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ElasticBeamDeflectionEquations.pdf
APPENDIX 44.A Elastic Beam Deflection Equations (w is the load per unit length.)
(y is positive downward.)
Case 1: Cantilever with End Load Case 2: Cantilever with Uniform Load
–
–
L
x
V
Mmax
yx
P reactions: Rl ¼ 0 Rr ¼ P shear:
V ¼ �P ðconstantÞ moments:
Mx ¼ �Px Mmax ¼ �PL end slope:
�l ¼ þ PL 2
2EI
�r ¼ 0 deflection:
yx ¼ P 6EI
� �
� ð2L3 � 3L2x þ x3Þ
ymax ¼ PL 3
3EI at x ¼ 0
–
–
L
x
Vr
Mr
yx
w reactions: Rl ¼ 0 Rr ¼ wL shear:
V x ¼ �wx V max ¼ �wL ¼ V r moments:
Mx ¼ � wx 2
2
Mmax ¼ � wL 2
2 ¼ Mr
end slope:
�l ¼ þ wL 3
6EI
�r ¼ 0 deflection:
yx ¼ w 24EI
� �
� ð3L4 � 4L3x þ x4Þ
ymax ¼ wL 4
8EI at x ¼ 0
Case 3: Cantilever with Triangular Load Case 4: Propped Cantilever with Uniform Load
L
–
–
x
Vmax
Mmax
yx
w reactions: Rl ¼ 0 Rr ¼ wL
2 shear:
V x ¼ � wx 2
2L
V max ¼ wL 2
at x ¼ L moments:
Mx ¼ � wx 3
6L
Mmax ¼ wL 2
6 at x ¼ L
end slope:
�l ¼ þ wL 3
24EI
�r ¼ 0 deflection:
yx ¼ w 120EIL
� �
� ð4L5 � 5L4x þ x5Þ
ymax ¼ wL 4
30EI at x ¼ 0
+
–
–
L x
Vr
Mr
Vl
θl
Rl
L 4+
reactions:
Rl ¼ 38wL Rr ¼ 58wL shear:
V l ¼ �38wL V r ¼ þ58wL
V x ¼ w 3L 8
� x � �
moments:
Mr ¼ Mmax ¼ � wL 2
8
Mx ¼ � w 8
� �
ð3xL � 4x2Þ
M ¼ 9wL 2
128
at x ¼ 0:375L end slope:
�l ¼ þ wL 3
48EI
�r ¼ 0 deflection:
yx ¼ w 48EI
� �
� ðL3x � 3Lx3 þ 2x4Þ
ymax ¼ wL 4
185EI
at x ¼ 0:4215L
(continued)
P P I * w w w . p p i 2 p a s s . c o m
A-98 C I V I L E N G I N E E R I N G R E F E R E N C E M A N U A L
S u p p o rt
M a te ria
l
APPENDIX 44.A (continued) Elastic Beam Deflection Equations (w is the load per unit length.)
(y is positive downward.)
Case 5: Cantilever with End Moment Case 6: Simple Beam with Center Load
+
L
x
M
yx
M0 reactions: Rl ¼ 0 Rr ¼ 0 shear:
V ¼ 0 moments:
M ¼ M0 ¼ Mmax end slope:
�l ¼ � M0L
EI
�r ¼ 0 deflection:
yx ¼ � M0 2EI
� �
� ðL2 � 2xL þ x2Þ
ymax ¼ � M0L
2
2EI
at x ¼ 0 +
–
L x
Vr
Vl
Rl Rr
1 2
P
+
L 2
L 2
θl θr
reactions:
Rl ¼ Rr ¼ P=2 shear:
V l ¼ P=2 V r ¼ �P=2 moments:
Mx1 ¼ Px 2
Mx2 ¼ P 2 ðL � xÞ
Mmax ¼ PL 4
end slope:
�l ¼ � PL 2
16EI
�r ¼ þ PL 2
16EI deflection:
yx1 ¼ P 48EI
� �
3xL2 � 4x3
at x < L=2
ymax ¼ PL 3
48EI
� �
at x ¼ L=2
