ADVANCED STRUCTURAL MECHANICS

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VSB– TECHNICAL UNIVERSITY OF OSTRAVA F ACULTY OF CIVIL ENGINEERING ADVANCED STRUCTURAL MECHANICS Lecture 1 Jir ˇı ´ Broz ˇovsky ´ Office: LP – H 406/3 Phone: 597 321 321 E-mail: [email protected] WWW: http://fast10.vsb.cz/brozovsky/

Transcript of ADVANCED STRUCTURAL MECHANICS

Page 1: ADVANCED STRUCTURAL MECHANICS

VSB– TECHNICAL UNIVERSITY OF OSTRAVA

FACULTY OF CIVIL ENGINEERING

ADVANCED STRUCTURAL

MECHANICS

Lecture 1

Jirı BrozovskyOffice: LP – H 406/3Phone: 597 321 321

E-mail: [email protected]

WWW: http://fast10.vsb.cz/brozovsky/

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Topics

1. Basic equations

2. Plane structures

3. Energetical principles, variational methods

4. Finite element method

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Recommended books

• Timoshenko. S. – Gere, J.: Mechanics of Materials, Van

Nostrand Company, New York, 1972 (or newer edition)

• Boresi, A. – Schmidt, R.: Advanced Mechanics of Materials,

John Wiley & Sons, 2003

• Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.: Con-

cepts and Applications of Finite Element Analysis, John

Wiley and Sons, 1995

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Topic 1:Basic equations of elasticity

• Basic unknowns

• Geometrical relations

• Equilibrium equations

• Material law

• Compatibility equations

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Basic unknowns

σ

σ

τ

τ

τ

ττ

τ

x

yy

z

yx

yz xy

xz

zx

zy

σz x

Stress vector

σ = {σx, σy, σz, τxy, τyz, τzx}T

(1)

Strain vector

ε = {εx, εy, εz, γxy, γyz, γzx}T

(2)

Displacement vector

u = {ux, uy, uz}T (3)

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Geometrical relations (1)

� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �0

y, v

x, u

A B

D C

β

αA’

D’

uv

dxux

dx

dy

−x

v

B’

− dxx

y

v

u− dy

dy

C’

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Geometrical relations (2)

� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �

0

y, v

x, u

A B

D C

β

αA’

D’

uv

dxux

dx

dy

−x

v

B’

− dxx

y

v

u− dy

dy

C’

εx =A′B′ −AB

AB=(x + dx + u + ∂u

∂xdx)− (x + u)− dx

dx=

∂u

∂x

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Geometrical relations (3)

Normal deformations

εx =∂u

∂xεy =

∂v

∂yεz =

∂w

∂z, (4)

Shear deformations

γyz = γzy =∂v

∂z+

∂w

∂y(5)

γzx = γxz =∂w

∂x+

∂u

∂z(6)

γxy = γyx =∂u

∂y+

∂v

∂x. (7)

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Geometrical relations (4)

Uvedene vztahy obecne neplatı:

γyz = γzy,

γzx = γxz,

γxy = γyx.

The the assumption of equality of shear stresses is based

on approximative fulfillment of moment equilibrium equati-

ons of differential element. Shear deformations are assumed

to be equal in the same way.

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Differential equilibrium conditions(1)

yxyx

zxzx xzxz

yz

yz

zy

τ

τ

τ

ττ

τ

τ

ττ

dx

zy

xy

σ’

y

x

xy

’τ

dz

dy

’τ

σ

σ

σ

σ

x

τ

z

z

y

σ

σx‘ = σx +∂σx

∂xdx, τxy‘ = τxy +

∂τxy

∂xdy, ... (8)

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Diff. equilibrium conditions (2)

σx‘ = σx +∂σx

∂xdx, τxy‘ = τxy +

∂τxy

∂ydy, ...

Fi,y = (σ′x−σx) dx dy+(τxy−τ ′xy) dx dz+(τxz−τxz‘) dx dy = 0

(σx−σx−∂σx

∂xdx) dy dz+(τxy−τxy−

∂τxy

∂ydy) dx dz+(τxz−τxz−

∂τxz

∂zdz) dx dy = 0

After simplification:

∂σx

∂x+

∂τxy

∂y+

∂τxz

∂z= 0 (9)

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Diff. equilibrium conditions (3)

∂σx

∂x+

∂τxy

∂y+

∂τxz

∂z+X = 0

∂τxy

∂x+

∂σy

∂y+

∂τyz

∂z+ Y = 0 (10)

∂τzx

∂x+

∂τzy

∂y+

∂σz

∂z+ Z = 0

where X,Y , Z are volume forces.

