WHAT STRENGTH OF MATERIALS IS IT ABOUT?

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WHAT STRENGTH OF MATERIALSIS IT ABOUT?
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Why You Dont Fall Through the Floor
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Linguistics
Moc materialów?
Strength of materials
SM is about the resistance of materials(and
structures) against external environmental
actions (forces, deformations, temperatures
etc.) which may lead to the loss of load
bearing capacity
Wytrzymalosc materialów
Wytrzymac co? Ile? Jak dlugo?
????????? ????????
Opór materialów?
Résistance des matériaux
Opór materialów?
Festigkeitlehre
Nauka o sile materialów?
Hållfasthetslära
Nauka o spójnosci materialów?
????
No, to juz zupelna chinszczyzna!
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Origin of SM

• Differential calculus
• Matrix algebra
• Calculus of variations
• Numerical methods

• Theory of elasticity
• Theory of plasticity
• Material Science

SM (of a deformable body)
HYPOTHESES
EXPERIMENTS
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Modelling scheme
Idealisation of
Material

• Continuous matter distribution (material
  continuum)
• Continuous mass distribution ?(x)
• Intact, unstressed initial state of a material

Loadings

• Permanent versus movable
• Constant versus variable in time (static versus
  dynamic)

Structure geometry

• Bulk structures (H LB)
• Surface structures (HLB)
• Bar structures (LHB)

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Mechanical loadings
External surface forces
Surface distributed force p N/m2
Line distributed force q N/m
Point force P N
b
Point moment M Nm
H
External volume forces
M Nm
(gravitational forces, inertia forces,
electromagnetic forces etc.) X N/m3
X N/m3
Displacements u(u,v,w) m (e.g. supports, forced
shift of structural members)
u0, v0
v0


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Internal forces
Fundamental observations

• A body (structure) under external loadings
  changes its shape (material points of this body
  are subjected to the displacement)
• This change in material points position
  influences forces of interaction and results in
  creation of internal forces
• If a body (structure) is in equilibrium each
  point of this body is also in mechanical
  equilibrium i.e. resultant of forces and moments
  is equal to zero.

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Internal forces
wi, i1,2 8
P2
P3
P1
A
Coulomb particle interaction assumed(convergent
set of internal forces)
Pn
Pi
A body in equilibrium
wi convergent, infinite, zero valued set of
internal forces
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Internal forces
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r
II
I
r point position vector
n
n - outward normal vector
n
w f(r,n)
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Internal forces
n
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Z ZI ZII 0
ZI wI 0
Body in equilibrium
ZII wII 0
wI wII 0
ZI - wI
ZII - wII
wII ZI
wI ZII
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Internal forces
wI ZII
wII ZI
The set of internal forces in part I is equal to
the set of external forcces acting on II
The set of internal forces in part II is equal to
the set of external forcces acting on I
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O is assumed to be the reduction point of
internal and external forces
Cross-sectional forces
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wII ZI
wI ZII
SwI SzII MwI MzII
SwII SzI MwII MzI
SwI - SwII MwI - MwII
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Cross-sectional forces
The components of the resultants of internal
forces reduced to the point O will be called
cross-sectional forces
Sw Sw(rO , n) MwI Mz(rO , n)
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Cross-sectional forces

• The immediate goal of SM is to evaluate internal
  forces
• These forces will define the conditions of
  material cohesion and its deformation
• As the first step the components of the sum and
  moment of cross-sectional forces will be
  evaluated as a function of chosen reduction point
  O, and cross-section plane n
• In what follows we will limit ourselves to bar
  structures, as the simplest approximation of 3D
  bodies (structures).

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• stop