2-Dimensional Stress AnalysisJayadeep U. B.M.E.D., NIT CalicutRef.: 1. Finite Elements and Approximation, Zienkiewicz, O. C., and Morgan, K., John Wiley & Sons.2. Zienkiewicz, O. C., The Finite Element Method, Tata McGraw Hill Edition.

Department of Mechanical Engineering, National Institute of Technology Calicut

IntroductionProblems in thin plate (no transverse loads or deflection, except for Poissons effect) Plane Stress (P-) problem.

2-D Problems in Stress Analysis:Problems in thick plate Plane Strain (P-) problem.

Axisymmetric problems (in both geometry and loads).

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Plane Strain ProblemsThe governing differential equations are the equilibrium equations in 2D.

Essential or Dirichlet B.C. Specified Displacements on the boundary:

Natural or Neumann B.C. Specified Tractions on boundary:

Lecture - 01Note: A boundary could be with mixed B.C., with the displacement specified in one direction and force in the other.

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. The displacement vector at a point:

Strain vector:

The stress vector:

Some Conventions:The body force vector:

The surface traction vector:

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. The approximation for displacement at any point inside an element e (assuming the element to be bi-linear rectangle):

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. Rewriting:

Note: Since we have a displacement or primary variable vector (of two values in this case) defined at any point, we get a 2x2 shape function matrix associated with each node, instead of a single value of shape function as in scalar field problems.

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. The approximation for strain at any point inside an element e:

In case of a bi-linear rectangular element, matrix [B] is 3x8 matrix, with derivatives of shape functions.

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. Stress-Strain Relations for Plane-Stress:

Stress-Strain Relations for Plane-Strain:

In general:

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. Strong form of W.R. Statement:

Greens Lemma:

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. Using Greens Lemma, the Weak Form is obtained as:

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. If there is no special reason to use different weighting functions in x & y directions, using Galerkin Method we get:

The term corresponding to Dirichlet B.C. is not normally considered, since this term is not included in the algebraic system, which is solved.

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. Combining the equations:

Rewriting this equation:

Substituting the expressions derived earlier, we get the elemental system:

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress & Strain Problems contd. Or the complete elemental system:

This stiffness matrix will be of size 8x8.

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

2D Example Plane Stress AnalysisFind the stress distribution in the 2D domain (ABCD), for the problem described below:

B.C.: Fixed in all directions on side AD and distributed force in x-direction (100 N/m2) on BC.

Youngs Modulus and Poissons ratio of the material are 2x1011 N/m2 and 0.3 respectively.

Assume unit thickness in the z-direction.

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.FE Mesh using bi-linear rectangle elements:

Approximation:

Weighted Residual Formulation is used. Elemental System:

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.Considering the symmetry of the system, only bottom half needs to be analyzed.

Symmetry B.C. nodes 4, 5 & 6: No displacement in y-direction for these nodes.

Other boundary conditions:

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.Consider element 1: The nodes of this element are 1, 2, 5 & 4. These become the local node numbers 1, 2, 3 & 4 respectively.

Shape functions, assuming a local node numbering scheme:

Shape function matrices can be written with these as the diagonal elements and 0 at the off-diagonal locations.

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.Stiffness matrix calculation for element 1:

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.Constitutive matrix for plane-stress:

This matrix will be same for both the elements, since the material is same.

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.Stiffness matrix calculation for element 1:

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.Stiffness matrix calculation for element 1:

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.Stiffness matrix calculation for element 1:

Calculate all the stiffness matrix and force vector elements by similar integrations. H.W.

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.By formulating the algebraic system as discussed above and solving after applying the essential B.C., we get the nodal displacements.

In general, this will not be sufficient and for the analysis to be complete, we have to determine the secondary parameters like reaction forces, stresses and strains.

The reactions can be easily calculated, if we have the original algebraic system by substituting the displacements, obtained from solution, in the equations corresponding to the unknown reactions.

However, in most of the cases the original matrix will not be retained after solution, due to computational reasons, especially in larger problems.

In Lagrange Multiplier method of enforcing the B.C., the reactions are calculated in the solution process.

