A simply-supported equilateral triangle is subjected to a linear thermal gradient. The deflections and bending moments are calculated.
Reference
Timoshenko, S. and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw Hill, 1959, pp. 92-97.
Problem
The purpose of this example is to demonstrate the ability of STAAD to calculate the response of a structure when thermally-loaded plate elements are utilized. The STAAD.Pro model considers one-half of the plate, modeled using 49 quad elements and seven tri plate elements. For specifying inclined supports, an auxiliary coordinate system is defined along the edge (see second figure below) not parallel to a global axis.
ΔT = Ttop – Tbottom = 4500 | |
Simply-supported, equilateral
triangle with thermal load
Finite element model of a
simply-supported, equilateral triangle
Theoretical Solution
From page 96 of the reference:
The origin of the x' axis in the reference is located at a/3, or 1 inch from the edge. Therefore:
x' = x-1
Upon substitution:
w = 0.00585 (x3 – 6x2 + 9x – 3y2x) | |
Comparison
Table 1. Comparison of results
Result Type |
Theory |
STAAD.Pro
|
Difference |
Z deflection at center of element along x-axis (in) |
Node 6 |
0.02331 |
0.02331 |
none |
Maximum (magnitude) bending moment along x-axis (in·lb/in) |
Mx at Element 8 |
-36.16 |
-36.15 |
none |
Mx at Element 14 |
-64.29 |
-65.64 |
2.1% |
My at Element 1 |
-42.59 |
-42.59 |
none |
My at Element 8 |
-8.839 |
-8.838 |
none |
Twisting moment along y-axis at center of elements (in·lb/in) |
Mxy at Element 55 |
36.19 |
36.06 |
none |
The results show excellent agreement for displacements and for quad plate element stresses except in the few instances where the magnitude of the answers is very small. The tri plate elements are lower order elements, and it is therefore not unusual for these elements to show poorer agreement than quads where mesh size is smaller.
STAAD Output
JOINT DISPLACE LIST 2
:SIMPLY SUPPORTED PLATE THERMAL -- PAGE NO. 5
JOINT DISPLACEMENT (INCH RADIANS) STRUCTURE TYPE = SPACE
------------------
JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN
2 1 0.00000 0.00000 0.00973 0.00000 -0.03853 0.00000
3 1 0.00000 0.00000 0.01658 0.00000 -0.02590 0.00000
4 1 0.00000 0.00000 0.02090 0.00000 -0.01489 0.00000
5 1 0.00000 0.00000 0.02303 0.00000 -0.00549 0.00000
6 1 0.00000 0.00000 0.02331 0.00000 0.00230 0.00000
7 1 0.00000 0.00000 0.02211 0.00000 0.00848 0.00000
8 1 0.00000 0.00000 0.01975 0.00000 0.01305 0.00000
9 1 0.00000 0.00000 0.01658 0.00000 0.01600 0.00000
