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Applied Mechanics Lab

Mechanics of Continua and Structures

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Mechanics of 1D Continua:

Reduction of the shear deformable, finite deformation, 1D continua theory to a single nonlinear ODE

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/, where is an arbitrary unit of length

\[d()ds1d()dξ\]

in (5.0) we get

\[((1+P0AGcos(ψ))ψ)=P02EIsin(ψ)\]

Defining

\[ˆP0:=P02EIˆμ:=GA2EI \label{def.2} \tag{def.2}\]

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

\[ˆL:=L\]

Failure stresses at the root

\[M=EI(ψ+sec2(γ)γ)MEI=(ψ+sec2(γ)γ)MEI=(1+ˆP0ˆμcos(ψ))ψ\]

Multiple solutions and their energies

Alt text

Total potential energy

\[ˆΠ:=ΠEI\] \[ˆΠ=12ˆL0(1+ˆP0cos(ψ))2ψ2dˆs+12ˆP20ˆμˆL0sin2(ψ)dˆsˆP0ˆμˆL0(cos(ψ)1)dˆs\]

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.

Numerical solution using the shooting method

See this mathematics code for a sample numerical solution of the equation (7.0), subject to the boundary conditions (???) and (???).

Alt text

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.

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