Case 7: Simple Beam with Intermediate Load Case 8: Simple Beam with Two Loads
+
–
L
x
Vr
Vl
Rl Rr
a b
+
21
P
θl θr
reactions:
Rl ¼ Pb L
Rr ¼ Pa L
shear:
V l ¼ þ Pb L
V r ¼ � Pa L
moments:
Mx1 ¼ Pbx L
Mx2 ¼ PaðL � xÞ
L
Mmax ¼ Pab L
at x ¼ a end slope:
�l ¼ � Pab 1 þ b
L
� �
6EI
�r ¼ Pab 1 þ a
L
� �
6EI deflection:
yx1 ¼ Pb 6EIL
� �
L2x � b2x � x3
at x < a
yx2 ¼ Pb 6EIL
� �
L b
� �
ðx � aÞ3
þðL2 � b2Þx � x3
0
@
1
A
at x > a
y ¼ Pa 2b2
3EIL at x ¼ a
ymax ¼ 0:06415Pb EIL
� �
ðL2 � b2Þ3=2
at x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aðL þ bÞ 3
r
+
–
L
x
Vr
Mx2
Vl
Rl Rr
+
2 31
P P b aa
θl θr
reactions:
Rl ¼ Rr ¼ P shear:
V l ¼ þP V r ¼ �P moments:
Mx1 ¼ Px Mx2 ¼ Pa Mx3 ¼ PðL � xÞ end slope:
�l ¼ � Paða þ bÞ
2EI
�r ¼ þ Paða þ bÞ
2EI deflection:
yx1 ¼ P 6EI
� �
3Lax � 3a2x � x3ð Þ at x < a
yx2 ¼ P 6EI
� �
3Lax � 3ax2 � a3ð Þ at a < x < a þ b
ymax ¼ P 24EI
� �
ð3L2a � 4a3Þ at x ¼ L=2
(continued)
P P I * w w w . p p i 2 p a s s . c o m
A P P E N D I C E S A-99
S u p p o rt
M a te ri a l
APPENDIX 44.A (continued) Elastic Beam Deflection Equations (w is the load per unit length.)
(y is positive downward.)
Case 9: Simple Beam with Uniform Load Case 10: Simple Beam with Triangular Load
(w is the maximum loading per unit length at the right end, not the total load,
W ¼ 1 2 Lw.)
+
+
–
L
x
Vr
Vl
Rl Rr
w
θl θr
reactions:
Rl ¼ Rr ¼ wL 2
shear:
V l ¼ þ wL 2
V r ¼ � wL 2
moments:
M ¼ w 2
� �
ðLx � x2Þ
Mmax ¼ wL 2
8 end slope:
�l ¼ �wL 3
24EI
�r ¼ þ wL 3
24EI deflection:
yx ¼ w 24EI
� �
� ðL3x � 2Lx3 þ x4Þ
ymax ¼ 5wL 4
384EI
at x ¼ L 2
+
+
–
L
x
Vr
Vl
Rl Rr
w
θl θr
reactions:
Rl ¼ wL 6
Rr ¼ wL 3
shear:
V l ¼ þ wL 6
V r ¼ � wL 3
V x ¼ wL 6
� �
1 � 3 x L
� �2 � �
moments:
Mx ¼ w 6
� �
Lx � x 3
L
� �
Mmax ¼ wL 2
9 ffiffiffi
3 p ¼ 0:0642wL2
at x ¼ 0:577L end slope:
�l ¼ �7wL 3
360EI
�r ¼ þ wL 3
45EI deflection:
yx ¼ w 360EI
� �
� 7L3x � 10Lx3 þ 3x 5
L
� �
ymax ¼ 0:00652ð Þ wL 4
EI
� �
at x ¼ 0:519L
(continued)
P P I * w w w . p p i 2 p a s s . c o m
A-100 C I V I L E N G I N E E R I N G R E F E R E N C E M A N U A L
S u p p o rt
M a te ria
l
APPENDIX 44.A (continued) Elastic Beam Deflection Equations (w is the load per unit length.)
(y is positive downward.)
