beam_can To determine deflection of a cantilev By Alex Slocum, 1/1/04, last mo
Enters numbers in BO Schematic
L af aw am
wa
M x A
Beam dimensions and properties Length, L (mm) Width, W (mm) Height, H (mm) Length increment, Linc (mm) Modulus of elasticity, E (N/mm^2) Moment of inertia, I (mm^4) Distance from farthest fiber to neutral axis, cc (mm) Loading Point load, F (N) Location of point load, af (mm) Distributed load amplitude, wa, (N/mm) Distributed load amplitude, wL, (N/mm) Starting point of distributed load, aw (mm) Moment load, M (N-mm) Location of moment load, am (mm) Maximum deflection (microns) Maximum slope (milli radians) Reactions at beam ends R A, Ra (N)
Values 100 25 6 1 200000 450 3 10 0 1 1 50 10 25 -56.771 0.779 0.000
R B, Rb (N)
60.000
MA, Ma (N)
0.000
MB, Mb (N-mm)
-2240.000
θA, ta (radians)
0.001
θB, tb (radians)
0.000
δA, da (mm)
-0.057
δB, db (mm)
0.000
Equations
−
−
V = −F x −a f
−w a x −aw
M A x R A x + − θ = θ A + EI 2 EI
2
( L − aw ) 2
w a x − aw
M = M A + R Ax −F x −a f 2
a
− −
(
−
2 2
F x − a f
−
EI
2
wa x − a EI
6 3
F x − a f M A x RAx + − δ = δ A + θ A x + 2 EI 6 EI 6 EI 2
R B =
θ A
δ A
=
( w L + wa ) ( L −aw )
+F
F ( L − a f )
(
2
EI
3
+
( L − aw ) 6
wa EI
− F 2 L −3a f L +a f 3
2
6
EI
3
)
−
= −( F L
M B
2
2
=
3
( w L −wa )
+
4 3
( L − aw )
−
24
wa ( 3 EI
Note: For other types of distributed loads, use principl e of superposition
ilever.xls er beam under superimposed loads dified 06/11/04 by Xue'en Yang LD, Results in RED
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
2
w
−wa ) x −aw
L
6
( w L −wa ) x −aw − 24 EI( L− aw )
w
4
x− aw
( L −
−
a− w 2
M (L − aw
4
M x − am
+
( w L −wa ) x − aw − 120 EI( L− a w)
24 EI
a− f )
0
M+ x a m−
( L − aw )
3
a
3
2
)
wa
EI 5
M x − am
+
2
EI
wa − )
( w L
+
M +
3
)
EI a+w )
( w L −wa ) ( 4 L a+w ) +
5
(
2 M L − am
+
2 EI
2
)
2
Deflection 0.0
0.900 0.800
-10.0
0.700
)
m( n oi t c el f e D
s n or c
)
s n ai d ra
-20.0
il
-30.0
li
m( e p ol S
-40.0
0.600 0.500 0.400 0.300 0.200
-50.0
0.100 0.000
-60.0 0
10
20
30
40
50
60
70
80
90
100
Distance from left end of beam (mm)
Moment 0.0
0
-2.0 -500 -4.0
) mN(
m
-6.0
-1000 )
a P( s s er
t
n e m o M
-1500 t
S
-8.0 -10.0 -12.0
-2000 -14.0 -2500
-16.0 0
10
20
30
40
50
60
70
Distance from left end of beam (mm)
Transverse Shear 0
-10
-20
)
(N
80
90
100
r S e sr e v s n ar T
a e h
-30
-40
-50
-60
-70
0
10
20
30
40
50
60
70
Distance from left end of beam (mm)
80
90
100
Slope
0
10
20
30
40
50
60
70
80
90
100
70
80
90
100
Distance from left end of beam (mm)
Stress
10
20
30
40
50
60
Distance from left end of beam (mm)
Distance along beam, x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
Shear (N) 0 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10
