Stability Functions

 Development of method of analysis for more extensive structures require some convenient means of summarizing the properties of individual  member when subjected to bending moment in the presence of axial loads ,while this is difficult to achieve for members loaded beyond the elastic limit , it is relatively easy in elastic range and stability function are used for analysis of stability of structure's. ,the analysis of elastic structures in which axial loads have negligible effect is facilitated by application of principle of superposition 

                                                                      Fig 1

Let member AB in fig 1 has uniform rigidity EI for bending in plane of diagram

and that axial load P=0  , if as shown , end B  is kept fixed point and direction while end A is rotated  about a fixed point , the terminal moments M ab and M ba and rotation θa are linearly related as under  

                                           Mab =  4 (EI/L)  θa

                                  Mba = 0.5 Mab

the ratio of Mab/ θa  =  4 (EI/L)  is the stiffness of member AB for rotation at A and is proportional to I/L  , This is rotational stiffness is used in the analysis of structures in method of moment distribution .

the ratio Mab/ Mba =  0.5  is also used in moment distribution as carry over factor .

the rotation of end B while A is kept fixed in postion and direction (see fig 1 (b) ) , similarly gives 

Mba =  4 (EI/L)  θb

Mab = 0.5 Mba

moment distribution consists of step by step deformation of structures by superposition of operation such as those shown fig 1 (a) and fig 1 (b) the operation being systematically directed towards a satisfaction of equilibrium requirements , superposition is justified by linearly of relationships between applied forces and deformation i.e by the constancy of the stiffness .

another type of deformation is traslation of one end of member relative to other through some distance Δ   ( see fig 1 (c)) , Member retain its orginal  direction at end A and B  thus inducing  terminal Moment Mab and Mba 

Mab = Mba =  -6 (EI/L)  .  Δ /L

Translating force F is given by 

  F  =   -   {(Mab + Mba) } /  L ) =  12 (  EL/ L³    )   .Δ

12 (  EL/ L³    )   is stiffness of member with respect to traslation 

the effect of transverse load is  introduced by considering fixed end moments

 Mab (f)  and Mba (f)  induced when both end are kept in fixed position and direction )fig 1 (d)) 

the state of a laterally loaded member in which rotation have occurred at both end together with translation (1 (e))  is derived by superposition from the four seprate operation in fig 1 (a)  to (d)  , giving the complete slope deflection equation for a member with no axial load in the form 

Mab (f)  =  Mba (f)  +  k {  4 θa +  2 θb  -   6  Δ /L  } .

where k = EI/L