1. Propagation of error Ignoring higher order terms, the error in function, based on the uncertainty in its inputs, can be estimated by where  is the symbol for uncertainty or error

2. Can be extended to as many variables as needed Useful for determining uncertainty of result based on measurement error

3. Example: required cross-sectional area for a beam Equation relating shear, load and area P=A f where P is load A is area f is shearing stress

4. Given: a beam that spans 28ft  1/12 ft Spacing between beams 9 ft Supporting live load 30 lb/ft 2  5 lb/ft 2 dead load 12.5 lb/ft 2  1 lb/ft 2 beam weight 22.5 lb/lin ft = 2.5 lb/ft 2 allowable horizontal shear f =230 psi total weight is 45 lb/ft 2  6 lb/ft 2

5. beam carries w = 45 lb/ft 2 * 9 ft = 405 lb/ft plus or minus 6 lb/ft 2 * 9 ft = 54 lb/ft load P is wL/2 (because we have two supports) maximum load is 3/2*load so for design A=3/4 * wL/ f

6. Using given numbers Now do propagation of error

8. Your first real numerical method - Root finding finding the value x where a function f(x)=0 You will encounter this process again and again

9. Example problem From water resources, Manning’s equation for open channel flow where Q is volumetric flow (m 3 /3) A is channel cross-sectional area R is hydraulic radius S is energy gradient n is Manning’s n

10. Hydaulic radius is defined as where P is the wetted perimeter So

11. For a rectangular channel h b Wetted area = bh Wetted perimeter = b+2h

12. Assume you are given the slope of the channel (roughly equivalent to the energy gradient), Manning’s n, the width of the channel, and the flow rate. Can you tell how deep the water runs? No, because this equation is not analytically solvable for h Can solve it graphically or numerically

13. Graphical technique basically consists of plot of h (or other unknown) vs known function of h. Where line crosses is root. Example: Given n, Q, S and b, plot lhs vs h and see when it is 0

16. Graphical method useful for getting an idea of what’s going on in a problem, but depends on eyeball. Consider a root finding method called Bisection

17. Bisection algorithm 1) Choose upper and lower bounds for x interval. The sign of f(x upper ) and f(x lower ) must be different. 2) Estimate root with 3) If f(x b ) has the same sign as f(x u ), then replace x u with x b If f(x b ) has the same sign as f(x l ), then replace x l with x b Repeat from 2) until convergence is reached.

18. Graphic example of how bisection works And some Matlab code to illustrate it