Understanding Stress and Strain in Structural Mechanics

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. dx xdxdy 2 1 B z z A zdzdy zdzdy 2 1 dz TORSION xdxdy SHEAR STRESS 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 • 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)

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 Copper Mild steel 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 • 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. P DOUBLE INTEGRATION METHOD 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. 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. 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 applythe forces, Fi=1 to n. • As these forces are being applied, dFn goes through displacementDn.(NotedFnisconstant) and does work: dU = dFnDn • Hence the total work done is: U+ dFnDn

29. 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 A2 A1 * * 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