Case 11: Simple Beam with Overhung Load Case 12: Simple Beam with Uniform Load Distributed over Half of Beam
+
–
xa
Vr
Vl
P Rra b
–
xbRl
reactions:
Rl ¼ P b
� �
ðb þ aÞ
Rr ¼ �Pa b
shear:
V l ¼ �P V r ¼ Pa
b moments:
Ma ¼ Pxa Mb ¼ Pa
b
� �
ðb � xbÞ Mmax ¼ Pa at xa ¼ a deflection:
ya ¼ P 3EI
� �
�
�
a2 þ abÞða � xaÞ
þ xa 2
� �
ðx2a � a2Þ
0
B
@
1
C
A
yb ¼ Paxb 6EI
� �
3xb � x2b b
� �
� 2b � �
ytip ¼ Pa 2
3EI
� �
ða þ bÞ ½max down�
ymax ¼ 0:06415ð Þ Pab 2
EI
� �
at xb
¼ 0:4226b ½max up�
+
+
–
L
x
Vr
Vl
Rl Rr
w
L 2
reactions:
Rl ¼ 3wL 8
Rr ¼ wL 8
shear:
V l ¼ þ 3wL 8
V r ¼ wL 8
V x ¼ w 3L 8
� x � �
x < L 2
� �
moments:
Mx ¼ w 8
� �
3Lx � 4x2ð Þ x < L 2
� �
Mx ¼ wL 2
8
� �
1 � x L
� �
x > L 2
� �
Mmax ¼ 9wL 2
128 x ¼ 3L
8
� �
deflection:
yx ¼ wx 384EI
� �
ð9L3 � 24Lx2 þ 16x3Þ x < L
2
� �
yx ¼ wLðL � xÞ 384EI
� �
ð16xL � 8x2 � L2Þ
x > L 2
� �
P P I * w w w . p p i 2 p a s s . c o m
A P P E N D I C E S A-101
S u p p o rt
M a te ri a l
153CIVIL ENGINEERING
Area Depth Web Flange Compact r ts
h o
Tors. Prop. Axis X-X Axis Y-Y
Shape A d t w b f t f section
J Cw I S r Z I r
In. 2
In. In. In. In. bf/2tf h/tw in. in. in.
4 in
6 In.
4 In.
3 In. In.
3 In.
4 In.
W24X68 20.1 23.7 0.415 8.97 0.585 7.66 52.0 2.30 23.1 1.87 9430 1830 154 9.55 177 70.4 1.87 W24X62 18.2 23.7 0.430 7.04 0.590 5.97 49.7 1.75 23.2 1.71 4620 1550 131 9.23 153 34.5 1.38 W24X55 16.3 23.6 0.395 7.01 0.505 6.94 54.1 1.71 23.1 1.18 3870 1350 114 9.11 134 29.1 1.34 W21X73 21.5 21.2 0.455 8.30 0.740 5.60 41.2 2.19 20.5 3.02 7410 1600 151 8.64 172 70.6 1.81 W21X68 20.0 21.1 0.430 8.27 0.685 6.04 43.6 2.17 20.4 2.45 6760 1480 140 8.60 160 64.7 1.80 W21X62 18.3 21.0 0.400 8.24 0.615 6.70 46.9 2.15 20.4 1.83 5960 1330 127 8.54 144 57.5 1.77
W21X55 16.2 20.8 0.375 8.22 0.522 7.87 50.0 2.11 20.3 1.24 4980 1140 110 8.40 126 48.4 1.73 W21X57 16.7 21.1 0.405 6.56 0.650 5.04 46.3 1.68 20.4 1.77 3190 1170 111 8.36 129 30.6 1.35 W21X50 14.7 20.8 0.380 6.53 0.535 6.10 49.4 1.64 20.3 1.14 2570 984 94.5 8.18 110 24.9 1.30 W21X48 14.1 20.6 0.350 8.14 0.430 9.47 53.6 2.05 20.2 0.803 3950 959 93.0 8.24 107 38.7 1.66 W21X44 13.0 20.7 0.350 6.50 0.450 7.22 53.6 1.60 20.2 0.770 2110 843 81.6 8.06 95.4 20.7 1.26 W18X71 20.8 18.5 0.495 7.64 0.810 4.71 32.4 2.05 17.7 3.49 4700 1170 127 7.50 146 60.3 1.70 W18X65 19.1 18.4 0.450 7.59 0.750 5.06 35.7 2.03 17.6 2.73 4240 1070 117 7.49 133 54.8 1.69 W18X60 17.6 18.2 0.415 7.56 0.695 5.44 38.7 2.02 17.5 2.17 3850 984 108 7.47 123 50.1 1.68 W18X55 16.2 18.1 0.390 7.53 0.630 5.98 41.1 2.00 17.5 1.66 3430 890 98.3 7.41 112 44.9 1.67 W18X50 14.7 18.0 0.355 7.50 0.570 