Facebook – The Truth You cannot be blind and ignorant. Use Microsoft Excel in derived equations to discover anything. Error was caused by non-rotation of axes by authors and professors. Expediency and compliance is not the approach. Interpretation of lab data and the calculation for yield strength are made incorrect by the non-rotation of axes. Basic mathematics and physics do not lie. Introduce a variable to rotate the axes and get the correct solution. Business as usual cannot be justified without a new paradigm to rotate the axes and computer usage in equations. All these years, brilliant minds in engineering have been misled by the NON-ROTATION OF AXES. DO YOUR MATH & STOP COPYING. Practitioners should stay within elastic limit where the behavior of stress/strain is predictable by Hooke’s Law. The column capacity axis for any rectangular section is at the diagonal where the minimum yield capacity is developed. The column capacity axis for any reinforced concrete circular column section is a diameter thru the center of any bar. You have been fixated with the horizontal & vertical axes as the X & Y axes and forgot to rotate these axes to do your math. The resultant yield capacity by Pythagorean Theorem decreases from the horizontal axis to the diagonal of a rectangular section. The yield capacity for a RC circular section varies between 2 bars. Minimum yield is at a diameter thru center of any bar. If basic math & physics are not followed, interpretations of measurements from modeling are slanted & conclusions incorrect. Microsoft Excel gives the graph of yield capacity where the axial load is on vertical axis & bending moment on the horizontal axis. External load which varies due to location is now plotted on this graph to get the real factor of safety vs. yield capacity. USA must adopt a new paradigm for the rotation of X & Y axes and lead the world. End this ignorance. Coordinates of reinforcing bars in a circular & rectangular section must be included in the analysis for yield capacity. Short column is 1 stress (compression or tension) and long column is 2 stresses (tension & compression) in section. Plot external load on the column capacity curve & obtain factor of safety. Column defines itself short or long. Equilibrium of internal (yield) & external load (locality) occurs between horizontal axis & diagonal of a rectangle. Equilibrium of internal & external load lies in the central angle between any 2 adjacent bars in a circle. Factor of safety = ratio between yield and external load should be greater than 1.00 to insure absence of failure. Embedment (development) length into concrete = {(tension)/(bond strength of reinforcing bars)}. Development length includes the geometric lengths of hooks from cut off points of tension bars. Property of parabola: y = {m(x)(L – x)/(L/2)2}; y & m are the ordinate and middle ordinate respectively. From analytic geometry, derive the equation of a rectangle using straight line equations for the sides. Equilibrium of internal & external forces is made by rotation of axes X&Y. Position of external load =M/P. Minimum yield capacity of rectangular & circular section including coordinates of steel bars should be used in design. Basic math & physics do not lie! Now that we are in the age of computer, Euler’s and Hooke’s Law can be easily applied. Discard all literature not rotating X&Y axes because you cannot perform basic math without rotation of axes. Consider the conclusions arrived at by experts without rotating X&Y axes. Where is basic mathematics? Consider interpretations of measurements from modelling w/ X&Y axes not rotated. Consistent w/ Euler’s & Hooke’s Law? XZ for stress/strain equations: Parabola for concrete & straight line for steel. XY for rectangular & circular equations. Case 1: Moment maximum: Compression = tension at edges of section. For steel: P = 0. Long column zone. Case 2. Full compression: Tension at edge = 0. Short & long column separation. Strain triangular. Case 3. Compression zone greater than depth of section. Strain is trapezoidal. Short column zone. Case 4. For steel: Compression strain is uniform & “c” becomes infinite. M = 0 & P = maximum. Short column zone. Case 5. Concrete: “c’ is moved from max M to P = 0. Steel: (see case 1) M = maximum & P = 0! Long column zone. For concrete: Pivot is at strain 0.003 for compression zone greater than balanced condition. For steel: Pivot is at fy/29000 for compression zone less than ½ depth of section as concrete is pivoted at 0.003. Special consideration for compressive steel in concrete as strain is less than 0.003 & therefore stress is fixed to fy. Rotate X&Y axes to diagonal for minimum yield capacity as XZ plane moves w/ stress/strain equations. Apply basic mathematics (algebra, trigonometry, analytic geometry & calculus) to Euler’s & Hooke’s Law. Column capacity curve by Excel. Concrete: graph is curvilinear. Steel: compressive graph is triangular. Plot external load (varies locally) on capacity graph for factor of safety. Depth inside chart depends on designer. Internal yield capacity equations will remain forever as long as Euler’s & Hooke’s Law are valid. …………………………………………………………………………………………………………………….. Details are shown in my book entitled “The Analytical Method in Reinforced Concrete” published by Universal Publishers of Boca Raton, Florida in 2004. Subsequent books, one published by CRC Press/Taylor and Francis in 2007 and another by Xlibris in 2012 reflect the analytical method as applied in steel sections. However, CRSI method of pivoting the entire section especially when compressive depth in concrete is less than balanced condition (see Case 5) is shown for comparison with the preferred method. The yield capacity is slightly more since more steel forces from the concrete edge is calculated at fy.
Posted on: Tue, 27 Jan 2015 03:10:54 +0000
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