{"id":2604,"date":"2024-08-07T19:06:49","date_gmt":"2024-08-07T13:36:49","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=2604"},"modified":"2025-07-16T15:25:25","modified_gmt":"2025-07-16T09:55:25","slug":"transformation-equations","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/transformation-equations","title":{"rendered":"Transformation Equations for Plane Strain"},"content":{"rendered":"

\"transformation-equations\"<\/p>\n

Principal Strains<\/h2>\n

These are strains on planes on which shear strains vanishes and values of shear strains are maximum and minimum. The planes are known as principals planes.<\/p>\n

Expression for principal strain<\/strong><\/h3>\n

\u03b51<\/sub>\/\u03b52<\/sub> = \u03b5x<\/sub> + \u03b5y<\/sub> \/2 \u00b1 \u221a (\u03b5x<\/sub> – \u03b5y<\/sub>\/2)2<\/sup> + (\u03c6xy<\/sub>\/2)2<\/sup><\/p>\n

Angle of orientation for principal plane,<\/p>\n

tan 2\u03b8p<\/sub> = \u03c6xy <\/sub>\/\u03b5x<\/sub> – \u03b5y<\/sub><\/p>\n

STRAIN ROSETTE<\/h3>\n

Strain rosette is an arrangement of three linear strain gauges in which linear strains are measured in any three directions.
\nIf among the three strains gauges two of them are mutually perpendicular then it is called rectangular rosette<\/p>\n

If all three strain gauges are at equal angle from each other, then it is called delta ( delta (\u2206) rosette.<\/p>\n

Case I: Rectangular Rosette<\/strong><\/h4>\n

\"Rosette\"<\/p>\n

Special case 1:<\/strong><\/p>\n

If\u00a0 \u03b8 = 45\u00b0<\/p>\n

then \u2208x<\/sub> = \u2208o<\/sub><\/p>\n

\u2208y<\/sub> = \u220890<\/sub><\/p>\n

and \u2208x’<\/sub>= \u220845<\/sub><\/p>\n

So, Eq. (i) can be written as<\/p>\n

\"Delta\"<\/p>\n

Case II: Delta (\u2206) Rosette<\/strong><\/p>\n

Let \u2208y<\/sub> is normal strain along y-direction and \u03c6xy <\/sub>is shear strain in x-y plane.<\/p>\n

\"Strain<\/p>\n

From Eq. (i) and (ii) \u2208y<\/sub> and \u03c6xy <\/sub>can be calculated.
\nHence, \u22081<\/sub> and \u22082<\/sub> can be calculated as<\/p>\n

\"Principal<\/p>\n

Case III: Star Rosette<\/strong><\/p>\n

Star rosette is special case is which all three strain gauges remainat an angle 120\u00b0 from adjacent strain gauge.<\/p>\n

\"Star<\/p>\n

THEORIES OF ELASTIC FAILURE<\/h3>\n

As we know a material is said to be failed if elastic limit is exceeded or permanent deformation takes place i.e., when the elastic limit is reached or when material starts yielding. In tensile test, yield stress (design stress) can be easily determined but if member is subjected to various complex stresses then it is very difficult to know the point of yielding or fracture. So, to improve the design of machine component, various theories of failure are developed considering physical behavior of material.<\/p>\n

Generally, there are two modes of failure:<\/strong><\/p>\n

1. Yielding or ductile failure
\n2. Brittle failure or fracture failure<\/p>\n

Maximum Principal Stress Theory (Rankine\u2019s Theory)<\/h4>\n

According to this theory, the material subjected to complex stresses will fail when principal stress induced in material reaches to yield stress<\/p>\n

\u03c31<\/sub> = \u03c3y<\/sub> … for tension<\/p>\n

and failure can occur in compression when least principal stress \u03c33 reaches the elastic limit stress (i.e., yield stress) in compression.<\/p>\n

\u03c33<\/sub> = \u03c3y<\/sub> … for compression<\/p>\n

For no failure, maximum principal stress developed in strained material should be less than equal to yield stress in uniaxial loading<\/p>\n

\u03c31<\/sub> \u2264 \u03c3y<\/sub><\/p>\n

For design stress, suitable factor of safety is introduced.<\/p>\n

So design stress \u03c31d<\/sub> \u2264 \u03c3y<\/sub>\/FOS<\/p>\n