DAY 6

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

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

TYPES OF STRESSES COMPRESSIVE TENSILE BENDING SHEAR TORSION

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

TWO DIMENSIONAL STRESS Plane stress Principal stress

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

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

VON-MISES STRESS Based on distortional energy

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

VOLUMETRIC STRAIN Volumetric strain

TWO DIMENSIONAL STRAIN Plane strain Principal strain

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

STRESS-STRAIN CURVE Mild steel Copper Thermoplastic

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

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

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

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

BEAM TYPES OF BENDING Hogging Sagging

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)

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)

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

DOUBLE INTEGRATION METHOD P Right of load Left of load L At x=L/2, dy/dx=0 At x=0, y=0 At x=L, y=0

MOMENT AREA METHOD First method Second method

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

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

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:

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

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:

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

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.

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

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

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

DEFLECTIONS OF BEAMS

DEFLECTIONS OF BEAMS

THREE MOMENT EQUATION

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

THREE MOMENT THEOREM