spacecraft materials and structures two dimensional elements consider an infinitesimally small cube...

Spacecraft Materials and Structuresمواد وهياكل المركبات الفضائيه

Code 494Instructor: Mohamed Abdou Mahran Kasem

Aerospace Engineering Department

Cairo University, Egypt

Two dimensional solidsPlane stress problems

Two dimensional elements

Consider an infinitesimally small cube volume surrounding a point within a material.

The application of external forces creates

internal forces and subsequently stresses within

the element.

The state of stress at a point can be defined

In terms of nine components on positive

Faces and their counterparts on the negative faces.

Two dimensional elements

• Because of equilibrium requirements only six independent stress components are needed.

• Thus the general state of stress at a point is defined by

Two dimensional elements

• In most aerospace applications, there is no forces acting in the Z-direction and subsequently no internal forces acting in the z-direction.

• We refer to this situation as plane stress situation.

Two dimensional elements

As forces applied to the body, the body will deform.

The displacement vector in terms of Cartesian coordinates has the form

Two dimensional elements

Two dimensional elements

• These components provide information about the size and shape changes that

occur locally in a given material due to loading.

• If no displacement in the z-direction, we call the situation plane strain.

• The strain-displacement relation has the form

Two dimensional elements

The strain-stress relation which known as Hook’s Law has the form

Two dimensional elements

For plane stress problems, Hook’s Law has the form

Two dimensional elements

For plane strain problems, Hook’s Law has the form

Two dimensional elements

Using the minimum potential energy approach

Two dimensional elements

Two dimensional elements

Linear triangular element

Two dimensional elements

Linear triangular element

Two dimensional elements

Linear triangular element

Two dimensional elements

Linear triangular element in terms of natural coordinates

Two dimensional elements

Linear triangular element in terms of natural coordinates

Two dimensional elements

Two dimensional elements

Two dimensional elements

Two dimensional elements

Two dimensional elements

Load Matrix

Two dimensional elements

Load Matrix

Linear Triangular element

Linear Triangular element

Linear Triangular element - Example

Linear Triangular element - Example

Linear Triangular element - Example

Linear Triangular element - Example

Linear Triangular element - Example

Linear Triangular element - Example

Linear Triangular element - Example

Isoperimetric formulation of quadrilateral element

• Isoparametric formulation means to

use single set of parameters to

represent any point within the

element.

• We call this set of parameters –

reference coordinates (natural

coordinates).

Isoperimetric formulation of quadrilateral element

Isoperimetric formulation of quadrilateral element

Isoperimetric formulation of quadrilateral element

Isoperimetric formulation of quadrilateral element

Isoperimetric formulation of quadrilateral element

Isoperimetric formulation of quadrilateral element

Isoperimetric formulation of quadrilateral element

Isoperimetric formulation of quadrilateral element