# V. 2D Cantilever Beam End Deflection 1

To find the free end deflection due to a joint load on a Cantilever beam modeled using Plate/shell elements.

## Reference

Hand calculation.

## Problem

Using the finite element method calculate the deflection of the free end of the cantilever beam.

### Cantilever beam and section

 E = 4,278 ksi

 h = 10 in

 b = 5 in

 P = 2 kips

 L = 60 in

## Theoretical Solution

Moment of Inertia:

I = (5 in)(10 in)3/12 = 416.7 in4

Deflection at free end:

## Comparison

Table 1. Comparison of results
Deflection at node B (in) 0.0808 0.08245 2.1% The expression for deflection for beam is used to compare the results of a model with Plate Elements with FE formulation, hence the difference in results.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\04 Plates Shells\2D Cantilever Beam End Deflection 1.STD is typically installed with the program.

STAAD SPACE :A CANTILEVER BEAM
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
*REFERENCE : HAND CALCULATION
*
UNIT INCHES KIP
JOINT COORDINATES
1 0 0 0; 2 0 2 0; 3 0 4 0; 4 0 6 0; 5 0 8 0; 6 0 10 0; 7 2.5 0 0;
8 2.5 2 0; 9 2.5 4 0; 10 2.5 6 0; 11 2.5 8 0; 12 2.5 10 0; 13 5 0 0;
14 5 2 0; 15 5 4 0; 16 5 6 0; 17 5 8 0; 18 5 10 0; 19 7.5 0 0;
20 7.5 2 0; 21 7.5 4 0; 22 7.5 6 0; 23 7.5 8 0; 24 7.5 10 0; 25 10 0 0;
26 10 2 0; 27 10 4 0; 28 10 6 0; 29 10 8 0; 30 10 10 0; 31 12.5 0 0;
32 12.5 2 0; 33 12.5 4 0; 34 12.5 6 0; 35 12.5 8 0; 36 12.5 10 0;
37 15 0 0; 38 15 2 0; 39 15 4 0; 40 15 6 0; 41 15 8 0; 42 15 10 0;
43 17.5 0 0; 44 17.5 2 0; 45 17.5 4 0; 46 17.5 6 0; 47 17.5 8 0;
48 17.5 10 0; 49 20 0 0; 50 20 2 0; 51 20 4 0; 52 20 6 0; 53 20 8 0;
54 20 10 0; 55 22.5 0 0; 56 22.5 2 0; 57 22.5 4 0; 58 22.5 6 0;
59 22.5 8 0; 60 22.5 10 0; 61 25 0 0; 62 25 2 0; 63 25 4 0; 64 25 6 0;
65 25 8 0; 66 25 10 0; 67 27.5 0 0; 68 27.5 2 0; 69 27.5 4 0;
70 27.5 6 0; 71 27.5 8 0; 72 27.5 10 0; 73 30 0 0; 74 30 2 0; 75 30 4 0;
76 30 6 0; 77 30 8 0; 78 30 10 0; 79 32.5 0 0; 80 32.5 2 0; 81 32.5 4 0;
82 32.5 6 0; 83 32.5 8 0; 84 32.5 10 0; 85 35 0 0; 86 35 2 0; 87 35 4 0;
88 35 6 0; 89 35 8 0; 90 35 10 0; 91 37.5 0 0; 92 37.5 2 0; 93 37.5 4 0;
94 37.5 6 0; 95 37.5 8 0; 96 37.5 10 0; 97 40 0 0; 98 40 2 0; 99 40 4 0;
100 40 6 0; 101 40 8 0; 102 40 10 0; 103 42.5 0 0; 104 42.5 2 0;
105 42.5 4 0; 106 42.5 6 0; 107 42.5 8 0; 108 42.5 10 0; 109 45 0 0;
110 45 2 0; 111 45 4 0; 112 45 6 0; 113 45 8 0; 114 45 10 0;
115 47.5 0 0; 116 47.5 2 0; 117 47.5 4 0; 118 47.5 6 0; 119 47.5 8 0;
120 47.5 10 0; 121 50 0 0; 122 50 2 0; 123 50 4 0; 124 50 6 0;
125 50 8 0; 126 50 10 0; 127 52.5 0 0; 128 52.5 2 0; 129 52.5 4 0;
130 52.5 6 0; 131 52.5 8 0; 132 52.5 10 0; 133 55 0 0; 134 55 2 0;
135 55 4 0; 136 55 6 0; 137 55 8 0; 138 55 10 0; 139 57.5 0 0;
140 57.5 2 0; 141 57.5 4 0; 142 57.5 6 0; 143 57.5 8 0; 144 57.5 10 0;
145 60 0 0; 146 60 2 0; 147 60 4 0; 148 60 6 0; 149 60 8 0; 150 60 10 0;
ELEMENT INCIDENCES SHELL
1 7 1 2 8; 2 8 2 3 9; 3 9 3 4 10; 4 10 4 5 11; 5 11 5 6 12; 6 13 7 8 14;
