Load stiffness for beam elements

Abaqus provides for loads per unit length in the beam cross-sectional directions as distributed load options for the beam elements (load types P1, P2). Since these are follower forces, they have a load stiffness; and this stiffness can sometimes be important especially in the case of buckling prediction by eigenvalue extraction. The symmetric form of this load stiffness is included in Abaqus/Standard (see Hibbitt, 1979, and Mang, 1980). This form is developed below.

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Distributed loads

ProductsAbaqus/Standard

The external virtual work on the beam is

δ⁢We=∫Sp⋅δ⁢u⁢d⁢S,

where the pressure load, p, is given by the externally prescribed pressure magnitude, p, as p=p⁢nα, where α=1or⁢ 2 defines the particular cross-sectional direction of the load. Therefore, nα    =    (-1)βnβ×(dx/dS), where β=2 when α=1, and β=1 when α=2 so that

δ⁢We=(-1)β⁢∫Sp⁢(nβ×d⁢xd⁢S)⋅δ⁢u⁢d⁢S,

where S is the material coordinate along the beam. Now assuming that the load magnitude, p, is externally prescribed so that it does not change with position, the rate of change of δ⁢We with change in position, d⁢u, is

d⁢δ⁢We=(-1)β⁢∫Sp⁢(d⁢nβ×d⁢xd⁢S+nβ×d⁢d⁢ud⁢S)⋅δ⁢u⁢d⁢S.

Now

d⁢nβ=d⁢ω×nβ,

and so

d⁢nβ×d⁢xd⁢S=(d⁢ω×nβ)⋅d⁢xd⁢S=(d⁢ω⋅d⁢xd⁢S)⁢nβ,    neglecting    nβ⋅d⁢xd⁢S.

Thus,

d⁢δ⁢We=(-1)β⁢∫Sp⁢[d⁢ω⋅d⁢xd⁢S⁢nβ⋅δ⁢u+(nβ×d⁢d⁢ud⁢S)⋅δ⁢u]⁢d⁢S.

This load stiffness is not symmetric, except in the case of a beam in a plane with fixed ends (or no ends, such as a ring), in which case the first term is exactly zero and the second gives the symmetric form

(-1)β⁢∫S12⁢p⁢nβ⋅(d⁢d⁢ud⁢S×δ⁢u+d⁢ud⁢S×d⁢δ⁢u)⁢d⁢S.

In Abaqus, even for the general beams in three dimensions, the load stiffness is introduced as the symmetric part of d⁢δ⁢We above.