Transcript
Dr. A. Vinoth JebarajVIT University, Vellore.
Stresses vs. Resisting Area’s(Fundamentals of stress analysis)
For Direct loading or Axial loading
For transverse loading
For tangential loading or twisting
Where I and J Resistance properties of cross sectional area
I Area moment of inertia of the cross section about the axes lying on the section (i.e. xx and yy)
J Polar moment of inertia about the axis perpendicular to the section
Varying cross section Constant cross section vertical position
Design for Bending
Design for Bending & Twisting
When a member is subjected to pure rotation, then it has to be designed for bending
stress which is induced due to bending moment caused by self weight of the shaft.
When a gear or pulley is mounted on a shaft by means of a key, then it has to be designed for
bending stress (induced due to bending moment) and also for torsional shear stress which is
caused due to torque induced by the resistance offered by the key .
Example: Rotating axle between two bearings.
Example: gearbox shaft
Beam
Radius of curvature Bending moment
Dimensions of a cross section Bending stress
Bending stresses or Longitudinal stresses ( out of plane stresses)
Pure Bending
If the length of a beam is subjected to a constant bendingmoment and no shear force ( zero shear force) then thestresses will be set up in that length of the beam due tobending moment only then it is said to be in pure bending.
Under bending, top fibers subjected to compressivestresses and bottom fiber subjected to tensile stresses andvice versa.
In the middle layer (neutral axis), there is no stress due toexternal load.
Assumptions in the Evaluation of Bending stress
Why Bending Stress is more Important than axial ?
Stiffness
Axial stiffness =
; Bending stiffness =
; Torsional stiffness =
Stiffness Stiffness
=
y
Is this equation is correct for the below beam?
P
Is it a straight beam? So What?
Stress Concentration near the hole
Curved beam
Nonlinear (hyperbolic) stress distribution
Neutral axis and centroidal axis are notsame
Practical Application of Bending Equation
In actual situation , when you consider any structure bendingmoment varies from point to point and it also accompaniedby shearing force.
In large number of practical cases, the bending moment ismaximum where shear force is zero.
It seems justifiable that to apply bending equation at thatpoint only.
Hence our assumptions in pure bending (zero shear force) is avalid one.
Plane of Bending
X – Plane
Y - Plane Z - Plane
Under what basis Ixx, Iyy and Izz have to be selected in bending
equation?
Bending
Bending Twisting
Transverse loading Beam Element (Bending)
Bending stress
FE Model
Why I – section is better?
Torque Applied
Reaction Torque
Shaft
Gear
Key
Resisting Tangential force
R = Radius of shaft, L = Length of the shaft T = Torque applied at the free endC = Modulus of Rigidity of a shaft materialτ = torsional shear stress induced at the cross sectionØ = shear strain, θ = Angle of twist
Torsional Equation
Polar moment of inertia [J][Area moment of inertia about the axis perpendicular to the section of the shaft]
Shaft circular cross section
Shear stress distribution in solid & hollow shafts
Shear stress
Shear stress
11.02 MPa
11.3 MPa
89.9 MPa
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