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13
2.3. SOLUTIONS TO EQUILIBRIUM EQUATIONS
and obtain the equilibrium equations for the cross section
M
∂2u
∂ψ
T ∂u
+
C
−
C
+L
− Eu − Rψ = 0
∂z 2
∂z
∂z
∂u
+ RT u + Aψ = θ
LT
∂z
∂θ
= TTr θ
∂z
(2.19)
This concludes the derivation of the equilibrium equations of the cross section.
2.3
Solutions to equilibrium equations
The set of equations in (2.19) admits two types of solutions – a particular integral
which depends on the boundary conditions, or the internal force resultants θ in
this case, and a homogeneous integral which corresponds to the eigensolutions when
θ = 0. The homogeneous and particular solutions will be henceforth referred to as
the extremity and the central solutions (cf. Giavotto et al. [6]), respectively. A
more detailed discussion on this topic is presented in Section 2.4.
2.3.1
Extremity solutions
The extremitiy solutions owe their name to the fact that they correspond to the
solutions at the extremities of the beam where the loads are applied. These are the
self-balanced, θ = 0, eigensolutions of (2.19), that is, the solutions to the following
set
M
∂2u
∂ψ
T ∂u
+
C
−
C
+L
− Eu − Rψ = 0
∂z 2
∂z
∂z
∂u
+ RT u + Aψ = 0
LT
∂z
or in matrix form
" ∂2u # M 0
C − CT
L
2
∂z
+
∂2ψ
0 0
−LT
0
2
∂z
∂u
∂z
∂ψ
∂z
−
E
RT
R
A
u
ψ
=0
=
0
(2.20)
0
Assuming solutions of the type
u = ũeλz
ψ = ψ̃eλz
and introducing the previous equation in (2.20) yields
E
M 0
C − CT
L
2
+λ
−
λ
T
0 0
RT
−L
0
R
A
ũ
ψ̃
The solution to (2.20) can be stated in terms of a linear combination of the solutions
to the eigenvalue problem
E R
M 0
C − CT
L
2
λ
+λ
−
=0
0 0
RT A
−LT
0