The Slope Deflection Method: Frames Without Sidesway - 1 | Structural Analysis - Civil Engineering (CE) PDF Download

 Page 1


Instructional Objectives 
After reading this chapter the student will be able to 
1. State whether plane frames are restrained against sidesway or not.  
2. Able to analyse plane frames restrained against sidesway by slope-deflection 
equations. 
3. Draw bending moment and shear force diagrams for the plane frame. 
4. Sketch the deflected shape of the plane frame. 
 
 
16.1 Introduction 
In this lesson, slope deflection equations are applied to solve the statically 
indeterminate frames without sidesway. In frames axial deformations are much 
smaller than the bending deformations and are neglected in the analysis. With 
this assumption the frames shown in Fig 16.1 will not sidesway. i.e. the frames 
will not be displaced to the right or left. The frames shown in Fig 16.1(a) and Fig 
16.1(b) are properly restrained against sidesway. For example in Fig 16.1(a) the 
joint can’t move to the right or left without support A also moving .This is true 
also for joint . Frames shown in Fig 16.1 (c) and (d) are not restrained against 
sidesway. However the frames are symmetrical in geometry and in loading and 
hence these will not sidesway. In general, frames do not sidesway if 
D
 
1) They are restrained against sidesway. 
2) The frame geometry and loading is symmetrical 
 
 
 
 
 
Page 2


Instructional Objectives 
After reading this chapter the student will be able to 
1. State whether plane frames are restrained against sidesway or not.  
2. Able to analyse plane frames restrained against sidesway by slope-deflection 
equations. 
3. Draw bending moment and shear force diagrams for the plane frame. 
4. Sketch the deflected shape of the plane frame. 
 
 
16.1 Introduction 
In this lesson, slope deflection equations are applied to solve the statically 
indeterminate frames without sidesway. In frames axial deformations are much 
smaller than the bending deformations and are neglected in the analysis. With 
this assumption the frames shown in Fig 16.1 will not sidesway. i.e. the frames 
will not be displaced to the right or left. The frames shown in Fig 16.1(a) and Fig 
16.1(b) are properly restrained against sidesway. For example in Fig 16.1(a) the 
joint can’t move to the right or left without support A also moving .This is true 
also for joint . Frames shown in Fig 16.1 (c) and (d) are not restrained against 
sidesway. However the frames are symmetrical in geometry and in loading and 
hence these will not sidesway. In general, frames do not sidesway if 
D
 
1) They are restrained against sidesway. 
2) The frame geometry and loading is symmetrical 
 
 
 
 
 
 
 
 
Page 3


Instructional Objectives 
After reading this chapter the student will be able to 
1. State whether plane frames are restrained against sidesway or not.  
2. Able to analyse plane frames restrained against sidesway by slope-deflection 
equations. 
3. Draw bending moment and shear force diagrams for the plane frame. 
4. Sketch the deflected shape of the plane frame. 
 
 
16.1 Introduction 
In this lesson, slope deflection equations are applied to solve the statically 
indeterminate frames without sidesway. In frames axial deformations are much 
smaller than the bending deformations and are neglected in the analysis. With 
this assumption the frames shown in Fig 16.1 will not sidesway. i.e. the frames 
will not be displaced to the right or left. The frames shown in Fig 16.1(a) and Fig 
16.1(b) are properly restrained against sidesway. For example in Fig 16.1(a) the 
joint can’t move to the right or left without support A also moving .This is true 
also for joint . Frames shown in Fig 16.1 (c) and (d) are not restrained against 
sidesway. However the frames are symmetrical in geometry and in loading and 
hence these will not sidesway. In general, frames do not sidesway if 
D
 
1) They are restrained against sidesway. 
2) The frame geometry and loading is symmetrical 
 
 
 
 
 
 
 
 
 
 
For the frames shown in Fig 16.1, the angle ? in slope-deflection equation is 
zero. Hence the analysis of such rigid frames by slope deflection equation 
essentially follows the same steps as that of continuous beams without support 
settlements. However, there is a small difference. In the case of continuous 
beam, at a joint only two members meet. Whereas in the case of rigid frames two 
or more than two members meet at a joint. At joint  in the frame shown in Fig 
16.1(d) three members meet. Now consider the free body diagram of joint C as 
shown in fig 16.2 .The equilibrium equation at joint C is 
C
 
 
  
?
= 0
C
M ?         0 = + +
CD CE CB
M M M 
 
Page 4


Instructional Objectives 
After reading this chapter the student will be able to 
1. State whether plane frames are restrained against sidesway or not.  
2. Able to analyse plane frames restrained against sidesway by slope-deflection 
equations. 
3. Draw bending moment and shear force diagrams for the plane frame. 
4. Sketch the deflected shape of the plane frame. 
 