If there are concentrated loads, there should be an additional force vector corresponding to them

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Plane Stress Analysis Example Contd.However the method of choice in FEM, for enforcing the essential B.C., is the Penalty Parameter method.

While using the penalty parameter method, we can get the reactions as the product of the Penalty parameter and the error in satisfying the essential B.C.

However, as mentioned earlier this will have some errors associated in the calculated reactions.

For calculating the stresses and strains, the elemental calculations will have to be performed, using the relations:

Generally, the stresses and strains calculated in different elements will have to be averaged to have smooth transition at the inter-element boundaries.

For example, the stress is constant inside a linear triangle, which is usually assigned at centroid and a weighted average is used at the nodal points & inter-element boundary.

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Axisymmetric ProblemsThe displacement vector:

Strain vector:

The stress vector:

Lecture - 03

Department of Mechanical Engineering, National Institute of Technology Calicut

Axisymmetric Problems contd. Stress-strain Relation:

Strain-Displacement Relation:

The other things are similar to plane stress/strain problems.

Lecture - 03

Department of Mechanical Engineering, National Institute of Technology Calicut

Coupled Thermal-Stress ProblemsAn interesting class of problems are the Thermal Stress problems when the temperature distribution causes stresses to be developed in the body.

We will be considering the problems with only one-way coupling, i.e., the temperature distribution influences the stress/deformation, but it does not happen the other way around.

An increase in temperature causes expansion of any material, and we can assume this expansion is same in all directions and depends only on the temperature increase (T) and not on the absolute value of temperature.

A common approach is to consider the effect of temperature by an initial strain distribution.

There are other approaches, but we will not be discussing them in this course.

Lecture - 03

Department of Mechanical Engineering, National Institute of Technology Calicut

Coupled Thermal-Stress Problems contd.In case plane stress, we can assume a initial strain vector, due to temperature as:

Note 1: The temperature change causes only normal strains in an isotropic material. Shear strains are absent.

Note 2: In plane strain, the initial strain vector will be different due to the Poisson effect.

Let us assume that the total strain, which is the sum of thermal and mechanical strains is given by:

Lecture - 03

Department of Mechanical Engineering, National Institute of Technology Calicut

Coupled Thermal-Stress Problems contd.We know that the thermal strains do not cause any stresses. Hence, the stress vector (using the notations defined earlier):

Note: The thermal stresses are not due to the free thermal expansion, but due to the constraints (either internal or external) imposed on it.

Substituting in the final expression of WR formulation, we get:

Lecture - 03

Department of Mechanical Engineering, National Institute of Technology Calicut

Coupled Thermal-Stress Problems contd.This equation can be re-written as:

Hence, the elemental system is obtained as:

In other words, the effect of thermal strains appear as a contribution to the load vector, in addition to the body forces and surface forces.

Lecture - 03

Department of Mechanical Engineering, National Institute of Technology Calicut

Coupled Thermal-Stress Problems contd.However, we need to take further care, while processing the results. From the displacement results, the total strain can be directly obtained.

It is best to decompose the total strain into mechanical and thermal components.

This system can be solved to obtain the displacements, after the assembly and application of essential B.C.

To calculate the stresses, we need to consider only the mechanical component of strain.

The initial strain method could be useful in many analyses other than thermal stress problems, like bolted joints.

Lecture - 03

Department of Mechanical Engineering, National Institute of Technology Calicut

Concluding RemarksWhile considering problems like this, we get multiple number of degrees of freedom (d.o.f.) per node, and the size of the elemental stiffness matrix becomes the product of number of nodes per element and d.o.f. per node.

There are three classes of 2D stress analysis problems Plane stress, Plane strain & Axisymmetric problems.

The Stress Analysis in 2D was considered in this topic, to introduce the concept of FEM in vector fields, like a displacement field.

Thermal-stress problems and one method of handling such problems (initial strain method) were discussed.

The method can be easily extended to 3D, with proper choice of finite elements (tetrahedron or brick elements), B.C., stress/strain/displacement vectors, stress-strain relations etc.

In the same node, there could be natural and essential B.C. like a displacement constraint in x & force in y directions.

Lecture - 03