1/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

COMPOSED BENDING

(Eccentric tension/compression)

2/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

zJ

M

y

yx

x y

zz

My

)(zx

maxz

Neutral axis for bending

yM

000

000

00x

T

Plane bending

3/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

A

Nx

x y

zz )(zx Neutral axis for

bending

yM

000

000

00x

T

N

Neutral axis for tension

+

-

Tension

4/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

000

000

00x

T

zJ

M

y

yx

y

zz

My

)(zx Neutral axis

yM

N + =

+A

N

0xNeutral axis equation:

zi

zz

N

A

J

M

A

N

A

Nz

J

M

yy

y

y

y20110

0zN

M y

2y

y iA

J

„Eccentric”

Squared inertia radius

Plane bendingCombined bending

z0

5/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

yJ

Mz

J

M

A

N

z

z

y

yx Normal stress

0zNM y 0yNM z

yJ

yNz

J

zN

A

N

zyx

00

20

201/

zy

x

i

yy

i

zzAN

0xNormal stress at neutral axis

20

2010

zy i

yy

i

zz

AN/

Eccentrics

Normal stress in terms of normal force and eccentrics y0 ,z0

Non-dimensional normal stress

Neutral axis equation

Bi-plane combined loading

6/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

NEUTRAL AXIS MOVEMENTin (y,z) plane

7/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

Side view Stress distribution Cross-section view

8/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

A+

N/A

Neutral axis in an „infinity”

NN

Side view Stress distribution Cross-section view

9/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

NA+

N/A

Side view Stress distribution Cross-section view

Neutral axis already being „seen”

10/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

NA

N/A

+

Side view Stress distribution Cross-section view

Neutral axis outside of cross-section

11/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

NA+

N/A

NN

Side view Stress distribution Cross-section view

Neutral axis touching cross-section contour

12/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

NA+

-

N/A

Side view Stress distribution Cross-section view

Neutral axis within cross-section

13/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

20

2010

zy i

yy

i

zz

Dual interpretation of neutral axis equation

Neutral axis co-ordinates

Eccentric co-ordinates

14/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

20

2010

zy i

yy

i

zz

1. If neutral axis coincides with cross-section contour edge given by the equation:

yz then from the transformed equation of neutral axis:

yi

y

z

i

z

iz

z

yy

20

0

2

0

2

one can find co-ordinates of normal force position (eccentricity):

0

2

z

iy

2

0yiz

20

0

2

z

y

i

y

z

i

2

0ziy

Dual interpretation of neutral axis equation

15/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

Kzz Kyy

02

22

0 yi

y

z

i

z

iz

yz

K

K

y

K

y

then by inserting these co-ordinates into neutral axis equation one can obtain eqaution of a straigth line defining the position of a normal force:

20

2010

zy i

yy

i

zz

2. If a number of neutral axis touches cross-section corner of given co-ordinates:

Dual interpretation of neutral axis equation

16/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

4 cm

14 cm

2 cm

8 cm

2 cm3 cm

8 cm

3 cm

Example of cross-section kernel finding

17/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

7,22 cm

y

z

a

b

c

d

D

B

A

C

Example of cross-section kernel finding

Mode 1: Finding eccentrties

18/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

7,22 cm

y

z

e

E

Example of cross-section kernel finding

Mode 2: Finding normal force acting lines

19/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

7,22 cm

y

z

f

F

Example of cross-section kernel finding

Mode 2: Finding normal force acting lines

20/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

7,22 cm

y

z

G

g

Example of cross-section kernel finding

Mode 2: Finding normal force acting lines

21/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

7,22 cm

y

z

Example of cross-section kernel finding

22/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

7,22 cm

y

z

Example of cross-section kernel finding

23/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

Cross-section kernel defines normal force positions (eccentrities) such that resulting normal stress in the whole cross-section area has the same sign (plus for N>0 and minus dla N<0).

Cross-section kernel has always a convex form.

Cross-section kernel

24/23M.Chrzanowski: Strength of Materials

SM2-05: Composed bending

stop