Simple Stresses & Strains

1. Unit 1
Simple Stresses and Strains

2. Materials Under Load
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3. Bodies subjected to Load
• All bodies are deformable
• Subjected to loads, there is change is shape
and size
• Loads may act in all three cardinal directions
• Straining actions may be tension, compression
or shear
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4. General Stress System
• On a three dimensional body, straining actions
can be reduced to three normal stresses in
cardinal directions and three shear stresses
• Normal stresses are - x, y, z
• Shear stresses - xy,xz, yx, yz, zx, zy
• In many elements, this can be reduced to two
dimensions or one dimension
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5. Stress Systems
• In a beam, the stress system can be reduced
to a plane stress system of two dimensions
• A beam has bending stresses, tensile and
compressive, and shear stress
• The stress system of an element of a beam is
thus x, xy
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6. Materials Under Load
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7. Stress Systems
• A truss member is subjected to tension or
compression only
• Truss member is subjected to uniaxial stress
• The basic stress conditions are thus tension,
compression and shear
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8. Materials Under Load
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9. Normal Stress
• Normal stress acts normal to the cross section
of the member
• A member subjected to tension or
compression
• A member subjected to tension
• To find the stress, we take a section some
where in the middle
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10. Stress
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11. Tensile stress
• Separate the member at the section
• The member is in equilibrium after the
straining action
• So each part also is in equilibrium
• Equilibrium of each part requires a force equal
to P – the stress resultant
• Stress is uniformly distributed at X-X
• Near the load P, stress is not uniform
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12. Stress
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13. Tensile stress
• The tensile stress x area gives a force P equal
and opposite to P
• Normal stress is designated by 
• Normal stress  = P/A
• Unit of normal stress is load/area is N/mm²,
kN/m²
• Giga Pascal (GPa) is 109 N/m²
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14. Tensile Stress
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15. Compressive Stress
• Compressive force (thrust) represented by
two forces acting towards each other
• Thrust tends to reduce the length
• The stress resultant at section x-x are two
forces acting away from each other
• Compressive stress is normal to the cross
section
• Symbol for normal stress is .
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16. Compressive Stress
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17. Normal Stress
• A thin cylinder subjected to pressure has
stresses along the circumference called hoop
or circumferential stress
• A cylinder has also stress along its length
called longitudinal stress
• Both these are normal stresses
• Shear stress acts tangential to the cross
section; symbol is ; Average shear stress =
load by area; has the same units as normal
stress
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18. Normal Stresses & shear stress
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19. Strain
• Tensile force elongates a member;
compressive force causes decrease in length
• Strain is elongation/shortening per unit length
• Strain is a ratio and has no unit
• Shear strain is measured as the angle 
• Tan  = L/L; tan =  in radians
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20. Modulii
• Strain is represented by the symbol ;
• For most material ratio of stress to strain is
constant up to a certain magnitude of stress
• Ratio of stress to strain is called modulus
• For normal stress and strain, this ratio is called
Young’s modulus of Elasticity with symbol E
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21. Modulii
• The ratio of shear stress to shear strain
• (/) is called modulus of rigidity; symbols G,
N or C
• E for some materials (GPa) – Steel – 210;
Aluminium – 70; brass – 95
• Modulus of rigidity (GPa) – Steel – 85;
Aluminium – 25; brass – 35
• Modulii have the same units as stress
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22. Elongation/Shortening
• Stress = ; Strain = ; E = /
• Elongation /shortening (L) =  L =  L/E
•  = P/A; L = PL/AE
• For a tapering round bar, Elongation
• L = 4PL/(πEd1d2)
• For a tapering bar of rectangular section
• L = PL log(w2/w1)/[Et(w2-w1)]
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23. Tapering sections
[Add figures 3.9(b) and 3.10(b) in this frame]
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24. Normal, Tensile, Compressive, &
Tangential Strain
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25. Composite sections
• Two or more material elements rigidly joined
to act together
• P = P1 + P2
• Second condition of equal strains
• P1L/A1E1 = P2L/A2E2; 1/E1= 2/E2
• Modular ratio m = E1/E2; 1 = 2 m
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26. Composite Sections
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27. Stresses due to Temperature Change
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28. Stress due to temperature
• Thermal stresses due to restraining
expansion/contraction due to change in
temperature
• Bar of single material, fully restrained
• Stress = E t
• Partial yielding by ‘a’;
• Stress = E (Lt – a)/L
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29. Thermal Stress in Bar of Single
Material
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30. Temperature stress in composite bars
• Two bars rigidly joined together, subjected to
temperature change
• No external restraint
• Thermal expansion coefficients 1, 2
• 1< 2; Free expansion of 1 = L1T
• Free expansion of 2 = L2T
• Material 1 expands more than L1T and in tension
• Material 2 expands less than L2T and in compression
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31. Thermal Stress in Bar of Composite
