Advanced Methods of Structural Analysis

7.4 Computation of Deflections of Redundant Structures 245

ql^216

8

M ql^2

P

l

A

P= 1 l/^2 l

M

q

A B

8

ql^2

RA ql

8

=^30

16 2 8

00 4

6 EI

⎟⎟=

⎠

⎞

⎜

⎜

⎝

⎛

= ⋅ + ⋅ ⋅ − ⋅

×

Δ =

Simpson rule

ver

A l

l ql^2 l ql^2

EIi

MP MP

a

Version 1

l ql^2 ql^2 ql^3

EI

MP M

16 8 48 EI

0 ⋅ 1 + 4 ⋅

6 EI

⎟⎟=

⎠

⎞

⎜⎜⎝

⎛

= ⋅ 1 − ⋅ 1

×

θA=

Version 2

l ql^2 ql^2 ql^3

EI

MP M

A 2 8 48 EI

1

6 EI 16

⎟⎟=

⎠

⎞

⎜⎜

⎝

⎛

θ = × = ⋅ − ⋅ 0

M= 1 A

1

1

1/2

M= 1 A

M

M

0 ⋅ 1 + 4 ⋅

b

Fig. 7.16 (a) Computation of vertical displacement at supportA.(b) Computation of slope at
support A. Two versions of unit state

Ve r s i o n 1 :

AD

MPMN

EI

D

l

6 EI



0  1 C 4 

ql^2

16

 1 

ql^2

8

 1



D

ql^3

48 EI

:

Ve r s i o n 2 :

AD

MPMN

EI

D

l

6 EI



0  1 C 4 

ql^2

16



1

2



ql^2

8

 0



D

ql^3

48 EI

:

It easy to check that superposition principle leads to the same result. Indeed, in case
of simply supported beam subjected to uniformly distributed loadqand support
momentMBDql^2 =8, the slope at supportAequals

ADAqCAMBD

ql^3

24 EI



MBl

6 EI

D

ql^3

24 EI



ql^2

8

l

6 EI

D

ql^3

48 EI

:

The reader is invited to check that slope at supportBis zero. For multiplication
of both bending moment diagrams, it is recommended to apply the Simpson’s rule.