Euler-Bernoulli beam elements

Euler–Bernoulli beam theory is a simplification of the linear theory of elasticity and covers the case for small deflections of a beam that is subjected to lateral loads only.

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In Euler–Bernoulli beam elements it is assumed that the internal virtual work rate is associated with axial strain and torsional shear only. Further, it is assumed that the cross-section does not deform in its plane or warp out of its plane, and that this cross-sectional plane remains normal to the beam axis. These are the classical assumptions of the Euler-Bernoulli beam theory, which provides satisfactory results for slender beams.

Let (S,S1,S2) be material coordinates such that S locates points on the beam axis and (S1,S2) measures distance in the cross-section. In addition, let n1,n2 be unit vectors normal to the beam axis in the current configuration: n1=n1(S),n2=n2(S). Then the position of a point of the beam in the current configuration is

xf=x+S1n1+S2n2,

where x=x(S) is the point on the beam axis of the cross-section containing xf. Then

dxfdS=dxdS+S1dn1dS+S2dn2dS,

and so length on the fiber at (S,S1,S2) is measured in the current configuration as

(dlf)2=dxfdSdxfdS(dS)2=(dxdS+S1dn1dS+S2dn2dS)(dxdS+S1dn1dS+S2dn2dS)(dS)2.

Now since the beam is slender, we will neglect terms of second-order in S1 and S2, the distance measuring material coordinates in the cross-section. Thus,

(1)(dlf)2=(dxdSdxdS+2S1dxdSdn1dS+2S2dxdSdn2dS)(dS)2.