Building Frames - 3 Civil Engineering (CE) Notes | EduRev

Structural Analysis

Civil Engineering (CE) : Building Frames - 3 Civil Engineering (CE) Notes | EduRev

 Page 1


 
 
Hence it is required to make 12 assumptions to reduce the frame in to a statically 
determinate structure. From the deformed shape of the frame, it is observed that 
inflexion point (point of zero moment) occur at mid height of each column and 
mid point of each beam. This leads to 10 assumptions. Depending upon how the 
remaining two assumptions are made, we have two different methods of 
analysis: Portal method and cantilever method. They will be discussed in the 
subsequent sections. 
) i ) ii
 
36.3.1 Portal method 
In this method following assumptions are made.  
1) An inflexion point occurs at the mid height of each column. 
2) An inflexion point occurs at the mid point of each girder.  
                                                                             
                                                         
Page 2


 
 
Hence it is required to make 12 assumptions to reduce the frame in to a statically 
determinate structure. From the deformed shape of the frame, it is observed that 
inflexion point (point of zero moment) occur at mid height of each column and 
mid point of each beam. This leads to 10 assumptions. Depending upon how the 
remaining two assumptions are made, we have two different methods of 
analysis: Portal method and cantilever method. They will be discussed in the 
subsequent sections. 
) i ) ii
 
36.3.1 Portal method 
In this method following assumptions are made.  
1) An inflexion point occurs at the mid height of each column. 
2) An inflexion point occurs at the mid point of each girder.  
                                                                             
                                                         
3) The total horizontal shear at each storey is divided between the columns 
of that storey such that the interior column carries twice the shear of exterior 
column.  
The last assumption is clear, if we assume that each bay is made up of a portal 
thus the interior column is composed of two columns (Fig. 36.6). Thus the interior 
column carries twice the shear of exterior column. This method is illustrated in 
example 36.2. 
 
Example 36.2  
Analyse the frame shown in Fig. 36.7a and evaluate approximately the column 
end moments, beam end moments and reactions. 
 
Solution:  
The problem is solved by equations of statics with the help of assumptions made 
in the portal method. In this method we have hinges/inflexion points at mid height 
of columns and beams. Taking the section through column hinges we 
get, (ref. Fig. 36.7b). 
O N M , .
 
20 2 0 = + + ? =
?
V V V F
X
 
                  
  or  kN 5 = V
Taking moment of all forces left of hinge R about R gives,  
 
0 5 . 2 5 . 1 = × - ×
y
M V 
( ) ? = kN 3
y
M 
 
Column and beam moments are calculates as, 
 
kN.m 5 . 7
kN.m 5 . 7 ; kN.m 5 . 7 5 . 1 5
- =
+ = = × =
CF
IH CB
M
M M
 
 
Taking moment of all forces left of hinge  about gives, S S
 
()
0
kN 3
0 5 . 2 5 . 1 5
=
? =
= × - ×
y
y
y
N
O
O
 
 
Taking a section through column hinges we get, (ref. Fig. 36.7c). L K J , ,
                                                                             
                                                         
Page 3


 
 
Hence it is required to make 12 assumptions to reduce the frame in to a statically 
determinate structure. From the deformed shape of the frame, it is observed that 
inflexion point (point of zero moment) occur at mid height of each column and 
mid point of each beam. This leads to 10 assumptions. Depending upon how the 
remaining two assumptions are made, we have two different methods of 
analysis: Portal method and cantilever method. They will be discussed in the 
subsequent sections. 
) i ) ii
 
36.3.1 Portal method 
In this method following assumptions are made.  
1) An inflexion point occurs at the mid height of each column. 
2) An inflexion point occurs at the mid point of each girder.  
                                                                             
                                                         
3) The total horizontal shear at each storey is divided between the columns 
of that storey such that the interior column carries twice the shear of exterior 
column.  
The last assumption is clear, if we assume that each bay is made up of a portal 
thus the interior column is composed of two columns (Fig. 36.6). Thus the interior 
column carries twice the shear of exterior column. This method is illustrated in 
example 36.2. 
 
