1. DAY 6

2. STRESS Stress is a measure of force per unit area within a body.
It is a body's internal distribution of force per area that reacts to external applied loads.
STRESS

3. ONE DIMENSIONAL STRESS
Engineering stress / Nominal stress
The simplest definition of stress, σ = F/A, where A is the initial cross-sectional area prior to the application of the load
True stress
True stress is an alternative definition in which the initial area is replaced by the current area
Relation between Engineering & true stress

4. TYPES OF STRESSES
COMPRESSIVE
TENSILE
BENDING
SHEAR
TORSION

5. SHEAR STRESS TORSION dx z D dz C Taking moment about CD, We get
zdzdy
xdxdy
TORSION 1 2 B A z D C
Taking moment about CD, We get
This implies that if there is a shear in one plane then there will be a shear in
the plane perpendicular to that

6. TWO DIMENSIONAL STRESS
Plane stress
Principal stress

7. THREE DIMENSIONAL STRESS
Cauchy stress
Force per unit area in the deformed geometry
Second Piola Kirchoff stress
Relates forces in the reference configuration to area in the reference configuration
X – Deformation gradient

8. 3D PRINCIPAL STRESS Stress invariants of the Cauchy stress
Characteristic equation of 3D principal stress is
Invariants in terms of principal stress

9. VON-MISES STRESS
Based on distortional energy

10. STRAIN
Strain is the geometrical expression of deformation caused by the action of stress on a physical body.
Strain – displacement relations
Normal Strain
Shear strain (The angular change at any point between two lines crossing this point in a body can be measured as a shear (or shape) strain)
Strain

11. VOLUMETRIC STRAIN
Volumetric strain

12. TWO DIMENSIONAL STRAIN
Plane strain
Principal strain

13. 3D STRAIN Strain tensor Green Lagrangian Strain tensor
Almansi Strain tensor

14. STRESS-STRAIN CURVE
Mild steel
Copper
Thermoplastic

15. BEAM
A STRUCTURAL MEMBER WHOSE THIRD DIMENSION IS LARGE COMPARED TO THE OTHER TWO DIMENSIONS AND SUBJECTED TO TRANSVERSE LOAD
A BEAM IS A STRUCTURAL MEMBER THAT CARRIES LOAD PRIMARILY IN BENDING
A BEAM IS A BAR CAPABLE OF CARRYING LOADS IN BENDING. THE LOADS ARE APPLIED IN THE TRANSVERSE DIRECTION TO ITS LONGEST DIMENSION

16. TERMINOLOGY SHEAR FORCE BENDING MOMENT CONTRAFLEXURE
A shear force in structural mechanics is an example of an internal force that is induced in a restrained structural element when external forces are applied
BENDING MOMENT
A bending moment in structural mechanics is an example of an internal moment that is induced in a restrained structural element when external forces are applied
CONTRAFLEXURE
Location, where no bending takes place in a beam

17. TYPES OF BEAMS CANTILEVER BEAM SIMPLY SUPPORTED BEAM FIXED-FIXED BEAM
OVER HANGING BEAM
CONTINUOUS BEAM

18. BEAMS (Contd…)
STATICALLY DETERMINATE
STATICALLY INDETERMINATE B A C D

19. BEAM
TYPES OF BENDING
Hogging
Sagging

20. DEFLECTION OF BEAMS
A loaded beam deflects by an amount that depends on several factors including:
the magnitude and type of loading
the span of the beam
the material properties of the beam (Modulus of Elasticity)
the properties of the shape of the beam (Moment of Inertia)
the beam type (simple, cantilever, overhanging, continuous)

21. DEFLECTION OF BEAMS Deflections of beam can be calculated using
Double integration method
Moment area method
Castiglianos theorem
Stiffness method
Three moment theorem (Continuous beam)

22. DOUBLE INTEGRATION METHOD
From Flexure formula
Radius of curvature
Ignoring higher order terms
From (1) & (3)

23. DOUBLE INTEGRATION METHODP
Right of load
Left of load L
At x=L/2, dy/dx=0
At x=0, y=0
At x=L, y=0

24. MOMENT AREA METHOD
First method
Second method

25. MOMENT AREA METHOD P Area of the moment diagram (1/2 L) L P/2 P/2
Taking moments about the end
PL/4

26. CASTIGLIANO’s THEOREM
Energy method derived by Italian engineer Alberto Castigliano in 1879.
Allows the computation of a deflection at any point in a structure based on strain energy
The total work done is then:
U =½F1D1+ ½F2D2 ½F3D3+…. ½FnDn F1 Fn F3 F2

27. Increase force Fn by an amount dF
CASTIGLIANO’s THEOREM (Contd …)
Increase force Fn by an amount dF
This changes the state of deformation and increases the total strain energy slightly:
Hence, the total strain energy after the increase in the nth force is:

28. Now suppose, the order of this process is reversed;
CASTIGLIANO’s THEOREM (Contd …)
Now suppose, the order of this process is reversed;
i.e., Apply a small force dFn to this same body and observe a deformation dDn; then apply the forces, Fi=1 to n.
As these forces are being applied, dFn goes through displacement Dn.(Note dFnis constant) and does work:
dU = dFnDn
Hence the total work done is:
U+ dFnDn

29. The end results are equal
CASTIGLIANO’s THEOREM (Contd …)
The end results are equal
Since the body is linear elastic, all work is recoverable, and the two systems are identical and contain the same stored energy:

30. CASTIGLIANO’s THEOREM (Contd …)
The term “force” may be used in its most fundamental sense and can refer for example to a Moment, M, producing a rotation, q, in the body. M q

31. CASTIGLIANO’s THEOREM (Contd …)
If the strain energy of an elastic structure can be expressed as a function of generalised displacement qi; then the partial derivative of the strain energy with respect to generalised displacement gives the generalised force Qi.
If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Qi; then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement qi in the direction of Qi.

32. CASTIGLIANO’s THEOREM
Strain energy P L
P/2
P/2
According to Castigliano’s theorem
PL/4

33. UNIT LOAD METHOD (VIRTUAL WORK METHOD)
Deflection (Translation) at a point:
Rotation at a point:

34. UNIT LOAD METHOD * * Unit load method Q=1
Area of the moment diagram (1/2 L) L
Q/2
Q/2
QL/4 A1 * * A2 d2 d1

35. DEFLECTIONS OF BEAMS

36. DEFLECTIONS OF BEAMS

37. THREE MOMENT EQUATION

38. THREE MOMENT EQUATION (Developed by clapeyron) Continuity condition
Using second moment-area theorem
Equating the above equations

39. THREE MOMENT THEOREM