Modelling of the P-δ effect using interpolating functions

Resumo

P-Delta is a second-order effect that arises from the consideration of loads acting on the deflected
configuration of the structure. This effect is especially relevant in slender structures, which present lateral
displacements large enough to significantly increase the bending moment caused by an axial load P acting upon
a displacement Delta (hence P-Delta). There are typically two sources of P-Delta, known as P-Δ (P-"big-delta")
and P-δ (P-"small-delta"). The P-big-delta result is easier to obtain in any geometrically nonlinear analysis, as it
is a global effect associated with displacements of the member ends. On the other hand, the P-small-delta effect
is associated with local displacements relative to the original shape of the element. The usual way to capture this
behavior is to subdivide the elements, thus transforming the problem into a P-Δ effect within each segment.
Since discretization can sometimes be unwanted, especially when dealing with students who still do not grasp
this concept, a solution to overcome it is interesting from a didactic point of view. This work proposes the use of
different sets of shape functions to interpolate the bending moment along the element’s length, to account for the
P-small-delta effect. Shape functions obtained directly from the solution of the differential equation of an axially
loaded deformed infinitesimal element and traditional Hermitian polynomials are used. Comparisons were made
with analytical and numerical solutions. Initial results for Euler Bernoulli beam theory indicate the ability of the
formulation to capture the P-δ effect successfully.