Moment (N-mm) 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -120 -130 -140 -150 -160 -170 -180 -190 -200 -210 -220 -230 -240 -250 -250 -260 -270 -280 -290 -300 -310 -320 -330 -340 -350 -360 -370 -380 -390 -400 -410 -420 -430 -440 -450 -460 -470 -480
Stress (Pa) 0.0 -0.1 -0.1 -0.2 -0.3 -0.3 -0.4 -0.5 -0.5 -0.6 -0.7 -0.7 -0.8 -0.9 -0.9 -1.0 -1.1 -1.1 -1.2 -1.3 -1.3 -1.4 -1.5 -1.5 -1.6 -1.7 -1.7 -1.7 -1.8 -1.9 -1.9 -2.0 -2.1 -2.1 -2.2 -2.3 -2.3 -2.4 -2.5 -2.5 -2.6 -2.7 -2.7 -2.8 -2.9 -2.9 -3.0 -3.1 -3.1 -3.2
Slope (mrad) 0.779 0.779 0.778 0.778 0.778 0.777 0.777 0.776 0.775 0.774 0.773 0.772 0.771 0.769 0.768 0.766 0.764 0.763 0.761 0.759 0.756 0.754 0.752 0.749 0.747 0.744 0.741 0.738 0.735 0.732 0.729 0.726 0.723 0.719 0.715 0.712 0.708 0.704 0.700 0.696 0.691 0.687 0.683 0.678 0.673 0.668 0.663 0.658 0.653 0.648
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
-10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 -24 -25 -26 -27 -28 -29 -30 -31 -32 -33 -34 -35 -36 -37 -38 -39 -40 -41 -42 -43 -44 -45 -46 -47 -48 -49 -50 -51 -52 -53 -54 -55 -56 -57 -58 -59 -60
-490 -501 -512 -525 -538 -553 -568 -585 -602 -621 -640 -661 -682 -705 -728 -753 -778 -805 -832 -861 -890 -921 -952 -985 -1018 -1053 -1088 -1125 -1162 -1201 -1240 -1281 -1322 -1365 -1408 -1453 -1498 -1545 -1592 -1641 -1690 -1741 -1792 -1845 -1898 -1953 -2008 -2065 -2122 -2181 -2240
-3.3 -3.3 -3.4 -3.5 -3.6 -3.7 -3.8 -3.9 -4.0 -4.1 -4.3 -4.4 -4.5 -4.7 -4.9 -5.0 -5.2 -5.4 -5.5 -5.7 -5.9 -6.1 -6.3 -6.6 -6.8 -7.0 -7.3 -7.5 -7.7 -8.0 -8.3 -8.5 -8.8 -9.1 -9.4 -9.7 -10.0 -10.3 -10.6 -10.9 -11.3 -11.6 -11.9 -12.3 -12.7 -13.0 -13.4 -13.8 -14.1 -14.5 -14.9
0.643 0.637 0.631 0.626 0.620 0.614 0.608 0.601 0.595 0.588 0.581 0.574 0.566 0.558 0.550 0.542 0.534 0.525 0.516 0.506 0.497 0.487 0.476 0.465 0.454 0.443 0.431 0.419 0.406 0.393 0.379 0.365 0.351 0.336 0.320 0.305 0.288 0.271 0.254 0.236 0.217 0.198 0.179 0.159 0.138 0.116 0.094 0.072 0.048 0.025 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 Length, L (mm) Width, W (mm) Height, H (mm) Length increment, Linc (mm) Modulus of elasticity, E (N/mm^2) Moment of inertia, I (mm^4) Distance from farthest fiber to neutral axis, cc (mm) Loading Condition Point load, F (N) Location of point load, af (mm) Distributed load amplitude, wa, (N/mm) Distributed load amplitude, wL, (N/mm) Starting point of distributed load, aw (mm) Moment load, M (N-mm) Location of moment load, am (mm) Maximum deflection (microns) Maximum slope (milli radians) Reactions at beam ends R A, Ra (N)
Values 100 25 6 1 200000 450 3
ProE Mechanica 100 25 6 1 200000 450 3
10 0 1 1 50 10 25 -56.771 0.779
10 0 1 1 50 10 25 -56.02
0.000
0
R B, Rb (N)
60.000
60
MA, Ma (N)
0.000
0
-2240.000
-2238
MB, Mb (N-mm) θA, ta (radians)
0.001
θB, tb (radians)
0.000
0
δA, da (mm)
-0.057
-0.06
δB, db (mm)
0.000
0
