Final exam PHƯƠNG PHÁP SỐ

Ouestion

1

(L.o.1.3; L.O.2.2)

(2 marks)

Consider the following function:

f{x}: -0.6xa- 0.8rB -Ex? -x + L.g

"rp: S.$,Ii:

{1}

p.t$

1/ Calculate analytically the first derivative and the second derivative of function (t) at xo
2l Estimate numerically the flrst derivative and the second derivative of function (1) at xo using
centered finite-divided difference formula Q) e (3). Compare the numerical results with analytical
ones to determine the errors

F,{ j.,,. : -.f(*o*r} + fi,ftxr+r} - I f{xr-r} +gqx,-r}

._.

-i- \-.

f

"(;r1)

-"i'{x;+r}

-

*

16

f{x6ar}

-

sn

f{x,} +

16

f{ni:r.}

L4,J

- f,(xc-a}

La hr

Ouestion 2 (L.O.1.3; L.O.2.1)

i/

t3)
(2 marks)

Compute numerically the following integral using Newton- C6tes method:
+l

J

(*' +9x2 * 7x-rrs ) dx

-1

The abscissae and factors of the 3 points of integration are given as follows

xr:-1,

@1

: *t *r:o, cDr:!rx3:1, -r:1

2/ Estimate the error (compare with the exact solution given by the analytical integration NewtonLeibnitz)
Ouestion

3

(L.O.4.3 ;

L.o.4.4; L.O.6. 1 )

(2 marks)

Consider a 3-node triangular (1- 2- 3) with node 1(x: 0; y: 0), node 2(x: 1; y: 0); node 3(x: 0; y: 1).
1/ Determine the three interpolation functions Nr(x, y), Nz(x, y), Nl(x, y)
2l Given
800; t2:830i t3: 810 are temperatures at node 1, node 2, node3, respectively.
Interpolate temperature at point M(x: 0.2;y:0.2)

tr:

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.-t -a

Ouestion 4 (L.O.4.3L.O .4.4; L.O.6.2)

(2 marks)

Consider the plane truss in Fig. 4-l with the given: Et: E2: 70,000 MPa (Young's modulus of
aluminium); Ar: A2: 64 mm2 (area of the cross section of the 2bars);Lr: Lz:800 mm (length of two

bars); P: 50,000 N; element l(l- 2); element 2(2- 3)

v

81.

l,

t

&,.I0

L

ffi
s
Figure 4- I
Use the finite element method to
1/ Determine:

2l

a/ Stiffness matrices of element I [Kr] and of
b/ Stiffrress matrix of structure [K]

elemett2lKzT

CaLcrlate the nodal displacements uz (horizontal component
component at node 2).

Question 5 (L.O.1.3; L.O.2.2)

at node 2)

afi

vz (vertical

(2 marks)

Consider a cantilevered beam (fixed at one end and free at the other end) with the given parameters E

(Young's modulus of beam material
(concentrated load) as shown in fig. 5-1

,I

(inertia moment of cross section), L(length of beam)

,P

Fig.5-1
With the finite element method, use one-element model:
1/ Determine the stiffness matrix of this element [Kr], the stiffness matrix of the whole structure
[K]. Notice that this beam structure consists of only one element.
2l Calculate nodal displacements: linear displacment vz and angular displacement #3
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