P-delta effect

In structural engineering, the P-Δ or P-delta effect refers to the abrupt changes in ground shear, overturning moment, and/or the axial force distribution at the base of a sufficiently tall structure or structural component when it is subject to a critical lateral displacement. A distinction can be made between P-delta effects on a multi-tiered building, written as P-Δ, and the effects on members deflecting within a tier, written as P-δ. P-delta is a second-order effect on a structure which is loaded laterally. One first-order effect is the initial deflection of the structure in reaction to the lateral load. The magnitude of the P-delta effect depends on the magnitude of this initial deflection. P-delta is a moment found by multiplying the force due to the weight of the structure and applied axial load, P, by the first-order deflection, Δ or δ. NUMERICAL EXAMPLE OF P DELTA EFFECT ON A CALCULATOR You have a 1 meter tall rigid vertical rod that rotates on a hinge at the bottom of the rod. There is a 1 newton load on the top of the rod. The rod has a hinge with a rotational stiffness of 0.8 newton meters per radian of rotation. So you input any initial rotational angle on the rod. The following table shows that the rod will iterate to 1.13 radians where the rod will be in stable equilibrium. The formula for this table is next radians rotation=sin(last radians rotation)/.8 In the table from the formula you can see the rod starts at .1 radians and iterates to 1.13 radians where it is in stable equilibrium. 1 .124 .156 .194 .241 .300 .367 .448 .542 .645 .751 .853 .942 1.01 1.06 1.09 1.11 1.12 1.12 1.13 1.13 and so on as it converges to 1.13 radians where the rod is stable. The P DELTA effect finds the stable final deformed shape of a structure just like how the rod rotates to a final deformed position at 1.13 radians. The idea is that iteratively repeated linear structural analyses can solve a non linear structural analysis problem. It takes multiple iterations of a linear analysis to compute the final deformed shape of a structure where the P DELTA effect is significant.