Civil Engineering Computation. Initial Meeting and Class Setup EXCEL Review Problems

Transcription

1 Civil Engineering Computation Initial Meeting and Class Setup EXCEL Review Problems

2 Class Meeting Times The class runs from 12:30 until 2:20 on Monday with the lab following at 2:30 This was the original plan but somehow the University didn t make the change in the schedule The class does not meet on Wednesdays. 2

3 First Issue Our first task is to complete a short survey on GIS. Please go to 3

4 First Issue Our first task is to complete a short survey on GIS. Please go to 4

5 Next Task 5 Now we can setup the submission site for you homework and projects.

6 Next Task 6 Now we can setup the submission site for you homework and projects.

7 Next Task 7 Now we can setup the submission site for you homework and projects.

8 Next Task 8

9 Next Task 9

10 Next Task Substitute your file name for DrP. 10

11 Next Task 11

12 Next Task 12

13 Computation 13 What tool do you need for the computation?

14 Computation What tool do you need for the computation? Hopefully, most of you can do this type of calculation in your head, or at least with pen and paper. You really don t need a calculator even though you could use on to make the calculation.

15 Computation What tool do you need for the computation? Now the computation is more complex and will require a calculator or some type of calculation device. No matter what you use for the computation, the question would be how many significant figures should you report.

16 Computation Significant Figures 16

17 Computation Significant Figures How to determine? How do you determine significant figures in a number? The significant figures are digits used to establish the value of the number. Zeros shown to merely locate the decimal (regardless if the decimal is shown) are typically not included as significant. 17

18 Computation Significant Figures 18

19 Computation Significant Figures What do they mean? Since virtually all numbers used in engineering are based on measurements, they have inherent uncertainty. l The number of significant figures implies the magnitude of that uncertainty. l 19

20 Computation Significant Figures What do they mean? Reporting the weight of an object as 4572 pounds implies an uncertainty on the order of a few pounds. l Reporting the weight as 4570 pounds implies an uncertainty on the order of a few 10 s of pounds. l 20

21 Computation Significant Figures No number is perfect, only good enough. Typically, in civil engineering, 2 or 3 significant figures would be appropriate. RARELY, does an engineer work with more precision than that! 21

22 Computation Significant Figures To illustrate how precise a number with three significant figures is, consider two numbers (3 sig figs, compared to 8 sig figs): 12,300,000 and 12,345,678. Compare the difference between these: (12,345,678-12,300,000) / 12,345,678 = = 0.37 % 22

23 Computation Significant Figures These numbers differ by less than one-half of one percent! Think of anything designed by civil engineers where everything is know to a higher degree of accuracy than 1 percent. 23

24 Computation Significant Figures What does it hurt to report a number with excessive significant figures? First of all, it shows that the engineer doesn t understand what numbers are. l Numbers are merely our way of quantifying the world. l Excessive significant figures communicate a level of precession that is not present. l 24

25 Computation Significant Figures So we will almost never present a solution to a civil engineering problem with more than two decimal places shown We won t always use two decimal places but that will be the upper limit usually 25

26 Computation 26 Now we will go to another problem. You are supervising the construction on a site and you have to estimate the payroll cost for the previous week. You have a 15 person crew and each has worked a different number of hours and is paid at a different rate. You also have an overtime rule that states time over 37.5 hours is paid at 1.5 times base rate and time over 48 hours is paid at 2.0 times base rate.

27 Computation ID Hours Pay Scale $17.26 $14.04 $31.81 $30.85 $27.22 $26.10 $27.61 $38.68 $28.65 $10.81 $17.07 $28.28 $27.38 $16.68 $12.43 Go ahead and work this out with you calculator. Write down your stand and end times and calculate how long it takes you to work this out.

28 Computation ID Hours Pay Scale $17.26 $14.04 $31.81 $30.85 $27.22 $26.10 $27.61 $38.68 $28.65 $10.81 $17.07 $28.28 $27.38 $16.68 $12.43 Now I am going to assume that this took a bit of time and was rather tedious. The idea isn t to work longer, it is to work harder. I am sure that most of you knew or guessed that a spreadsheet would have make this much easier and you would be correct.

