per ACI limitations. The numerical confirmation of the minimum column yield capacities of these bars is done using Microsoft Excel. This procedure is not currently done worldwide.

5. There will be hundreds of integral equations derived in invoking Euler’s law that any section is subjected to an axial load and a bending moment. For a circle the diameter is the axis for a minimum column yield capacity of a concrete section for an even number of bars. For an odd number of reinforcing bars, the minimum column yield capacity occurs between any 2 reinforcing bars.  For a rectangle the diagonal is the axis for minimum column yield capacity of a concrete section. This condition can be numerically established by Microsoft Excel. This is not done currently in the approximate method adopted worldwide. For a rectangular section there is a reduction of column yield capacity from the horizontal axis to the diagonal of a rectangle that is not known by the current method of approximation.

6. In this procedure, 2 pivot points for balanced condition and other cases are implemented while the current one pivot point adopted worldwide is obviously incorrect including non-rotation of XYZ axes. Without rotation of orthogonal XYZ axes, basic mathematics and physics cannot be applied. 2 pivot points are used because steel has its own strain limitation at yield capacity. Current procedure uses only one pivot point - that of concrete.

7. Beyond balanced condition of strain, compressive strain for steel remains at yield capacity while the tensile capacity is limited at yield. This is the 2nd pivot point to the neutral axis to limit steel’s yield capacity. This is not done in practice.

8. Measurements in the laboratory for modelling reinforced concrete should reconcile with the derived equations shown in my books and proceedings of ISEC, SEMC, ASCE, RILEM and Global concrete from 2001 to 2011. Otherwise wrong conclusions will be made by the researchers not knowing the truth above-mentioned. I have also submitted an article using 2 pivot points in  reinforced concrete column to HKIE structural engineers which they published in 2005 but no one rebutted me about the accuracy of this exposure as of this writing.

9. The details and free body diagrams for 2 pivot points and applying basic mathematics and physics mentioned above can be seen in my book entitled “Analytical Method in Reinforced Concrete”, published by Universal Publishers of Boca Raton, Florida in 2004. Correct software in reinforced concrete can be done using the derived equations shown in this book.

10. The proof of the true analytical method in structural analysis for concrete is shown in my software prepared in Microsoft Excel 95 and registered in the Library of Congress as well as advertised in an ASCE magazine in 1996 but ignored worldwide. I still have a copy of that software developed by my friend, William R. Mulford who was so gracious to let me use it for my personal research.

11. The equations for steel as shown in AISC Manual have been published by CRC Press/Taylor and Francis in 2007 in a book entitled “Structural Analysis: The Analytical Method”. Chapter 2 of that book showed the tabulated values for I-sections where external load can be plotted to determine the factor of safety in design. No one is using these exact values for design purposes. Current practice is still using the approximate methods recommended by experts, which are of course erroneous in this age of electronic digital computers. Chapter one of this book showed the actual integration of Boussinesq’s elastic equation which eluded the talents of W. Steinbrenner and Karl Terzaghi and therefore Newmark charts and approximate methods of finite-element is currently used. Chapter one also show the correct approach for foundation analysis not shown in any book.

12. Xlibris of Indiana published in 2012 a book entitled “The Analytical Method in Structural Analysis” showing the correct one pivot point analysis and recent addition to the 2nd book (one article per chapter) by CRC Press.

13. I have completed my 4th book book entitled “The True Analytical Method in Structural Analysis” and a CD of Microsoft Excel 95 workbook for numerical confirmations of all my derived exact equations for structural problems. This book is using 2 pivot points with rotation of XYZ axes instead of only one for concrete without rotation of orthogonal XYZ axes as done in practice. These 2 CD’s are now in my possession and my proposal was sent to several USA refutable publisher of technical literature to contact me. So far no refutable USA publisher is willing to circulate these 2 CD’s. Their reasoning becomes obvious when all published books are compared to this book showing the exact analysis using the readily available computer and Microsoft Excel. All literature using approximate methods will have to be discarded and relegated to posterity.

14. America can be great again if this truth is published. Then we can proudly state to “Keep America Great”.

The world is currently buried or shall I say brainwashed by the tons of existing literature done before the advent of electronic digital computers. No global expert is willing to prove that the current practice is erroneous because either they are ignorant or lazy to derive the hundreds of integral calculus equations to implement the well-established principles of Euler’s, Hooke’s law and Pythagorean Theorem known to science. Microsoft Excel of electronic digital computer can numerically confirm these derived equations. I exposed these equations in my technical books published in 2004, 2007 and 2012 as well as the proof of these equations in my software prepared in Microsoft Excel 95 and registered in the Library of Congress in 1996. These experts have been relying on copying existing literature for expediency and compliance.

Ramon V. Jarquio, P.E.