beam_cantilever.xls
To determine deflection of a cantilever beam under superimposed loads
By Alex Slocum, 1/1/04, last modified 06/11/04 by Xue'en Yang
Enters numbers in BOLD, Results in RED
Schematic
L
af F
wL
aw
am wa
M
x
A B
Beam dimensions and properties Values
Length, L (mm) 100
Width, W (mm) 25
Height, H (mm) 6
Length increment, Linc (mm) 1
Modulus of elasticity, E (N/mm^2) 200000
Moment of inertia, I (mm^4) 450
Distance from farthest fiber to neutral axis, cc (mm) 3
Loading
Point load, F (N) 10
Location of point load, af (mm) 0
Distributed load amplitude, wa, (N/mm) 1
Distributed load amplitude, wL, (N/mm) 1
Starting point of distributed load, aw (mm) 50
Moment load, M (N-mm) 10
Location of moment load, am (mm) 25
Maximum deflection (microns) -56.771
Maximum slope (milli radians) 0.779
Reactions at beam ends
RA, Ra (N) 0.000
RB, Rb (N) 60.000
MA, Ma (N) 0.000
MB, Mb (N-mm) -2240.000
qA, ta (radians) 0.001
qB, tb (radians) 0.000
dA, da (mm) -0.057
dB, db (mm) 0.000
Equations
0
V F x a f w a x aw
w L w a x aw 2
2 L aw
M M A R Ax F x a f
w a x aw
2
w L w a x aw 3
M x am
0
2 6 L aw
0
V F x a f w a x aw
w L w a x aw 2
2 L aw
M M A R Ax F x a f
w a x aw
2
w L w a x aw 3 M x a 0
2 6 L aw m
w L w a x aw 4 M
2
2
M Ax R A x F x af w a x aw
3
q q A
EI 2 EI 2 EI 6 EI 24 EI L aw
w L w a x aw
3
2 3 F x af w a x aw
4
d d A q Ax M A x R A x
2 EI 6 EI 6 EI 24 EI 120 EI L aw
w L w a L aw L aw 2 wL w
RB F
2
M B F L a f
2
wa
3
L a w L w a M L aw
2
F L af 3
qA wa
w
2 EI 6 EI
4
EI
dA
F 2 L 3 3a f L 2 a f
3
L a w 3L a w w 4L a
w
3
a
L a w
w
6 EI 24 EI
5
Note: For other types of distributed loads, use principle of superposition
=
+
beam_cantilever.xls
n of a cantilever beam under superimposed loads
/1/04, last modified 06/11/04 by Xue'en Yang
umbers in BOLD, Results in RED
L
F
wL
B
Instructions
Enter total length of beam in mm
Enter width of beam in mm
Enter height of beam in mm
Enter length increment to be used in finite difference calculation
Enter elastic modulus in N/mm^2
=1/12*W*H^3
= H/2
See schematic for definitions of loads and positions
Enter amplitude of the point load, in N
Enter location of the point load, in mm
Enter amplitude of the distributed load near the cantilever end, in N/mm
Enter amplitude of the distributed load at the clamped end, in N/mm
Enter location for wa, in mm
Enter amplitude of the applied moment, in N*mm
Enter location of the moment load, in mm
Return maximum deflection along the beam in microns
Return maximum slope along the beam in milli radians
