Advanced Mathematical Methods for Civil Engineering Applications Wonsiri Punurai, PhD

Transcription

1 Advanced Mathematical Methods for Civil Engineering Applications Wonsiri Punurai, PhD Department of Civil Engineering Room 69, EG Building Faculty of Engineering, Mahidol University Personal Web:

2 MATLAB 0 MATLAB M-files (script files) MATLAB Mat-files (binary files) Importing Data into /Eporting Data from MATLAB Displaying output on the command window Input and Output

3 MATLAB M-files Semi-interactive mode commands can be saved, refined and reused in MATLAB editor as needed. These files end in.m in all platforms. Any word processor can also be used to edit and save files as tet. Good for many problem types (ranging from small simple to large and comple models). Open built-in editor, select from menu: File > New > M-file

4 Sample M-files - I Try this eample, In MATLAB command window, type This single line defines two matrices (a and b) and computes their product (c). This can be done in the same fashion in m-file format. Open new m-file using editor. Now type the following % A sample file to compute a product of two matrices - Save the file as sample.m - Go back to Command Window and type sample to eecute file.

5 Sample M-files - II Try another eample, In MATLAB command window, type

6 Sample M-files - III.5 Plot of simple trigonometric functions sin(t) cos(t) sin(t)*cos(t) 0.5 Simple trigonometry plot θ (radius)

7 Function m-files

8 Eample function m-files

9 MATLAB Mat-files These files are convenient to store information that needs to be reused. These binary files end in.mat. Use the load and save commands at the MATLAB command line. They are platform independent.

10 Sample MAT-files From previous eample, we have created and eecuted m-file called sample.m % A sample file to compute a product of two matrices Type whos in the command window, ones will see a, b, c matrices. Let say we want to save these matrices (or variables) for later. Type save sample.mat Now try typing clear and then type load sample.mat Type whos again and you will see matrices a,b and c in workspace.

11 Importing Data into MATLAB Suppose that we have a data file (called road_elevation_centerline.tt) containing road elevation data at center line. The information provided includes: ) column station no. (ranging from to 5) 2) column 2 elevation at each station

12 Importing Data (I) Method in MATLAB command window, use load command >> load road_elevation_centerline.tt An array name road_elevation_centerline which has dimensions 52 was just being loaded. We know that first and second column corresponding to station # and elevation at centerline respectively. One can produce two new array variables station no road_elevation_centerline (:,) ; center_elev road_elevation_centerline (:,2) ;

13 Importing Data (II) Method 2 To import data go to the Editor Window Select Import from the File pull-down menu

14 Importing Data (III) Method use a function called dlmread >> road_elevation_centerline dlmread( road_elevation_centerline.tt ) >> station no road_elevation_centerline(:,); >> center_elev road_elevation_centerline(:,2);

15 Eporting data from MATLAB Save data into a file in any specific format. Use a function called dlmwrite to specify any delimiters needed. >> load road_elevation_centerline.tt >> dlmwrite( road_elevation_centerline.tt,ans) >> save road_elevation_centerline2.tt

16 Displaying output Use function disp to display to output screen. Typically used in conjunction with num2str to convert numerical to string variables

17 Input and Output

18 Tet Output

19 System of linear equations BASIC

20 System of linear equations ( unknowns) In mathematics, a system of linear equations is a collection of linear equations involving the same set of variables. is a system of three equations in a three variables, 2,

21 Solution of linear equations (by hand-i) A solution to a linear system is an assignment of numbers to the variables such that all the equation are simultaneously satisfied. Given In the matri form, we have A b To obtain [], one must compute det(a) and A det(a)

22 Solution of linear equations (by hand-ii) From A b Multiply both sides by A - A - A A - b I* A - b A product of A - bgives

23 Solution of linear equations (by MATLAB I) inv(a)*b;

24 Solution of linear equations (by MATLAB II)

25 Problems in relation to CE work Eample: The design of a steel truss Structural analysis in which the equations of equilibrium are applied at each node and written in terms of the unknown element internal forces [f] and known eternally applied load [P]. Both terms are related by [B][f] [P] where [B], in this case, represents statics matri.

26 System of linear equations (n unknowns)

27 Eample: Member forces in truss. Write down matrices [B], [f], and [P] at each node. 2. Solve for [f] matri for each node..

28 Solution: Member forces in truss (I) cos60 2 F F 750 sin 60 cos60 2 F F F sin 0 cos0 2 F F sin 60 cos60 2 F F sin 60 sin 0 cos60 cos0 F F F F F

29 What we just did is so called Matri Formulation

30 Requirements for a solution Later