Machanics of Solids
-
Upload
kishlaya-kashyap -
Category
Documents
-
view
223 -
download
0
Transcript of Machanics of Solids
-
8/14/2019 Machanics of Solids
1/71
Lecture Notes
Mechanics Of Solids
ByProf. Y. Nath
-
8/14/2019 Machanics of Solids
2/71
DEFINITION
-
8/14/2019 Machanics of Solids
3/71
Cordinate System
( x , y , z ) or
----
right hand tria
Displacement (u , v , w ) or
Stress = Linear , stress vector (force / area)
T Stress vector = Force / Area with outward drawn normal
Similarly for other plane
-
8/14/2019 Machanics of Solids
4/71
Infinite plane exist at a point P Infinite vectors
Plane/ Area is defined by outward drawn normal.Infinite planes 3mutually outward drawn planes at point P
(x1, x2 , x3,) Planes
at P
Stressvector
State of stress at a point
State of stress at a point =
-
8/14/2019 Machanics of Solids
5/71
Hypothesis
Body is continuous and remain continuous under the action of external forces -
(H1) Elastic Continuum Neighbor remain neighbor under external force No cracks
/gaps or voids(H2) There exists a unique unstressed state of the solid body to which the bodyreturns whenever all the external forces are removed.
(H3) Principle of super position holds good
-
8/14/2019 Machanics of Solids
6/71
Newtons Law
....................... (1)
is mass density
-
8/14/2019 Machanics of Solids
7/71
Similarly
...................... (2)
....................(3)
Euler's Equation
and Components would contribution all other components are either to x orcross it.
Neglecting Higher order terms
(No body couple)
-
8/14/2019 Machanics of Solids
8/71
-
8/14/2019 Machanics of Solids
9/71
PABC Tetrahedron at point P formed by 3 surfaces parallel to coordinate axises with unit normalLet h is the height of the surface between P & ABC
Body forces along(x1, ,x2 x3), vol =
and
i=1,2,3 are called Cauchy formulae are sufficient to define the traction on any plane.
-
8/14/2019 Machanics of Solids
10/71
NORMAL & SHEAR
STRESSES
Let and are the normal & shear comps. On the oblique plan
Substitution of
i =1,2,3 j =1,2,3
-
8/14/2019 Machanics of Solids
11/71
Cross section =2 cm * 3 cm
F = 6000N
Find the normal & shear stress components on
a equally inclined/plane relative to X1,X2,X3
Normal stresscomponent
-
8/14/2019 Machanics of Solids
12/71
Show that
-
8/14/2019 Machanics of Solids
13/71
-
8/14/2019 Machanics of Solids
14/71
-
8/14/2019 Machanics of Solids
15/71
-
8/14/2019 Machanics of Solids
16/71
-
8/14/2019 Machanics of Solids
17/71
-
8/14/2019 Machanics of Solids
18/71
-
8/14/2019 Machanics of Solids
19/71
-
8/14/2019 Machanics of Solids
20/71
-
8/14/2019 Machanics of Solids
21/71
-
8/14/2019 Machanics of Solids
22/71
3 equations with 6 Unknowns
Displacement Components(u , v , w )
3 Unknowns
Stress Tensor
-
8/14/2019 Machanics of Solids
23/71
6 Unknowns
6 equations
No. of Unknowns = 6 + 3 + 6 = 15
No. of equations = 3 + 6 =9
Need to have more equations for the unique solution
Simplest Relationship(LINEAR)
Superposition Principal ( the law of independence of effects of forces ( ) on the deformation (
) )
is developed in the presence of (combined
effect)
Each of stresses caused by each component of
-
8/14/2019 Machanics of Solids
24/71
Hookes Law for Isotropic Material
Two material cons tants.
Isotropy No direct ional property (same in al l).
Direction of principal ( ) stress and Principal strain ( ) must
coincide
-
8/14/2019 Machanics of Solids
25/71
State of stress and strain in terms of Principal components
Has same effect along 2 and 3 orthogonal
directions
Youngs Modulus E
E Modulus of elasticity
orYoungs Modulus
-
8/14/2019 Machanics of Solids
26/71
Poissons Ratio
Uniaxial1D state of stress
Isotropy
Hooks Law inShear
-
8/14/2019 Machanics of Solids
27/71
Cauchy Formula ( Transformation) we Know
Bulk Modulus
k = (Hydros tatic Stress ( Pressure) / Volumetric Strain
-
8/14/2019 Machanics of Solids
28/71
-
8/14/2019 Machanics of Solids
29/71
-
8/14/2019 Machanics of Solids
30/71
Equation of
Equilibrium -
Isotropic Materials
-
8/14/2019 Machanics of Solids
31/71
-
8/14/2019 Machanics of Solids
32/71
Coupled ode not easy to solve them but are cons istent and will give the Unique so lution provided , we are able to
integrate them.
