Strain Energy. Variational Theorems. Concept of Minimum Min imum Potential Energy E nergy.. The Ritz Method. Strain Energy
capacity to do work energy – capacity energy – stored stored energy as a result of work previously done e.g. wind-up toys, potential energy – spinning tops strain energy is a form of potential energy that is stored in materials that have undergone/ been subjected to strain when the strained material returns to its original dimensions, it does work
energy stored within a material when work has been done on the material str ain ener ener gy – energy
assumption: material remains within elastic region so that all the energy is recovered i.e. no permanent deformations thus
with reference to the diagram above: work done by (a gradually applied load) in straining material = area under the load-extension curve the area above the curve is known as the complementary energy
Strain Energy due to Normal (direct) Stress
consider the elemental length of bar,
, with coss-sectional area
Young’s modulus,
From which we get,
Substituting this into
For a bar of length , total strain energy is
If the bar has constant c/s area
Normal stress,
gives
so
Strain Energy due to Shear
Strain energy
Shear modulus
From which we get
Putting the above expressions together gives shear strain energy,
Total energy from shear
Now
Strain Energy due to Bending
Bending theory equation:
Strain energy =work done=
now
So
Strain energy becomes
Total strain energy from bending
For constant bending moment
Strain Energy due to Torsion
Torsion theory
Angle of twist
Strain energy = work done =
From simple torsion theory
Total strain energy from torsion
is in radians
Since T is constant in most practical applications
So we have
Also from simple torsion theory
Now the polar moment of inertia for a shaft
Also the volume for a bar is
Substituting these into the above expression for strain energy gives
FEM- take a complex problem that is difficult to solve break it down into a simple one that can be solved
Physical model or Model based simulation
Break down into Elements and nodes
Governing DE which often Cannot be solved directly
Approximate solution* BUT enforcing BCs and substituting back into DE does not yield sufficient # of linearly independent equations to solve for unknown coefficients
Solve for displacements at nodes and interpolate across element to get strains and stresses
∑
*
Generate sufficient linearly independent equations to solve for coefficients
to FEM
Variational Theorems
As stated before there are two ways of approaching FE Method
One is fairly straight-forward and easy to understand
Physical appr oach (a.k.a. model based simulation)
Variational Methods e.g. Ritz
start with a physical system/ structure break it down into elements reconstruct by assembly process using nodes find the solution (deformations, etc.)
What do you do when you have something like thi s? how do you ‘break up’ a ODE/PDE so you can solve it?
Here we have a Differential Equation (DE) or governing equation
This equation must be ‘manipulated’ in order to generate a discret e FE model
Formulation of a variational or weak form of the governing equation yield FEM equations which can then be solved
This is the M athematical appr oach
Because DEs are difficult to solve so we seek approximate solutions using various methods that must minimise error in solution
In fact FEM assumes that the solution of the DE is a piecewise continuous function
Two formulations (i.e. formulation of FE equations) which are widely used are the Galerkin Method and the Ritz Method
We will consider the Ritz Method which is derived using the Principle of Minimum Potential Energy
Formulation- process by which governing equation is ‘broken down’ into a system of equations (represented in matrix form) which can then be solved by computers Equations are solved element by element at the nodes
Quick Review of Differential Equations
Linear Differential Equations have the general form:
⏞
⏟
e.g.
this DE will have two boundary conditions (BCs)
two types of BCs
⏞
(i) Neumann/ Natural Boundary Conditions (NBCs) & (ii) Dirichlet/ Essential Boundary Conditions (EBCs) NBCs consist of higher order derivatives and are not sufficient to solve the DE completely EBCs are sufficient to solve the DE completely
Thus, the DE above can be solved completely if one of the two conditions is met: (a) EBC prescribed at both ends i.e. , where *is the prescribed value (b) EBC prescribed at one end and NBC at the same/other end i.e. and
Now
will have four BCs of which two must be EBCs Can be solved completely if: (a) at both ends or (b)
at one end and
In this case the NBCs are
at the other
and
Potential Energy
A system which is constrained at certain portions will deform when subjected to loading These deformations are unknowns and are precisely what we seek in FEM i.e. deformations are linked to strains, and strains are linked to stresses After loading the structure attains a new state and the Potential Energy (PE) of the s ystem (elastic) is given by where is the PE, is the strain energy and is the work done on the body by external forces (-ve sign implies energy lost by external forces)
There are many possible deformed/deflected shapes of the loaded struct ure so the question becomes: How do we know which is the correct deflected shape?
Total potential energy attains stationary value (maximum or minimum) at the actual displacement.
