Graphics Programs Reference

In-Depth Information

A uniform beam forms the semiparaboliccantileverarch
AB
. The vertical dis-

placementof
A
due to the force
P
can be shown to be

E I
C
h

Pb
3

δ
A
=

b

where
E I
is the bending rigidity of the beam and

C
h

b

z
2

2
h

b
z
2

1

=

+

1

dz

0

Write aprogram thatcomputes
C
(
h

/

b
)for any givenvalueof
h

/

b
to four decimal

places. Use the program to compute
C
(0

.

5),
C
(1

.

0) and
C
(2

.

0).

=
π/
0
ln(sin
x
)
dx
. A“brute force”method

that works istosplit the integral into several parts: from
x

15.

There is no elegantway to compute
I

=

.

.

0to0

01,from 0

01

to 0

.

2 and from
x

=

0

.

2 to

π/

2. In the first part wecan use the approximation

sin
x

x
, which allows ustoobtain the integral analytically. The other two parts

can beevaluatedwith Gauss-Legendrequadrature. Use this method to evaluate

I
to six decimal places.

≈

16.

h
(m)

620

612

112

80

575

530

425

310

52

35

15

p
(Pa)

0

The pressureofwindwas measuredat variousheights onavertical wall, as shown

on the diagram. Find the height of the pressurecenter, which is definedas

112 m

0
hp
(
h
)
dh

112 m

0

h

=

p
(
h
)
dh

Hint:
fit a cubic polynomial

to

the data and then apply Gauss-Legendre

quadrature.

∗
6.5
Multiple Integrals

Multiple integrals, such as the areaintegral
A

y
)
dx dy
,can also beevaluated

by quadrature. The computations arestraightforward if the region of integration has a

simple geometricshape, such as a triangle oraquadrilateral.Duetocomplications in

f
(
x

,

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