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Analytical Method vs. Approximate Method in Structural Analysis

 

by Ramon V. Jarquio

 

Author

Ramon V. Jarquio, P.E.

 

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My email: parabola5@aol.com

 

INSTRUCTIONS FOR INPUT OF DATA IN EXCEL SPREADSHEETS 

 

1. Rectangular Footing

 

In Spreadsheet “A” you enter English units for dimensions “b” and “d” as measured in inches.  Enter allowable soil bearing pressure, “q”, in kips per square inch. See Capacity Curve in English and SI units.

 

Sheet “B” shows the Capacity Curve of the footing when θ = 0.

Sheet “C” shows the Capacity Curve of the footing when θ = π/2.

 

2. Circular Footing

 

You enter in the spreadsheet English unit for dimension “R” as measured in inches. Enter allowable soil bearing pressure, “q”, in kips per square inch. See Capacity Curve in English and SI units.

 

3. Uniform Surcharge Load

 

You enter in the spreadsheets dimensions “a” and “b” as measured in inches. For dimension “h” enter the English unit of measurement of feet. For allowable soil bearing pressure, “q”, enter kips per square foot. See graph of the vertical stresses as this surface loading is distributed through depth.

 

4. Triangular Surcharge Load

 

For dimensions “a” and “b” enter measurement in inches. For dimension “h” enter measurement in feet. For allowable soil bearing pressure, “q”, enter kips per square foot. See graph of the vertical stresses as this surface loading is distributed through depth.

 

5. Steel Pipe

 

You enter in the spreadsheet the values of radius, “R” and thickness, “t” as measured in inches. Enter the value of steel yield stress, “fy” in kips per square inch. See the Capacity curve of the steel pipe.

 

NOTE: For further details and other solutions in structural analysis, please see the book entitled “Structural Analysis: The Analytical Method” published by Taylor and Francis, Boca Raton, 2007.

 

 

CIRCULARFOOTING RECTANGULARFOOTING
  STEELPIPECOLUMNCAPACITY TRIANGULARLOAD
  UNIFORMLOAD  

 

 

 

 

The Analytical Method in Structural Analysis

 

 
Expose’

 

We have computers now! We do not have to “count our fingers” anymore nor copy someone else’s work! We can find the truth by using basic mathematics, physics of the problem and Microsoft Excel to do all the calculations. Structural engineers that I know would like to find out the truth if only they can. Well there is now a way to find the truth as I have enumerated the steps above.

Apply basic mathematics, physics of the problem and Microsoft Excel to do all the numerical calculations in the equations derived by the analyst. If you cannot do these basic steps with your computers, then you are relegated to copying others and not knowing the truth about the veracity of the articles you are copying.

If you like to see how these steps worked for me you may consult my 3rd book published by Xlibris Corporation dated April, 2012. Here is the book trailer if interested.

Recommendations for “The Analytical Method”



Apply the 3 tools that I mentioned i.e., basic mathematics (algebra, trigonometric identities, analytic geometry and differential and integral calculus), physics (laws derived from observed facts) and Microsoft Excel of the computers to find the truth in anything that you meet in your profession. There is now a new paradigm required from the approximate to the exact method because of the availability of personal computers. If you have to copy existing literature because you have no choice just like me when I was still active as a professional engineer, you can always find the truth about the article you are copying and make the right decision as a professional.

For structural engineers, you have to apply Leonhard Euler’s principle which states that “any section is subjected to an axial load and a moment” using basic mathematics as well as physics of the problem. Superimpose the stress strain ratio of the material in the problem. Then you can integrate the axial forces and bending moments about the centroid. You can then compare your external load with the yield capacity of the material. Then you can get your real factor of safety versus the yield capacity. There is no more “factor of ignorance” as practiced worldwide using the standard interaction formula. You can do this because you were trained in school to have a working knowledge of the tools above-mentioned.

For your information I am the only one in the world doing the “rotation of axes” to a given concrete and steel section. These terms were never mentioned in my books but it is evident that this principle was indicated in all the figures and equations shown in my books.


You may follow my lead as exposed in my books or you can apply the 3 tools above-mentioned to find the truth about the article you are copying and determine whether it is true or not. We do not have to apply approximate methods anymore nor copy existing literature or someone else’s work. There is a saying “seek ye the truth and the truth shall make you free” that should be followed by a serious researcher.
 