Reaction force at A
Reaction force at B
Reaction moment at A
Reaction moment at B
Rotation at A
Rotation at B
Deflection at A
Deflection at B
w a x aw
2
L aw
aw
2
w L w a x aw 3
M x am
0
6 L aw
w a x aw
2
L aw
aw
2
w L w a x aw 3 M x a 0
6 L aw m
wa x aw
3
w L w a x a w 4 M x am
6 EI 24 EI L aw EI
w L w a x aw 5 M
3
w a x aw x am
4 2
af
I 24 EI 120 EI L aw 2 EI
L aw 2 wL wa M
F L a f
2
wa
3
w a M L aw
4
EI
3
w a 3 L aw
wL w a 4 L aw M L am
2
2
5
2 EI
=
+
Deflection
10.0
0.0
0 10 20 30 40 50 60 70 80 90 100
-10.0
Deflection (mcrons)
-20.0
-30.0
-40.0
-50.0
-60.0
Distance from left end of beam (mm)
Moment
0
0 10 20 30 40 50 60 70 80 90 100
-500
Moment (N-mm)
-1000
-1500
-2000
-2500
Distance from left end of beam (mm)
Transverse Shear
0
0 10 20 30 40 50 60 70 80 90 100
-10
Transverse Shear (N)
-20
-30
Transverse Shear (N)
-30
-40
-50
-60
-70
Distance from left end of beam (mm)
Slope
0.900
0.800
0.700
Slope (milli radians)
0.600
0.500
0.400
0.300
0.200
0.100
0.000
0 10 20 30 40 50 60 70 80 90 100
Distance from left end of beam (mm)
Stress
0.0
0 10 20 30 40 50 60 70 80 90 100
-2.0
-4.0
-6.0
Stress (Pa)
-8.0
-10.0
-12.0
-14.0
-16.0
Distance from left end of beam (mm)
Distance along beam, x Shear (N) Moment (N-mm) Stress (Pa) Slope (mrad)
0 0 0 0.0 0.779
1 -10 -10 -0.1 0.779
2 -10 -20 -0.1 0.778
3 -10 -30 -0.2 0.778
4 -10 -40 -0.3 0.778
5 -10 -50 -0.3 0.777
6 -10 -60 -0.4 0.777
7 -10 -70 -0.5 0.776
8 -10 -80 -0.5 0.775
9 -10 -90 -0.6 0.774
10 -10 -100 -0.7 0.773
11 -10 -110 -0.7 0.772
12 -10 -120 -0.8 0.771
13 -10 -130 -0.9 0.769
14 -10 -140 -0.9 0.768
15 -10 -150 -1.0 0.766
16 -10 -160 -1.1 0.764
17 -10 -170 -1.1 0.763
18 -10 -180 -1.2 0.761
19 -10 -190 -1.3 0.759
20 -10 -200 -1.3 0.756
21 -10 -210 -1.4 0.754
22 -10 -220 -1.5 0.752
23 -10 -230 -1.5 0.749
24 -10 -240 -1.6 0.747
25 -10 -250 -1.7 0.744
26 -10 -250 -1.7 0.741
27 -10 -260 -1.7 0.738
28 -10 -270 -1.8 0.735
29 -10 -280 -1.9 0.732
30 -10 -290 -1.9 0.729
31 -10 -300 -2.0 0.726
32 -10 -310 -2.1 0.723
33 -10 -320 -2.1 0.719
34 -10 -330 -2.2 0.715
35 -10 -340 -2.3 0.712
36 -10 -350 -2.3 0.708
37 -10 -360 -2.4 0.704
38 -10 -370 -2.5 0.700
39 -10 -380 -2.5 0.696
40 -10 -390 -2.6 0.691
41 -10 -400 -2.7 0.687
42 -10 -410 -2.7 0.683
43 -10 -420 -2.8 0.678
44 -10 -430 -2.9 0.673
45 -10 -440 -2.9 0.668
46 -10 -450 -3.0 0.663
47 -10 -460 -3.1 0.658
48 -10 -470 -3.1 0.653
49 -10 -480 -3.2 0.648
50 -10 -490 -3.3 0.643
51 -11 -501 -3.3 0.637
52 -12 -512 -3.4 0.631
53 -13 -525 -3.5 0.626
54 -14 -538 -3.6 0.620
55 -15 -553 -3.7 0.614
