6
H1 Deflections and Slopes of Beams H TABLE H-1 DEFLECTIONS AND SLOPES OF CANTILEVER BEAMS v deflection in the y direction (positive upward) v dv/dx slope of the deflection curve d B v(L) deflection at end B of the beam (positive downward) u B v(L) angle of rotation at end B of the beam (positive clockwise) EI constant 1 v 2 q 4 x E 2 I (6L 2 4Lx x 2 ) v 6 q E x I (3L 2 3Lx x 2 ) d B 8 q E L 4 I u B 6 q E L 3 I 2 v 2 q 4 x E 2 I (6a 2 4ax x 2 ) (0 x a) v 6 q E x I (3a 2 3ax x 2 ) (0 x a) v 2 q 4 a E 3 I (4x a) v 6 q E a 3 I (a x L) At x a: v 8 q E a 4 I v 6 q E a 3 I d B 2 q 4 a E 3 I (4L a) u B 6 q E a 3 I (Continued) q a b q y x A B u B d B L © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Deflections and Slopes of Beams - cengage.com and Slopes of Beams H ... (L/2) deflection at midpoint C of the beam (positive downward) x 1 distance from support A to point of maximum

Embed Size (px)

Citation preview

Page 1: Deflections and Slopes of Beams - cengage.com and Slopes of Beams H ... (L/2) deflection at midpoint C of the beam (positive downward) x 1 distance from support A to point of maximum

H1

Deflections and Slopesof Beams

H

TABLE H-1 DEFLECTIONS AND SLOPES OF CANTILEVER BEAMS

v � deflection in the y direction (positive upward)v� � dv/dx � slope of the deflection curvedB � �v(L) � deflection at end B of the beam (positive downward)uB � �v�(L) � angle of rotation at end B of the beam (positive clockwise)EI � constant

1 v � ��2

q

4

x

E

2

I� (6L2 � 4Lx � x2) v� � ��

6

q

E

x

I�(3L2 � 3Lx � x2)

dB � �8

q

E

L4

I� uB � �

6

q

E

L3

I�

2 v � ��2

q

4

x

E

2

I� (6a2 � 4ax � x2) (0 � x � a)

v� � ��6

q

E

x

I�(3a2 � 3ax � x2) (0 � x � a)

v � ��2

q

4

a

E

3

I�(4x � a) v� � ��

6

q

E

a3

I� (a � x � L)

At x � a: v � ��8

q

E

a4

I� v� � ��

6

q

E

a3

I�

dB � �2

q

4

a

E

3

I�(4L � a) uB � �

6

q

E

a3

I�

(Continued)

q

a b

q

y

xA B

uB

dB

L

36025_09_online_appH_pH1-H6.qxd 3/3/11 8:18 AM Page H1

© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Page 2: Deflections and Slopes of Beams - cengage.com and Slopes of Beams H ... (L/2) deflection at midpoint C of the beam (positive downward) x 1 distance from support A to point of maximum

H2 APPENDIX H Deflections and Slopes of Beams

3 v � ��1

q

2

b

E

x2

I�(3L � 3a � 2x) (0 � x � a)

v� � ��q

2

b

E

x

I�(L � a � x) (0 � x � a)

v � ��24

q

EI�(x4 � 4Lx3 � 6L2x2 � 4a3x � a4) (a � x � L)

v� � ��6

q

EI�(x3 � 3Lx2 � 3L2x � a3) (a � x � L)

At x � a: v � ��1

q

2

a

E

2b

I�(3L � a) v� � ��

q

2

a

E

b

I

L�

dB � �24

q

EI�(3L4 � 4a3L � a4) uB � �

6

q

EI�(L3 � a3)

4 v � ��6PExI

2

�(3L � x) v� � ��2PExI

�(2L � x)

dB � �P3E

LI

3

� uB � �P2E

LI

2

5 v � ��6PExI

2

�(3a � x) v� � ��2PExI

�(2a � x) (0 � x � a)

v � ��6PEaI

2

�(3x � a) v� � ��2PEaI

2

� (a � x � L)

At x � a: v � ��3PEaI

3

� v� � ��2PEaI

2

dB � �6PEaI

2

�(3L � a) uB � �2PEaI

2

6 v � ��M

2E0x

I

2

� v� � ��M

E0

I

x�

dB � �M

2E0L

I

2

� uB � �M

E0

I

L�

M0

P

a b

P

q

a b

36025_09_online_appH_pH1-H6.qxd 3/3/11 8:18 AM Page H2

© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Page 3: Deflections and Slopes of Beams - cengage.com and Slopes of Beams H ... (L/2) deflection at midpoint C of the beam (positive downward) x 1 distance from support A to point of maximum

