Mechanics of Continua and Structures
The equations derived by Wenqiang are as follows:
\\begin{align}
\gamma
&=
\theta-\psi,
\label{eq:def:gamma}
\tag{1.1}
\\
M
&=
\mathsf{E} \mathsf{I} \left(\psi'+\sec^2(\gamma\gamma'\right),
\tag{1.2}
\label{eq:GEBM}
\\
V&=-\mathsf{A} \mathsf{G}\tanγ,
\tag{1.3}
\label{eq:1.3}
\\
M'&=-V,
\tag{1.4}
\label{eq:1.4}
\\
V&=-P_0\sinψ
\tag{1.5}
\label{eq:1.5}
\end{align}\)
From (???) and (???) we get that
\[AGtan(γ)=P0sin(ψ)tan(γ)=P0AGsin(ψ)\]Differentiating 2.2 we get
\[sec2(γ)γ′=P0AGcos(ψ)ψ′\]Replacing the term sec2(γ)γ′ that appears in (???) with the right hand side of (3.0), we get
\\begin{align}
M
&=
\mathsf{E} \mathsf{I} \left(\psi'+\frac{P_0}{\mathsf{A}\mathsf{G}}\cos(\psi\psi'\right),
\tag{1.2b}
\label{eq:1.2b}
\\
&=\mathsf{E} \mathsf{I} \left1+P0AGcos(ψ\right)\psi',
\tag{1.2c}
\label{eq:1.2c}\\
\end{align}\)
Differentiating (???) we get
\\begin{align}
M'&=\mathsf{E}\mathsf{I}
\left(\left(1+\frac{P_0}{\mathsf{A}\mathsf{G}}\cos(\psi\right)\psi'\right)'
\tag{4.0}
\label{eq:4.0}
\end{align}\)
Substituting V in (???) with the right hand side of (???) and M′ in (???) with the right hand side of (???) we get
\[EI((1+P0AGcos(ψ))ψ′)′=P0sin(ψ)\]Introducting the non-dimensional variable ˆs=s/ℓ, where ℓ is an arbitrary unit of length
\[d(⋅)ds→1ℓd(⋅)dξ\]in (5.0) we get
\[((1+P0AGcos(ψ))ψ′)′=P0ℓ2EIsin(ψ)\]Defining
\[ˆP0:=P0ℓ2EIˆμ:=GAℓ2EI \label{def.2} \tag{def.2}\]Equation (6.1) can be written as
\[((1+ˆP0ˆμcos(ψ))ψ′)′=ˆP0sin(ψ)\]we get
\[((1+ˆP0cos(ψ))ψ′)′=ˆP0ˆμsin(ψ)\]The nonlinear differential equation (7.0) is subject to the boundary conditions
\\begin{align}
\psi(0&=0,
\label{eq:7.1a}
\tag{7.1}
\\
\psi'ˆL&=0,
\label{eq:7.1b}
\tag{7.2}
\end{align}\)
where
The energies of the red, blue, and green deformed configuration are, respectively, -5.39304, -11.0318, -4.91333. The relatively straight shape has the lowest energy.
See this mathematics code for a sample numerical solution of the equation (7.0), subject to the boundary conditions (???) and (???).
Some sample results from this mathematics code.
Remark: For a given set of boundary conditions, there are several solutions to the nonlinear differential equation. We need to slightly perturb the boundary conditions or guide the solution through the initial guess to numerically arrive at the solution that has the physical characteristics e.g.,containingaloop that we are most interested.