# ELLIPSE OF STRESS & STRAIN CLASS NOTES FOR MECHANICAL ENGINEERING

**ELLIPSE OF STRESS & STRAIN **

**CLASS NOTES FOR MECHANICAL **

**ENGINEERING**

## Ellipse of stress finds resultant stress

## and the angle of obliquity. In 2-D, it

## is ellipse of stress . In 3-D it is ellipsoid

## of stress. Actually all bodies are 3-D. All

## stress systems are also 3-D. But to make

## study easy to understand, first 2-D stress

## system is studied. The stress is different

## on each plane passing through a point.

## Therefore, it is not justified to study any

## one particular plane. There are actually

## infinite planes. In order to fully understand

## a 2D stress system, study the stresses

## across all possible plane orientations.

## The axis of the ellipse are the two principal

## stresses.

**ELLIPSE OF STRESS**

#### It is the locus of the resultant stress on the infinite inclined planes. It is a graphical method to find the resultant stress and the angle of obliquity on a given inclined plane.

**CONSTRUCTION**

#### Being a graphical method, there are a few steps of construction.

#### (i) Taking O as center. Draw two concentric circles with radii equal to σ_{1 }and σ_{2} respectively. Take σ_{1 } greater of the _{ }principal stresses.

#### (ii) Draw x-axis and y-axis as usual. Take X-axis as the plane of σ_{2} and y-axis as the plane of σ_{1}.

#### (iii) Draw a line LM at an angle of θ with the plane of σ_{1 }i.e. with the y-axis.

#### (iv) From O draw perpendicular to LM. It cuts the inner circle at P and the outer circle at N respectively.

#### (v) From N draw a line NR perpendicular to X-axis to meet at point R.

#### (vi) From point P draw a perpendicular to meet NR at point Q.

#### (vii) Join OQ.

#### (viii) OQ is resultant stress on the inclined plane. Point Q traces its locus. Locus is an ellipse for different inclined planes.

#### (ix) Measure angle QON which is the angle of obliquity.

#### (x) Major principal stress becomes the semi major axis of the ellipse.

#### (xi) Minor principal stress becomes the semi minor axis of the ellipse.

**Utility of the Ellipse of stress**

#### (i) Ellipse of stress determines the state of stress on any inclined plane (any point) within the stressed body.

#### (ii) It determines the angle of obliquity on any inclined plane.

#### (iii) Higher angle of obliquity represents high shear stress on the inclined plane under consideration. It is in case of a ductile material.

#### (iv) Lower angle of obliquity represents lower shear stress on the inclined plane. It is a case of a brittle material.

**ELLIPSE OF STRAIN**

**It is the locus of the resultant strain on the infinite inclined planes within a body under principle strains. It is a graphical method to find the resultant strain and the angle of obliquity on a given inclined plane (with the plane of major principal strain).**

**CONSTRUCTION**

Being a graphical method, there are a few steps of construction.

(x) Taking O as center, draw two concentric circles with radius equal to ε_{1 }and radius equal ε_{2} respectively. ( ε_{1 }is the greater _{ }principal strain). Principal strains are the axis of the ellipse.

(xi) Draw x-axis and y-axis as usual. X-axis as the plane of ε_{2} and y-axis as the plane of ε_{1}.

(xii) Draw a line LM at an angle of θ with the plane of ε_{1 }i.e. with the y-axis.

(xiii) From O draw perpendicular to LM to cut the inner circle at P and the outer circle at N respectively.

(xiv) From N draw a line NR perpendicular to X-axis.

(xv) From point P draw a perpendicular to meet NR at point Q.

(xvi) Join OQ.

(xvii) OQ Will be resultant strain on the inclined plane and point Q will trace its locus which will be an ellipse for different inclinations of the inclined plane.

(xviii) Measure angle QON which is the angle of obliquity.

https://www.mesubjects.net/wp-admin/post.php?post=1136&action=edit Mohr’s stress Circle

https://www.mesubjects.net/wp-admin/post.php?post=7556&action=edit Principal stresses