Slope deflection method - Structure Analysis for Civil engineering design

 Slope Deflection Method - To analysis of Structures

• Slope Deflection Method is useful to analyse indeterminate structures.
• Intdeterminate strutures like continuous beams and plane frames.
• The unknowns in slope deflection method are degree of freedom i.e. slope and deflection.
• Combinedly these slope and deflections are known as displacements. Thus slope-deflection method is a displacement method.
• It is classical method on whick moment distribution method, kani's method and stiffness matrix method are based.

Slope Deflection Method - force method or displacement method

•  It is Displacement method.

 Assumptions in Slope Deflection Method - Structure analysis

• Analysis of a beam and frame means determination of moment and shear force throughout the length of the member
• i.e. determination of BMD and SFD for the member.
• BMD and SFD for a member of structure can be drawn if we know the internal end moments of a member
• Thus, in slope deflection method we establish a relationship between the degree of freedom and the member end moments. This relationship is called slpe deflection relationship.
• Finally by using equillibriums equation we find the slpe deflection relationship to obtain the member moments
• To find out the slope deflection relationship method of superposition is used.
  
• Axial deformation are neglected.
• Shear Deformation are neglected.
• Clockwise moment is take as Positive (+)ve
• Clockwise rotation is take as Psitive (+)ve
• In slope deflection equation, deformation are considered to be caused by bending moments only.
  

Important points about Slope Deflection Method - 

• 1879 -1880 Heinrich Manderla invented similar to slope deflection method.
• 1882 - later Otto Mohr introduced improved methods of it called slpe deflection method.
• 1914 - Axel Bendixen gives great detail of slpe deflection method
• 1915 - George A mancy also gives details about slpe deflection
  

Step for Analysis in Slope Deflection Method

1. Computing of fixed end moment

    the formulae for fixed end actions for various load cases are given in many books and tables from where you can use.
   

2. Relate member and moments to joint displacement

    

   

   In slope deflection equation, deformation are considered to be caused by bending moments only.
   these are also known as solpe deflection equation 
   

3. Formulate equilibrium equations

    These are obtained by making algebraic sum of moments at each joint as zero.In case of frames with sway, additional equations are obtained consiering shear condition

4. Solve the equation

   This will give displacements (Primary unknowns) i.e. slope at joints and deflection.

5. Back-substitution

   In the expression for end moments formed in step 2 substitute value of known dispalcements as obtaind in step 4. this gives final end moments for each member form this,the support reactions can also be calculated.

6. Sketch shear force disarem and bending moment diagram .
   

Fixed Eng Momets Table with diagram

Fixed End Momets

 

 

Graphical Presentation of steps in the determination of slope deflection relationship

Consider all joints to be fixed and determine moments

Fixed support

Allow the one support to settle with respect to other support and find moments on support because of it

End moments caused by settlement of fixed support

 

Allow one end to roatate and find end moment because of it.

End momets caused by ration at hinge suport

 

Allow other end to rotate and find member moments because of it.

 

 

End moments caused by rotation at other supports

 

After adding all end moments for one supports we get

slope deflece equation

 

Frequently asked questions related to slope deflection methods based on ( FAQ'S )

Kani's methods is based on which principle?

• Kanis method is based on application of slope deflection method.
  
• Kanis method offers an iterative approach for applying slope deflection method, it is also known as rotation contribution method.
• In this method if an end of a meber is fixed the rotation at he end being zero, the rotation contribution at the end is also zero
• if an end of a member is hinged or pinned, it will be convenient to consider the end as fixed and to take the relative stiffness as 3/4th 

 what is slope deflection method on the basis of stiffness method or flexibility method?

• Slope deflection method is stiffness method in which unknown joint displacement are found out by applying the equilibriup condition at end
• In slope deflection eqution we use the principple of superposition by considering seperately the moments developed at each support due to each of the displacement ( slopes and displacements) and then loads, so displacement at joint are Independents.

What are the displacement methods other than slope deflection method to analyse indeterminate structures?

• Some displacement methods for anlysis of indeterminate structure.
  

1. Slope deflection method
2. Moment distribution method
3. Stiffness matrix method
4. Kani's method
   

When we use displacement method?

• Displacement method is suitable when Dk < Ds . Number of degree of kinetic indeterminacy is less than degree of static indeterminacy.