8 beam deflection 2015 bach khoa

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CHAPTER 8: BEAM DEFLECTION

8.1 Deflection of a beam under transverse loading

8.2 Equation of the elastic curve

8.3 Direct determination of the elastic curve from the loading

8.4 Statically indeterminate beams

8.5 Examples

8.6 Statically indeterminate beams

8.1 DEFLECTION OF A BEAM UNDER TRANSVERSE

LOADING

• Relationship between bending moment and curvature for pure bending remains valid for general transverse loadings

• Curvature varies linearly with x

• At the free end A,  A  

8.1 DEFLECTION OF A BEAM UNDER TRANSVERSE

LOADING

• Overhanging beam

• Reactions at A and C

• Bending moment diagram

• Curvature is zero at points where the bending

moment is zero, i.e., at each end and at E

• Maximum curvature occurs where the moment magnitude is a maximum

• An equation for the beam shape or elastic curve

is required to determine maximum deflection and slope

8.2 EQUATION OF THE ELASTIC CURVE

• From elementary calculus, simplified for beam parameters,

2

2 2

3 2 2 2

1

1

dx

y d

dx dy dx

y d

1 0

2

2 1

C x C dx x M dx y

EI

C dx x M dx

dy EI EI

x

M dx

y d EI EI

x x

8.2 EQUATION OF THE ELASTIC CURVE

0 0

C x C dx x M dx y

EI

x x

8.3 DIRECT DETERMINATION OF THE ELASTIC CURVE

FROM THE LOAD DISTRIBUTION

• For a beam subjected to a distributed load,

dx

dV dx

M d x

V dx

dM    

2 2

• Equation for beam displacement becomes

 x

w dx

y d EI dx

M

d   

4

4 2

2

4 3

2 2 2 1

3 1 6

1C x C x C x C

dx x w dx dx dx x

y EI

• Integrating four times yields

• Constants are determined from boundary conditions

8.4 STATICALLY INDETERMINATE BEAMS

• Consider beam with fixed support at A and roller support at B

• From free-body diagram, note that there are four unknown reaction components

0 0

C x C dx x M dx y

EI

x x





  

• Also have the beam deflection equation,

which introduces two unknowns but provides three additional equations from the boundary conditions:

0 ,

At 0

0 ,

0

At x    y  x  L y 

• Conditions for static equilibrium yield

0 0

8.5 EXAMPLES

EXAMPLE 9.01

ft 4 ft

15 kips

50

psi 10 29 in

723 68

P

E I

W

For portion AB of the overhanging beam,

(a) derive the equation for the elastic curve,

(b) determine the maximum deflection,

SOLUTION:

•Develop an expression for M(x) and derive differential equation for elastic curve

•Integrate differential equation twice and apply boundary conditions to obtain elastic curve

•Locate point of zero slope or point of maximum deflection

•Evaluate corresponding maximum deflection

R L

a P

M   0  

x L

a P dx

y d

EI  

2 2

-The differential equation for the elastic curve,

EXAMPLE 9.01

x EI

PaL

y

L

L x

L

x EI

PaL dx



EI

PaL y

EI

PaL y

6

0642

0

2 max 

2 max

in 723 psi

10 29 6

in 180 in

48 kips 50 0642

max 

y

8.5 EXAMPLES

EXAMPLE 9.02

For the uniform beam, determine the

reaction at A, derive the equation for

the elastic curve, and determine the

slope at A (Note that the beam is

statically indeterminate to the first

degree)

SOLUTION:

•Develop the differential equation for the elastic curve (will be functionally

dependent on the reaction at A)

•Integrate twice and apply boundary

conditions to solve for reaction at A and

to obtain the elastic curve

•Evaluate the slope at A

x L

x w x

R M

A

A D

6

0 3

2 1 0

3 0

2 0

M dx

y d

6

3 0 2

8.5 EXAMPLES

EXAMPLE 9.02

L

x w x R

M dx

5 0 3

1

4 0 2

120 6

1

24 2

1

C x C L

x w x

R y

EI

C L

x w x

R

EI dx

dy EI

6

1 : 0 ,

at

0 24

2

1 : 0 ,

at

0 :

0 ,

0

at

2 1

4 0 3

1

3 0 2

w L

R y

L x

C L

w L

R L

x

C y

1 3

8.5 EXAMPLES

EXAMPLE 9.02

x L

w L

x w x

L w y

3 0

120

1 120

10

1 6 1





•Differentiate once to find the slope,

at x = 0,

8.6 METHOD OF SUPERPOSITION

Principle of Superposition:

•Deformations of beams subjected to

combinations of loadings may be

obtained as the linear combination of

the deformations from the individual

wL EI

wL

y B II

384

7 2

48 128

4 3

II B I

B B

48 6

3 3

wL y

y

y B B I B II

384

7 8

4 4

8.6 METHOD OF SUPERPOSITION

APPLICATION OF SUPERPOSITION TO STATICALLY

INDETERMINATE BEAMS

•Method of superposition may be applied

to determine the reactions at the supports

of statically indeterminate beams

•Designate one of the reactions as

redundant and eliminate or modify the

•Release the “redundant” support at B, and find deformation

•Apply reaction at B as an unknown load to force zero displacement at B

L L

L L

L EI

w

y B w

4

3 3

4

01132

0

3

2 3

2 2 3

2 24

L EIL

R

3 2

2

01646

0 3

wL y

3 4

01646

0 01132

0

w A

3 3

04167

L

L EIL

wL

R A

3 2

2

03398

0 3

3 6

0688

wL

R A w

A A

3 3

03398

0 04167