10 1 0.00000 0.00000 0.01295 0.00000 0.01735 0.00000
11 1 0.00000 0.00000 0.00921 0.00000 0.01709 0.00000
12 1 0.00000 0.00000 0.00570 0.00000 0.01518 0.00000
13 1 0.00000 0.00000 0.00276 0.00000 0.01165 0.00000
14 1 0.00000 0.00000 0.00075 0.00000 0.00672 0.00000
************** END OF LATEST ANALYSIS RESULT **************
77. PRINT ELEMENT JOINT STRESSES LIST 1 8 14 55
ELEMENT JOINT STRESSES LIST
:SIMPLY SUPPORTED PLATE THERMAL -- PAGE NO. 6
ELEMENT STRESSES FORCE,LENGTH UNITS= POUN INCH
----------------
STRESS = FORCE/UNIT WIDTH/THICK, MOMENT = FORCE-LENGTH/UNIT WIDTH
ELEMENT LOAD SQX SQY MX MY MXY
VONT VONB SX SY SXY
TRESCAT TRESCAB
1 1 -0.01 -0.01 -2.41 -42.59 2.78
25029.83 25029.83 0.00 0.00 0.00
25668.88 25668.88
TOP : SMAX= -1331.23 SMIN= -25668.88 TMAX= 12168.82 ANGLE= 3.9
BOTT: SMAX= 25668.88 SMIN= 1331.23 TMAX= 12168.82 ANGLE=-86.1
JOINT -0.00 -0.01 -1.67 -45.50 1.93
1 26888.39 26888.39 0.00 0.00 0.00
TOP : SMAX= -951.54 SMIN= -27351.53 TMAX= 13200.00 ANGLE= 2.5
BOTT: SMAX= 27351.53 SMIN= 951.54 TMAX= 13200.00 ANGLE=-87.5
JOINT -0.00 -0.00 -3.15 -39.68 1.93
2 23007.69 23007.69 0.00 0.00 0.00
TOP : SMAX= -1829.24 SMIN= -23867.71 TMAX= 11019.23 ANGLE= 3.0
BOTT: SMAX= 23867.71 SMIN= 1829.24 TMAX= 11019.23 ANGLE=-87.0
JOINT -0.02 -0.00 -3.15 -39.68 3.64
18 23229.73 23229.73 0.00 0.00 0.00
TOP : SMAX= -1675.24 SMIN= -24022.00 TMAX= 11173.38 ANGLE= 5.6
BOTT: SMAX= 24022.00 SMIN= 1675.24 TMAX= 11173.38 ANGLE=-84.4
JOINT -0.02 -0.01 -1.67 -45.50 3.64
17 27078.78 27078.78 0.00 0.00 0.00
TOP : SMAX= -822.50 SMIN= -27480.66 TMAX= 13329.08 ANGLE= 4.7
BOTT: SMAX= 27480.66 SMIN= 822.50 TMAX= 13329.08 ANGLE=-85.3
8 1 2.23 1.01 -36.15 -8.84 2.79
19799.74 19799.74 0.00 0.00 0.00
21861.08 21861.08
TOP : SMAX= -5134.01 SMIN= -21861.08 TMAX= 8363.53 ANGLE= 84.2
BOTT: SMAX= 21861.08 SMIN= 5134.01 TMAX= 8363.53 ANGLE= -5.8
JOINT -4.50 -1.02 -35.37 -11.73 1.92
8 18829.74 18829.74 0.00 0.00 0.00
TOP : SMAX= -6945.23 SMIN= -21315.87 TMAX= 7185.32 ANGLE= 85.4
BOTT: SMAX= 21315.87 SMIN= 6945.23 TMAX= 7185.32 ANGLE= -4.6
JOINT -4.50 3.05 -36.95 -5.96 1.93
9 20715.56 20715.56 0.00 0.00 0.00
TOP : SMAX= -3504.10 SMIN= -22244.13 TMAX= 9370.01 ANGLE= 86.4
BOTT: SMAX= 22244.13 SMIN= 3504.10 TMAX= 9370.01 ANGLE= -3.6
JOINT 8.97 3.05 -36.79 -5.89 3.65
25 20883.86 20883.86 0.00 0.00 0.00
TOP : SMAX= -3281.80 SMIN= -22330.46 TMAX= 9524.33 ANGLE= 83.4
BOTT: SMAX= 22330.46 SMIN= 3281.80 TMAX= 9524.33 ANGLE= -6.6
JOINT 8.97 -1.02 -35.50 -11.77 3.64
24 19168.57 19168.57 0.00 0.00 0.00
