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

    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