APPENDIX H Deflections and Slopes of Beams H3

7 v � ��M

2E0x

I

2

� v� � ��M

E0

I

x� (0 � x � a)

v � ��M

2E0a

I�(2x � a) v� � ��

M

E0

I

a� (a � x � L)

At x � a: v � ��M

2E0a

I

2

� v� � ��M

E0

I

a�

dB � �M

2E0a

I�(2L � a) uB � �

M

E0

I

a�

8 v � ��12

q

00

L

x2

EI�(10L3 � 10L2x � 5Lx2 � x3)

v� � ��24

q

L0x

EI�(4L3 � 6L2x � 4Lx2 � x3)

dB � �3

q

00L

E

4

I� uB � �

2

q

40L

E

3

I�

9 v � ��12

q

00

L

x2

EI�(20L3 � 10L2x � x3)

v� � ��24

q

L0x

EI�(8L3 � 6L2x � x3)

dB � �1

1

1

2

q

00

E

L

I

4

� uB � �q

80

E

L

I

3

10 v � ��3

q

p

04

L

EI� �48L3 cos �

p

2L

x� � 48L3 � 3p3Lx2 � p3x3�

v� � ��p

q30

E

L

I��2p 2Lx � p2x2 � 8L2 sin �

p

2L

x��

dB � �3

2

p

q04

L

E

4

I� (p 3 � 24) uB � �

p

q03

L

E

3

I� (p2 � 8)

(Continued)

q = q0 cos �x—2L

q0

q0

q0

M0

a b

36025_09_online_appH_pH1-H6.qxd 3/3/11 8:18 AM Page H3

© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Page 4: Deflections and Slopes of Beams - cengage.com and Slopes of Beams H ... (L/2) deflection at midpoint C of the beam (positive downward) x 1 distance from support A to point of maximum

H4 APPENDIX H Deflections and Slopes of Beams

TABLE H-2 DEFLECTIONS AND SLOPES OF SIMPLE BEAMS

v � deflection in the y direction (positive upward)v� � dv/dx � slope of the deflection curvedC � �v(L/2) � deflection at midpoint C of the beam (positive downward)x1 � distance from support A to point of maximum deflectiondmax � �vmax � maximum deflection (positive downward)uA � �v�(0) � angle of rotation at left-hand end of the beam

(positive clockwise)uB � v�(L) � angle of rotation at right-hand end of the beam

EI � constant (positive counterclockwise)

1 v � ��24

qx

EI�(L3 � 2Lx2 � x3)

v� � ��24

q

EI�(L3 � 6Lx2 � 4x3)

dC � dmax� �3

5

8

q

4

L

E

4

I� uA � uB � �

2

q

4

L

E

3

I�

2 v � ��38

q

4

x

EI�(9L3 � 24Lx2 � 16x3) �0 � x � �

L2

��v� � ��

38

q

4EI�(9L3 � 72Lx2 � 64x3) �0 � x � �

L2

��v � ��

38

q

4

L

EI�(8x3 � 24Lx2 � 17L2x � L3) ��

L2

� � x � L�v� � ��

38

q

4

L

EI�(24x2 � 48Lx � 17L2) ��

L2

� � x � L�dC � �

7

5

6

q

8

L

E

4

I� uA � �

1

3

2

q

8

L

E

3

I� uB � �

3

7

8

q

4

L

E

3

I�

3 v � ��24

q

L

x

EI�(a4 � 4a3L � 4a2L2 � 2a2x2 � 4aLx2 � Lx3) (0 � x � a)

v� � ��24

q

LEI�(a4 � 4a3L � 4a2L2 � 6a2x2 � 12aLx2 � 4Lx3) (0 � x � a)

v � ��24

q

L

a

E

2

I�(�a2L � 4L2x � a2x � 6Lx2 � 2x3) (a � x � L)

v� � ��24

q

L

a

E

2

I�(4L2 � a2 � 12Lx � 6x2) (a � x � L)

uA � �24

q

L

a

E

2

I�(2L � a)2 uB � �

24

q

L

a

E

2

I�(2L2 � a2)

q

a

q

L—2

L—2

q

y

x

uA uBA B

L

36025_09_online_appH_pH1-H6.qxd 3/3/11 8:18 AM Page H4

© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Page 5: Deflections and Slopes of Beams - cengage.com and Slopes of Beams H ... (L/2) deflection at midpoint C of the beam (positive downward) x 1 distance from support A to point of maximum