6.57 45.2 1.98 17.4 1.24 3040 800 88.9 7.38 101 40.1 1.65 W18X46 13.5 18.1 0.360 6.06 0.605 5.01 44.6 1.58 17.5 1.22 1720 712 78.8 7.25 90.7 22.5 1.29 W18X40 11.8 17.9 0.315 6.02 0.525 5.73 50.9 1.56 17.4 0.810 1440 612 68.4 7.21 78.4 19.1 1.27 W16X67 19.7 16.3 0.395 10.2 0.67 7.70 35.9 2.82 15.7 2.39 7300 954 117 6.96 130 119 2.46 W16X57 16.8 16.4 0.430 7.12 0.715 4.98 33.0 1.92 15.7 2.22 2660 758 92.2 6.72 105 43.1 1.60 W16X50 14.7 16.3 0.380 7.07 0.630 5.61 37.4 1.89 15.6 1.52 2270 659 81.0 6.68 92.0 37.2 1.59 W16X45 13.3 16.1 0.345 7.04 0.565 6.23 41.1 1.88 15.6 1.11 1990 586 72.7 6.65 82.3 32.8 1.57 W16X40 11.8 16.0 0.305 7.00 0.505 6.93 46.5 1.86 15.5 0.794 1730 518 64.7 6.63 73.0 28.9 1.57 W16X36 10.6 15.9 0.295 6.99 0.430 8.12 48.1 1.83 15.4 0.545 1460 448 56.5 6.51 64.0 24.5 1.52 W14X74 21.8 14.2 0.450 10.1 0.785 6.41 25.4 2.82 13.4 3.87 5990 795 112 6.04 126 134 2.48 W14X68 20.0 14.0 0.415 10.0 0.720 6.97 27.5 2.80 1303 3.01 5380 722 103 6.01 115 121 2.46 W14X61 17.9 13.9 0.375 9.99 0.645 7.75 30.4 2.78 13.2 2.19 4710 640 92.1 5.98 102 107 2.45 W14X53 15.6 13.9 0.370 8.06 0.660 6.11 30.9 2.22 13.3 1.94 2540 541 77.8 5.89 87.1 57.7 1.92 W14X48 14.1 13.8 0.340 8.03 0.595 6.75 33.6 2.20 13.2 1.45 2240 484 70.2 5.85 78.4 51.4 1.91 W12X79 23.2 12.4 0.470 12.1 0.735 8.22 20.7 3.43 11.6 3.84 7330 662 107 5.34 119 216 3.05 W12X72 21.1 12.3 0.430 12.0 0.670 8.99 22.6 3.40 11.6 2.93 6540 597 97.4 5.31 108 195 3.04 W12X65 19.1 12.1 0.390 12.0 0.605 9.92 24.9 3.38 11.5 2.18 5780 533 87.9 5.28 96.8 174 3.02 W12X58 17.0 12.2 0.360 10.0 0.640 7.82 27.0 2.82 11.6 2.10 3570 475 78.0 5.28 86.4 107 2.51 W12X53 15.6 12.1 0.345 9.99 0.575 8.69 28.1 2.79 11.5 1.58 3160 425 70.6 5.23 77.9 95.8 2.48 W12X50 14.6 12.2 0.370 8.08 0.640 6.31 26.8 2.25 11.6 1.71 1880 391 64.2 5.18 71.9 56.3 1.96 W12X45 13.1 12.1 0.335 8.05 0.575 7.00 29.6 2.23 11.5 1.26 1650 348 57.7 5.15 64.2 50.0 1.95 W12X40 11.7 11.9 0.295 8.01 0.515 7.77 33.6 2.21 11.4 0.906 1440 307 51.5 5.13 57.0 44.1 1.94 W10x60 17.6 10.2 0.420 10.1 0.680 7.41 18.7 2.88 9.54 2.48 2640 341 66.7 4.39 74.6 116 2.57 W10x54 15.8 10.1 0.370 10.0 0.615 8.15 21.2 2.86 9.48 1.82 2320 303 60.0 4.37 66.6 103 2.56
Table 1-1: W-Shapes Dimensions
and Properties
bf
d
tf Y
tw
X X
W10x49
14.4 10.0 0.340 10.0 0.560 8.93 23.1 2.84 9.42 1.39 2070 272
54.6 4.35 60.4 93.4 2.54
W10x45
13.3 10.1 0.350 8.02 0.620 6.47 22.5 2.27 9.48 1.51 1200 248 49.1 4.32 54.9 53.4 2.01
W10x39
11.5
9.92 0.315 7.99 0.530 7.53 25.0 2.24 9.39 0.976 992 209 42.1 4.27 46.8 45.0 1.98
Adapted from Steel Construction Manual, 13th ed., AISC, 2005.
