1/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

SHEAR&

NON-UNIFORM BENDING

2/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

Mx = 0, My = 0, Mz = 0

Qx=N= 0, Qy ≠ 0, Qz = 0

Qx=N=0, Qy= 0, Qz ≠ 0

y

z

xQy

x

z

y

Qzx

z

y

Qz

My

Formal definition: the case when set of internal forces reduces solely to the shear force vector perpendicular to the bar axis

or

Shear is associated with bending !

NON-UNOFRM BENDING PURE SHEAR

Mx = 0, My ≠ 0, Mz = 0

Qx= N=0, Qy= 0, Qz ≠ 0

3/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

M

Q

Reactions

?Mu0

Q=const

Pure shear

4/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

P P

P P

A

AA

A

4A= P = P/4A Mean shear stress

Pure shear

5/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

Z

X

Z

X

Bernoulli hypothesis of plane cross-sections does not hold!!

For h/l<<1 distorsion is small and we will use the formula for normal stress derived from this assumption :

z

J

M

y

yx

h

l

Non-uniform bending

6/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

Z

X

Z

X

zx

zx

xz xz

„point” imageZ

X

P

xz

Qz = P = A

xzdA

xz= xz(z,y)

?

P

Non-uniform bending

7/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

xz

y

yzzx zbJ

zSxQ ~~

*

dxx A*

Z

y

xx dxxx

xM y

dxxM y xxz

xQz dxxQz

A*

z

dAxxNA

x*

*

dxxxz

xz~~b (z)

b (z)

C,F

D,F

B,D

B,C

F

CB

D

dxzbdxxNxN zxdx

)(~~**lim

0

zJ

M

y

yx

zJ

dxxMdxx

y

yx

Non-uniform bending

dAdxxdxxNA

x *

*

Prismatic bar!

8/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

zb

zSxQ

zzxxz maxmax~max

zb

zSxQ

J zy

xz

1~Formula holds for prismatic bars only!

Distribution along z-axis

and is given in main principal axes of cross-section inertia.

z

y

For A: S*(zmax)=0 since A*=0 0~ xzFor B: S*(zmax)=0 since A*=A 0~ xz

zmax

zmin

A

B

A*

A

Kinematic Boundary Conditions in A and B:

jijiq

0~1~000 xzxzxyxxq

01000 yzyzyyxyq

zzzzyzxz qq 100~0

Also:

z

Non-uniform bending

9/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

Distribution along z-axis; special cases

b(z) = b =const|z

y

z

h/2

h/2

b

Parabola 2o

max= 3Q/2bh

b(z) – linear function of zz

y

2h/3

h/3

b

Parabola 3o

max= ?

b(z) – step-wise change

h/2

h/2

b

y

z

cb/ c=c/b

max

Parabola 2o

y

z

h/2

h/2

b

cb

Non-uniform bending

10/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

Stress distribution in beams – trajectories of main principal stresses

z

J

xM

y

yx

xz

y

yzzx zbJ

zSxQ

*

222,1 4

2

1

2 xzxx

2,12,1

z

xztg

zx

zx

xz

xz

x

xx

z

x

1

2

2

1

1

2z

x

2

1

0z2,1

2,1 xztg

Non-uniform bending

For z=0: 0x xz 2,1

12,1 tg o452,1

11/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

1

2

2

1

1

2z

x

2

1

2=

1

2z

x

2=45o

1=45o

2=

1=

1=

0yM0zQ

l/2

Principal stress trajectories

Non-uniform bending

1

2=90º

1=0º2=0

1= x 1

2

z

x

12/11M.Chrzanowski: Strength of Materials

SM2-07: Shear

stop