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Mechanics of Solids
BH IR V TH KK R
ASSOCIATE PROFESSOR,NAVRACHANA UNIVERSITY
If I have seen further than others, it is by standing upon the shoulders of giants – Sir I
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Mechanics of Solids
Solid mechanics is the branch of continuum mechanics that studies the behavior ofmaterials, especially their motion and deformation under the action of forces, tempchanges, phase changes, and other external or internal agents.*
* Wikipedia
Build Structures that do NOT fail under applied
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Force and Moment
A force arises from the action (or reaction) of one body on a
The SI unit of force is the Newton (N).
The moment of a force about a point is equal to the producmagnitude of the force and the perpendicular distance frompoint to the line of action of the force.
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Body Under System of Forces
Stress: Respo
material to ap
forces on a giv
in a given dire
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Stress at a Point
3D State of Stress
sx
sy
sy
tyx
txy
txy
tyx
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Transformation of Stress
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Effect of ForceWhen a force is applied to an elastic body, the body deforms. The way in which the deforms depends upon the type of force applied to it.
Compression force makes the body shorter.
A tensile force makes the body longer
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Bar Under Axial ForceThe usual case – Normal Stress or Direct Stress
A
F
Area
ForceStress s
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Shear Stress
The stress that tends to CUT a member
tXY
tXY
tXY
tXY
tXY
tXY
Complimentary Stre
Appl
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Strain
P
Original length Change in lengthShear St
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Stress-Strain Relationship
Proportionality Limit
Elasticity LimitLower Yield Point
Ruptur
Upper Yield Point
Ultimate Stress Point
S t r e s s
Strain
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Hooke’s Law
Proportionality Limit
Elasticity LimitLower Yield Point
Rupture
Upper Yield Point
Ultimate Stress Point
S t r e s s
Strain
ds
de
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Hooke’s Law - RelevanceRelates stress with strain
Load carrying capacity with deformation
P
Aluminum
Steel
If Areas are equal f
and Aluminum, whmaterial will take m
load? Steel or Alum
ESteel = 2x105 MEAluminum = 70,0
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Hooke’s Law to Rescue
Since the top will move equally for both materials Deformations can be equated
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Stress Components
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Principal Planes and Principal StressePlanes on which shear stress component vanishes Principal Planes
Principal Stresses are the stresses on Principal Planes Principal Stresses
sx
sx
sy
sy
tXY
tXY
tXY
tXY
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Hydrostatic (Mean) Stress
Mean of three Principal Stresses
“Hydrostatic stress is simply the average of thenormal stress components of any stress tenso
Remaining stress is “Deviatoric Stress”
* Wikipedia
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Failure Envelope
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Example: Stress at crack tip
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Examples: von Mises Stresses
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Forces
Self WeightLive Loads
Wind
Earthquakes
Etc…
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Idealization
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1D StructuresBeamsMembers under bending
ColumnsMembers under axial compression
Beam-ColumnsMembers under bending and axial comp
TieMembers under axial tension
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Equilibrium
Every action has an
equal and opposite
reaction
Sir Isaac Newton
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Conditions of Equilibrium (2D)
Sum of all the forces acting in x-direction MUST be zero, i.e. ∑Fx = 0.
Sum of all the forces acting in y-direction MUST be zero, i.e. ∑Fy = 0.
Sum of all the moments about any point must be zero, i.e. ∑Mz = 0.
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Beams
We determine reactions and internal actions usingconditions of static equilibrium
P
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Euler-Bernoulli Theory of Bending
Plane section before bending remains plane after bMaterial is homogeneous and isotropic
Beam bends in an arc of circle
Radius of curvature is large compared to beam dim
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Strain Deformations
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Beam Under Bending
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TorsionMoment is applied about the longitudinal axis of a member Torsio
Deformation Twist
Extremely common in Civil Engineering structures
Usually considered as equivalent moment and shear acting additiona
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Effect of Torsion
• The existence of the axial shear components is
demonstrated by considering a shaft made up
of axial slats.
The slats slide with respect to each other whenequal and opposite torques are applied to the
ends of the shaft.
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SummaryWe discussed the following
Basic effect of loads on bodies
Stresses and Strains
Hooke’s Law
Failure Theories
Euler-Bernoulli’s Theory of Pure bending of beams
Torsion
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Thank You
Questions?