Blog / WordPress : http:/www.ramonjarquio.com/blog

Facebook : www.facebook.com/public/Ramon-Jarquio

Dailymotion : http://www.dailymotion.com/video/44784056

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My email: parabola5@aol.com

 
Update

 

We are now in the age of computers and there are no more justifications for approximate methods. Every engineer who has a working knowledge of basic mathematics and physics can now update his information in engineering accurately using the analytical method I started since 1980.

My goal in becoming a member of Linkdin is to inform my fellow engineers that it is now time to implement the analytical method in structural analysis because we have computers to do all our numerical calculations.

Unfortunately, we have been buried for many years in copying existing literature and others in the approximate methods given to us by experts, well intentioned authors and professors that we thought expediency and practicality as well as credentials is the proof of the veracity of everything in our profession. Computers can now expose the truth in our profession and basic math and physics of a problem can produce the correct solution in anything that we do.

We can no longer copy nor blame others or credentials for any mistakes discovered when we apply our basic principles and knowledge that we learned in college. The practice of expediency and practicality worldwide can no longer be relied on if the methods employed are approximate. Tests results maybe slanted or not following the correct theory of structures.

We can now easily discover any mistake by analyzing the standard interaction formula copied worldwide. Use the Leonard Euler’s principle that any section is subjected to an axial load and a moment. This principle and basic mathematics will facilitate the calculations of the yield capacity of any given concrete or steel section for all eccentricity using Microsoft Excel. All that is needed by the analyst is the determination of the external load which can vary locally and compared to the yield capacity determined by Microsoft Excel.

We can now determine the real factor of safety versus yield capacity of the material and not rely on what I call ‘factor of ignorance’ using the standard interaction formula and the principal axes. I am the only one in the world who has done the ‘rotation of axes’ to obtain the correct solution of a structural problem. If an engineer cannot do the free body diagrams or the required mathematics to solve a problem, he or she may refer to my books where the analytical method is illustrated and the required parameters considered in the solution. The discovery of the truth by the engineer may be shared with others to spare them the trouble of doing the math or thinking of the correct solution. Computers can remember all the equations and any shortcuts developed by the analyst can be stored for reference.

Experts, authors and professors are welcome to comment and give a plausible reason for being silent all these years in the use of the analytical method in structural analysis. You may check the international proceedings of ISEC (Honolulu, Rome, Shunan, Melbourne and Zurich), SEMC (Cape Town, South Africa), ASCE (Cancun, New York, Nashville and Montreal), RILEM (Moscow 2005) Global Concrete, Scotland 2005 and HKIE 2005 Journal. These were my efforts in exposing the truth in engineering since I retired in 2000 from the NYCTA. I am not expected to solve everything in life but I have shown the way for the analytical method. You may follow my lead to discover other truths.

Email: parabola5@aol.com

The approximate method in structural analysis has been in practiced for so long world wide that it has shown itself in codes especially, ACI, AISC, ASCE and AASHTO as well as codes of other countries in some form or another. Basically, it is based on an assumption that the sum of the ratios of biaxial stress and axial load should equal unity or less. This ratio has been adjusted to account for results of tests. This is based on the standard flexure formula and essentially approximates a point in the capacity of a given section in the orthogonal axes X and Y. The true free body diagram of a given section as a result cannot be drawn properly for an external biaxial bending and axial load. This method has been copied over and over since the slide-rule era that it has become true to the copier.

 

The exact method is the true method to follow given stress-strain property of the material and given geometry of the section. For steel, it is known that the stress-strain is a straight line. For concrete, the stress-strain is parabolic. The equations of the stress-strain curve and the equations of the rectangle and circle can be written from analytic geometry. Once these equations are known, we can use calculus to compute the forces and moments that can be developed in a section. The solutions might take hundreds of equations but we have computers now to program these equations. The Excel program is very much suited to this task. All the given parameters of a section can be substituted and the minimum capacity due to biaxial bending and axial load can be shown in a curve known as the minimum capacity curve. The minimum capacity for a rectangular section is defined when the axis falls on the diagonal of the enclosing rectangle of the section. Most of the resultant falls inside the sector defined by the horizontal axis and the diagonal. The analyst can easily determined the truth about this procedure if he just applies basic mathematics. He does not need the mediocre approximate method above to verify the truth.