56 -16 -568 -3.8 0.608
57 -17 -585 -3.9 0.601
58 -18 -602 -4.0 0.595
59 -19 -621 -4.1 0.588
60 -20 -640 -4.3 0.581
61 -21 -661 -4.4 0.574
62 -22 -682 -4.5 0.566
63 -23 -705 -4.7 0.558
64 -24 -728 -4.9 0.550
65 -25 -753 -5.0 0.542
66 -26 -778 -5.2 0.534
67 -27 -805 -5.4 0.525
68 -28 -832 -5.5 0.516
69 -29 -861 -5.7 0.506
70 -30 -890 -5.9 0.497
71 -31 -921 -6.1 0.487
72 -32 -952 -6.3 0.476
73 -33 -985 -6.6 0.465
74 -34 -1018 -6.8 0.454
75 -35 -1053 -7.0 0.443
76 -36 -1088 -7.3 0.431
77 -37 -1125 -7.5 0.419
78 -38 -1162 -7.7 0.406
79 -39 -1201 -8.0 0.393
80 -40 -1240 -8.3 0.379
81 -41 -1281 -8.5 0.365
82 -42 -1322 -8.8 0.351
83 -43 -1365 -9.1 0.336
84 -44 -1408 -9.4 0.320
85 -45 -1453 -9.7 0.305
86 -46 -1498 -10.0 0.288
87 -47 -1545 -10.3 0.271
88 -48 -1592 -10.6 0.254
89 -49 -1641 -10.9 0.236
90 -50 -1690 -11.3 0.217
91 -51 -1741 -11.6 0.198
92 -52 -1792 -11.9 0.179
93 -53 -1845 -12.3 0.159
94 -54 -1898 -12.7 0.138
95 -55 -1953 -13.0 0.116
96 -56 -2008 -13.4 0.094
97 -57 -2065 -13.8 0.072
98 -58 -2122 -14.1 0.048
99 -59 -2181 -14.5 0.025
100 -60 -2240 -14.9 0.000
Deflection (microns)
-56.8
-56.0
-55.2
-54.4
-53.7
-52.9
-52.1
-51.3
-50.6
-49.8
-49.0
-48.2
-47.5
-46.7
-45.9
-45.2
-44.4
-43.6
-42.9
-42.1
-41.3
-40.6
-39.8
-39.1
-38.3
-37.6
-36.8
-36.1
-35.4
-34.6
-33.9
-33.2
-32.5
-31.7
-31.0
-30.3
-29.6
-28.9
-28.2
-27.5
-26.8
-26.1
-25.4
-24.7
-24.1
-23.4
-22.7
-22.1
-21.4
-20.8
-20.1
-19.5
-18.8
-18.2
-17.6
-17.0
-16.4
-15.8
-15.2
-14.6
-14.0
-13.4
-12.8
-12.3
-11.7
-11.2
-10.6
-10.1
-9.6
-9.1
-8.6
-8.1
-7.6
-7.1
-6.7
-6.2
-5.8
-5.4
-4.9
-4.5
-4.2
-3.8
-3.4
-3.1
-2.8
-2.4
-2.2
-1.9
-1.6
-1.4
-1.1
-0.9
-0.7
-0.6
-0.4
-0.3
-0.2
-0.1
0.0
0.0
0.0
Beam dimensions and properties Values ProE Mechanica
Length, L (mm) 100 100
Width, W (mm) 25 25
Height, H (mm) 6 6
Length increment, Linc (mm) 1 1
Modulus of elasticity, E (N/mm^2) 200000 200000
Moment of inertia, I (mm^4) 450 450
Distance from farthest fiber to neutral axis, cc (mm) 3 3
Loading Condition
Point load, F (N) 10 10
Location of point load, af (mm) 0 0
Distributed load amplitude, wa, (N/mm) 1 1
Distributed load amplitude, wL, (N/mm) 1 1
Starting point of distributed load, aw (mm) 50 50
Moment load, M (N-mm) 10 10
Location of moment load, am (mm) 25 25
Maximum deflection (microns) -56.771 -56.017
Maximum slope (milli radians) 0.779
Reactions at beam ends
RA, Ra (N) 0.000 0
RB, Rb (N) 60.000 60
MA, Ma (N) 0.000 0
MB, Mb (N-mm) -2240.000 -2238
qA, ta (radians) 0.001
qB, tb (radians) 0.000 0
dA, da (mm) -0.057 -0.056
dB, db (mm) 0.000 0