Advanced Package Training Scaffolding 2011.1 - Scia-Software GbR

10. The second order calculation
Timoshenko
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Advanced Training
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 2 nd 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 than 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-∆ effect) are taken into account in this method.
This principle is illustrated in the following figure.
M(x) = Hx
M(L) = HL
First Order Theory
The local P-δ effect will be regarded further in this course.
M(x) = Hx + P δ + P Δx /L
M(L)= HL + P Δ
Second Order Theory
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 Engineer 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.
This technique is illustrated in the following diagram.