 

By using the exact method the capacity of a given section is determined using basic mathematics and the given parameters including the angle for minimum capacity for biaxial bending and axial load. The resulting curve is the capacity of a section to resist external loads. It is computed only once by the Excel method.

 

Knowing the capacity curve requires the analyst to determine the external load and plot against this curve. If inside the curve, the given section is adequate. If outside the curve, a bigger section is assumed by the analyst. He can determine his factor of safety within the curve and the decision to accept or reject is his decision.

 

You can see from above arguments that the approximate method has no relation to the analytical method. It is pure guesswork and has no relation to the true capacity of a section. This method has been copied by authors for many years; it is now time to apply the true method of analysis because we have computers now to integrate all the forces in a given section without guessing.

 

The author has produced two books defining the analytical approach outline above. These books are:

 Jarquio, R. V., Structural Analysis: The Analytical Method, 1st ed.,CRC Press/Taylor & Francis, Boca Raton, Florida, 2007

Jarquio, R. V., Analytical Method in Reinforced Concrete, 1st ed., Universal Publishers,  Boca Raton, Florida, 2004.

 

Anyone interested in the truth and the analytical approach may consult these books and questions maybe emailed to the author as desired.

 

“The Analytical Method in Structural Analysis”

I am still waiting for the questions that researchers would like to ask me about my books. They may not know what questions to ask since we have been brainwashed and buried for many years accepting literature done by authors and professors using the approximate method. We are now in the age of computers and this practice is no longer correct. I may not solve everything in life but I showed the way in my books to find the truth in anything. Note that I am the only one in the world doing the ‘rotation of axes’ while everyone is copying the principal axes to any section. The following are my tips for their understanding of the things exposed in my web site at http://www.ramonjarquio.com and the equations in my books which were published by Universal Publishers in 2004, the second one was by CRC Press/Taylor and Francis in 2007 and the third one published by Xlibris in March, 2012.

For structural analysis, apply the Leonhard Euler’s principle which states that any section is subjected to ‘an axial load and a moment’. Note: This refers to the capacity of any rectangular or circular section at all eccentricities.

Apply basic mathematics and physics learned in school to derive the equations involved in any problem. From analytic geometry, the equation of a circle and rectangular section is known.

From physics we know the equation of parabola to define the stress/strain relation for concrete. For steel the stress/strain ratio is linear.

Draw the free body diagrams with ‘rotation of axes’ to enable the engineer to analyze for equilibrium of internal and external forces.

Use our knowledge of differential and integral calculus to derive the equations for axial load and moment to any rectangular or circular section. The stress/strain diagram over the given section for any position is given in the free body diagram and therefore we can integrate the volume generated for axial load and a moment at the centroid of the section.

The Microsoft program can now do the numerical calculations of the equations derived above at all eccentricities. The graph produced is now what we call capacity of any section. There was no need to define ‘short’ or ‘long’ column as others do. However, if you want to do that, a short column is that which the given section is at full compression. A long column can then be defined as that where the stress at the section is now compression and tension. The external load when plotted in this envelope will define itself whether it is short or long. Therefore, we do not need the short or long column terms.

My discovery is that the minimum capacity of a rectangular section is at a diagonal. For a circular section reinforced with steel bars the minimum capacity is between reinforcement. However, for a circular section, assume a diameter through the center of any bar as the capacity axis since the variation between any two reinforcing bars is very small.

Now that the capacity of any section in rectangular and circular section is known from Microsoft Excel solutions, we can now determine the real factor of safety from the yield capacity versus the external load which can vary due to local regulations and applicable codes. Worldwide the practice is still what I call the ‘factor of ignorance’ in determining the factor of safety.

Apply the 3 tools of basic mathematics, physics and the computer that we learned in college to find the truth in anything that we encounter in our profession. The educational system must be revamped for a student to be familiar in the use of these 3 tools for discovery of the truth in anything that we do.

I started the exact analysis in 1980 and I was able to finish my research and produced my books after retirement from the TA. I would like all engineers to know the truth now that we have computers to perform all the numerical calculations. A new paradigm from the approximate to the exact method can now be implemented if one is interested in the truth. Expediency and compliance with regulations is not always the best approach in finding the truth. ‘Seek ye the truth and the truth shall make you free’ should be followed for enlightenment.

