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109_CEG 4012 Notes Fall 2011 CEG 4012 Notes Fall 2011

109_CEG 4012 Notes Fall 2011 CEG 4012 Notes Fall 2011 - a 3...

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Then BC20 Useful Equations o Design problems to optimize footing size often end up with a quadratic or cubic equation in B. Solutions for both are given below (from CRC Math Handbook). Most of the time only one of the roots is practical. QUADRATIC EQUATIONS Any quadratic equation may be reduced to the form ax 2 + bx + C ... 0 -b :!: Vb 1 - 4ac x - 20 If a, b, and c are real then: If6 z - 4QC is positive, the roots are real and uncqual: If 6 z - 4ac is zero. thc roots are real and equal; If 6 z - 4ac is negativc, the roots are imaginary and unequal. CUBIC EQUATIONS A cubic equation, y3 + pyz + qy + , = 0 may be reduced to the form x" + ax + b ::t 0 by substituting for y the value, x - ~ . Here a - i (3q - p2) and b = * (2 p 3 - 9pq + 27,). _3/ b &/b 2 a j _3/ 6 _ Jr.2- For solution let, A - Y - "2 + V 4' + 27' B "-y + "2 + V 4" + 2f' then the values of x will be given by, A B _A+B A-B_q _A+B_A-B_q +, 2 + 2 v -,). 2 2 v -,). and y::o X - ~ - (Don't forget this step) If " q, , are real, then: b 2 a 3 If '4 + 27 > 0, there will
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Unformatted text preview: a 3 If '4 + 27 > 0, there will be one real root and two conjugate complex roots; 6 2 a J If 4' + 27 -0, there will be three real roots of which at least two are equal: b 2 a J If "4 + 27 < 0, there will be three real and unequal roots. o There are other methods to find roots. The Newton-Ralphson iterative formula uses the first derivative and normally converges nicely in just a few iterations. Excel, Quattro, and Lotus have built-in "solver" functions. MathCAD software will do this easily (probably overkill). Many scientific calculators also have built-in "solver" functions. o After obtaining an answer, the wise engineer plugs it back into the original equations to verify the solution....
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