spacecraft materials and structures two dimensional elements consider an infinitesimally small cube...
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Spacecraft Materials and Structuresمواد وهياكل المركبات الفضائيه
Code 494Instructor: Mohamed Abdou Mahran Kasem
Aerospace Engineering Department
Cairo University, Egypt
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Two dimensional solidsPlane stress problems
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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.
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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
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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.
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Two dimensional elements
As forces applied to the body, the body will deform.
The displacement vector in terms of Cartesian coordinates has the form
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Two dimensional elements
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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
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Two dimensional elements
The strain-stress relation which known as Hook’s Law has the form
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Two dimensional elements
For plane stress problems, Hook’s Law has the form
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Two dimensional elements
For plane strain problems, Hook’s Law has the form
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Two dimensional elements
Using the minimum potential energy approach
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Two dimensional elements
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Two dimensional elements
Linear triangular element
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Two dimensional elements
Linear triangular element
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Two dimensional elements
Linear triangular element
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Two dimensional elements
Linear triangular element in terms of natural coordinates
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Two dimensional elements
Linear triangular element in terms of natural coordinates
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Two dimensional elements
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Two dimensional elements
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Two dimensional elements
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Two dimensional elements
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Two dimensional elements
Load Matrix
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Two dimensional elements
Load Matrix
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Linear Triangular element
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Linear Triangular element
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Linear Triangular element - Example
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Linear Triangular element - Example
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Linear Triangular element - Example
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Linear Triangular element - Example
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Linear Triangular element - Example
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Linear Triangular element - Example
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Linear Triangular element - Example
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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).
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Isoperimetric formulation of quadrilateral element
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Isoperimetric formulation of quadrilateral element
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Isoperimetric formulation of quadrilateral element
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Isoperimetric formulation of quadrilateral element
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Isoperimetric formulation of quadrilateral element
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Isoperimetric formulation of quadrilateral element
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Isoperimetric formulation of quadrilateral element
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Isoperimetric formulation of quadrilateral element