Structural Analysis: The Analytical Method

This the title of my second book published by CRC Press/tandf of Boca Raton, Florida in 2007. One pivot point per Hooke’s Law and rotation of orthogonal XYZ axes in a 3D structural analysis to validate the use of basic mathematics and physics has been employed. The hundreds of integral calculus equations required to implement the established principles of Euler’s, Hooke’s Law and Pythagorean Theorem were authenticated by Microsoft Excel 95. A software was used by the undersigned for this purpose of an analytical solution.

Current practice of one pivot point for concrete with rotation of orthogonal XYZ axes as well as the coordinates of reinforcing steel bars for a circular and rectangular section were included in this software that only the undersigned possessed. There is a reduction of about 25% in the minimum column yield capacities of the rectangular section from the horizontal axis to the diagonal of the rectangle. This is not known by world experts because finite-element methods on fixed orthogonal XYZ axes is wrong but copied by everyone in the world. Analytic geometry showed the equation of a parabola and the sides of a rectangle is defined by the straight-line equations. From these properties we can derive the hundreds of analytical integral calculus equations to satisfy the established principles of Euler’s, Hooke’s Law and Pythagorean Theorem.

Measurements in the laboratory from modelling must be reconciled to the derived integral calculus equations from the established principles of Euler’s, Hooke’s Law and Pythagorean Theorem. Unfortunately, this is not currently done and therefore wrong conclusions of the researchers became the results of this ignorance.

The chapters of this book are described for the reader to digest and implement the EXACT METHOD of structural analysis. The reader must exercise his knowledge of basic mathematics and physics learned in school to follow the free body diagrams and integral calculus equations derived and shown in this book.

Chapter 1 - Deals with the prediction of circular and rectangular concrete foundation bearing capacity using one pivot point for compressive stresses in concrete. For a rectangular section, the diagonal is the bearing capacity axis to obtain the minimum bearing capacity. Planar distribution of stress under the footing foundation is assumed and graphs of minimum bearing capacity is provided. The boundaries of the different cases of loading on footing foundations are analyzed for bearing capacities. For surface loading as the roadway embankment imposes vertical stress on the underlying soil foundation, the Boussinesq’s elastic equation was integrated (using 3 levels of integration) by the author and applied the pressures underneath roadway embankments using superposition of geometric elements. The integration of Boussinesq’s elastic equation eluded the talents of W. Steinbrenner and Carl Terzaghi and therefore approximate methods using finite-element procedures on fixed orthogonal XYZ axes and Newmark charts were used to approximate the pressures underneath embankments. This is the current practice even today because researchers are either ignorant or very lazy to integrate the Boussinesq’s elastic equations. Therefore for expediency and compliance experts have been copying each other from the tons of literature on approximate methods.

You may see the proceedings of ISEC-02 (Sept. 23-26, 2003 Rome, Italy) entitled “System-based Vision for Strategic and Creative Design” by Franco Bontempi pp. 1997-2002.

You can also refer to the proceedings of an international conference on global concrete entitled “Application of Codes, Design and Regulations” by Ravindra K. Dhir, Moray D Newlands and Andrew Whyte.

Chapter 2 - Shows the yield capacity of steel sections whose properties are indicated in the AISC Steel Manual. One pivot point for compressive and tensile stresses for steel is introduced such that steel’s stress/strain is a straight line. Again, analytic geometry and rotation of orthogonal XYZ axes were used to derive the equations involved in the preparation of tables for yield capacities of these sections. For rectangular sections, the diagonal of the enclosing rectangle is the column capacity axis for yield capacity calculation for sections mentioned in the AISC steel manual. Superposition of the rectangular sections have been employed to derive the minimum column yield capacities of these sections. Tabular charts for these sections in the AISC steel manual are provided for the designer to plot the external loads in order to determine his real factor of safety in design. Practitioners can easily plot the external load on the envelope of minimum column yield capacity defined in these charts. Current practice uses the interaction formula and guesswork which is obviously incorrect for the calculation of factor of safety in design. DELIBERATE IGNORANCE is still being practiced even with the presence of these charts in this book.

Chapter 3 - Illustrates the correct execution of the current method of one pivot point for reinforced concrete and rotation of XYZ axes plus the parabolic behavior of concrete ultimate stress as well as the use of the coordinates of reinforcing rods at the center of the sections. Current practice is not doing this procedure and using incorrect equivalent parabola for concrete without even the rotation of XYZ axes. Do not be confused in the term plasticity since the steel reinforcement is embedded into concrete. This condition is inherent in reinforced concrete when the elastic strain of steel is different from the elastic strain of concrete (0.003 in USA and 0.0035 in Canada). The yield capacity of a reinforced concrete section is limited to the elastic stress of Fy (yield stress of steel) and Fc’ (ultimate stress of concrete) and therefore the condition of plasticity is automatically considered in the analysis for yield capacity of a given section. An example of a reinforced concrete column is shown for verification of a structural expert.

Refer to the proceedings of ISEC-04 (September, 2007, Melbourne, Australia) on page 727 entitled “Innovations in Structural Engineering and Construction” by Y. M. Xie and I. Patnaikuni.

Chapter 4 - Shows the yield capacity of CFT columns and one pivot point in concrete filled tubes. So far no one in the world has done this exact analysis except the author because it will take time to derive all the integral calculus equations applicable to this problem. Microsoft Excel 95 was used in the workbook to confirm the derived analytical equations.

Refer to the Joint International Conference on Computing and Decision Making in Civil and Building Engineering (June 14-16, 2006) in Montreal, Canada. The title of the article (pp. 503-512) is “Limitations of the standard Interaction Formula for Biaxial Bending as Applied to Rectangular Steel Tubular Columns”.

You may also refer to ISEC-03 proceedings in Shunan, Japan in 2005 entitled “Collaboration and Harmonization in Creative Systems” by T. Hara.

RILEM also included in their proceedings (2005, Moscow) my article entitled “Limitations of the Standard Interaction Formula for Rectangular Columns”.

Note: Electronic digital computers using Microsoft Excel can verify the numerical simplification of all derived equations in this book. Worldwide experts should be able to confirm the veracity of this statement although it will take time for several of them in structural mechanics to prove this statement. Microsoft Excel 95 workbooks were used by the author to verify the statements made in this book.

Without this book, DELIBERATE IGNORANCE will be the global practice since reference books do not rotate the orthogonal XYZ axes to implement basic mathematics and physics on well-established Euler’s, Hooke’s Law and Pythagorean Theorem. Book 1 and the finished but unpublished Book 4 recommends the 2 pivot points for an EXACT METHOD of analysis.

Ramon V. Jarquio, P.E.

Email: [email protected] or [email protected]

Engineering website: https:/www.ramonjarquio.com