Global Error in non-rotation of orthogonal XYZ axes

Rotation of orthogonal XYZ axes not done by any expert in the world except the undersigned.

The first thing that anybody wanting to know the truth is to apply basic mathematics and physics known to everyone. To implement this concept in structural analysis, a researcher should know the basic principles applicable in a problem. For structural analysis, the principles that are applicable are Euler’s, Hooke’s Law and Pythagorean Theorem.

Free body diagrams involved in a problem has to be drawn representing the variable parameters to get to the solution. Euler states that any rectangular or circular section is subjected to an axial load and a bending moment. The orthogonal X and Z axes represents the fixed axes to draw a rectangular section of dimension “a” and “b” for instance.

From analytic geometry, the equations of the sides of a rectangle can be derived using straight line equations. The X and Z axes are in the center of a rectangle and represent the horizontal and vertical axes of the section.

Superimposed on this rectangular section on the Y and X axes is the known behavior of the material in use as defined by Hooke’s Law. For concrete, the nature of stress in the breaking of concrete cylinders is parabolic. The equation of a parabola is known in analytic geometry, therefore a diagram of compressive stresses in the concrete can be represented.

Rotation of the orthogonal XYZ axes in a 3D structural analysis is effected by designating these fixed axes as 1-1, 2-2 and 3-3. By doing so, an angular rotation of X and Z axes at the center of the rectangle can be represented in the diagram. The minimum yield capacity of a rectangular section can be seen occurring at the diagonal (X and Z axes rotated to this position). Microsoft Excel of the digital computer can numerically show this condition.

The Pythagorean Theorem is required since applying integral calculus to the forces that are developed along the X and Z axes must have a resultant value and thus this principle is required. Once these hundreds of derived equations are programmed using Microsoft Excel 95, the numerical values of these equations are obtained in no time.

The yield capacity envelope of a rectangular section is a graph produced by the software designed by the undersigned and a close friend using Microsoft Excel 95 to program these hundreds of equations involved in a rectangular and circular section reinforced with any number of bars. This envelope shows the axial load in a vertical axis and the bending moment at the horizontal axis. The limits or boundaries of this yield envelope includes the standard key points of the current method.

The external load at any locality is variable and estimated by the designer or copy those used by others if available. The yield envelope is only a function of the strength of the material used and thus is constant in value. When the external load is plotted inside this envelope, the factor of safety is determined and failure is impossible to occur. Also (not mentioned by anybody except the undersigned) plotting the external load inside the envelope will determine whether the column is short or long automatically. The real factor of safety in the design is determined in this fashion. Unfortunately, this approach is not even done by any expert in the world.

Please note that I have a software programmed in Microsoft Excel since 1996 and registered in the Library of Congress and advertised in an ASCE magazine in 1996. This software signifying the exact approach was ignored by experts worldwide. Instead these experts were copying existing literature of approximate methods made to look accurate by finite-element procedures and fixed orthogonal XYZ axes to obtain the structural solution. This approach of maintaining the status quo of expedience and compliance is already proven erroneous by the undersigned 2 decades ago. You can now see the global consternation if this expose’ is proven by a refutable expert.

For any researcher to see how the above exact method is implemented for any rectangular section reinforced with any number of bars according to ACI regulations, please read my first book published by Universal Publishers in 2004. The researcher will note that contrary to the current practice of one pivot point favored by ACI and CRSI, the undersigned is using 2 pivot points – one for concrete and the other for steel. The case of 2 pivot points occur in concrete when the compressive section is less than balanced condition. At this condition the tensile steel stress is limited to the yield at the outermost point.

The title of the book illustrating the exact method is ‘Analytical Method in Reinforced Concrete’. Nobody will accept the error committed in the non-rotation of axes that the exact method illustrated in this book was ignored by almost everyone since it was published in 2004.

Subsequently, CRC Press/tandf published in 2007 my second book illustrating in chapter one the governing equations which eluded K. Terzaghi and W. Steinbrenner in integrating Boussinesq’s elastic equation. Chapter 2 tabulated the column capacities of I-sections in the AISC manual to be used by any designer. Chapter 3 illustrate the one-pivot point current practice but with rotation of orthogonal XYZ axes missed by experts. Chapter 4 includes CFT columns following the current practice of one pivot point.

My third book published by Xlibris of Indiana in 2012 includes the yield capacities of channel sections in the AISC manual and the bearing capacity factor of cohesion less sand or rock.

My fourth book entitled “The True Analytical Method in Structural Analysis” is now in a CD and a CD of Microsoft Excel workbook for structural problems encountered by the undersigned since becoming a Professional Engineer in the state of New York in 1978. This book illustrates the use of 2 pivot points instead of only one as the current practice for reinforced concrete. These CD’s are now in the safekeeping of my sons until a refutable USA publisher will publish this blockbuster book for world consumption. The proof of the exact method is in a software kept by the undersigned and also entrusted to my sons.