Page 132 - Marine Structural Design
P. 132
108 Part I Structural Design Principles
where the subscript is the number of the nodal points. The undetermined coefficients for v in
Eq. (6.4) can be expressed as the four deformations of nodal points, and expressed as
By solving @. (6.6), we get a solution for {a},
{a1 = [AI-’ Me
and the strain is expressed as,
where K, is the curvature and may be expressed as
d2v
k, = -- = 2a, + 6a,n = 10 0 2 6x11.)
dw2
= 10 0 2 6x1 [A]-’ {d), 3 [B]{d},
The stress 0, is then given as,
a, = E&, = -EYK,
The principle of virtual work for this beam element may be expressed as,
~[jl’gxaxdh={6 dj, (6.8)
where A is the cross-section area, and {f), is the nodal force vector corresponding to the
nodal displacement,
where F, , M, (a = 1,2) is the shear force and bending moment for nodal point a . Equation (6.8)
may be rewritten as,
where I is the second order inertia moment for the cross-section, and expressed as,
I= Ly2dA
