1 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending INCLINED BENDING

1/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

INCLINED BENDING

2/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

If bending moment does not coincide with any of principal central axis of cross-section inertia we have to deal with inclined bending.

y

z

M

Plane of load acting

yM

zM

yy MM

Direction of bending moment vector

cosMM z zz MM

MM

sinMM y

y

z

MyM

zM

y

z

yyz cot)90tan(

+

3/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

y

z

zM

yM

y

z

yJ

M

z

zx y

J

Mz

J

M

z

z

y

yx

We use superposition principle

For neutral axis:

cosMM z sinMM y

yJ

zJ

Mzy

x

cossin

0x yJ

Jz

z

y cot

y

z

zJ

M

y

yx

Neutral axis

M

Jy>Jz

Jy<Jz

y

z

y

z

4/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

yJ

Mz

J

M

z

z

y

yx Stress distribution:

0x

yJ

Jz

z

y cot

y

z

M

Neutral axis

is represented by the equation of the plane

max

minTension

zone

Compression zone

max

This plane cuts thorough cross-section plane at the neutral axis

5/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

Animation of neutral axis movement dependent on bending moment

direction

6/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

y

z

80o

70o

60o

50o

40o

30o

20

10o

00o

h/b=1,41

Jy/Jz =2 Loading plane

Neutral axis

Bending moment vector

Angle

7/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

y

z

h/b=1,41

Jy/Jz =2

Loading plane

Neutral axis

Bending moment vector

80o

70o

60o

50o

40o

30o

20

10o

00o

8/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

y

z

b/h=1,41

Jy/Jz =1/2

Loading plane

Neutral axis

Bending moment vector

80o

70o

60o

50o

40o

30o

20

10o

00o

9/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

y

z

Płaszczyzna obciążeniaOś obojętna

Wektor momentu

h=b=a

Jy/Jz =1

y

zZmax =a/2=0,5a

Zmax

=a/2½=0,7a

aJ

M5,0max a

J

M7,0max

Ideal shape: independent of plane of loading angle maximum stress is the same!

y

z

10/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

stop

11/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

y

z

h/b=1,41

Jy/Jz =2 Loading plane

Neutral axis

Bending moment vector

Angle

80o

70o

60o

50o

40o

30o

20

10o

00o

12/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

y

z

h/b=1,41

Jy/Jz =2 Loading plane

Neutral axis

Bending moment vector

Angle 80o

70o

60o

50o

40o

30o

20

10o

00o

13/10M.Chrzanowski: Strength of Materials

SM2-04: Inclined bending

y

z

Płaszczyzna obciążeniaOś obojętna

Wektor momentu

h=b=a

Jy/Jz =1

y

z

α=45°

Zmax =a/2=0,5a

Zmax

=a/2½=0,7a

aJ

M5,0max a

J

M7,0max

Ideal shape: independent of plane of loading angle maximum stress is the same!