C07 design 14 AZ

background image

Adam Paweł Zaborski

Project “The development of the didactic potential of Cracow University of Technology in the range of

modern construction” is co-financed by the European Union within the confines of the European Social Fund

and realized under surveillance of Ministry of Science and Higher Education.

14. Stress state

Introduction

Definitions

The stress vector

i

p

is internal forces’ density in the plane with outer normal

i

n

, parallel to the axis of the

coordinate system.
The stress tensor

ij

σ

is a matrix of stress vector components written in the given coordinate system:

j

ij

i

n

p

σ

=

The components of the stress tensor in the new coordinate system may be computed by the transformation
formula:
– in index notation:

kl

jl

ik

ij

a

a

σ

=

σ

– in matrix notation:

T

A

Σ

A

Σ

=

'

where:

ij

σ

is a stress matrix with normal stress on the diagonal and shear stress elsewhere:

4

4

4

3

4

4

4

2

1

4

4

4

3

4

4

4

2

1

notation)

ng

(engineeri

z

zy

zx

yz

y

yz

xz

xy

x

notation)

c

(scientifi





σ

τ

τ

τ

σ

τ

τ

τ

σ

=

σ

σ

σ

σ

σ

σ

σ

σ

σ

=

σ

33

32

31

23

22

21

13

12

11

ij

The first index specifies the direction of the outer unit normal of the section plane. The second index
specifies the stress direction.
A matrix can be the stress matrix if it fulfills the partial differential equations (PDE) of internal equilibrium
(Navier’s equations):

0

0

0

=

+

σ

+

τ

+

τ

=

+

τ

+

σ

+

τ

=

+

τ

+

τ

+

σ

z

z

yz

xz

y

yx

y

xy

x

xx

xy

x

P

z

y

x

P

z

y

x

P

z

y

x

,

(

)

x

y

x

j

i

P

i

j

ij

,

,

,

,

0

,

=

=

+

σ

and the static boundary conditions:

z

z

y

yz

x

xz

z

z

yz

y

y

x

xy

y

z

xz

y

xy

x

x

x

n

n

n

q

n

n

n

q

n

n

n

q

σ

+

τ

+

τ

=

τ

+

σ

+

τ

=

τ

+

τ

+

σ

=

,

(

)

z

y

x

j

i

n

q

j

ij

i

,

,

,

,

=

σ

=

Stress state analysis

The main goal of the stress state analysis is to find such directions of the sections where the stresses values
are extreme. This happens when the outer normal and the stress vector are parallel, and leads to the
eigenvalues problem:

(

)

0

=

λδ

σ

λ

=

σ

j

ij

ij

i

j

ij

v

v

v

.

The set of algebraic linear equation system has non-zero (non-trivial) solution if and only if the main
determinant of the system is zero. From this condition we get a cubic equation for the principal stress
values:

background image

Adam Paweł Zaborski

Project “The development of the didactic potential of Cracow University of Technology in the range of

modern construction” is co-financed by the European Union within the confines of the European Social Fund

and realized under surveillance of Ministry of Science and Higher Education.

0

3

2

2

1

3

=

σ

+

σ

σ

I

I

I

.

The stress matrix in the principal directions is diagonal and the shear components are zero. From algebra
we know that the principal normal stresses are extreme and the principal directions are perpendicular:

4

4

4

3

4

4

4

2

1

4

4

4

3

4

4

4

2

1

)

3

,

2

,

1

(

)

,

,

(

0

0

0

0

0

0

in

3

2

1

in

z

zy

zx

yz

y

yz

xz

xy

x

σ

σ

σ





σ

τ

τ

τ

σ

τ

τ

τ

σ

z

y

x

Mohr’s circles

The domain of possible results due to the transformation of a coordinate system is illustrated by Mohr’s
circles, Fig. 14.1 with the shaded region.

τ

xy

σ

2

σ

τ

σ

1

σ

y

σ

x

σ

3

Fig. 14.1 Mohr’s circles

Examples

Three-dimensional state of stress

The stress matrix and the outer normal vector of a section are given. Determine the principal directions and
stresses, the normal and tangential components of the stress vector for the section.

=

σ

120

60

40

60

85

17

40

17

240

ij

,

(

)

2

,

2

,

7

n

Solution
We calculate the invariants:

205

120

85

240

)

(

1

=

+

=

σ

=

ij

I

tr

24089

120

40

40

240

120

60

60

85

85

17

17

240

2

=

+

+

=

I

3494920

60

40

85

17

40

120

40

60

17

17

120

60

60

85

240

)

det(

3

=

+

+

=

σ

=

ij

I

The cubic equation:

0

3494920

24089

205

2

3

=

σ

σ

σ

has the roots:

1

.

141

,

0

.

101

,

1

.

245

3

2

1

=

σ

=

σ

=

σ

.

The principal directions are given by the transformation matrix:

background image

Adam Paweł Zaborski

Project “The development of the didactic potential of Cracow University of Technology in the range of

modern construction” is co-financed by the European Union within the confines of the European Social Fund

and realized under surveillance of Ministry of Science and Higher Education.

3

2

1

z

y

x

a

ij

=

9583

.

0

2686

.

0

9583

.

0

2627

.

0

9624

.

0

0689

.

0

1123

.

0

0404

.

0

9929

.

0

.

We calculate the length of the normal vector:

8730

.

3

4

4

7

=

+

+

=

n

and we normalize the vector (to obtain unit normal vector)

)

5164

.

0

,

5164

.

0

,

6831

.

0

(

n

The stress vector is:

)

3

.

120

,

30

.

1

,

5

.