Theories of failure
Ultimate Aim DESIGN
Failure material failure
Uncertainties
MaterialLoading ,Geometry
Ignorance
Mathematical limitations etc.
Simplified Criteria Theories ofyielding
-
8/14/2019 Machanics of Solids
33/71
Principal stressPrincipal strain
are extremum
More useful for Brittle material ,is also called Rankine criteriafor failure.
2. Maximum-Principal Strain Theory
In case of compressive state of stress
1. Maximum-Principal Stress Theory
-
8/14/2019 Machanics of Solids
34/71
3. Maximum Shear Stress Theory
is called Tresca theory useful for ductile material--- cup and cone in simple tension 450
Tresca theory fails in hydro static state of stresses
4. Total strain energy theory --( Beltrami &
Haigh)
5. Shear or Distortion Strain Energy Theory
-
8/14/2019 Machanics of Solids
35/71
-
8/14/2019 Machanics of Solids
36/71
Energy Method
First law of thermodynamics
Adiabatic Process
-
8/14/2019 Machanics of Solids
37/71
Work of Gravity
-
8/14/2019 Machanics of Solids
38/71
Strain energy due to shear stress
Similarly by other components of stresses
Super position
-
8/14/2019 Machanics of Solids
39/71
Ex.1 Bar in tension
-
8/14/2019 Machanics of Solids
40/71
Ex.2 Torsion of circular shaft (r , L)
-
8/14/2019 Machanics of Solids
41/71
Ex.3 Bending strain EnergyNormal stress
-
8/14/2019 Machanics of Solids
42/71
Strain energy due to shear in Beam
Castigliano Theorem
-
8/14/2019 Machanics of Solids
43/71
-
8/14/2019 Machanics of Solids
44/71
Castig liano Theorem
Are corresponding deflection,twists
and rotation due to
Fict itions loads are applied at the point where
there is no load and deflection is songat
at that point where there is no load
Ex.1
-
8/14/2019 Machanics of Solids
45/71
Ex.2
-
8/14/2019 Machanics of Solids
46/71
Ex.3
-
8/14/2019 Machanics of Solids
47/71
Ex.4
-
8/14/2019 Machanics of Solids
48/71
-
8/14/2019 Machanics of Solids
49/71
Ex.5
-
8/14/2019 Machanics of Solids
50/71
Uniaxial
ENGG. Prob. Definition
A. Everyone makes / one who works makes the mistakesB. Never accept the single way solution
Equation of equilibriumNavie rs equation
-
8/14/2019 Machanics of Solids
51/71
Are coupled p.d.e. and it is extremely difficult to solve
Are 3D equations
Simplifications
Most of the simple engineering problems belongs to 1D problems
(i) Uniaxial deformationBars
(ii) Torsion of shaftShaft
(iii) One axis symmetric bendingBeams
(iv) Combined state of stress problems
-
8/14/2019 Machanics of Solids
52/71
-
8/14/2019 Machanics of Solids
53/71
-
8/14/2019 Machanics of Solids
54/71
-
8/14/2019 Machanics of Solids
55/71
-
8/14/2019 Machanics of Solids
56/71
-
8/14/2019 Machanics of Solids
57/71
-
8/14/2019 Machanics of Solids
58/71
-
8/14/2019 Machanics of Solids
59/71
-
8/14/2019 Machanics of Solids
60/71
-
8/14/2019 Machanics of Solids
61/71
-
8/14/2019 Machanics of Solids
62/71
-
8/14/2019 Machanics of Solids
63/71
-
8/14/2019 Machanics of Solids
64/71
-
8/14/2019 Machanics of Solids
65/71
-
8/14/2019 Machanics of Solids
66/71
-
8/14/2019 Machanics of Solids
67/71
-
8/14/2019 Machanics of Solids
68/71
-
8/14/2019 Machanics of Solids
69/71
-
8/14/2019 Machanics of Solids
70/71
-
8/14/2019 Machanics of Solids
71/71