SpECTRE  v2022.09.02
Elasticity::Solutions::BentBeam Class Reference

A state of pure bending of an elastic beam in 2D. More...

#include <BentBeam.hpp>

## Classes

struct  BendingMoment

struct  Height

struct  Length

struct  Material

## Public Types

using constitutive_relation_type = Elasticity::ConstitutiveRelations::IsotropicHomogeneous< 2 >

using options = tmpl::list< Length, Height, BendingMoment, Material >

## Public Member Functions

BentBeam (const BentBeam &)=default

BentBeamoperator= (const BentBeam &)=default

BentBeam (BentBeam &&)=default

BentBeamoperator= (BentBeam &&)=default

std::unique_ptr< elliptic::analytic_data::AnalyticSolutionget_clone () const override

BentBeam (double length, double height, double bending_moment, constitutive_relation_type constitutive_relation)

double length () const

double height () const

double bending_moment () const

const constitutive_relation_typeconstitutive_relation () const

double potential_energy () const
Return potential energy integrated over the whole beam material.

template<typename DataType , typename... RequestedTags>
tuples::TaggedTuple< RequestedTags... > variables (const tnsr::I< DataType, 2 > &x, tmpl::list< RequestedTags... >) const

void pup (PUP::er &p) override

virtual std::unique_ptr< AnalyticSolutionget_clone () const =0

## Static Public Attributes

static constexpr Options::String help

## Detailed Description

A state of pure bending of an elastic beam in 2D.

### Details

This solution describes a 2D slice through an elastic beam of length $$L$$ and height $$H$$, centered around (0, 0), that is subject to a bending moment $$M=\int T^{xx}y\mathrm{d}y$$ (see e.g. [124], Eq. 11.41c for a bending moment in 1D). The beam material is characterized by an isotropic and homogeneous constitutive relation $$Y^{ijkl}$$ in the plane-stress approximation (see Elasticity::ConstitutiveRelations::IsotropicHomogeneous). In this scenario, no body-forces $$f_\mathrm{ext}^j$$ act on the material, so the Elasticity equations reduce to $$\nabla_i T^{ij}=0$$, but the bending moment $$M$$ generates the stress

\begin{align} T^{xx} &= \frac{12 M}{H^3} y \\ T^{xy} &= 0 = T^{yy} \text{.} \end{align}

By fixing the rigid-body motions to

$\xi^x(0,y)=0 \quad \text{and} \quad \xi^y\left(\pm \frac{L}{2},0\right)=0$

we find that this stress is produced by the displacement field

\begin{align} \xi^x&=-\frac{12 M}{EH^3}xy \\ \xi^y&=\frac{6 M}{EH^3}\left(x^2+\nu y^2-\frac{L^2}{4}\right) \end{align}

in terms of the Young's modulus $$E$$ and the Poisson ration $$\nu$$ of the material. The corresponding strain $$S_{ij}=\partial_{(i}\xi_{j)}$$ is

\begin{align} S_{xx} &= -\frac{12 M}{EH^3} y \\ S_{yy} &= \frac{12 M}{EH^3} \nu y \\ S_{xy} &= S_{yx} = 0 \end{align}

and the potential energy stored in the entire infinitesimal slice is

$\int_{-L/2}^{L/2} \int_{-H/2}^{H/2} U dy\,dx = \frac{6M^2}{EH^3}L \text{.}$

## ◆ get_clone()

 std::unique_ptr< elliptic::analytic_data::AnalyticSolution > Elasticity::Solutions::BentBeam::get_clone ( ) const
inlineoverridevirtual

## ◆ help

 constexpr Options::String Elasticity::Solutions::BentBeam::help
staticconstexpr
Initial value:
{
"A 2D slice through an elastic beam which is subject to a bending "
"moment. The bending moment is applied along the length of the beam, "
"i.e. the x-axis, so that the beam's left and right ends are bent "
"towards the positive y-axis. It is measured in units of force."}

The documentation for this class was generated from the following file:
• src/PointwiseFunctions/AnalyticSolutions/Elasticity/BentBeam.hpp