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
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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.
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.
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.
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.
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
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