TOP : SMAX= -6732.25 SMIN= -21626.52 TMAX= 7447.13 ANGLE= 81.5
BOTT: SMAX= 21626.52 SMIN= 6732.25 TMAX= 7447.13 ANGLE= -8.5
14 1 -57.76 58.83 -65.64 18.37 1.02
45910.85 45910.85 0.00 0.00 0.00
50421.84 50421.84
TOP : SMAX= 11032.26 SMIN= -39389.58 TMAX= 25210.92 ANGLE= 89.3
BOTT: SMAX= 39389.58 SMIN= -11032.26 TMAX= 25210.92 ANGLE= -0.7
:SIMPLY SUPPORTED PLATE THERMAL -- PAGE NO. 7
ELEMENT STRESSES FORCE,LENGTH UNITS= POUN INCH
----------------
STRESS = FORCE/UNIT WIDTH/THICK, MOMENT = FORCE-LENGTH/UNIT WIDTH
ELEMENT LOAD SQX SQY MX MY MXY
VONT VONB SX SY SXY
TRESCAT TRESCAB
JOINT -57.76 58.83 -65.22 18.13 1.02
14 45571.45 45571.45 0.00 0.00 0.00
TOP : SMAX= 10886.75 SMIN= -39142.11 TMAX= 25014.43 ANGLE= 89.3
BOTT: SMAX= 39142.11 SMIN= -10886.75 TMAX= 25014.43 ANGLE= -0.7
JOINT -57.76 58.83 -66.46 18.13 1.02
15 46298.03 46298.03 0.00 0.00 0.00
TOP : SMAX= 10886.64 SMIN= -39884.58 TMAX= 25385.61 ANGLE= 89.3
BOTT: SMAX= 39884.58 SMIN= -10886.64 TMAX= 25385.61 ANGLE= -0.7
JOINT -57.76 58.83 -65.22 18.86 1.02
16 45864.36 45864.36 0.00 0.00 0.00
TOP : SMAX= 11323.39 SMIN= -39142.05 TMAX= 25232.72 ANGLE= 89.3
BOTT: SMAX= 39142.05 SMIN= -11323.39 TMAX= 25232.72 ANGLE= -0.7
55 1 63.71 -48.81 -2.49 -42.60 36.06
44968.46 44968.46 0.00 0.00 0.00
49519.82 49519.82
TOP : SMAX= 11232.03 SMIN= -38287.79 TMAX= 24759.91 ANGLE= 30.5
BOTT: SMAX= 38287.79 SMIN= -11232.03 TMAX= 24759.91 ANGLE=-59.5
JOINT -130.44 -11.74 -0.31 -45.48 35.28
67 45646.34 45646.34 0.00 0.00 0.00
TOP : SMAX= 11394.66 SMIN= -38869.58 TMAX= 25132.12 ANGLE= 28.7
BOTT: SMAX= 38869.58 SMIN= -11394.66 TMAX= 25132.12 ANGLE=-61.3
JOINT -17.92 11.58 -3.49 -38.96 35.48
68 43142.03 43142.03 0.00 0.00 0.00
TOP : SMAX= 11063.97 SMIN= -36532.57 TMAX= 23798.27 ANGLE= 31.7
BOTT: SMAX= 36532.57 SMIN= -11063.97 TMAX= 23798.27 ANGLE=-58.3
JOINT 165.93 -53.39 -2.02 -39.46 36.61
70 44500.96 44500.96 0.00 0.00 0.00
TOP : SMAX= 12222.83 SMIN= -37112.28 TMAX= 24667.55 ANGLE= 31.5
BOTT: SMAX= 37112.28 SMIN= -12222.83 TMAX= 24667.55 ANGLE=-58.5
JOINT 237.25 -141.67 -5.26 -48.10 36.69
71 46961.48 46961.48 0.00 0.00 0.00
TOP : SMAX= 9479.68 SMIN= -41498.48 TMAX= 25489.08 ANGLE= 29.9
BOTT: SMAX= 41498.48 SMIN= -9479.68 TMAX= 25489.08 ANGLE=-60.1
**** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES ****
MAXIMUM MINIMUM MAXIMUM MAXIMUM MAXIMUM
PRINCIPAL PRINCIPAL SHEAR VONMISES TRESCA
STRESS STRESS STRESS STRESS STRESS
4.149848E+04 -4.149848E+04 2.548908E+04 4.591085E+04 5.042184E+04
PLATE NO. 55 55 55 14 14
CASE NO. 1 1 1 1 1
:SIMPLY SUPPORTED PLATE THERMAL -- PAGE NO. 8