Transcript Title

SM1-01: About SM
WHAT
STRENGTH OF MATERIALS
IS IT ABOUT?
M.Chrzanowski: Strength of Materials
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SM1-01: About SM
Why You Don’t Fall Through the Floor
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SM1-01: About SM
Linguistics
Strength of materials
Moc materiałów?
材料力学
No, to już zupełna „chińszczyzna”!
SM is about the
resistance
of materials
Wytrzymałość materiałów Wytrzymać:
co? Ile?
Jak długo?
(and structures)
against
Соротивле матеиаов
Opór materiałów?
external environmental
actions
Résistance des matériaux Opór materiałów?
(forces, deformations,
temperatures etc.)
Festigkeitlehre
Nauka o sile materiałów?
which may lead to
the loss
of
Hållfasthetslära
Nauka o spójności
materiałów?
load bearing capacity
M.Chrzanowski: Strength of Materials
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SM1-01: About SM
Origin of SM
Physics
Mathematics
Theoretical
mechanics
(of a rigid body)
• Theory of elasticity
• Theory of plasticity
• Differential calculus
SM
(of a deformable body)
• Material Science
• Matrix algebra
• Calculus of variations
• Numerical methods
HYPOTHESES
EXPERIMENTS
M.Chrzanowski: Strength of Materials
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SM1-01: About SM
Modelling scheme
Idealisation of:
Material
• Continuous matter distribution (material continuum)
• Continuous mass distribution ρ(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 ~ L~B)
• Surface structures (H«L~B)
• Bar structures (L»H~B)
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SM1-01: About SM
Mechanical loadings
External surface forces
P [N]
Surface distributed force p [N/m2]
l
Line distributed force q [N/m]
p [N/m2]
Point force P [N]
Point moment M [Nm]
q [N/m]
b
l
B
H
External volume forces
L X [N/m3]
(gravitational forces, inertia forces,
electromagnetic forces etc.) X [N/m3]
M [Nm]
Displacements u(u,v,w) [m] (e.g. supports, forced shift of structural members)
≡
M.Chrzanowski: Strength of Materials
u=0, v=0
v
u
v=0
+
w
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SM1-01: About SM
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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SM1-01: About SM
Internal forces
P2
P1
w2
P3
A
w1
w3
Pn
n
A body in equilibrium
P  0
i
1
Pi
n
M
1
i
0
{wi}, i=1,2 …∞
wi
Coulomb particle interaction
assumed
(convergent set of internal
forces)


w  0
i
1
m
i
0
1
{wi} – convergent, infinite, zero valued set of internal forces
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SM1-01: About SM
n
∞
Internal forces
P2
P1
∞
P3
A
r
A
w1
wi w
w
II
I
w2
w3
r – point position vector
Pn
n
Pi
n
n
n - outward normal vector
n
P2
P3
P1
w
A
w
I
Pn
M.Chrzanowski: Strength of Materials
A
II
w= f(r,n)
Pi
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SM1-01: About SM
Internal forces
n
{wI}
∞
P1
n
{ZII}
P2
P3
A
A
∞{w }
I
Pn
{ZI}
II
II
Pi
{Z} = {ZI} + {ZII} ≡ {0}
{ZI} + {wI} ≡ {0}
Body in equilibrium
{wI} + {wII} ≡ {0}
{ZI} ≡ - {wI}
{wII} ≡{ZI}
M.Chrzanowski: Strength of Materials
{ZII} + {wII} ≡ {0}
{ZII} ≡ - {wII}
{wI} ≡{ZII}
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SM1-01: About SM
Internal forces
n
{wI}
n
{ZII}
P2
P3
P1
A
A
II
I
Pn
{wII}
Pi
{ZI}
{wI} ≡ {ZII}
{wII} ≡ {ZI}
The set of internal forces in
The set of internal forces in
part I is equal to the set of
part II is equal to the set of
external forcces acting on II
external forcces acting on I
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SM1-01: About SM
O is assumed to be
Cross-sectional forces
SwI
MzI
∞
P3
P2 I}
{w
P1
MwI
SzI
n
O O
the reduction point of
internal and external
forces
∞
SzII
MwII
{wII}
MzII
PiS
wII
n
{wI} ≡ {ZII}
Pn
{wII} ≡ {ZI}
SwI ≡ SzII
SwI ≡ - SwII
SwII ≡ SzI
MwI ≡ MzII
MwI ≡ - MwII
MwII ≡ MzI
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SM1-01: About SM
Cross-sectional forces
n
P2
P1
P3
O
rO
Pi
n
Pn
Sw ≡Sw(rO , n)
MwI ≡Mz(rO , n)
M.Chrzanowski: Strength of Materials
The components of the
resultants of internal
forces reduced to the
point O will be called
cross-sectional forces
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SM1-01: About SM
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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SM1-01: About SM
stop
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