Chapter 9 Deflection of Beams - Online Lessons Using ... · Chapter 9 Deflection of Beams ......

9-1 Introduction

Chapter 9 Deflection of Beams

INTRODUCTION Deflection of Beams using Integration

Statically Indeterminate Beams using Integration

Deflection of Beams using Superposition

Statically Indeterminate Beams using Superposition

B A

L 0.5L

w

A

50 lb/in

48

B

9-2 Equation of the Elastic Curve

EQUATION OF THE ELASTIC CURVE

P

VdxdM

=

dyTANdx

θ θ = ≅

1 ( )d d M xds dx EIθ θ

ρ = = =

3

3

( )d y dM V xdx EIdx EI

= =

4

4

( )d y dV w xdx dxEI EI

= = −

Review,

Noting, ρ ρ

x

y

x dx

2

2

( )d d y M xdx dx EIθ

∴ = =

4

4

( )d y w xdx EI

= −

dV wdx

= −

9-3

Example a) Determine the equation for the vertical displacement and slope at any point. b) Find the displacement and slope at B. Use the second order differential equation to solve. EI is constant.

A

P

L

B

P

9-4

A

w

L

B

B

Example a) Determine the equation for the vertical displacement and slope at any point. b) Find the displacement and slope at B. Use the second order differential equation to solve. EI is constant.

9-5

A Mo

L

B

Example a) Determine the equation for the vertical displacement and slope at any point. b) Find the displacement and slope at B. Use the second order differential equation to solve. EI is constant.

B

Mo

9-6

A

L

0.5L

B C

Example Determine the vertical displacement and slope at point c. Use the second order differential equation to solve. EI is constant.

A

A

0.5L

B

Mo

Mo

9-7

Example Determine the vertical displacement at the center of the beam. Use the second order differential equation to solve. EI is constant.

B A

L 0.5L

w

A

9-8

Example Determine the maximum vertical displacement of the beam. Use the second order differential equation to solve. EI is constant.

A

B A

L

ow

9-9

Example Determine the maximum vertical displacement of the beam. Use the second order differential equation to solve. EI is constant. Units: kN, m.

B A

L

A

46

20022.2 10 mm

E GPaI x

= =

Mo

Mo

9-10

Example Determine the vertical displacement at C. Use the second order differential equation to solve. EI is constant. Units: kN, m.

B A

10

C

1 2

A

A

B 2

46

20022.2 10 mm

E GPaI x

= =

M

9-11

A

L

B

P

P

Example a) Determine the equation for the vertical displacement and slope at any point. b) Find the displacement and slope at B. Use the fourth order differential equation to solve. EI is constant.

9-12

A

w

L

B

B

Example a) Determine the equation for the vertical displacement and slope at any point. b) Find the displacement and slope at B. Use the fourth order differential equation to solve. EI is constant.

9-13

A Mo

L

B

Example a) Determine the equation for the vertical displacement and slope at any point. b) Find the displacement and slope at B. Use the fourth order differential equation to solve. EI is constant.

Mo

9-14

B A

L 0.5L

w

A

Example a) Determine the equation for the vertical displacement and slope at any point. b) Find the displacement at L/2 and the slope at A. Use the fourth order differential equation to solve. EI is constant.

9-15

Example Determine the reactions at A and B. Use the fourth order differential equation to solve. EI is constant. Units: lb, in.

STATICALLY INDETERMINATE BEAMS USING INTEGRATION

A

w

L

B

9-16

Example Determine the reactions at A and B. Use the second order differential equation to solve. EI is constant. Units: lb, in.

B

A

L

B

w

9-17

A

w

L

B

Example Determine the reactions at A and B. Use the second order differential equation to solve. Neglect the effect of any axial reactions. EI is constant. Units: lb, in.

A

w

9-17.1

ExampleDetermine the reactions at A and B. Use the fourth order differential equation to solve. Neglect the effect of any axial reactions. EI is constant. Units: lb, in.

A

L

B

sinoxw w

Lπ⎛ ⎞= ⎜ ⎟⎝ ⎠

9-17.2

9-18

DEFLECTION OF BEAMS USING SUPERPOSITION

ExampleUsing superposition, determine the displacement at C. EI is constant. Units: lb, in.

A

160 lb/in

120

B

A

120

B

8000

4

629 1053.4

psi

in

E xI==

A

160 lb/in

120

B

60

8000

C

9-19

ExamplePost AC is made of steel and has a diameter of 18 mm, and BD is made of copper and has a diameter of 42 mm. Determine the displacement of point E on the steel beam AB. E(steel)= 200 GPa, E(copper)= 120 GPa. Units: mm, kN

1000

4000

A BE

601000

4000

2200

A B

C D

E

601000

4000

A BE

E

4645.5 10 mmI x=

9-20

ExampleKnowing that each beam has a rectangular cross section as shown, determine the displacement at E. E= 200GPa. EI is constant.Units: kN, m.

AB

C D

1.5 1.51.41.4

4

E

160

200

Beam AB80

100

Beam CD

C D

4

E

AB

1.5 4.3

AB

4.3 1.5

9-21

ExampleThe horizontal beam AB rests on the two short springs with the same length. The spring at A has stiffness of 250 kN/m and the spring at B has a stiffness of 150 kN/m. Determine the displacement under the load. Units: kN, mm. 3.7220

A B

900

C

A B

900

C

AB

2315 10 mEI x=

3.7

9-22

ExampleUsing superposition, determine the displacement at C. EI is constant. Units: kN, m.

A

8

B

10

2

C

A

B

2621.4 10 mEI x=

C

10

C

9-23

ExampleThe 160x200 mm rectangular beam ABC rests on a spring at B. The spring at B has stiffness of 2500 kN/m. Determine the displacement at C. Units: kN, m.

AB

C

1082

2621.4 10 mEI x=From a previous solution with point B being a rigid roller:yc= 4.99 mm

AB

C

9-24

STATICALLY INDETERMINATE BEAMS USING SUPERPOSITION

ExampleUsing superposition, determine the reactions at A and B.EI is constant. Units: lb, in.

A

w

L

B

A

w

B

AB

9-25

BA

9648

C

50 lb/in

ExampleDue to the loading and poor construction, support B settles 1/16". Using superposition, determine the reactions at A, B, and C.EI is constant. Units: lb, in.

AC

50 lb/in

AC

4

629 1011.3

psi

in

E xI==

9-26

A

4000

A

ExampleThe W6x20 is supported at B by a 0.25" rod. Using superposition, determine the force in the rod.EI is constant. E=29E6 psi for both members.Units: lb, in.

A

120

B

60

4000

C

80

9-27

A

30

73

ExampleUsing superposition, determine the reactions at A and the force in the spring at B. The spring constant is 1 kN/mm. EI is constant.Units: kN, m.

A

30

A

26100 10 mEI x=

9-28

ExampleUsing superposition, determine the reactions at A and B. Neglect the effect of any axial reactions. EI is constant. Units: kN, m.

A

2 kN/m

4

B2

A

2 kN/m

A

A

9-29Summary

Deflection of Beams using Integration

Statically Indeterminate Beams using Integration

Deflection of Beams using Superposition

Statically Indeterminate Beams using Superposition

BA

L0.5L

w

A

50 lb/in

48

B

SUMMARY