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Geometric non linearity-peymann et all

Consider attached file.It lists the details of the Timoshinko method for geometric non linear analysis.

Can anyone tell me how is it diiferent from incremental analysis conceptually?

Is it used commercially?Which method is more sophisticated-Timoshinko,incremental or Newton Raphson?

Please help

If you are unable to see the attachment: 

Timoshenko
The first method is the so called Timoshenko method (Th.II.O) which is based on the exact Timoshenko solution for members with known normal force. It is a 2nd order theory with equilibrium on the deformed structure which assumes small displacements, small rotations and small strains.

When the normal force in a member is smaller then the critical buckling load, this method is very solid. The axial force is assumed constant during the deformation. Therefore, the method is applicable when the normal forces (or membrane forces) do not alter substantially after the first iteration. This is true mainly for frames, buildings, etc. for which the method is the most effective option.

The influence of the normal force on the bending stiffness and the additional moments caused by the lateral displacements of the structure (the P-Δ effect) are taken into account. This principle is illustrated in the following figure.

 

 

The local P-δ effect will be regarded further in this course.

If the members of the structure are not in contact with subsoil and do not form ribs of shells, the finite element mesh of the members must not be refined.

The method needs only two steps, which leads to a great efficiency. In the first step, the axial forces are solved. In the second step, the determined axial forces are used for Timoshenko's exact solution. The original solution was generalised in SCIA-ESA PT to allow taking into account shear deformations.

The applied technique is the so called ‘total force method' or ‘substitution method'. In each iteration step, the total stiffness of the structure is adapted and the structure is re-calculated until convergence.

 

Hi-I have read that the convergence criteria for displacement for solving a geometric non linear problem being given by:

Σ(uxi2 + uyi2 +uzi2)- Σ(ux,i-12 + uy,i-12  +uz,i-12  )- / Σ(uxi2 + uyi2 +uzi2) < = 0.05

Can anyone tell me the basis of this formula?

Please help

where:

uxi , uyi and uzi are displacements in directions x,y,z respectively in ith iteration

Jayadeep U. B.'s picture

Hi Kajal,

I think it is just the ratio of the norm (L2 norm, to be precise) of error in displacement vector  (defined as the difference in the displacement vector between (i)th and (i-1)th iterations) and the norm of the displacement vector at (i)th iteration.  The tolerance on the right hand side of the equation should be a variable parameter (the value you have given is 5%, which is really high in my opinion).

Regards,

Jayadeep U.B.

Hi Jayadeep,

Thanks for the reply.

Suppose we are using Newton Raphson for carrying out geometric non linear analysis-in that case at the end of every iteration we check whether external force = internal force or not and within a certain tolerance we stop.

That is, in such a case (Newton Raphson) ca we use the above formula-because tolerance there is related to equality of external and internal forces and not displacement?

Then, I wou;ld be grateful, if you tell me the meaning of 'norm' terminology used above?Why do you call 'norm 2' and what?

Thanks,

kajal

Jayadeep U. B.'s picture

Hi Kajal,

There can be different criteria for checking convergence, and the force criterion you mentioned is only one among them.  Examples include displacement criterion (discussed above) and energy criterion.  Somewhere, I have seen a discussion on the relative merits and demerits of different criteria, though I am not competent to talk about them at present.  May be some experts in iMechanica will provide more information in this regard.  ANSYS uses both force and displacement criteria together for checking convergence, to avoid some pitfalls of using only one convergence criterion (I am not familiar with other software).

As you know, the fundamental result of an FE structural analysis is a displacement vector. Hence we need to develop the concept of the magnitude of the vector (say, error vector) for making comparisons (say, with the tolerance).  L2 norm (Euclidean norm) is the most commonly used measure for this.  It is the square root of the sum of squares of all the elements of the vector, which can be correlated with the distance between two points in the Euclidean space (n-dimensional).  Of course, taking the square root is not a must, since the convergence tolerance can be modified suitably. I think the summation is notwritten properly in your earlier comment (The error vector should be calculated before finding norm).

Hope it is useful. Regards,

Jayadeep U.B.

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