GUJARAT TECHNOLOGICAL UNIVERSITY B. E Semester: 3 Civil Engineering Structural Analysis-1 Faculty: Chandra Bhakuni chandrabhakuni@gmail. com
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 (A 4 Grid or Plain) – Use pencils (6 B to 6 H) only – Evaluation: Internal Exam, External exam, Tutorial submissions + Oral viva • Give group names
Week 1: Torsion • • • 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 (k. Nm) • 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 (k. N/m 2) • 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 (k. N/m 2) • 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 - R 2
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%20 of%20 materials/lects%20&%20 picts/image/lect 19/lecture 19. htm • Wikipedia http: //en. wikipedia. org/wiki/Torsion_%28 mechanics%29 • http: //emweb. unl. edu/negahban/em 325/06 -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.
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