Engineering maths

Detail of the task

Answer all the following questions. You must include sufficient working to demonstrate your understanding of the solution.

1. You are to calculate the forces in a 7 member truss, simply supported at A and C, and subject to a force of 180kN at point B, as shown in the diagram below. Note that the base of the bridge (the members with lengths AB and BC) is horizontal.  You are given sufficient details (angles a and k, and lengths AB, AD, BC and DE) for you to calculate the geometry of your bridge ñ see the list attached to the assignment details on Blackboard.   Note that k is the angle between the member DE and the horizontal. Each student has a different set of data.  You must show your working. 

a) Using ONLY the sine and cosine rules, and the fact that you know what the 3 angles of a triangle add up to, find all the lengths and angles in your truss.  Produce a list of lengths, and a list of angles, each correct to 4 significant figures.  You will need to work to at least 5 significant figures.   25%

b) You are also to calculate the resultant force at each support, and the force in each member (correct to 4 significant figures). 25%

2. Use the Newton-Raphson method to find ALL 3 roots, correct to 8 significant figures, for the following equation. Use the same AB and BC values from question 1. You must use Excel to create a table showing the calculated iterations for each root. Include a printed version of the table in your report.

  6%

3. The Colebrook White equation for calculating pipe friction can be expressed as shown below.  Rearrange the equation to the form ëk =í so that k could be easily found if all other variables were known.  

  3%

4. Manningís equation for flow in Open Channels can be written as 

   where R, the hydraulic mean radius, is expressed as

 

For a Rectangular channel of width w (which you can take as being of the same width as the base of your truss, which is length AB + length BC) and a flow depth of y, R can therefore be expressed as:

 

a) Calculate R for at least 10 values of y (starting with y = w and reducing y each time until you reach a value of y which is less than w/20).  Use Excel to produce both a table and a graph of your results. You should include the working for one sample calculation; you do not need to show your working for all values of R. Make sure you include axes titles and units, and use the horizontal axis for the independent variable. 8%

b) On the basis of your results, do you think it reasonable to use the approximation of R = y for ëwide channelsí (i.e. when w is very much larger than y)?  Explain your answer, and add the line R = y to your graph. 4%

c) Use Manningís equation, together with the formula Q = VA, to find the normal depth for a channel of width w, with a slope of 0.0012, Manningís n = 0.015 and a flow rate of 40m3s-1.  You should assume that the channel is not a wide channel, and you should use trial and error to find the solution. 6%

5.

a) Calculate the Units for Manningís n, showing your working. 4%

b) If a layer of fluid is trapped between two parallel plates, one of which is stationary and the other is moving at a speed of V (m/s), then the force (N) which must be exerted to keep the plate moving is proportional to the speed of the plate (m/s), multiplied by the area of the plate (m2), and divided by the distance between the plates.  The constant of proportionality is known as the dynamic viscosity.  Calculate the units for the dynamic viscosity, showing your working. 6%

6 If  ,   and  find the matrices for:

a) 2%

b) 3%

c) 2%

d) 6%

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