Numerical Integration and Differentiation - Numerical Structural Analysis - page 127

0 00912
0 00781
0 00443
0 0000

.
.
.

q
q
q
q

0

1

radians =

2

.

3

The exact solution at the end is q 0 = 0 . 00936 radians, which is a 2.51% error,
while the value in the center is exact. The relationship for moment is as follows:

′′ =− = −−

M
EI

y 2 y y

i + 1 i i − 1

y

i

h

2

− 2 000

1210
0121
0022

y
y
y
y

M
M
M
M

0 0

−

EI
h

1 1

− =

2

−

2 2

−

3 3

− 2 000

11210
0121
0022

y
y
y
y

M
M
M
M

0 0

−

36 EI

L

1 1

− =

2

−

2 2

−

3 3

2
2
2
22

−
−+
−+
−

y M

M
M
M

0 0

y y y

36 EI

L

0 1 2 1

− =

2

y y y

1 2 3 2

y y

2 3 3

000
2083 4
3333 3
3750 0

. M

M
M
M

0

.
.
.

1

kip-inches =

2

3

The exact solution at the center is M 3 = 3750 k-in which is exact, while the
values at the ends is also exact. The relationship for shear is as follows:

V
EI

y − 2 y + 2 y − y

i + 2 i + 1 i − 1 i − 2

y ′′′ =− =

i

2 h

3

0420
2121
1212
0000

− y

y
y
y

V
V
V
V

0 0

−

EI
h

1 1

− =

2

3

− −

2 2

3 3