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Mohr Circle Problems

The document discusses stress and strain analysis in three dimensions. It covers topics like triaxial stress states, principal stresses, normal and shear stresses on oblique planes, and Mohr's circle for representing the state of stress. Several examples of stress states are analyzed and illustrated using Mohr's circle.

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21je0444
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42 views17 pages

Mohr Circle Problems

The document discusses stress and strain analysis in three dimensions. It covers topics like triaxial stress states, principal stresses, normal and shear stresses on oblique planes, and Mohr's circle for representing the state of stress. Several examples of stress states are analyzed and illustrated using Mohr's circle.

Uploaded by

21je0444
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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1.

A long cylindrical pressure vessel with closed ends is to be made by rolling a strip of plastic
of thickness d and width w into a helix and making a continuously fused joint, as illustrated. It
is desired to subject the fused joint to a tensile stress only 80 percent of the maximum in the
parent plastic. What is the minimum tilt of strip of plastic?
Y

X
Google images
A cylindrical pressure vessel with an inside diameter 480 mm is constructed by spirally
wrapping a 8 mm thick steel plate and butt-welding the mating edges of the plate. The butt-
welded seams form an angle of 50° with a transverse plane through the cylinder. Determine
the largest gage pressure that can be used inside the vessel if the allowable normal stress
perpendicular to the weld is 100 MPa and the allowable shear stress parallel to the weld is 25
MPa.

50°
State of stress in three dimensions Triaxial state of stress

The components of σ along each of the axes:


-
𝑆𝑥 = 𝜎𝑙 𝑆𝑦 = 𝜎𝑚 𝑆𝑧 = 𝜎𝑛

Area KOL = Al Area JOK = Am Area JOL = An

Taking summation of forces in the X direction results in

In the other two directions results in

Mechanical Metallurgy - Dieter


Solution of the determinant gives a cubic eqn. in terms of σ

Three roots - ? Three principal stress: σ1 , σ2 and σ3

Their directions?

The coefficients of the cubic equations are invariant.


-
Normal and shear stress on any oblique plane whose
normal has the direction cosines l, m, n with the X, Y, Z axes
respectively.

S – total stress 𝑆2 = 𝜎2 + 𝜏2
𝑆 2 = 𝑆𝑥2 + 𝑆𝑦2 + 𝑆𝑧2

Taking the summation of the


forces in the X, Y and Z directions,
we get the orthogonal 𝜏𝑥𝑦 = 𝜏𝑦𝑥
components of the total tress: 𝜏𝑦𝑧 = 𝜏𝑧𝑦
𝜏𝑥𝑧 = 𝜏𝑧𝑥
Normal stress on the oblique
plane:

Magnitude of shear stress on the oblique plane: 𝜏 2 = 𝑆 2 − 𝜎 2


Magnitude of shear stress on the oblique plane: 𝜏2 = 𝑆2 − 𝜎2
Magnitude and direction of two components lying in the oblique plane : resolve the
stress components Sx , Sy , Sz into the Y’ and Z’ directions lying in the oblique plane.

Maximum shear stress

The principal shear stresses occur for the


following combinations of direction
cosines that bisect the angle between
two of the three principal axes:

Maximum shear stress??


Planes of principal shear stresses:

Dieter
Mohr’s Circle – 3D state of stress
Uniaxial tension
Uniaxial compression
Biaxial tension
Triaxial tension
Uniaxial tension + biaxial compression
Equal triaxial tension: Mohr Circle ???

𝜎1 = 𝜎2 = 𝜎3

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