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Lecture 3

The document discusses Mohr's Circle for plane stress, detailing the relationships between stress components and their geometric representation. It explains how to derive the principal stresses and maximum shear stresses using the circle, emphasizing the relationship between angles in the physical plane and their representation in the circle. Key points include the calculation of average stress, the coordinates of significant points on the circle, and the orientation of elements under stress.

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engineerhamza27
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10 views11 pages

Lecture 3

The document discusses Mohr's Circle for plane stress, detailing the relationships between stress components and their geometric representation. It explains how to derive the principal stresses and maximum shear stresses using the circle, emphasizing the relationship between angles in the physical plane and their representation in the circle. Key points include the calculation of average stress, the coordinates of significant points on the circle, and the orientation of elements under stress.

Uploaded by

engineerhamza27
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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MOHR’S CIRCLE – PLANE STRESS

σx’ - (σx + σy ) = (σx-σy) cos 2 θ + τxysin 2θ


2 2

τx’y’ = - σx –σy sin 2 θ + τxy cos 2θ


2
Squaring and adding these two equations will give:
[σx - ((σx + σy )/2)] 2 + τ2x’y’ = [(σx-σy)/2]2 + τ2xy

(σ – σavg)2 + τ2x’y’ = R2 σavg = (σx + σy )/2

R = √ [(σx-σy)/2]2 + τ2xy
(σ – σavg)2 + τ2x’y’ = R2 σavg = (σx + σy )/2

R = √ [(σx-σy)/2]2 + τ2xy
Mohr Circle
The stress components are usually known (σx, σy τxy).
The center of Mohr circle is at σavg on the normal stress axis.
Consider this case :
X’ axis is coinciding with
the x axis ( θ=0)
σx’ = σx , τx’y’ = τxy

This will give reference


point A which have
coordinates of (σx , τxy)
This will give reference point A which have coordinates of
(σx , τxy) ( θ=0)
σx’ = σx , τx’y’ = τxy
Consider rotating the
x axis 90o counter
clockwise.

σx’ =σy , τx’y’ = -τxy


These values will give
the coordinates of point
G (σy , -τxy)
Point A (σx, τxy) , point G (σy, -τxy), point C is the center located
at σ avg, R is the radius of the Mohr circle.
Line CG is 180o from line CA which , point corresponds to =0
where point G came about from rotation of 90o. This means that a
rotation of θ on the real element will correspond to 2θ on the circle
in the same direction.
Principle stress

•The principle stresses σ1 and σ2 (σ1> σ2) are represented by point B


and D where the circle intersects the s axis (where τ =0)
•The angle between CA and CB represent 2 θ p1, that is the
angle the principle axis do with the x axis, in this case the angle
is measured counter clockwise.
Orientation of an element showing the principle directions

N.B. 2θP on the


circle is actually
θ having the
same direction
Maximum In-Plane Shear Stress

Angle CA-CE IS
2 θ s1, measured
clockwise from
the reference
point A.

Point F and E represent the point of maximum shear stress


and the average normal stress
Orientation of an element having maximum in-
plane shear stress and average normal stress
Stress on an arbitrary plane, inclined by an known
angle θ

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