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3/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
Linguistics
Соротивле матеиаов
Résistance des matériaux
Festigkeitlehre
Hållfasthetslära
材料力学
Wytrzymałość materiałów Wytrzymać: co? Ile? Jak długo?
Opór materiałów?
Opór materiałów?
Nauka o sile materiałów?
Nauka o spójności materiałów?
No, to już zupełna „chińszczyzna”!
Moc materiałów? 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
4/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
SM
(of a deformable body)
Theoretical mechanics
(of a rigid body)
EXPERIMENTS
Physics Mathematics
HYPOTHESES
• Theory of elasticity
• Theory of plasticity
• Material Science
• Differential calculus
• Matrix algebra
• Calculus of variations
• Numerical methods
Origin of SM
5/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
Idealisation of:
Material
Loadings
Structure geometry
• Continuous matter distribution (material continuum)
• Continuous mass distribution ρ(x)
• Intact, unstressed initial state of a material
• Permanent versus movable
• Constant versus variable in time (static versus dynamic)
• Bulk structures (H ~ L~B)
• Surface structures (H«L~B)
• Bar structures (L»H~B)
Modelling scheme
6/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
B
L
External surface forces
Point force P [N]
Line distributed force q [N/m]
Point moment M [Nm]
Surface distributed force p [N/m2]
H
b
External volume forcesM [Nm]
P [N]
p [N/m2]
l
l q [N/m]
(gravitational forces, inertia forces, electromagnetic forces etc.) X [N/m3]
Displacements u(u,v,w) [m] (e.g. supports, forced shift of structural members)
Mechanical loadings
X [N/m3]
u=0, v=0 v=0
+
≡u
v
w
7/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
• 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.
Internal forces
Fundamental observations
8/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
AP1
P3
Pn
P2
Pi
wi
w1
w2
w3
01
n
iP 01
n
iM 01
iw 01
im
A body in equilibrium
Coulomb particle interaction assumed(convergent set of internal forces)
{wi} – convergent, infinite, zero valued set of internal forces
Internal forces
{wi}, i=1,2 …∞
9/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
n
n
wi
w2
w1
w3
A
Pi
AP1
P3
Pn
P2
III
AP1
Pn
I
ww
w A
P3
P2
Pi
IIw
n
r
w= f(r,n)
Internal forces
n - outward normal vector
nr – point position vector
n
∞∞
10/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
n
AP1
Pn
I
A
P3
P2
Pi
II
n{wI}
{ZI}
{wII}
{ZII}
{ZI} + {wI} ≡ {0} {ZII} + {wII} ≡ {0}
{Z} = {ZI} + {ZII} ≡ {0}
{wI} + {wII} ≡ {0}{ZI} ≡ - {wI} {ZII} ≡ - {wII}
{wII} ≡{ZI} {wI} ≡{ZII}
Body in equilibrium
Internal forces
∞
∞
11/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
n
AP1
Pn
IA
P3
P2
Pi
II
{wI}
{ZI}
{wII}
{ZII}n
{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
Internal forces
12/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
MwII
SwII
SwI
MwI
P3
Pi
O
n
OP1
Pn
P2
n
{wI}{wII}
{wII} ≡ {ZI}{wI} ≡ {ZII}
SwII ≡ SzI
MwII ≡ MzI
SwI ≡ - SwII
MwI ≡ - MwII
SwI ≡ SzII
MwI ≡ MzII
SzII
MzII
MzI
SzI
Cross-sectional forces O is assumed to be the reduction point of internal and external forces
∞ ∞
13/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
P3
Pi
O
n
n
Sw ≡Sw(rO , n)
MwI ≡Mz(rO , n)
rO
P1
Pn
P2
Cross-sectional forces
The components of the resultants of internal forces reduced to the point O will be called cross-sectional forces
14/14M.Chrzanowski: Strength of Materials
SM1-01: About SM
• 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).
Cross-sectional forces