08-16-2017, 09:11 PM
[attachment=8474]
CHAPTER-1
STRESS TENSOR
INTRODUCTION:
Two planes from infinitesimal distance away and parallel to each other were made to pass through the body, an elementary slice would be isolated. Then, if an additional two pairs of planes were passed normal to the first pair, a cube of infinitesimal dimensions would be isolated from the body. Such a cube is as shown in the Fig.1.1 (a). On the near faces of the cube, i.e., on the faces away from the origin, the directions of stress are positive if they coincide with the positive direction of the axes. On the faces of the cube towards the origin, from the action-reaction equilibrium concept, positive stresses act in the direction opposite to the positive directions of the axes. The designations for stresses shown in Fig.1.1(a) are widely used in the mathematical theories of elasticity and plasticity.
PURE SHEAR:
As shown in the figure below lets assume a cylindrical shaft subjected to Torsion (T).At a point on the outer surface on a plane perpendicular to the axis of the shaft there exists a state of pure shear.
PRINCIPAL PLANE:
A state of pure shear as shown above can be translated in to a principal plane at an angle 45 deg to the pure shear plane.
A plane which carries only normal stresses are called principal planes.
CHAPTER-2
THEORIES OF FAILURE
YIELD AND FAILURE CRITERIA IN DUCTILE MATERIAL
Experiments made on the flow of ductile metals under biaxial stress have shown that maximum distortion energy theory express well the condition under which the ductile metals at normal temperatures start to yield.
Further as remarked earlier, the purely elastic deformation of a body under hydrostatic pressure ( _oct)is also supported by this theory.
Energy of Distortion Theory
This theory is based on the work of Hubber, von Mises Hencky. According to this theory, it is not the total energy which is the criterion for failure; in fact the energy absorbed during the distortion of an element is responsible for failure. The energy of distortion can be obtained by subtracting the energy of volumetric expansion from the total energy. It was shown in the analysis of stress(section1.22) that any given state of stress can be uniquely resolved into an isotropic state and a pure shear (or deviatoric) state. If 1, 2,and 3 are the principal stresses at a point then it consists of two additive component tensors. The element of one component tensor are defined as the mean hydrostatic stress:
MAXIMUM SHEAR STRESS THEORY
Observations made in the course of extrusion tests on the flow of soft metals (ductile)through orifices lend support to the assumption that the plastic state in such metals is created when the maximum shearing stress just reaches the value of the resistance of the metal against shear.
.
.
.
CHAPTER-6
1D-element Rod, truss (Linear and non linear),beam
One Dimensional Elements
In the finite element method elements are grouped as 1D, 2D and 3D elements. Beams and plates are grouped as structural elements. One dimensional elements are the line segments which are used to model bars and truss. Higher order elements like linear, quadratic and cubic are also available. These elements are used when one of the dimension is very large compared to other two. 2D and 3D elements will be discussed in later chapters.
Seven basic steps in Finite Element Method
These seven steps include
Modeling
Discretization
Stiffness Matrix
Assembly
Application of BC s
Solution
Results
Let s consider a bar subjected to the forces as shown