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