     # Spacecraft Materials and Structures Two dimensional elements Consider an infinitesimally small cube

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• Spacecraft Materials and Structures مواد وهياكل المركبات الفضائيه

Code 494 Instructor: Mohamed Abdou Mahran Kasem

Aerospace Engineering Department

Cairo University, Egypt

• Two dimensional solids Plane 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

• 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

• Two dimensional elements

• 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

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