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V. Warped Surface Displacements

To find the displacements at the free end of a warped cantilever plate due to in-plane load and out of plane loads.

Reference

MacNeal, R.H. and Harder, R.C., A Proposed Standard Set of Problems to Test Finite Element Accuracy, Finite Element in Analysis and Design 1, 1985.

Problem

The finite element model is as shown below: Find the displacements at the tip in the direction of the loads. Loading is unit forces at the free end: in plane and out of plane.

E = 29,000.0 ksi.

L = 12.0 in.

B = 1.1 in.

t = 0.22 in.

Twist = 90º (root to tip)

Poisson’s ratio = 0.22

Finite element model of warped, cantilever plate

Comparison

Table 1. Comparison of results
Result Type Theory STAAD.Pro Difference Comments
δ due to in-plane load (in) 5.424(10)-3 5.590(10)-3 3.1% Instead of using triangular element, a more advanced element type could to be used. Also the mesh size could be reduced to get closer result in comparison the theoretical value.
δ due to out-of-plane load (in) 1.754(10)-3 1.950(10)-3 11.2%

STAAD Input

Tip: You can copy and paste this content directly into a .std file to run in STAAD.Pro.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\04 Plates Shells\Warped Surface Displacements.STD is typically installed with the program.

STAAD SPACE :A WARPED CANTILEVER PLATE
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
* REFERENCE: MACNEAL AND HARDER, A PROPOSED STANDARD SET OF PROBLEMS
*            TO TEST FINITE ELEMENT ACCURACY,
*            FINITE ELEMENT IN ANALYSIS AND DESIGN 1, NORTH HOLLAND
*            1985
INPUT WIDTH 72
UNIT INCHES POUND
JOINT COORDINATES
1 0 -0.55 0; 2 1 -0.545 -0.072; 3 2 -0.531 -0.142; 4 3 -0.508 -0.21;
5 4 -0.476 -0.275; 6 5 -0.436 -0.335; 7 6 -0.389 -0.389;
8 7 -0.335 -0.436; 9 8 -0.275 -0.476; 10 9 -0.21 -0.508;
11 10 -0.142 -0.531; 12 11 -0.072 -0.545; 13 12 0 -0.55; 14 0 0 0;
15 1 0 0; 16 2 0 0; 17 3 0 0; 18 4 0 0; 19 5 0 0; 20 6 0 0; 21 7 0 0;
22 8 0 0; 23 9 0 0; 24 10 0 0; 25 11 0 0; 26 12 0 0; 27 0 0.55 0;
28 1 0.545 0.072; 29 2 0.531 0.142; 30 3 0.508 0.21; 31 4 0.476 0.275;
32 5 0.436 0.335; 33 6 0.389 0.389; 34 7 0.335 0.436; 35 8 0.275 0.476;
36 9 0.21 0.508; 37 10 0.142 0.531; 38 11 0.072 0.545; 39 12 0 0.55;
ELEMENT INCIDENCES SHELL
1 1 2 15; 2 15 14 1; 3 14 15 28; 4 28 27 14; 5 2 3 16; 6 16 15 2;
7 15 16 29; 8 29 28 15; 9 3 4 17; 10 17 16 3; 11 16 17 30; 12 30 29 16;
13 4 5 18; 14 18 17 4; 15 17 18 31; 16 31 30 17; 17 5 6 19; 18 19 18 5;
19 18 19 32; 20 32 31 18; 21 6 7 20; 22 20 19 6; 23 19 20 33;
24 33 32 19; 25 7 8 21; 26 21 20 7; 27 20 21 34; 28 34 33 20; 29 8 9 22;
30 22 21 8; 31 21 22 35; 32 35 34 21; 33 9 10 23; 34 23 22 9;
35 22 23 36; 36 36 35 22; 37 36 37 24; 38 24 23 36; 39 23 24 11;
40 11 10 23; 41 37 38 25; 42 25 24 37; 43 24 25 12; 44 12 11 24;
45 38 39 26; 46 26 25 38; 47 25 26 13; 48 13 12 25;
ELEMENT PROPERTY
1 TO 48 THICKNESS 0.32
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 2.9e+07
POISSON 0.22
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 14 27 FIXED
LOAD 1 UNIT LOAD AT TIP, OUT OF PLANE
JOINT LOAD
13 39 FY 0.25
26 FY 0.5
LOAD 2 UNIT LOAD AT TIP, IN PLANE
JOINT LOAD
13 39 FZ 0.25
26 FZ 0.5
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 13 26 39
FINISH

STAAD Output

   JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = SPACE
   ------------------
 JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
     13    1   -0.00015   0.00202  -0.00195  -0.00000   0.00022   0.00036
           2    0.00035  -0.00195   0.00559   0.00000  -0.00060  -0.00028
     26    1   -0.00000   0.00202  -0.00195  -0.00000   0.00022   0.00036
           2    0.00000  -0.00195   0.00559   0.00000  -0.00060  -0.00028
     39    1    0.00015   0.00202  -0.00195  -0.00000   0.00022   0.00035
           2   -0.00034  -0.00195   0.00559   0.00000  -0.00060  -0.00028