First sorry for my bade language,and thanks for answers.so my problem is i have a structure formed by beam and column (4 story and 2 bays)to include the geometric effect the geometric stiffness is added to the elastic stiffness but a geometric stiffness is a function of axial forces so at first step i divide the total load on a small load increment and i consider at first step (small load) the geometric effect is not taking account so just elastic stiffness is taking account and i obtain the internal axial force witch i us it at next step in the geometric stiffness witch is added to the elastic stiffness to obtain the global stiffness of the element (beam or column).
My problem is that the behavior of the structure is not linear so equation F(forces)=K(stiffness).D(displacement): is applicable just for a small load increment so the equation become dF=k.dD to resolve this equation i know that i need to use numerical method's in my case want use secant stiffness method's to resolve system
thanks
If you want to do get it right, then use a computer program that specializes in 2nd order analysis. Unless your structure is so simple that you can use a some theoretically derived solution. See AISC 13th or 14th edition commentary or Chen & Lui, Structural Stability: Theory and Implementation (1987).
If you're looking for an approximate hand calculation, then use either the B1-B2 method described in AISC, or the Theta method described in ASCE-7.
The theta method is the one I usually use:
Secondary Moment = P*Delta_0
where P = axial force in the frame and Delta is the 1st order deflection of the frame.
Apply this moment to your structure using a secondary shear
Secondary shear = P*Delta_0 / H
where H is the height of the frame.
Theta is the ratio of the secondary sehar to the origial story shear, Vo.
Theta = P*Delta_o / (H*Vo)
Your incremental deflection will then be Delta_o*Theta. This additional deflection would result in additional P-Delta mometn. The limit of the infinite series becomes:
Maybe you should search for the Newtion-Raphson and Modified Newton-Raphson methods. These will help you apply an incremental load and thus consider the equilibrium conditions in the deformed state.
Regards
Analysis and Design of arbitrary cross sections Reinforcement design to all major codes
Moment Curvature analysis