Ramon V. Jarquio, PE

Email: parabola5@aol.com

Website: http://www.ramonjarquio.com

 

 

The Analytical Method in Structural Analysis

There are 3 basic tools that you need to find the truth about anything! Here they are:

(1) Basic mathematics which includes Algebra, Trigonometry, Analytic Geometry and Differential and Integral Calculus.

(2) Physical principles involved in a problem - for structural the Euler's principle which states that "any section is subjected to an axial load and a moment" should be employed. The equilibrium equations and the stress/strain equation of the material are always included in the analysis.

(3) Microsoft Excel to do all the calculations required in the equations derived from (1) and (2) above.

The procedure looks so simple yet no expert in the world including all authors and professors have employed "the rotation of axes" for any section except me in this case! Everyone is copying from one another without knowing the truth about anything. It seems that credential is more important than the basic tools themselves.

The rotation of axes allows the analyst to employ the equilibrium equations so that the external load is made equal to the capacity of the section. The governing external load is different for each locale and analyst, but the centroid of the resultant external load is known from “ Pythagorean Theorem” when the moment about the orthogonal axes are known. However, the minimum capacity is at the diagonal of the rectangular section since the capacity decreases from the horizontal axis to the diagonal. It is this minimum capacity of the rectangular section that the author recommends when computing the real safety factor from the yield capacity of the member. The resultant external load can be compared to this resultant capacity of the rectangular section.

The model sample for testing and the results obtained therefore should be related to the minimum capacity of a rectangular section. This is not done by anyone and therefore the conclusions arrived by experts not related to the minimum capacity can be erroneous and therefore slanted to the ones they know from existing literature.

I am not expected to solve everything in life but what I did in my books was to show the proper way to follow my lead in finding the truth! People are satisfied to just copy literature including me when I was still active in the profession. I had no choice at the time but I vowed to find the truth myself if I can.

The derived equations are all in this book and Microsoft Excel can be used to do all the numerical calculations required by the equations. When you can program in Microsoft Excel, you may easily duplicate my spreadsheets mentioned at the back of my 3rd book published by Xlibris Corporation April, 2012.

 

 

 

 
Comment

 

Before you became an engineer, you have a working knowledge of basic mathematics and physics in your college years. Now that almost everyone has a personal computer, the truth can now be exposed using basics and the power of the computer to do the numerical calculations required by the equations you derived for a particular problem in your profession. A new paradigm from the approximate to the exact method is now possible because of the availability of personal computers. You do not have to copy the works of others without knowing the veracity of what you are copying. You do not have to blame others for any mistakes because you can prove whether what you are copying is true or not using your basic knowledge and the computer.

There is no more justification to use approximate method because we have computers now to do all the numerical calculations in no time after all the equations are entered and verified in the computer. The computer remembers all the equations required in the solution of a problem. To get a different answer we can vary any or all of the variables involved in our solution. We can use Microsoft Excel or any mathematical program we are familiar with.

If we have the talent and patience in basic mathematics and physics, we can derive all the equations required in a problem. We can find many truths in our profession if these 3 tools of math, physics and computer are employed in any problem encountered. Our conclusions will be tempered by the truths that we can discover by going back to basics that we learned in school. If you want to find out how the analytical method is applied you may consult my books.

 

To Whom It May Concern


There are no more justifications to use approximate methods in our structural analysis because of the availability of personal computers. Microsoft Excel or other mathematical program in the computer can now perform the numerical calculations required by the equations we derived for any problem encountered in practice using basic math and physics. Any truth can now be exposed using basic principles we learned in school.

“Slide rule” and “finite element” approximations are no longer valid and thus existing literature and well intentioned authors and professors should now apply the analytical method in any analysis. We can do the right approach by urging all engineers to use basic principles of math and physics harnessing the power of computers for all numerical calculations required. We can now do the updating of all information related to engineering using the exact method in our analysis. Our educational system should emphasize basic mathematics and physics for all students. It may take time to adopt a new paradigm from the approximate to the exact method because we have been buried for many years using the approximate methods. Computers are now readily available and are all over the place that it is relatively easy to harness the power of computers to do all the numerical calculations required in our equations for a particular problem.

Respectfully,

Ramon V. Jarquio

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