29 Computation 29

30 Computation Another advantage that a spreadsheet would give us is the repeatability of calculations. Next week, the only thing that would have to be changed would be the number of hours each individual worked (assuming the same crew was on site). 30

31 Computation It is also easy to check and see how, each of the calculations was made. This makes checking for errors much easier if you are reviewing work from someone else. 31

32 EXCEL Review Problems Create an EXCEL worksheet that simulates throwing a pair of dice 20 times. To do so, enter the formula =RANDBETWEEN(1,6) in the first 20 rows of columns A and B. Each value will be a random integer between 1 and 6, representing the outcome of one random die. 32

33 EXCEL Review Problems Within each row of column C, please the sum of the corresponding values in columns A and B. In column D, place one of the following labels within each row: 33

34 EXCEL Review Problems If the value in column C is 7 or 11, display You Win If the value in column C is 2, 3, or 12, display You Lose Any other value, display Inconclusive 34

35 EXCEL Review Problems 35 The following table describes an environmental reaction for three chemical compounds as a function of time.

36 Time (sec) Conc A (mg/l) Conc B (mg/l) Conc C (mg/l)

37 EXCEL Review Problems Enter the data into an EXCEL worksheet. Plot all three concentrations as a function of time on the same graph. Each plot should be an x-y scatter type plot with a smooth curve between the points. Show all the individual data points. Display each data set in a different color. Save the graph on its own worksheet (chart). 37

38 EXCEL Review Problems 38

39 EXCEL Review Problems 39

40 EXCEL Review Problems 40

41 Modeling a Physical System You re given the task of predicting the velocity of a bungee jumper as a function of time during the free-fall part of the jump. This information will be used as part of a larger analysis to determine the length and required strength of the bungee cord for jumpers of different mass. 41

42 Modeling a Physical System Based on this insight and your knowledge of physics and fluid mechanics, you develop the following mathematical model for the rate of change of velocity with respect to time: dv c 2 dt 42 = g d m v

43 Modeling a Physical System dv cd 2 = g v dt m v = downward velocity t = time g = acceleration due to gravity cd = drag coefficient m = mass of the jumper 43

44 Modeling a Physical System dv cd 2 = g v dt m There is a closed form solution for this problem available through calculus! gc $ gm d v (t ) = tanh # t& cd " m % 44

45 Modeling a Physical System dv cd 2 v (t ) = = g v dt m 45! gc $ gm d tanh # t& cd " m % We really aren t concerned in this class with how you get to this solution but we will use it to check our approximate numerical solution.

46 Modeling a Physical System dv cd 2 v (t ) = = g v dt m 46! gc $ gm d tanh # t& cd " m % It is possible to make an approximation to the derivative here by looking at the velocity at very small time steps.

47 Modeling a Physical System dv cd 2 v (t ) = = g v dt m! gc $ gm d tanh # t& cd " m % Mathematically the approximation would be dv Δv v (ti+1 ) v (ti ) = dt Δt ti+1 ti 47

48 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % The smaller the time step (Δt) usually the better the approximation dv Δv v (ti+1 ) v (ti ) = dt Δt ti+1 ti 48

49 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % Now we can rearrange the expression to allow us to predict the velocity at the next time step from the velocity and the derivative at the current time step. dv v (ti ) + (ti+1 ti ) = v (ti+1 ) dt 49

50 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % Substituting the derivative in the original model " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m 50

51 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % Knowing the initial conditions v=v0 at t=0 and the parameters of the system, g, cd, and m, we can select a step size for t and use the equation below to model the velocity changes with time " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m 51

52 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % If we are looking for the terminal velocity, we will need to calculate the model until the velocity no longer changes " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m 52

53 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % The smaller (within reason) the value for the time step, the better the model will be but the longer it will take the model to make the calculations " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m 53

54 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % Not a real problem here but it can be a problem when the models become more complicated " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m 54

55 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % This method of approximation is known as the Euler s method or Euler s approximation " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m 55

56 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % Use a mass of 68.1 kg, g = 9.81 m/s2, and a drag coefficient of 0.25 kg/m. Determine the terminal velocity that will be attained. Use your calculator and a step size of 5 sec. " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m 56

57 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % Use a mass of 68.1 kg, g = 9.81 m/s2, and a drag coefficient of 0.25 kg/m. Determine the terminal velocity that will be attained. Now we will set up a spreadsheet and look at the same model. 57 " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m

58 Modeling a Physical System dv cd 2 = g v dt m! gc $ gm d v (t ) = tanh # t& cd " m % Now we can look at both our numerical approximation and the closed form solutions. " 2% cd v (ti ) + $ g ( v (ti )) ' (ti+1 ti ) = v (ti+1 ) # & m 58

