This document discusses stresses and resisting areas for different types of loading on structural members. It covers direct/axial loading, transverse loading, and tangential/twisting loading. Key concepts include: - Area moment of inertia (I) and polar moment of inertia (J) describe a cross-section's resistance to bending and twisting stresses. - Beams must be designed to resist both bending stresses from applied moments and twisting stresses if external torques are present. - Bending stresses are induced by bending moments and cause compression on the top fibers and tension on the bottom fibers. Assumptions made in calculating bending stresses are discussed.

1. Dr. A. Vinoth Jebaraj
VIT University, Vellore.

2. 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

3. Varying cross section Constant cross section vertical position

4. 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

5. Beam
Radius of curvature Bending moment
Dimensions of a
cross section
Bending stress

6. Bending stresses or Longitudinal stresses
( out of plane stresses)

7. Pure Bending

8.  If the length of a beam is subjected to a constant bending
moment and no shear force ( zero shear force) then the
stresses will be set up in that length of the beam due to
bending moment only then it is said to be in pure bending.
 Under bending, top fibers subjected to compressive
stresses and bottom fiber subjected to tensile stresses and
vice versa.
 In the middle layer (neutral axis), there is no stress due to
external load.

9. Assumptions in the Evaluation of Bending stress

10. Why Bending Stress is more Important than axial ?

11. Stiffness
Axial stiffness = ; Bending stiffness = ; Torsional stiffness =

13. Stiffness Stiffness

18. = 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 not
same

19. Practical Application of Bending Equation
 In actual situation , when you consider any structure bending
moment varies from point to point and it also accompanied
by shearing force.
 In large number of practical cases, the bending moment is
maximum where shear force is zero.
 It seems justifiable that to apply bending equation at that
point only.
 Hence our assumptions in pure bending (zero shear force) is a
valid one.

20. 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

21. Transverse loading Beam Element (Bending)
Bending stress
FE Model
Why I – section is better?

22. Torque Applied
Reaction Torque
Shaft
Gear
Key
Resisting Tangential force

23. R = Radius of shaft, L = Length of the shaft
T = Torque applied at the free end
C = Modulus of Rigidity of a shaft material
τ = torsional shear stress induced at the cross section
Ø = shear strain, θ = Angle of twist
Torsional Equation

25. Polar moment of inertia [J]
[Area moment of inertia about the axis perpendicular to the section of the shaft]
Shaft circular cross
section

27. Shear stress distribution in solid & hollow shafts

28. Shear stress
Shear stress
11.02 MPa
11.3 MPa

29. 89.9 MPa