Make up Test: BE

The Make up test of EME-213 is sheduled on 13 Nov 2014, THURSDAY (12:00 -1:00 PM) in the drawing hall

Syllabus for Make up: Topics Covered in the Class uptill 8 Nov 2014 in both the classes (of Mr. Khwaja Zaheer uddin & M Azeem )

Make up Test: BTech

The Make up test of ME-213 is sheduled on 13 Nov 2014, THURSDAY (12:00 -1:00 PM) in the drawing hall

Syllabus for Make up Topics Covered in the Class uptill 8 Nov 2014 in both the classes (of Mr. Najeeb ur Rahman & M Azeem )

Stress Transformation: Lecture (important Figures)

https://drive.google.com/file/d/0B0hvU4t2vA-CU1NrMkpxdmJSQUE/view?usp=sharing

Regarding Make up Test of ME-213

Those Students who need to appear in the Make-Up Test for ME-213, may kindly discuss the schedule with me till Saturday (8/11/14) so that It may not have any clash with other Make Up Test Schedule. Otherwise it will be notified at any day in the upcoming week.

Please inform the back log students who are in the III, IV, V year of graduation and appearing in this Course.

Stress Transformation: Plane Stress

Plane State of Stress

A class of common engineering problems involving stresses in a thin plate or on the free surface of a structural element, such as the surfaces of thin-walled pressure vessels under external or internal pressure, the free surfaces of shafts in torsion and beams under transverse load, have one principal stress that is much smaller than the other two. By assuming that this small principal stress is zero, the three-dimensional stress state can be reduced to two dimensions. Since the remaining two principal stresses lie in a plane, these simplified 2D problems are called plane stressproblems. Assume that the negligible principal stress is oriented in the z-direction. To reduce the 3D stress matrix to the 2D plane stress matrix, remove all components with z subscripts to get, where txy = tyx for static equilibrium. The sign convention for positive stress components in plane stress is illustrated in the above figure on the 2D element.

Coordinate Transformations

The coordinate directions chosen to analyze a structure are usually based on the shape of the structure. As a result, the direct and shear stress components are associated with these directions. For example, to analyze a bar one almost always directs one of the coordinate directions along the bar's axis. Nonetheless, stresses in directions that do not line up with the original coordinate set are also important. For example, the failure plane of a brittle shaft under torsion is often at a 45° angle with respect to the shaft's axis. Stress transformation formulas are required to analyze these stresses. The transformation of stresses with respect to the {x,y,z} coordinates to the stresses with respect to {x',y',z'} is performed via the equations, where q is the rotation angle between the two coordinate sets (positive in the counterclockwise direction). This angle along with the stresses for the {x',y',z'} coordinates are shown in the figure below,

Principal Stresses and Principal Directions

The normal stresses (sx' and sy') and the shear stress (tx'y') vary smoothly with respect to the rotation angle q, in accordance with the coordinate transformation equations. There exist a couple of particular angles where the stresses take on special values. First, there exists an angle qp where the shear stress tx'y' becomes zero. That angle is found by setting tx'y' to zero in the above shear transformation equation and solving for q (set equal to qp). The result is, The angle qp defines the principal directions where the only stresses are normal stresses. These stresses are called principal stresses and are found from the original stresses (expressed in the x,y,z directions) via, The transformation to the principal directions can be illustrated as:

Maximum Shear Stress Direction

Another important angle, qs, is where the maximum shear stress occurs. This is found by finding the maximum of the shear stress transformation equation, and solving for q. The result is, The maximum shear stress is equal to one-half the difference between the two principal stresses, The transformation to the maximum shear stress direction can be illustrated as:

Generalized Hookes Law (Anisotropic material)

Recalling One-dimensional Hooke's Law

Robert Hooke, who in 1676 stated, "The power (sic.) of any springy body is in the same proportion with the extension." announced the birth of elasticity. Hooke's statement expressed mathematically is, where F is the applied force (and not the power, as Hooke mistakenly suggested), u is the deformation of the elastic body subjected to the force F, and k is the spring constant (i.e. the ratio of previous two parameters).

Generalized Hooke's Law (Anisotropic Form)

Cauchy generalized Hooke's law to three dimensional elastic bodies and stated that the 6 components of stress are linearly related to the 6 components of strain. The stress-strain relationship written in matrix form, where the 6 components of stress and strain are organized into column vectors, is,   ,      s = C·e or,   ,      e = S·s where C is the stiffness matrix, S is the compliance matrix, and S = C-1. In general, stress-strain relationships such as these are known as constitutive relations. In general, there are 36 stiffness matrix components. However, it can be shown that conservative materials possess a strain energy density function and as a result, the stiffness and compliance matrices are symmetric. Therefore, only 21 stiffness components are actually independent in Hooke's law. The vast majority of engineering materials are conservative. Please note that the stiffness matrix is traditionally represented by the symbol C, while S is reserved for the compliance matrix.