1 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM WHAT STRENGTH OF MATERIALS IS IT ABOUT?

1/14M.Chrzanowski: Strength of Materials

SM1-01: About SM

WHAT STRENGTH OF MATERIALS

IS IT ABOUT?

2/14M.Chrzanowski: Strength of Materials

SM1-01: About SM

Why You Don’t Fall Through the Floor

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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

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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

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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

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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

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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

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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 …∞

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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

∞∞

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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

∞

∞

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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

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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

∞ ∞

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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

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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

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SM1-01: About SM

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