APPENDIX H Deflections and Slopes of Beams H5

L—2

L—2

P4 v � ��

4P8E

xI

�(3L2 � 4x2) v� � ��16

PEI�(L2 � 4x2) �0 � x � �

L2

��dC � dmax � �

4P8LE

3

I� uA � uB � �

1P6LE

2

I�

5 v � ��6PLbExI

�(L2 � b2 � x2) v� � ��6PL

bEI�(L2 � b2 � 3x2) (0 � x � a)

uA � �Pab

6

(

L

L

E

I

b)� uB � �

Pab

6

(

L

L

E

I

a)�

If a b, dC ��Pb(3L

48

2

E�

I4b2)

� If a � b, dC ��Pa(3L

4

2

8E�

I4a2)

If a b, x1 � ��L2 �

3

b� 2

�� and dmax �

6 v � ��6PExI

�(3aL � 3a2 � x2) v� � ��2PEI�(aL � a2 � x2) (0 � x � a)

v � ��6PEaI

�(3Lx � 3x2 � a2) v� � ��2PEaI

�(L � 2x) (a � x � L � a)

dC � dmax � �2P4Ea

I�(3L2 � 4a2) uA � uB ��

Pa(

2

L

E

I

a)�

7 v � ��6

M

L0

E

x

I� (2L2 � 3Lx � x2) v� � ��

6

M

LE0

I� (2L2 � 6Lx � 3x2)

dC � �M

160

E

L

I

2

� uA � �M

3E0L

I� uB � �

M

6E0L

I�

x1 � L�1 � ��33��� and dmax � �

9

M

�0

3�L

E

2

I�

(Continued)

M0

P P

a a

Pb(L2 � b2)3/2

��9�3� LEI

P

a b

36025_09_online_appH_pH1-H6.qxd 3/3/11 8:18 AM Page H5

© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Page 6: Deflections and Slopes of Beams - cengage.com and Slopes of Beams H ... (L/2) deflection at midpoint C of the beam (positive downward) x 1 distance from support A to point of maximum

H6 APPENDIX H Deflections and Slopes of Beams

8 v � ��2

M

4L0

E

x

I�(L2 � 4x2) v� � ��

24

M

L0

EI�(L2 � 12x2) �0 � x � �

L2

��dC � 0 uA � �

2

M

40

E

L

I� uB � ��

2

M

40

E

L

I�

9 v � ��6

M

L0

E

x

I�(6aL � 3a2 � 2L2 � x2) (0 � x � a)

v� � ��6

M

LE0

I�(6aL � 3a2 � 2L2 � 3x2) (0 � x � a)

At x � a: v � ��M

3L0a

E

b

I�(2a � L) v� � ��

3

M

LE0

I�(3aL � 3a2 � L2)

uA � �6

M

LE0

I�(6aL � 3a2 � 2L2) uB � �

6

M

LE0

I�(3a2 � L2)

10 v � ��M

2E0x

I�(L � x) v� � ��

2

M

E0

I�(L � 2x)

dC � dmax � �M

80

E

L

I

2

� uA � uB � �M

2E0L

I�

11 v � ��36

q

00

L

x

EI�(7L4 � 10L2x2 � 3x4)

v� � ��360

q

L0

EI�(7L4 � 30L2x2 � 15x4)

dC � �7

5

6

q

80L

E

4

I� uA � �

3

7

6

q

00L

E

3

I� uB � �

4

q

50L

E

3

I�

x1 � 0.5193L dmax � 0.00652�q

E0L

I

4

12 v � ��96

q

00

L

x

EI�(5L2 � 4x2)2 �0 � x � �

L2

��v� � ��

192

q

L0

EI� (5L2 � 4x2)(L2 � 4x2) �0 � x � �

L2

��dC � dmax � �

1

q

20

0

L

E

4

I� uA � uB � �

1

5

9

q

20 L

E

3

I�

13 v � ��p

q04

L

E

4

I� sin �

p

Lx� v� � ��

p

q03

L

E

3

I� cos �

p

Lx�

dC � dmax � �p

q04

L

E

4

I� uA � uB � �

p

q03

L

E

3

I�

q0

L—2

L—2

q0

a b

M0

L—2

L—2

M0

M0M0

q = q0 sin�x—L

36025_09_online_appH_pH1-H6.qxd 3/3/11 8:18 AM Page H6

© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.