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