Example 36.2  
Analyse the frame shown in Fig. 36.7a and evaluate approximately the column 
end moments, beam end moments and reactions. 
 
Solution:  
The problem is solved by equations of statics with the help of assumptions made 
in the portal method. In this method we have hinges/inflexion points at mid height 
of columns and beams. Taking the section through column hinges we 
get, (ref. Fig. 36.7b). 
O N M , .
 
20 2 0 = + + ? =
?
V V V F
X
 
                  
  or  kN 5 = V
Taking moment of all forces left of hinge R about R gives,  
 
0 5 . 2 5 . 1 = × - ×
y
M V 
( ) ? = kN 3
y
M 
 
Column and beam moments are calculates as, 
 
kN.m 5 . 7
kN.m 5 . 7 ; kN.m 5 . 7 5 . 1 5
- =
+ = = × =
CF
IH CB
M
M M
 
 
Taking moment of all forces left of hinge  about gives, S S
 
()
0
kN 3
0 5 . 2 5 . 1 5
=
? =
= × - ×
y
y
y
N
O
O
 
 
Taking a section through column hinges we get, (ref. Fig. 36.7c). L K J , ,
                                                                             
                                                         
 
 
 
 
60 ' ' 2 ' 0 = + + ? =
?
V V V F
X
 
                  
  or '1 5 kN V =
 
                                                                             
                                                         
Page 4


 
 
Hence it is required to make 12 assumptions to reduce the frame in to a statically 
determinate structure. From the deformed shape of the frame, it is observed that 
inflexion point (point of zero moment) occur at mid height of each column and 
mid point of each beam. This leads to 10 assumptions. Depending upon how the 
remaining two assumptions are made, we have two different methods of 
analysis: Portal method and cantilever method. They will be discussed in the 
subsequent sections. 
) i ) ii
 
36.3.1 Portal method 
In this method following assumptions are made.  
1) An inflexion point occurs at the mid height of each column. 
2) An inflexion point occurs at the mid point of each girder.  
                                                                             
                                                         
3) The total horizontal shear at each storey is divided between the columns 
of that storey such that the interior column carries twice the shear of exterior 
column.  
The last assumption is clear, if we assume that each bay is made up of a portal 
thus the interior column is composed of two columns (Fig. 36.6). Thus the interior 
column carries twice the shear of exterior column. This method is illustrated in 
example 36.2. 
 
Example 36.2  
Analyse the frame shown in Fig. 36.7a and evaluate approximately the column 
end moments, beam end moments and reactions. 
 
Solution:  
The problem is solved by equations of statics with the help of assumptions made 
in the portal method. In this method we have hinges/inflexion points at mid height 
of columns and beams. Taking the section through column hinges we 
get, (ref. Fig. 36.7b). 
O N M , .
 
20 2 0 = + + ? =
?
V V V F
X
 
                  
  or  kN 5 = V
Taking moment of all forces left of hinge R about R gives,  
 
0 5 . 2 5 . 1 = × - ×
y
M V 
( ) ? = kN 3
y
M 
 
Column and beam moments are calculates as, 
 
kN.m 5 . 7
kN.m 5 . 7 ; kN.m 5 . 7 5 . 1 5
- =
+ = = × =
CF
IH CB
M
M M
 
 
Taking moment of all forces left of hinge  about gives, S S
 
()
0
kN 3
0 5 . 2 5 . 1 5
=
? =
= × - ×
y
y
y
N
O
O
 
 
Taking a section through column hinges we get, (ref. Fig. 36.7c). L K J , ,
                                                                             
                                                         
 
 
 
 
60 ' ' 2 ' 0 = + + ? =
?
V V V F
X
 
                  
  or '1 5 kN V =
 
                                                                             
                                                         
Taking moment of all forces about P gives (vide Fig. 36.7d) 
 
()
()
015 1.5 5 1.5 3 2.5 2.5 0
15 kN
15 kN
py
y
y
MJ
J
L
=× +× +× - × =
=?
=?
?
 
