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 = implementation defined

Public Member Functions
	BentBeam (const BentBeam &)=default
BentBeam &	operator= (const BentBeam &)=default
	BentBeam (BentBeam &&)=default
BentBeam &	operator= (BentBeam &&)=default
std::unique_ptr< elliptic::analytic_data::AnalyticSolution >	get_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_type &	constitutive_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
	NOLINTNEXTLINE(google-runtime-references)
virtual std::unique_ptr< AnalyticSolution >	get_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. [194], 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 $f_{\mathrm{e}\mathrm{x}\mathrm{t}}^j$ act on the material, so the Elasticity equations reduce to ${\mathrm{\nabla }}_i{T}^{ij}=0$, but the bending moment $M$ generates the stress

$$\begin{array}{}\text{(1)}& T^{xx}& =\frac{12M}{H^3}y\text{(2)}& T^{xy}& =0=T^{yy}\text{.}\end{array}$$

By fixing the rigid-body motions to

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

we find that this stress is produced by the displacement field

$$\begin{array}{}\text{(3)}& {\xi }^x& =-\frac{12M}{E{H}^3}xy\text{(4)}& {\xi }^y& =\frac{6M}{E{H}^3}(x^2+\nu y^2-\frac{L^2}{4})\end{array}$$

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

$$\begin{array}{}\text{(5)}& S_{xx}& =-\frac{12M}{E{H}^3}y\text{(6)}& S_{yy}& =\frac{12M}{E{H}^3}\nu y\text{(7)}& S_{xy}& =S_{yx}=0\end{array}$$

and the potential energy stored in the entire infinitesimal slice is

$${\int }_{-L/2}^{L/2}{\int }_{-H/2}^{H/2}Udydx=\frac{6M^2}{E{H}^3}L\text{.}$$

Member Function Documentation

get_clone()

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

inlineoverridevirtual

Implements elliptic::analytic_data::AnalyticSolution.

Member Data Documentation

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