MANE 4240 & CIVL 4240Introduction to Finite ElementsFEM Discretization of 2D ElasticityProf. Suvranu De

Reading assignment:

Lecture notesSummary:

FEM Formulation of 2D elasticity (plane stress/strain)Displacement approximationStrain and stress approximationDerivation of element stiffness matrix and nodal load vectorAssembling the global stiffness matrix Application of boundary conditions Physical interpretation of the stiffness matrix

Recap: 2D ElasticitySu: Portion of the boundary on which displacements are prescribed (zero or nonzero)

ST: Portion of the boundary on which tractions are prescribed (zero or nonzero)

Examples: concept of displacement field

xy321422ExampleFor the square block shown above, determine u and v for the following displacementsxy14Case 1: StretchCase 2: Pure sheary221/2

SolutionCase 1: StretchCheck that the new coordinates (in the deformed configuration)Case 2: Pure shearCheck that the new coordinates (in the deformed configuration)

Recap: 2D ElasticityFor plane stress(3 nonzero stress components)For plane strain(3 nonzero strain components)

Equilibrium equationsBoundary conditions1. Displacement boundary conditions: Displacements are specified on portion Su of the boundary2. Traction (force) boundary conditions: Tractions are specified on portion ST of the boundaryNow, how do I express this mathematically?Strong formulationBut in finite element analysis we DO NOT work with the strong formulation (why?), instead we use an equivalent Principle of Minimum Potential Energy

Principle of Minimum Potential Energy (2D)Definition: For a linear elastic body subjected to body forces X=[Xa,Xb]T and surface tractions TS=[px,py]T, causing displacements u=[u,v]T and strains e and stresses s, the potential energy P is defined as the strain energy minus the potential energy of the loads (X and TS)P=U-W

Strain energy of the elastic bodyUsing the stress-strain lawIn 2D plane stress/plane strain

Principle of minimum potential energy: Among all admissible displacement fields the one that satisfies the equilibrium equations also render the potential energy P a minimum.

admissible displacement field: 1. first derivative of the displacement components exist2. satisfies the boundary conditions on Su

Finite element formulation for 2D:

Step 1: Divide the body into finite elements connected to each other through special points (nodes)

xySuSTuvxpxpyElement e3214

Total potential energyPotential energy of element e:Total potential energy = sum of potential energies of the elementsThis term may or may not be present depending on whether the element is actually on ST

Step 2: Describe the behavior of each element (i.e., derive the stiffness matrix of each element and the nodal load vector).

Inside the element evu1234u1u2u3u4v4v3v2v1Displacement at any point x=(x,y)Nodal displacement vector(x1,y1)(x2,y2)(x4,y4)(x3,y3)whereu1=u(x1,y1)v1=v(x1,y1)etc

If we knew u then we could compute the strains and stresses within the element. But I DO NOT KNOW u!!

Hence we need to approximate u first (using shape functions) and then obtain the approximations for e and s (recall the case of a 1D bar)

This is accomplished in the following 3 Tasks in the next slideRecall

TASK 1: APPROXIMATE THE DISPLACEMENTS WITHIN EACH ELEMENT

TASK 2: APPROXIMATE THE STRAIN and STRESS WITHIN EACH ELEMENT

TASK 3: DERIVE THE STIFFNESS MATRIX OF EACH ELEMENT USING THE PRINCIPLE OF MIN. POT ENERGY

Well see these for a generic element in 2D today and then derive expressions for specific finite elements in the next few classesDisplacement approximation in terms of shape functionsStrain approximationStress approximation

Displacement approximation in terms of shape functionsTASK 1: APPROXIMATE THE DISPLACEMENTS WITHIN EACH ELEMENTDisplacement approximation within element e

Well derive specific expressions of the shape functions for different finite elements later

TASK 2: APPROXIMATE THE STRAIN and STRESS WITHIN EACH ELEMENTApproximation of the strain in element e

Compact approach to derive the B matrix:

Stress approximation within the element e

Potential energy of element e:TASK 3: DERIVE THE STIFFNESS MATRIX OF EACH ELEMENT USING THE PRINCIPLE OF MININUM POTENTIAL ENERGY Lets plug in the approximations

RearrangingFrom the Principle of Minimum Potential EnergyDiscrete equilibrium equation for element e

Element stiffness matrix for element eElement nodal load vectorDue to body forceDue to surface tractionSTeeFor a 2D element, the size of the k matrix is 2 x number of nodes of the elementQuestion: If there are n nodes per element, then what is the size of the stiffness matrix of that element?

If the element is of thickness tElement nodal load vectorDue to body forceDue to surface tractionFor a 2D element, the size of the k matrix is 2 x number of nodes of the elementtdAdV=tdA

The properties of the element stiffness matrix1. The element stiffness matrix is singular and is therefore non-invertible2. The stiffness matrix is symmetric3. Sum of any row (or column) of the stiffness matrix is zero! (why?)

The B-matrix (strain-displacement) corresponding to this element is We will denote the columns of the B-matrix as

Computation of the terms in the stiffness matrix of 2D elements

The stiffness matrix corresponding to this element iswhich has the following formThe individual entries of the stiffness matrix may be computed as follows

Step 3: Assemble the element stiffness matrices into the global stiffness matrix of the entire structureFor this create a node-element connectivity chart exactly as in 1Duv3v1243u2u4u3u1v1v4v2Element #1Element #2

ELEMENTNode 1Node 2Node 311232234

Stiffness matrix of element 1Stiffness matrix of element 2There are 6 degrees of freedom (dof) per element (2 per node)u2u3v2v3u4v4v2u3v3u4v4u2

88K=Global stiffness matrixHow do you incorporate boundary conditions?Exactly as in 1Du1v1u2v2u3v3u4v4u1v1u2v2u3u4v4v3

Finally, solve the system equations taking care of the displacement boundary conditions.

Physical interpretation of the stiffness matrixConsider a single triangular element. The six corresponding equilibrium equations ( 2 equilibrium equations in the x- and y-directions at each node times the number of nodes) can be written symbolically as

Choose u1 = 1 and rest of the nodal displacements = 0 Hence, the first column of the stiffness matrix represents the nodal loads when u1=1 and all other dofs are fixed. This is the physical interpretation of the first column of the stiffness matrix. Similar interpretations exist for the other columnsxyu1=1231

Now consider the ith row of the matrix equationThis is the equation of equilibrium at the ith dof

Consistent and Lumped nodal loadsRecall that the nodal loads due to body forces and surface tractionsThese are known as consistent nodal loads1. They are derived in a consistent manner using the Principle of Minimum Potential Energy2. The same shape functions used in the computation of the stiffness matrix are employed to compute these vectors

Example123xyp per unit areaTraction distribution on the 1-2-3 edgepx= ppy= 0Well see later that bbN1N2N3

The consistent nodal loads are123xybbpb/3pb/34pb/3

The lumped nodal loads are123xybbpb/2pb/2pbLumping produces poor results and will not be pursued further

Displacement approximation in terms of shape functionsStrain approximation in terms of strain-displacement matrixStress approximationSummary: For each elementElement stiffness matrixElement nodal load vector