bending of beams mecheng242 mechanics of materials 2.3 combined bending and axial loading 2.0...

Bending of BeamsMECHENG242 Mechanics of Materials

2.3 Combined Bending and Axial Loading

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)

P1

P2

☻

2.2 Stresses in Beams

2.1 Revision – Bending Moments

x

x

Mxz Mxz

x

P

☻☻


2.4 Beam Deflection (Refer: B, C & A–Sec 7.1, 7.2, 7.3, 7.4)

2.4.1 Moment-Curvature Equation

Recall: THE ENGINEERING BEAM THEORY

R

E

I

M

'y z

xzx

y

xNA

NA

A B

x

A’ B’s

xs If deformation is small (i.e. slope is “flat”):

v (Deflection)


A’

B’S

Alternatively: from Newton’s Curvature Equation

2

2

dx

vd

R

I

xS.R dx

d

R

I

x

v

and (slope is “flat”)

v

x

R

)x(fv

2

2

dx

vd

R

I

1dx

dv2

if

2

32

2

2

dxdv

1

dxvd

R

I

R

v


From the Engineering Beam Theory:

R

E

I

M

z

xz z

xz

EI

M

R

1

2

2

dx

vd

xz2

2

z Mdx

vdEI

Flexural Stiffness

Bending Moment

Curvature

Mxz

2

2

dx

vd

R

1

Flexural Stiffness

Recall, for Bars under axial loading:

LoaduK

Axial Stiffness

Extension


Curvature

Slope

Deflection

Since, xzz

2

2

MEI

1

dx

vd

Curvature

1xzz

CdxMEI

1

dx

dv

Slope

21xzz

CdxCdxdxMEI

1v

Deflection

Where C1 and C2 are found using the boundary conditions.

Rdx

dvv


x

yP

B

L

A

P

Mxz

Qxy

Example:

x

P

P.L

P.L

v = Deflection

v vMax

DeflectedShape

xz2

2

z Mdx

vdEI

dx

dvEIz

PLPxMxz

PLPx

vEIz

1

2

CPLx2

xP

21

23

CxC2

PLx

6

xP


21

23

z CxC2

PLx

6

xPvEI

P

To find C1 and C2:

Boundary conditions: (i) @ x=0 0dx

dv

(ii) @ x=0 0v

0C&0C 21

Equation of the deflected shape is:

2

PLx

6

xPvEI

23

z

vMax occurs at x=L

z

3

Max EI

PL

3

1v


a b

L

2.4.2 Macaulay’s Notation

y

x

Example:

Qxy

Mxz

P

L

Pa

L

Pb x

L

Pb

P

xz2

2

z Mdx

vdEI

12

2

z Cax2P

2

xL

Pbdx

dvEI

axPxLPbMxz

axPxLPb

2133

z CxCax6PxL6

PbvEI


2133

z CxCax6PxL6

PbvEI

Boundary conditions: (i) @ x=0 0v

(ii) @ x=L 0v

From (i): 0C2

From (ii): LCaL6PLL6

Pb0 133

221 LbL6

PbC Since (L-a)=b

Equation of the deflected shape is:

xLbL6Pbax6

PxL6Pb

EI

1v 2233

z


This value of x is then substituted into the above equation of the deflected shape in order to obtain vMax.

To find vMax:vMax occurs where (i.e. slope=0)0

dx

dv

2222

z LbL6Pbax2

P2

xL

Pb0EI.e.i

Assuming vMax will be at x<a, 0ax.e.i 2

when0dx

dv 222 Lb3

1x 22 bL31

z

3

Max EI48

PLv

P

vMax

2

L

2

LNote:

2

Lba if


2.4.3 Summary

After considering stress caused by bending, we have now looked at the deflections generated. Keep in mind the relationships between Curvature, Slope, and Deflection, and understand what they are:

• Curvature

• Slope

• Deflection

Apart from my examples and problems:

R

IM

EI

1

dx

vdxz

z2

2

dx

dv

v

• B, C & A Worked Examples, pg 185-201

Problems, 7.1 to 7.15, pg 207

bending of beams mecheng242 mechanics of materials 2.3 combined bending and axial loading 2.0...

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