7 14 8 9 15; 8 15 9 10 16; 9 16 10 11 17; 10 17 11 12 18;
11 19 13 14 20; 12 20 14 15 21; 13 21 15 16 22; 14 22 16 17 23;
15 23 17 18 24; 16 25 19 20 26; 17 26 20 21 27; 18 27 21 22 28;
19 28 22 23 29; 20 29 23 24 30; 21 31 25 26 32; 22 32 26 27 33;
23 33 27 28 34; 24 34 28 29 35; 25 35 29 30 36; 26 37 31 32 38;
27 38 32 33 39; 28 39 33 34 40; 29 40 34 35 41; 30 41 35 36 42;
31 43 37 38 44; 32 44 38 39 45; 33 45 39 40 46; 34 46 40 41 47;
35 47 41 42 48; 36 49 43 44 50; 37 50 44 45 51; 38 51 45 46 52;
39 52 46 47 53; 40 53 47 48 54; 41 55 49 50 56; 42 56 50 51 57;
43 57 51 52 58; 44 58 52 53 59; 45 59 53 54 60; 46 61 55 56 62;
47 62 56 57 63; 48 63 57 58 64; 49 64 58 59 65; 50 65 59 60 66;
51 67 61 62 68; 52 68 62 63 69; 53 69 63 64 70; 54 70 64 65 71;
55 71 65 66 72; 56 73 67 68 74; 57 74 68 69 75; 58 75 69 70 76;
59 76 70 71 77; 60 77 71 72 78; 61 79 73 74 80; 62 80 74 75 81;
63 81 75 76 82; 64 82 76 77 83; 65 83 77 78 84; 66 85 79 80 86;
67 86 80 81 87; 68 87 81 82 88; 69 88 82 83 89; 70 89 83 84 90;
71 91 85 86 92; 72 92 86 87 93; 73 93 87 88 94; 74 94 88 89 95;
75 95 89 90 96; 76 97 91 92 98; 77 98 92 93 99; 78 99 93 94 100;
79 100 94 95 101; 80 101 95 96 102; 81 103 97 98 104; 82 104 98 99 105;
83 105 99 100 106; 84 106 100 101 107; 85 107 101 102 108;
86 109 103 104 110; 87 110 104 105 111; 88 111 105 106 112;
89 112 106 107 113; 90 113 107 108 114; 91 115 109 110 116;
92 116 110 111 117; 93 117 111 112 118; 94 118 112 113 119;
95 119 113 114 120; 96 121 115 116 122; 97 122 116 117 123;
98 123 117 118 124; 99 124 118 119 125; 100 125 119 120 126;
101 127 121 122 128; 102 128 122 123 129; 103 129 123 124 130;
104 130 124 125 131; 105 131 125 126 132; 106 133 127 128 134;
107 134 128 129 135; 108 135 129 130 136; 109 136 130 131 137;
110 137 131 132 138; 111 139 133 134 140; 112 140 134 135 141;
113 141 135 136 142; 114 142 136 137 143; 115 143 137 138 144;
116 145 139 140 146; 117 146 140 141 147; 118 147 141 142 148;
119 148 142 143 149; 120 149 143 144 150;
ELEMENT PROPERTY
1 TO 120 THICKNESS 5
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 4278
POISSON 0.1889
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 2 5 6 FIXED BUT FY
3 4 FIXED
145 TO 150 FY -0.333333
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 145 TO 150
FINISH


  JOINT    DISPLACE LIST     145
:A CANTILEVER BEAM                                       -- PAGE NO.    4
JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = SPACE
------------------
JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
145    1   -0.01016  -0.08254   0.00000   0.00000   0.00000  -0.00189
146    1   -0.00605  -0.08248   0.00000   0.00000   0.00000  -0.00177
147    1   -0.00201  -0.08245   0.00000   0.00000   0.00000  -0.00175
148    1    0.00201  -0.08245   0.00000   0.00000   0.00000  -0.00175
149    1    0.00605  -0.08248   0.00000   0.00000   0.00000  -0.00177
150    1    0.01016  -0.08254   0.00000   0.00000   0.00000  -0.00189
************** END OF LATEST ANALYSIS RESULT **************