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Material law (1)

Defines relations between stresses and strains.

Hooke law for 1D problem (simple tension/compression):

εx =σx

E

F

L L∆

Αx

εx =∆L

L=

∂L

∂x

σx =F

A= E εx

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Material law (2)

Hooke law for 3D problems:

εx =1

E[σx − ν (σy + σz)] , γyz =

τyz

2 G

εy =1

E[σy − ν (σx + σz)] , γxz =

τxz

2 G(11)

εz =1

E[σz − ν (σx + σy)] , γxy =

τxy

2 G

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Conclusion

15 unknowns:

3 displacements u

6 stresses ε

6 strains σ

15 available equations:

6 geometrical relations

6 matrial law equations 3 equilibrium conditions

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Compatibility conditions (1)

They define continuity of deformations – the continuous vo-

lume must remain to be continuous also after deformation.

They can be obtained from geometrical relations by elimination

of displacements.

∂2εx

∂y2+

∂2εy

∂x2+ =

∂2εxy

∂x∂y

∂2εy

∂z2+

∂2εz

∂y2+ =

∂2εyz

∂y∂z(12)

∂2εz

∂x2+

∂2εx

∂z2+ =

∂2εzx

∂z∂x16

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Compatibility conditions (2)

∂2εx

∂y∂z=1

2

−∂γyz

∂x+

∂γxz

∂y+

∂γxy

∂z

∂2εy

∂z∂x=1

2

+∂γyz

∂x−

∂γxz

∂y+

∂γxy

∂z

(13)

∂2εz

∂x∂y=1

2

+∂γyz

∂x+

∂γxz

∂y−

∂γxy

∂z

For plane problems (2D, to be discussed later) it can be written:

∂2εx

∂y2+

∂2εy

∂x2=

∂2γxy

∂x∂y(14)

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Topic 2: Plane problems

2D plane structures supported and loaded in their middle plane

will be discussed.

Two main variants of plane problem:

• plane stress

• plane strain

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Plane strain

1

All deformations (strains) must

lie in the x− y plane:

ε = {εx, εy, γxy}T (15)

The structure can not freely move in z direction. As a result,

there is σz stress. σz:

σ = {σx, σy, σz, τxy}T (16)

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Plane strain

y

x

στ

x

xy

z

All stresses lie in the x−y plane:

σ = {σx, σy, τxy}T (17)

The wall can freely move in z direction, therefore there is on

σz stress but the εz strain is nonzero:

ε = {εx, εy, εz, γxy}T (18)

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Plane problem – conclusions (1)

Equilibrium equations

∂σx

∂x+

τxy

∂y+X = 0 (19)

∂τxy

∂x+

σy

∂y+ Y = 0

Geometrical relations

εx =∂u

∂x, εy =

∂v

∂y, γxy = γyx =

∂u

∂y+

∂v

∂x. (20)

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Plane problem – conclusions (2)

Material law: plane stress

σx =E

1− µ2(εx + µ εy)

σy =E

1− µ2(εy + µ εx) (21)

τx =E

2(1− µ)γxy

(22)

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Plane problem – conclusions (3)

Material law: plane strain

σx =E

(1 + µ)(1− 2µ)[(1− µ) εx + µ εy]

σy =E

(1 + µ)(1− 2µ)[µ εx + (1− µ) εy] (23)

τx =E

(1 + µ)(1− 2µ)γxy1

2(1− µ)

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Airy’s equation for plane problem

Airy function F – describes stress state of plane problem if:

σx =∂2F

∂x2, σy =

∂2F

∂x2, τxz = −

∂2F

∂x∂y. (24)

Airy equation – compatibility relation is (14) writtnen with use

of Airy function:

∂4F

∂x4+ 2

∂4F

∂x2∂y2+

∂4F

∂y4= 0 (25)

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Stress distribution on wall:N

S

N . . . beam theory,

S . . . plane stress theory.

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Forces in wall

Nx = σx h [N

m]

Ny = σy h [N

m]

Nxy = τxy h [N

m]

x

yx

xyn

z

n

n

n

y

τxy

yx

h

τ

σ

σx

y

y

�����������������������������

�����������������������������

x

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