The Principle of Minimum Potential Energy
Every applied load will have a unique deformed shape at equilibrium and this unique shape occurs when the PE is an extreme value (eit her maximum or minimum) When the PE is a minimum the equilibrium state is said to be ‘stable’
Among all the displacement states of a conservative system that satisfy compatibility and boundary restraints, those that also satisfy equilibrium make the potential energy stationary. or A system is in a state of equilibrium only when its potential energy is minimal. or For conservative structural systems, of all the kinematically* admissible deformations, those corresponding to the equilibrium state extremise (i.e. maximise or minimise) the total potential energy. If the extremum is a minimum, the equilibrium state is stable. (*satisfy geometric boundary conditions)
Assumptions:
(i) strains and displacements are small (ii) no loss of energy in static loading process i.e.
(ii) system is conservative
Principle of Minimum Potential Energy is the basis for displacement FEM
Example
Use the Principle of Minimum Potential Energy to solve for the equilibrium values of (DOFs) when forces are applied.
Solution
[ ] [ ] The deflection (elongation or contraction) of the springs can be written as follows
Now
so
For equilibrium, must be an extremum with respect to all DOF i.e.
These can be written in matrix form as
{}{} i.e
NB
, where K is the stiffness matrix
From this we can get the s : This set of simultaneous equations is now amenable to solution by computers
For elasticity problems
property
action
behaviour
Elastic Thermal Fluid
Property Stiffness Conductivity Viscosity
Behaviour Displacement Temperature Velocity
Action Force Heat source Body force
unknowns
Variational Methods- A General Discussion
∑ ∫
̅
A variational method is one in which an approximate solution to a DE, , is sought and coefficients are determined using a weighted integral statement
Variational methods:
1. provide means by which DEs can be solved; DE is put in weighted integral form and then an approximate solution over the domain assumed to be a linear combination of appropriately chosen approximation functions and unknown coefficients 2. differ from each other in the choice of the weight function and integral statement used which in turn dictates choice of approximate functions Advantages yield sufficient number of linearly independent equations for determination of coefficients
∑
Disadvantages solution is piecewise over entire domain difficult to construct approximate function for problems with arbitrary domains coefficients are arbitrary NOTE Variational Methods are not FEM- they are part of the pre-processing that leads to FEM
weak form variational form total potential energy there is a need to construct a weak form of DE and classify BCs weak form – weighted integral statement of DE in which the differentiation is distributed between the dependent variable and the weight function and includes BCs
a weak form exists for all problems
The Variational Principle
In variational calculus we look for a function that minimises a functional A functional is an equation which does not depend on coordinates but on functions. The variables of the equation are functions
Variational Formulation
The variational statement can take several forms
* +
Where
An example is
It is essentially describes the potential energy of the system under investigation
The Ritz Method
The Ritz Method is a variational method The exact solution of the governing equation is unknown but an approximate solution is sought The fundamental idea behind this method is to find an approximate solution which minimizes a certain functional and having the form
̅ ̅
Where are unknown parameters/ Ritz coefficients and functions
̅
,
are approximate/trial
The above Ritz function is a linear combination of N known functions that have unknown coefficients These coefficients are adjustable and can be varied until the lowest energy configuration is found i.e. are determined by requiring that is minimized with respect to
(̅)
The trial functions must satisfy the essential BCs
Ritz Equations for the Coefficients
(̅ )
When the approximate solution is substituted into the functional and integrated, we get as a function of the Ritz coefficents i.e.
The Ritz coefficient are adjusted such that parameters
i.e. is minimized with respect to the
This can be written as
The parameters are independent so
Are the N Ritz equations which give us the N Ritz parameters
If the functional
and
is a quadratic the variation is given as
are bilinear and linear forms
The Ritz approximation becomes
̅
and
So
The system of linear Ritz equations
Or
is the governing matrix
The Finite Element Method FEM seeks a continuous (often polynomial) approximation of DE solution over each element in terms of nodal values and assembly of element equations by imposing interelement continuity of solution and balance of forces
Approximate solution takes the form
provides a systematic procedure to the for derivation of approximation functions over sub-regions of the domain approximate functions often called interpolation functions and the degree of the function depends on number of nodes in element and order of DE being solved linear quadratic
Exact solution
nodes
Advantages geometrically complex domains of a problem are represented as a collection of geometrically simple sub-domains – finite elements an approximate solution is derived over each element using the idea that ay continuous function can be represented by linear combinations of algebraic polynomials algebraic relations among unknown coefficients obtained by satisfying the governing equation over each element unknown coefficients represent actual values of the solution at the nodes
So we can say that : FEM is essentially an element by element application of a variational method e.g. Ritz Method
although there is only one FEM, every problem solved using FEM can have several different models depending on the method used to generate equations
Exact solution
Ritz – piecewise continuous
FEM – element by element solution
