Transcript GUJARAT TECHNOLOGICAL UNIVERSITY B.E

GUJARAT TECHNOLOGICAL UNIVERSITY
B.E Semester: 3 Civil Engineering
Structural Analysis-1
Faculty: Chandra Bhakuni
[email protected]
Equip students with
• Sound understanding of underlying principles
related to mechanics of structural elements
and solids.
• Introductory fundamentals relating to analysis
of structural systems elements
Students to note
• Detailed list of topics are given in the course content
document. Available at stationer’s shop or the GTU
website
• One topic will take one week. Read the topic before you
come.
• Buy basic text books for reference
• For term work:
– Tutorial Notebook/File (A4 Grid or Plain)
– Use pencils (6B to 6H) only
– Evaluation: Internal Exam, External exam, Tutorial
submissions + Oral viva
• Give group names
Week 1: Torsion
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Torsion of solid and hollow circular shaft
Shear stress and strain due to Torsion
Angle of twist
Torsional Moment of Resistance
Power transmitted by shaft
• Keys and coupling
• Combined bending and torsion
• Close coiled helical springs
Torsion of solid and hollow circular
shaft
• Define Torsion: The action of twisting or the
state of being twisted, esp. of one end of an
object relative to the other
• Circular Shafts
– Hollow
– Solid
Simple torsion equation
•T = Torque (kNm)
•J = Torsion constant for the section . It is a geometric property similar to ‘polar Moment of
Inertia’
•τ = Maximum shear stress at the outer surface (kN/m2)
•r= R = Outer radius of shaft (m) : maximum strear stress occurs on the outer surface of the shaft
where r = R
•G = Shear modulus; or modulus of rigidity (kN/m2)
•φ = angle of twist (radians)
•L is the length of the object (m)
The product GJ is called the torsional rigidity.
Tutorial:
2. Compare Modulus of Elasticity, Modulus of Rigidity and Torsional Rigidity
Basic kinematics of torsion
(Shear stress and strain due to Torsion)
The value of maximum shearing stress in the solid circular shaft
can be determined as
It is assumed that when twisted, each cross-section in a circular bar
rotates as a rigid body.
The shear stress, through Hook’s law, is related to the shear
strain. Therefore, the shear stress is given by
τ= G ρ
Where ρ = Torsional Strain
Polar Moment of Inertia
Section Modulus (Z)
• Z = Polar Moment of Inertia
• Z=J/y=J/R
• For Solid Circular shaft
• For Hollow Shaft
• Moment of Inertia is also known as Moment of Area.
Power Transmitted by a shaft
• In practical applications, the diameter of the shaft must
sometimes be calculated from the power which it is required
to transmit.
• Given the power required to be transmitted, speed in rpm ‘N'
Torque T, the formula connecting
• These quantities can be derived as follows
Torsional Moment of Resistance
• For Solid Shaft
• For Hollow Shaft:
r = R 1 - R2
Torsional Stiffness ‘k’
is defined as the torque per radian twist .
Torsional flexibility = 1/ k = L/(G.J)
TORSION OF HOLLOW SHAFTS
• From the torsion of solid shafts of circular x – section , it is
seen that only the material at the outer surface of the shaft
can be stressed to the limit assigned as an allowable working
stresses.
• All of the material within the shaft will work at a lower
stress and is not being used to full capacity. Thus, in these
cases where the weight reduction is important, it is
advantageous to use hollow shafts.
• In discussing the torsion of hollow shafts the same
assumptions will be made as in the case of a solid shaft. The
general torsion equation as we have applied in the case of
torsion of solid shaft will hold good .
References
• IIT-ROORKEE:
http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-
ROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect19/lecture19.htm
• Wikipedia
http://en.wikipedia.org/wiki/Torsion_%28mechanics%29
• http://emweb.unl.edu/negahban/em325/06-Torsion/torsion.htm
Tutorials
1. Five sketches of Torsion in things you see daily
2. Compare Modulus of Elasticity, Modulus of
Rigidity and Torsional Rigidity Lab test report
3. Answer Q.1. from Rethaliya
4. Answer Q.2. from Rethaliya
5. Example Questions 1 to 15 from Rethaliya.
Marks will be given for method/approach you
use, not for final answer
6. Laboratory experiment on Torsion. Provide
calculation showing Torsion stress taken by
reinforcement steel bars.