Generalised Displacement control (can track both snap through and snap back)

I've been programming geometric non-linear analysis of beams.I'm done with load control approch,displacement control approach (whih can track snap through NOT snap back)and generalised displacement control (which can track both snapm through and snap back).

I have validated my results too for all these methods.Though, I think I understand what goes on in load control and displacement control (which is also clearly explained in the book by Zieman , Gallagher), I have a clarification in generalised displacemen control and will be grateful if any of you'll can help me with this .

I've been referring the paper :

Key Considerations in Tracing the Postbuckling Response of Structures

with Multi Winding Loops by Y. B. Yang a; L. J. Leu a; Judy P. Yang a

I've also attached a part of this paper (1 page).

My interpretation of the generalised stiffness parameter (defned and explained in the paper) and question is stated below:

 

 

Generalised stiffness parameter: Concept

 

Generalised stiffness parameter is defined as the ratio of the norm of the displacement at first increment to the displacement at current increment. It represents the stiffness of the structure at the ith (current) incremental step with respect to the  first incremental step.

 

I understand that in the denominator of equation 42 of the paper we have the product of the displacement vectors at i-1 and ith increment.

 

Now, if the product (dot product) of the vectors is +ve it means angle between them is <90 and hence snap-through has not yet occurred but if the dot product of the vectors is –ve it means angle between them > 90 and thus snap through has occurred, so when this is detected, we mutiply the load increment factor of equation 41 by a minus sign.

 

Right?

 

However, GSP is applied only for j (iteration no) equal to 1 for every increment.

 

However, I am not able to understand the physical interpretation of equation 43 of the paper as I’ve stated for equations 41 and 42 above.Also, I do not follow the significance of the minus sign in equation 43.I know the interpretation is somewhat similar to 41 and 42 but not able to extract the wording.Can anyonegive me some clue?

You can equations 41,42 and 43 in te attachment.

Anyone who has ever programmed/ used generalised displacement control-please please help (though I've finishd programming and got result, I feel I need to understand it better)!

Please help

 

Respects,

Kajal