 
 
 
                                                                             
                                                         
Page 5


 
 
Hence it is required to make 12 assumptions to reduce the frame in to a statically 
determinate structure. From the deformed shape of the frame, it is observed that 
inflexion point (point of zero moment) occur at mid height of each column and 
mid point of each beam. This leads to 10 assumptions. Depending upon how the 
remaining two assumptions are made, we have two different methods of 
analysis: Portal method and cantilever method. They will be discussed in the 
subsequent sections. 
) i ) ii
 
36.3.1 Portal method 
In this method following assumptions are made.  
1) An inflexion point occurs at the mid height of each column. 
2) An inflexion point occurs at the mid point of each girder.  
                                                                             
                                                         
3) The total horizontal shear at each storey is divided between the columns 
of that storey such that the interior column carries twice the shear of exterior 
column.  
The last assumption is clear, if we assume that each bay is made up of a portal 
thus the interior column is composed of two columns (Fig. 36.6). Thus the interior 
column carries twice the shear of exterior column. This method is illustrated in 
example 36.2. 
 
Example 36.2  
Analyse the frame shown in Fig. 36.7a and evaluate approximately the column 
end moments, beam end moments and reactions. 
 
Solution:  
The problem is solved by equations of statics with the help of assumptions made 
in the portal method. In this method we have hinges/inflexion points at mid height 
of columns and beams. Taking the section through column hinges we 
get, (ref. Fig. 36.7b). 
O N M , .
 
20 2 0 = + + ? =
?
V V V F
X
 
                  
  or  kN 5 = V
Taking moment of all forces left of hinge R about R gives,  
 
0 5 . 2 5 . 1 = × - ×
y
M V 
( ) ? = kN 3
y
M 
 
Column and beam moments are calculates as, 
 
kN.m 5 . 7
kN.m 5 . 7 ; kN.m 5 . 7 5 . 1 5
- =
+ = = × =
CF
IH CB
M
M M
 
 
Taking moment of all forces left of hinge  about gives, S S
 
()
0
kN 3
0 5 . 2 5 . 1 5
=
? =
= × - ×
y
y
y
N
O
O
 
 
Taking a section through column hinges we get, (ref. Fig. 36.7c). L K J , ,
                                                                             
                                                         
 
 
 
 
60 ' ' 2 ' 0 = + + ? =
?
V V V F
X
 
                  
  or '1 5 kN V =
 
                                                                             
                                                         
Taking moment of all forces about P gives (vide Fig. 36.7d) 
 
()
()
015 1.5 5 1.5 3 2.5 2.5 0
15 kN
15 kN
py
y
y
MJ
J
L
=× +× +× - × =
=?
=?
?
 
 
 
 
                                                                             
                                                         
 
 
Column and beam moments are calculated as, (ref. Fig. 36.7f)  
 
kN.m 5 . 22 5 . 1 15 ; . 5 . 7 5 . 1 5 = × = = × =
BA BC
M m kN M 
 
kN.m 30 - =
BE
M 
 
kN.m 45 5 . 1 30 ; kN.m 15 5 . 1 10 = × = = × =
ED EF
M M 
 
kN.m 30 kN.m 30 - = - =
EH EB
M M 
 
kN.m 5 . 22 5 . 1 15 ; kN.m 5 . 7 5 . 1 5 = × = = × =
HG HI
M M 
 
kN.m 30 - =
HE
M 
 
Reactions at the base of the column are shown in Fig. 36.7g. 
 
 
36.3.2 Cantilever method 
The cantilever method is suitable if the frame is tall and slender. In the cantilever 
method following assumptions are made.  
1) An inflexion point occurs at the mid point of each girder.  
2) An inflexion point occurs at mid height of each column. 
3) In a storey, the intensity of axial stress in a column is proportional to its 
horizontal distance from the center of gravity of all the columns in that storey. 
Consider a cantilever beam acted by a horizontal load as shown in Fig. 36.8. In 
such a column the bending stress in the column cross section varies linearly from 
its neutral axis. The last assumption in the cantilever method is based on this 
fact. The method is illustrated in example 36.3. 
P
 
 
 
 
 
                                                                             
                                                         
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