59 In-Lab Problem The amount of a uniformly distributed bacterial contaminant contained in a closed reactor is measured by its concentration c (milligram/liter or mg/l). The bacteria are exposed to a disinfectant which causes the population to decrease at a decay rate proportional to its concentration; that is dc = kc dt 59

60 In-Lab Problem Use Euler s method to solve this equation from t = 0 to 1d with k=0.175d-1. Employ a step size of t=0.01d. The concentration at t = 0 is mg/l. dc = kc dt 60

61 In-Lab Problem Plot the solution on a semilog graph (i.e., ln c versus t) and determine the slope. Interpret your results. dc = kc dt 61

62 Homework Problem cha01102_ch01_ qxd 12/17/10 7:58 AM Page 21 A storage tank contains a liquid at depth y where y = 0 when the tank is half full. PROBLEMS Liquid is withdrawn at a constant flow rate Q to meet demands. y Qin 0 62 FIGURE P1.9 FIGURE

63 Homework Problem cha01102_ch01_ qxd 12/17/10 7:58 AM Page 21 The contents are resupplied at a sinusoidal rate 3Q sin2(t). An equation can be written forproblems this system as y dv day = = 3Qsin 2 (t ) Q dt dt 63 Qin 0 FIGURE P1.9 FIGURE

64 Homework Problem cha01102_ch01_ qxd 12/17/10 7:58 AM Page 21 The surface area is constant so the expression can be rewritten as PROBLEMS y Qin 2 dy 3Qsin (t ) Q = dt A 64 0 FIGURE P1.9 FIGURE

65 Homework Problem Use Euler s method to solve for the depth y from t = 0 to 10 d with a step size of 0.5 d. The parameter values are A = 1250 m2 and PROBLEMS Q = 450 m3/d. Assume that the initial condition is y = 0. cha01102_ch01_ qxd 12/17/10 7:58 AM Page 21 y Qin 2 dy 3Qsin (t ) Q = dt A 65 0 FIGURE P1.9 FIGURE

66 Homework Problem cha01102_ch01_ qxd 12/17/10 7:58 AM Page 21 Plot the depth y as a function of t. PROBLEMS y Qin 2 dy 3Qsin (t ) Q = dt A 66 0 FIGURE P1.9 FIGURE

67 Homework Problem cha01102_ch01_ qxd 12/17/10 7:58 AM Page 21 Now assume that the outflow if a function of the depth as shown below PROBLEMS y Qin Qsin t ) α (1+ y) dy ( = dt A 67 0 FIGURE P1.9 FIGURE

68 Homework Problem Use Euler s method to solve for the depth y from t = 0 to 10 d with a step size of 0.5 d. The parameter values are A = 1250 m2, Q = 450 m3/d, and α = 150. Assume PROBLEMS that the initial condition is y = 0. cha01102_ch01_ qxd 12/17/10 7:58 AM Page 21 y Qin Qsin t ) α (1+ y) dy ( = dt A 68 0 FIGURE P1.9 FIGURE

69 Homework Problem cha01102_ch01_ qxd 12/17/10 7:58 AM Page 21 Plot the results. PROBLEMS y Qin Qsin t ) α (1+ y) dy ( = dt A 69 0 FIGURE P1.9 FIGURE

70 Homework Problem 70 In addition to the downward force of gravity (weight) and drag, an object falling through a fluid is also subject to a buoyancy force which is proportional to the displaced volume.

71 Homework Problem 71 For example, for a sphere with diameter d (m), the sphere s volume is V = π d 3/6, and its projected area is A = πd2/4.

72 Homework Problem The buoyancy force can then be computed as Fb =-ρvg. Buoyancy was neglected in the bungee problem because it is relatively small for an object like a bungee jumper moving through air. However, for a more dense fluid like water, it becomes more prominent. 72

73 Homework Problem Rewrite the differential equation for the velocity for the special case of a sphere. Use this equation developed in to compute the terminal velocity (i.e., for the steady-state case). 73

74 Homework Problem 74 Use the following parameter values for a sphere falling through water: sphere diameter = 1 cm, sphere density = 2,700 kg/m3, water density = 1,000 kg/m3, and Cd = Use Euler s method with a step size of Δt = s to numerically solve for the velocity from t = 0 to 0.25 s with an initial velocity of zero. Plot your results.