134

(

p

n

p

i

ij

j

σ

=

its length:

5

.

180

=

p

the normal component of the stress vector is:

45

.

30

=

=

=

σ

i

i

n

p

n

p

and the tangential component is:

9

.

177

2

2

=

σ

=

τ

p

Plane stress

For the given stress tensor, determine principal stresses and directions.





=

σ

10

16

16

40

T

.

Solution
The principal stresses for the plane state of stress are:

68

.

14

16

2

10

40

2

10

40

2

2

68

.

44

16

2

10

40

2

10

40

2

2

2

2

2

2

2

2

2

2

2

1

=

+

+

=

τ

+



σ

σ

σ

+

σ

=

σ

=

+

+

+

=

τ

+



σ

σ

+

σ

+

σ

=

σ

y

x

y

x

y

x

y

x

y

x

y

x

,

The principal directions:

)

7

.

73

(

,

286

.

1

4175

.

3

16

40

68

.

14

tan

)

3

.

16

(

,

285

.

0

2925

.

0

16

40

68

.

44

tan

2

2

2

1

1

1

°

=

α

=

=

τ

σ

σ

=

α

°

=

α

=

=

τ

σ

σ

=

α

y

x

x

y

x

x

The transformation matrix and the stress tensor in principal directions are, cf. Fig. 14.2:

9598

.

0

2808

.

0

2808

.

0

9598

.

0

2

y

x

x





=

σ

68

.

14

0

0

68

.

44

T

background image

Adam Paweł Zaborski

Project “The development of the didactic potential of Cracow University of Technology in the range of

modern construction” is co-financed by the European Union within the confines of the European Social Fund

and realized under surveillance of Ministry of Science and Higher Education.

14.68

14.68

44.68

44.68

x

16

10

10

16

16

16

40

40

2

1

y

Fig. 14.2 Plane state of stress

or, in 3D notation:

0

1

0

9598

.

0

0

2808

.

0

2808

.

0

0

9598

.

0

3

2

1

z

y

x

=

σ

68

.

14

0

0

0

0

0

0

0

68

.

44

T

Static boundary conditions

Given the stress tensor, determine the mass forces and the boundary loading of the shield.

.

x

y

4

3

3

2

1





=

σ

8

2

2

5

y

y

T

Fig. 14.3 Shield and stress state

Solution
The mass forces from Navier’s equations are: P (2, 0)



σ

+

τ

=

τ

+

σ

=

y

y

x

y

x

y

y

y

x

x

x

x

n

n

q

n

n

q

the boundary (1):

8

,

0

2

)

1

,

0

(

=

=

=

y

x

q

y

q

n

the boundary (2):

y

q

q

y

x

2

,

5

)

0

,

1

(

=

=

n

the boundary (3):

8

.

4

6

.

1

,

2

.

1

4

5

)

6

.

0

,

8

.

0

(

+

=

=

y

q

y

q

y

x

n

The loading is drawn in Fig. 14.4.

q

nx

q

ny

5

5

-8

-8

-8.8

-4

4.8

11.2

-8

Fig. 14.4 The loading of the shield

The checking of the solution:

0

4

3

2

1

2

4

5

5

)

4

8

.

8

(

2

1

=

+

+

+

=

X

,OK, similarly:

=

=

=

=

0

,

0

K

K

o

M

Y

background image

Adam Paweł Zaborski

Project “The development of the didactic potential of Cracow University of Technology in the range of

modern construction” is co-financed by the European Union within the confines of the European Social Fund

and realized under surveillance of Ministry of Science and Higher Education.

Workshop theme

1) The stress matrix is given:

=

σ

10

40

0

40

20

20

0

20

30

T

MPa.

Determine: a) normal stress on the section plane with the normal

(

)

3

1

3

2

3

2

,

,

v

, b) shear stress in the plane

parallel to the vectors:

(

)

0

,

,

2

1

2

1

1

v

and

(

)

2

3

4

2

3

1

2

3

1

2

,

,

v

. Interpret the obtained results graphically.

2) Write the boundary conditions on the side surface of floating a wooded column. Assume the draught
greater than the radius of the cylinder, Fig. 14.5

R

Fig. 14.5 Floating wooded column

Review problems

1) Given the plane stress matrix, find principal stresses and their directions. Illustrate the stress state before

the transformation and after this.





=

σ

0

150

150

0

T

[MPa]

2) Given the stress state and the shield shape with dimensions, Fig. 14.6, determine the mass forces and the

loading on the boundary.

y

x

3

4





=

σ

y

x

x

xy

T

5

7

7

2

Fig.

14.6 Plane stress state

3) There is uniaxial stress state in the cross-section plane of a bar,

150

=

σ

x

[MPa]. Determine the stress

components in the plane turned by 45 degrees along z axis, Fig. 14.7.

y

x

Fig. 14.7 Uniaxial stress state

background image

Adam Paweł Zaborski

Project “The development of the didactic potential of Cracow University of Technology in the range of

modern construction” is co-financed by the European Union within the confines of the European Social Fund

and realized under surveillance of Ministry of Science and Higher Education.

Addendum

Glossary

outer (exterior) unit normal – wersor normalnej zewnętrznej
stress vector – wektor naprężenia
load vector – wektor obciążenia
stress matrix – macierz naprężenia
stress tensor – tensor naprężenia
normal stress – naprężenia normalne
shear stress – naprężenia styczne
internal balance equations – równania równowagi wewnętrznej
static boundary conditions – statyczne warunki brzegowe
stress state analysis – analiza stanu naprężenia
Mohr’s circles – koła Mohra


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