12- 2-D Problems

Dr. Ahmet Zafer Şenalpe-mail: azsenalp@gmail.com

Mechanical Engineering DepartmentGebze Technical University

ME 520Fundamentals of Finite Element Analysis

In general, the stresses and strains in a structure consist of six components:

Under certain conditions, the state of stresses and strains can be simplified. A general 3-D structure analysis can, therefore, be reduced to a 2-D analysis.

Review of the Basic Theory

ME 520 Dr. Ahmet Zafer Şenalp 2Mechanical Engineering Department, GTU

12- 2-D Problems

Plane Stress :

A thin planar structure with constant thickness and loading within the plane of the structure (xy-plane).

Plane Problems

ME 520 Dr. Ahmet Zafer Şenalp 3Mechanical Engineering Department, GTU

12- 2-D Problems

Plane Strain :

A long structure with a uniform cross section and transverse loading along its length (z-direction).

Stress-Strain-Temperature (Constitutive) Relations:For elastic and isotropic materials, we have;

or

: Initial strainE : Young’s modulus : Poisson’s ratioG : Shear modulus

which means that there are only two independent materials constants for homogeneous and isotropic materials..

Plane Problems

ME 520 Dr. Ahmet Zafer Şenalp 4Mechanical Engineering Department, GTU

12- 2-D Problems

We can also express stresses in terms of strains by solving the above equation:

or:

: initial stress

The above relations are valid for plane stress case. For plane strain case, we need to replace the material constants in the above equations in the following fashion:

Plane Problems

ME 520 Dr. Ahmet Zafer Şenalp 5Mechanical Engineering Department, GTU

12- 2-D Problems

For example, the stress is related to strain by:

in the plane strain case.Initial strains due to temperature change (thermal loading) is given by:

: coefficient of thermal expansion : the change of temperature

Note that if the structure is free to deform under thermal loading, there will be no (elastic) stresses in the structure.

Plane Problems

ME 520 Dr. Ahmet Zafer Şenalp 6Mechanical Engineering Department, GTU

12- 2-D Problems

Strain and Displacement Relations:For small strains and small rotations, we have:

In matrix form:

Equilibrium Equations:In elasticity theory, the stresses in the structure must satisfy the following equilibrium equations:

Plane Problems

ME 520 Dr. Ahmet Zafer Şenalp 7Mechanical Engineering Department, GTU

12- 2-D Problems

where fx and fy are body forces (such as gravity forces) per unitvolume. In FEM, these equilibrium conditions are satisfied inan approximate sense.

Boundary Conditions:The boundary S of the body can be divided into two parts, Su and St. The boundary conditions (BC’s) are described as:

in which tx and ty are traction forces (stresses on the boundary) and the barred quantities are those with known values.In FEM, all types of loads (distributed surface loads, body forces, concentrated forces and moments, etc.) are converted to point forces acting at the nodes.Exact Elasticity Solution:The exact solution (displacements, strains and stresses) of a given problem must satisfy the equilibrium equations, the given boundary conditions and compatibility conditions (structures should deform in a continuous manner, no cracks and overlaps in the obtained displacement fields).

Plane Problems

ME 520 Dr. Ahmet Zafer Şenalp 8Mechanical Engineering Department, GTU

12- 2-D Problems

on Su

on St

A plate is supported and loaded with distributed force p as shown in the figure.

The exact solution for this simple problem can be found easily as follows: Displacement:

Strain:

Stress:

Exact (or analytical) solutions for simple problems are numbered (suppose there is a hole in the plate!). That is why we need FEM!

Plane ProblemsExample:

ME 520 Dr. Ahmet Zafer Şenalp 9Mechanical Engineering Department, GTU

12- 2-D Problems

Elastic modulus : EPoisson ratio :

Displacements (u, v) in a plane element are interpolated from nodal displacements (ui, vi) using shape functions Ni as follows:

N : shape function matrix,u : displacement vectord : nodal displacement vectorHere we have assumed that u depends on the nodal values of u only, and v on nodal values of v only.From strain-displacement relation, the strain vector is:

: B=strain-diplacement matrix

Finite Elements for 2-D Problems A General Formula for the Stiffness Matrix

ME 520 Dr. Ahmet Zafer Şenalp 10Mechanical Engineering Department, GTU

12- 2-D Problems

Consider the strain energy stored in an element:

From this, we obtain the general formula for the element stiffness matrix:

ME 520 Dr. Ahmet Zafer Şenalp 11Mechanical Engineering Department, GTU

Finite Elements for 2-D Problems A General Formula for the Stiffness Matrix

12- 2-D Problems

Note that unlike the 1-D cases, E here is a matrix which is given by the stress-strain relation.

The stiffness matrix k is symmetric since E is symmetric. Also note that given the material property, the behavior of k depends on the B matrix only, which in turn on the shape functions.

Thus, the quality of finite elements in representing the behavior of a structure is entirely determined by the choice of shape functions. Most commonly employed 2-D elements are linear or quadratic triangles and quadrilaterals.

ME 520 Dr. Ahmet Zafer Şenalp 12Mechanical Engineering Department, GTU

Finite Elements for 2-D Problems A General Formula for the Stiffness Matrix

12- 2-D Problems