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Analytical Method vs. Approximate Method in Structural Analysis

 

by Ramon V. Jarquio

 

Author

Ramon V. Jarquio, P.E.

 

 

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  

 

 

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.