Bending of BeamsMECHENG242 Mechanics of Materials

2.2 Stresses in Beams (Refer: B,C & A –Sec’s 6.3-6.6)

2.3 Combined Bending and Axial Loading (Refer: B,C & A –Sec’s 6.11, 6.12)

2.1 Revision – Bending Moments

2.0 Bending of Beams

2.4 Deflections in Beams

2.5 Buckling

(Refer: B,C & A –Sec’s 7.1-7.4)

(Refer: B,C & A –Sec’s 10.1, 10.2)

x

x

Mxz Mxz

x

P

P1

P2

☻

Bending of BeamsMECHENG242 Mechanics of Materials

Mxz Mxz

2.2 Stresses in Beams(Refer: B, C & A–Sec 6.3, 6.4, 6.5, 6.6)2.2.1 The Engineering Beam Theory

x

yMxz Mxz

A B

C D

Compression

Tension

No StressNA

Neutral Axis

z

y

y’ y’

A’ B’

C’ D’ y’

R

x

x=0 on the Neutral Axis. In general we must find the position of the Neutral Axis.

Bending of BeamsMECHENG242 Mechanics of Materials

Mxz Mxz

A B

C Dy’

A’ B’

C’ D’ y’

R

Mxz Mxz

Assumptions Rx'B'AAB

Plane surfaces remain plane

Beam material is elastic0zy 0x and only

Bending of BeamsMECHENG242 Mechanics of Materials

1

Geometry of Deformation:

Rx'B'AABCD0

x LL

'yR'D'C

R

R'yRx

R'y

Hookes Law: zyxx E1

0zy

Ex

x and 'y

RE

x

CDCD'D'C

Bending of BeamsMECHENG242 Mechanics of Materials

'yRE

x

1

x

y

y’

NA

Neutral Axis 0+ve-ve

Linear Distribution of x

(Eqn )1

x

Note:E is a Material Property

is Curvature

R1

x

x

y

Mxz Mxz

x

Bending of BeamsMECHENG242 Mechanics of Materials

z

y

x

Equilibrium:

x

z

y

y’

Ax

Mxz

Area, A

Let xxx FA

But 0Fx 0dA

A x

0dA'yRE

A

AdA'y First Moment of Area

,0dA'yIfA

Then y’ is measured from the centroidal axis of the beam cross-section.

“Neutral Axis” coincides with the XZ plane through the centroid.

y’y’

NA

Neutral Axis

Centroid

Bending of BeamsMECHENG242 Mechanics of Materials

2

Equilibrium:0M'yF xzxx 0Mz

A x dA'y

xzA2 MdA'y

RE

as 'y

RE

x

1

Let A

2Z dA'yI =The 2nd Moment of Area about Z-axis

xzz MIRE

R

EI

Mz

xz

THE SIMPLE BEAM THEORY:

1 2&RE

'yIM x

z

xz

xz

y

y’

Ax

Mxz

Area, AA xzM xzM

Bending of BeamsMECHENG242 Mechanics of Materials

RE

'yIM x

z

xz

E'yx

zIxzM

R

- Applied Bending Moment - Property of Cross-Sectional Area - Stress due to Mxz - Distance from the Neutral Axis - Young’s Modulus of Beam Material - Radius of Curvature due to Mxz

- N.m - m4 - N/m2 or Pa- m - N/m2 or Pa- m

z

y

y’y’NA

Neutral Axis

xo

'yI

Mz

xzx

Bending of BeamsMECHENG242 Mechanics of Materials

z

y

o

(Refer: B, C & A–Appendix A, p598-601)2.2.2 Properties of Area

y’

A

x

Mxz

x

z

y

o

RE

'yIM x

z

xz

y’ is measured from the Centroidal or Neutral Axis, z. Iz is the 2nd Moment of Area about the Centroidal or Neutral Axis, z.

Position of Centroidal or Neutral Axis:

y’

Centroidal Axis

zoy’

Area, Ay

n

,0dA'y.e.iA

(Definition)

y A

dA'yyA

A

dA'yA1y

AA

Bending of BeamsMECHENG242 Mechanics of Materials

y

n

Example:

zo

Centroidal Axis

A

dA'yA1y

200

10

20

120

(Dimensions in mm)

y mm6.89

20120102001y

000,144000,250400,41y

400,4000,394

mm55.89

m106.89 3

12510200 6020120

125

60

Bending of BeamsMECHENG242 Mechanics of Materials

Example:

z

y

o

2nd Moment of Area:

2

d

2d

2z yb'yI

Definition:

A

2Z dA'yI

A

2y dAzI,Also

z’

y’

y’y2

d

2d

2b

2b

2d

2d

3

3yb

12bd3

12dbI,Also

3

y

A

o

y

z

Bending of BeamsMECHENG242 Mechanics of Materials

The Parallel Axis Theorem:

z

y

o

d

0

2n yb'yI

Definition:2

zn yAII

Example:

y’y2

d

2d

2b

2b

d

0

3

3yb

3bd3

12bdI

3

z

n y

ny

2nz yAII

23

2dbd

3bd

z

o

y

Bending of BeamsMECHENG242 Mechanics of Materials

Example: (Dimensions in mm)

z

y

o

20010

20

120

89.6

30.4

89.6

20

2030.4

20010

1

23

3bdI

3

1,z

36.8920 3

46 mm1079.4

3bdI

3

2,z

34.3020 3

46 mm1019.0

23

3,z yA12bdI 2

3

4.351020012

10200 46 mm1028.3

2zn yAII

• What is Iz?• What is maximum x?

35.4

Bending of BeamsMECHENG242 Mechanics of Materials

Example: (Dimensions in mm)

z

y

o

20010

20

120

89.6

30.4

89.6

20

2030.4

20010

35.4

1

23

3,z2,z1,zz IIII

46z mm1026.8I 46 m1026.8

2zn yAII

• What is Iz?• What is maximum x?

Bending of BeamsMECHENG242 Mechanics of Materials

y

NA x

'yI

Mz

xzx

Maximum Stress:

89.6

40.4 Mxz

Maxz

xzMax,x y

IM

36

xzMax,x 106.89

1026.8M

(N/m2 or Pa)

Bending of BeamsMECHENG242 Mechanics of Materials

Example:

The Perpendicular Axis Theorem:

z

y

o

222 'z'yR

32d4

Az’

y’ R A

2

A

2

A

2 dA'zdA'ydAR

yzx IIJ

The Polar 2nd Moment of Area (About the X-axis)

R

R

A

2x dARJ

2d

0

2 dRR2R

From Symmetry, yz II

yyzx I2IIJ

2JI x

y

o

y

z

64d4

Bending of BeamsMECHENG242 Mechanics of Materials

2.2.3 SummaryThe Engineering Beam Theory determines the axial stress distribution generated across the section of a beam. It is applicable to long, slender load carrying devices.

RE

'yIM x

z

xz

Calculating properties of beam cross sections is a necessary part of the analysis.

• Neutral Axis Position, y

• 2nd Moments of Area, Iy, Iz, Jx

Properties of Areas are discussed in Appendix A of B, C & A.