Advanced Methods of Structural Analysis

230 7 The Force Method

ı 22 D

MN 2 MN 2

EI

D

1

1 EI



1

2

 8  8 

2

3

 8 D

170:67

EI

m

kN



; (b)

ı 12 Dı 21 D

MN 1 MN 2

EI

D

1

1 EI



3 C 8

2

 5  10 D

275

EI

m

kN



:

1PD

MN 1 M^0

P

EI

D

1

1 EI

 32  5  10 

1

1 EI



1

3

 25  5  10



1

2 EI



4

6

.2 10 C6/ 32 D

2; 294

EI

.m/;

2PD

MN 2 M^0

P

EI

D

1

1 EI



3 C 8

2

 5  32 

1

1 EI



5

6

.8 25 C 4 5:56:25C 3 0/D

1;161:25

EI

.m/: (c)

3.Verification of coefficients and free terms of canonical equations. The unit and
loaded displacement should be checkedbefore solving of canonical equations
(a). For this purpose, we need to construct the summary unit bending moment
diagramMN†DMN 1 CMN 2 (Fig.7.11f).
 First row control.The sum of coefficients in the first canonical equation must
be equal to the result of multiplication the summary unit bending moment
diagramMN†by a primary bending moment diagramMN 1. Indeed,

ı 11 Cı 12 D

666:67

EI

C

275

EI

D

941:67

EI

;

while

MN†MN 1

EI

D

1

1 EI

13 C 18

2

 5  10 C

1

2 EI

1

2

 10  10 

2

3

 10 D

941:67

EI

:

Therefore, the first row control is satisfactory. The second row control may be

performed similarly.

 Simultaneous control.The total sum of all coefficients and the result of multi-

plication of summary unit bending moment diagram by “itself” are

ı 11 Cı 12 Cı 21 Cı 22 D

1

EI

.666:67C 275 C 275 C170:67/D

1;387:34

EI

;

MN†MN†

EI

D

1

1 EI

1

2

 3  3 

2

3

 3 C

1

1 EI

5

6

.2 13  13 C 2  18  18

C 13  18 C 18 13/C

1

2 EI

1

2

 10  10 

2

3

 10 D

1;387:34

EI

:

 Free terms control.The sum of the loaded displacements is

1PC2PD

1

EI

.2;294C1;161:25/D

3;455:25

EI

;