 
16.1 Introduction 
In this lesson, slope deflection equations are applied to solve the statically 
indeterminate frames without sidesway. In frames axial deformations are much 
smaller than the bending deformations and are neglected in the analysis. With 
this assumption the frames shown in Fig 16.1 will not sidesway. i.e. the frames 
will not be displaced to the right or left. The frames shown in Fig 16.1(a) and Fig 
16.1(b) are properly restrained against sidesway. For example in Fig 16.1(a) the 
joint can’t move to the right or left without support A also moving .This is true 
also for joint . Frames shown in Fig 16.1 (c) and (d) are not restrained against 
sidesway. However the frames are symmetrical in geometry and in loading and 
hence these will not sidesway. In general, frames do not sidesway if 
D
 
1) They are restrained against sidesway. 
2) The frame geometry and loading is symmetrical 
 
 
 
 
 
 
 
 
 
 
For the frames shown in Fig 16.1, the angle ? in slope-deflection equation is 
zero. Hence the analysis of such rigid frames by slope deflection equation 
essentially follows the same steps as that of continuous beams without support 
settlements. However, there is a small difference. In the case of continuous 
beam, at a joint only two members meet. Whereas in the case of rigid frames two 
or more than two members meet at a joint. At joint  in the frame shown in Fig 
16.1(d) three members meet. Now consider the free body diagram of joint C as 
shown in fig 16.2 .The equilibrium equation at joint C is 
C
 
 
  
?
= 0
C
M ?         0 = + +
CD CE CB
M M M 
 
At each joint there is only one unknown as all the ends of members meeting at a 
joint rotate by the same amount. One would write as many equilibrium equations 
as the no of unknowns, and solving these equations joint rotations are evaluated. 
Substituting joint rotations in the slope–deflection equations member end 
moments are calculated. The whole procedure is illustrated by few examples. 
Frames undergoing sidesway will be considered in next lesson. 
 
Example 16.1 
Analyse the rigid frame shown in Fig 16.3 (a). Assume EI to be constant for all 
the members. Draw bending moment diagram and also sketch the elastic curve.  
 
Solution 
In this problem only one rotation needs to be determined i. e. .
B
? Thus the 
required equations to evaluate 
B
? is obtained by considering the equilibrium of 
joint B . The moment in the cantilever portion is known. Hence this moment is 
applied on frame as shown in Fig 16.3 (b).  Now, calculate the fixed-end 
moments by fixing the support B (vide Fig 16.3 c). Thus 
 
 
 
Page 5


Instructional Objectives 
After reading this chapter the student will be able to 
1. State whether plane frames are restrained against sidesway or not.  
2. Able to analyse plane frames restrained against sidesway by slope-deflection 
equations. 
3. Draw bending moment and shear force diagrams for the plane frame. 
4. Sketch the deflected shape of the plane frame. 
 
 
16.1 Introduction 
In this lesson, slope deflection equations are applied to solve the statically 
indeterminate frames without sidesway. In frames axial deformations are much 
smaller than the bending deformations and are neglected in the analysis. With 
this assumption the frames shown in Fig 16.1 will not sidesway. i.e. the frames 
will not be displaced to the right or left. The frames shown in Fig 16.1(a) and Fig 
16.1(b) are properly restrained against sidesway. For example in Fig 16.1(a) the 
joint can’t move to the right or left without support A also moving .This is true 
also for joint . Frames shown in Fig 16.1 (c) and (d) are not restrained against 
sidesway. However the frames are symmetrical in geometry and in loading and 
hence these will not sidesway. In general, frames do not sidesway if 
D
 
1) They are restrained against sidesway. 
2) The frame geometry and loading is symmetrical 
 
 
 
 
 
 
 
 
 
 
For the frames shown in Fig 16.1, the angle ? in slope-deflection equation is 
zero. Hence the analysis of such rigid frames by slope deflection equation 
essentially follows the same steps as that of continuous beams without support 
settlements. However, there is a small difference. In the case of continuous 
beam, at a joint only two members meet. Whereas in the case of rigid frames two 
or more than two members meet at a joint. At joint  in the frame shown in Fig 
16.1(d) three members meet. Now consider the free body diagram of joint C as 
shown in fig 16.2 .The equilibrium equation at joint C is 
C
 
 
  
?
= 0
C
M ?         0 = + +
CD CE CB
M M M 
 
At each joint there is only one unknown as all the ends of members meeting at a 
joint rotate by the same amount. One would write as many equilibrium equations 
as the no of unknowns, and solving these equations joint rotations are evaluated. 
Substituting joint rotations in the slope–deflection equations member end 
moments are calculated. The whole procedure is illustrated by few examples. 
Frames undergoing sidesway will be considered in next lesson. 
 
Example 16.1 
Analyse the rigid frame shown in Fig 16.3 (a). Assume EI to be constant for all 
the members. Draw bending moment diagram and also sketch the elastic curve.  
 
Solution 
In this problem only one rotation needs to be determined i. e. .
B
? Thus the 
required equations to evaluate 
B
? is obtained by considering the equilibrium of 
joint B . The moment in the cantilever portion is known. Hence this moment is 
applied on frame as shown in Fig 16.3 (b).  Now, calculate the fixed-end 
moments by fixing the support B (vide Fig 16.3 c). Thus