Material
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32. Shear stress and Strain
• Shear stress is found in riveted joints, keyed-
shafts, beams and rotating shafts
• Shear stress acts tangential to the surface
• Shear stress  shear strain ( = L/L)
•  = G 
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33. Shear Stress & Strain
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34. Complementary Shear Stress
• Shear stress on two opposite faces
• Accompanied by shear stress of opposite
nature on the other two opposite faces
• Shear stress is taken positive when the
moment of the forces about a point within the
element is clockwise
• Shear stress on opposite faces is
complementary shear stress
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35. Complementary Shear Stress
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36. State of Pure Shear
• When element is subjected to only shear
stress, it is state of pure shear
• The element elongates along one diagonal
and shortens along the other
• There is tensile stress along one diagonal and
compressive stress along the other
• Stresses along the diagonal is the same as that
acting on the sides
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37. Stress & Strain along Diagonals
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38. Poisson’s ratio
• Material stressed in one direction has stresses
of opposite nature in the other two directions
• Ratio of longitudinal strain to lateral strain is
Poisson’s ratio
• Symbol for Poisson’s ratio = 1/m, 
• Poisson’s ratio lies between o.25 & 0.34 and
cannot be more than 0.5
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39. Lateral Strain & POISSON’s Ratio
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40. Multi- axial Stresses & Generalized
Hooke’s Law
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41. Generalized Hooke’s Law
• Three stresses x, y, z acting on element
• Strain x = [x - y - z]/E
• x + y + z = (1-2) [x + z + z]/E
• x = z = z = ; x = y = z
• E = (1 - 2) ; 1- 2 is positive;  <0.5
• Shear strain xy = xy/G; yz = yz/G; zx =
zx/G
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42. Bulk Modulus
• Volumetric Strain v = V/V
• Bulk Modulus K = /v [x=y=z=]
• v = x + y +z
• x = x/E - [y + z]/E
• v = (x + y + z)(1 - 2)/E
• v = 3(1 - 2)/E [x = y = z = ]
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43. Bulk Modulus
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44. Elastic Constants
• Elastic constants – E, G, K and 
• Relationships between constants
• G = E / [2(1 + )]
• K = E / [3(1 - 2)]
• G = 3K (1- 2) / [2(1+ )]
• E = 9KG / (3K + G)
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45. Relationship between Elastic
Constants
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46. Mechanical Properties
• Elasticity –regaining shape and size on
removal of load
• Homogeneous – having same elastic
properties every where
• Isotropic – same elastic properties in all
directions at a point
• Plasticity – material undergoes deformation
but does not regain the shape and size on
removal of load
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47. Mechanical Properties
• Ductility – helps to draw thin wires
• Brittleness – tendency to shatter on impact
• Toughness – ability to undergo large plastic
deformations
• Resilience – ability to recover size and shape
after deformation
• Hardness – ability to resist wear, cutting,
scratching or intendation
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48. Mechanical Properties
• Fatigue – failure caused by repeated cycles of
loading even under low stress
• Creep – deformation under constant stress
over a long period of time
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49. Testing of mild steel
• Standard specimen and test procedure as per
BIS specification
• Standard test specimen with gauge length
marked
• Testing done on universal testing machine
• Extensometers used to measure extension
over gauge length
• Specimen tested to failure
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50. Stress Strain Diagram for Mild Steel
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51. Stress – Strain diagram
• Stress – strain diagram has an initial straight
portion where stress  strain
• Slope of the straight line is E
• Proportional limit – stress up to which the
diagram is linear
• Elastic limit – stress up to which when
stressed, shape and size cab regained
• Yield point – stress beyond which plastic
deformation takes place
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52. Stress-Strain diagram
• Ultimate stress – Maximum stress ordinate
• Breaking stress – stress at which material
breaks (lower than ultimate stress)
• As the area decreases when loaded, true
stress when plotted against strain shows a
different type of diagram
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53. Stress-Strain diagram
• Percentage elongation – is the increase in
length expressed as a percentage of original
length
• Percent reduction in area – area measured at
neck. Difference in area expressed as a
percentage of original area
• Value of E for all types of steels is nearly the
same
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54. Stress Strain Diagram for Mild Steel
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55. Stress Strain Diagram for Mild
Steel
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56.  -  diagrams
• Stress-strain diagram for other materials show
a lot of variation
• Different steels have an initial straight portion;
high strength steels do not show much plastic
deformation and hence brittle
• Concrete does not have a well-defined
straight part
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57. Stress Strain Diagram for Other
Material
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58. Stress Strain Diagram for Other
Material
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59. Factor of Safety
• Stress allowable under working loads
• Yield stress / working stress is factor of safety
• Factor of safety is less for steel as it is a
factory manufactured product
• For site-made product like concrete under
not-so-controlled conditions, factor safety is
high
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