V. 2D Triangular Surface with Thermal Load
A simply-supported equilateral triangle is subjected to a linear thermal gradient. The deflections and bending moments are calculated.
References
Timoshenko, S. and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw Hill, 1959, pp. 92-97.
Details
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.
E = 10×(10)6
lb/in2
υ = 0.3
α = 12 (10)-6 in/in
°F
ΔT = Ttop –
Tbottom = 450.0
t = 0.1 inches
a = 3.0 inches
Calculations
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:
Mx = -22.5xMy = -22.5 (2 - x)Mxy = 22.5yw = 0.00585 (x3 – 6x2 + 9x – 3y2x)
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.
Input
Tip: 您可以直接将此内容复制并粘贴到 .std 文件中，以在 STAAD.Pro 中运行。
The file C:\Users\Public\Public Documents\STAAD.Pro 2026\Samples \Verification Models\04 Plates Shells\2D Triangular Surface with Thermal Load.STD is typically installed with the program.
STAAD
STAAD SPACE :SIMPLY SUPPORTED PLATE THERMAL
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
* AN EQUILATERAL TRIANGLE WITH LINEAR THERMAL GRADIENT
*
* EXPECTED ANSWERS
* DEFLECTION OF NODE 6, IN THE X3 DIRECTION
* THEORY .02331
* STAAD .02331
*
* INTERNAL MOMENTS IN QUAD PLATE NO 8
* MX MY
* THEORY -36.16 -8.864
* STAAD -36.15 -8.84
*
INPUT WIDTH 79
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 0.214286 0 0; 3 0.428571 0 0; 4 0.642857 0 0; 5 0.857143 0 0;
6 1.07143 0 0; 7 1.28571 0 0; 8 1.5 0 0; 9 1.71429 0 0; 10 1.92857 0 0;
11 2.14286 0 0; 12 2.35714 0 0; 13 2.57143 0 0; 14 2.78571 0 0; 15 3 0 0;
16 2.78571 0.123718 0; 17 0 0.247436 0; 18 0.214286 0.247436 0;
19 0.428571 0.247436 0; 20 0.642857 0.247436 0; 21 0.857143 0.247436 0;
22 1.07143 0.247436 0; 23 1.28571 0.247436 0; 24 1.5 0.247436 0;
25 1.71429 0.247436 0; 26 1.92857 0.247436 0; 27 2.14286 0.247436 0;
28 2.35714 0.247436 0; 29 2.57143 0.247436 0; 30 2.35714 0.371154 0;
31 0 0.494872 0; 32 0.214286 0.494872 0; 33 0.428571 0.494872 0;
34 0.642857 0.494872 0; 35 0.857143 0.494872 0; 36 1.07143 0.494872 0;
37 1.28571 0.494872 0; 38 1.5 0.494872 0; 39 1.71429 0.494872 0;
40 1.92857 0.494872 0; 41 2.14286 0.494872 0; 42 1.92857 0.61859 0;
43 0 0.742307 0; 44 0.214286 0.742307 0; 45 0.428571 0.742307 0;
46 0.642857 0.742307 0; 47 0.857143 0.742307 0; 48 1.07143 0.742307 0;
49 1.28571 0.742307 0; 50 1.5 0.742307 0; 51 1.71429 0.742307 0;
52 1.5 0.866025 0; 53 0 0.989743 0; 54 0.214286 0.989743 0;
55 0.428571 0.989743 0; 56 0.642857 0.989743 0; 57 0.857143 0.989743 0;
58 1.07143 0.989743 0; 59 1.28571 0.989743 0; 60 1.07143 1.11346 0;
61 0 1.23718 0; 62 0.214286 1.23718 0; 63 0.428571 1.23718 0;
64 0.642857 1.23718 0; 65 0.857143 1.23718 0; 66 0.642857 1.3609 0;
67 0 1.48461 0; 68 0.214285 1.48461 0; 69 0.428571 1.48461 0;
70 0.214286 1.60833 0; 71 0 1.73205 0;
*72 3 1.73205 0
ELEMENT INCIDENCES SHELL
1 1 2 18 17; 2 2 3 19 18; 3 3 4 20 19; 4 4 5 21 20; 5 5 6 22 21; 6 6 7 23 22;
7 7 8 24 23; 8 8 9 25 24; 9 9 10 26 25; 10 10 11 27 26; 11 11 12 28 27;
12 12 13 29 28; 13 13 14 16 29; 14 14 15 16; 15 17 18 32 31; 16 18 19 33 32;
17 19 20 34 33; 18 20 21 35 34; 19 21 22 36 35; 20 22 23 37 36; 21 23 24 38 37;
22 24 25 39 38; 23 25 26 40 39; 24 26 27 41 40; 25 27 28 30 41; 26 28 29 30;
27 31 32 44 43; 28 32 33 45 44; 29 33 34 46 45; 30 34 35 47 46; 31 35 36 48 47;
32 36 37 49 48; 33 37 38 50 49; 34 38 39 51 50; 35 39 40 42 51; 36 40 41 42;
37 43 44 54 53; 38 44 45 55 54; 39 45 46 56 55; 40 46 47 57 56; 41 47 48 58 57;
42 48 49 59 58; 43 49 50 52 59; 44 50 51 52; 45 53 54 62 61; 46 54 55 63 62;
47 55 56 64 63; 48 56 57 65 64; 49 57 58 60 65; 50 58 59 60; 51 61 62 68 67;
52 62 63 69 68; 53 63 64 66 69; 54 64 65 66; 55 67 68 70 71; 56 68 69 70;
DEFINE MATERIAL START
ISOTROPIC ALUMINUM
E 1e+07
POISSON 0.3
ALPHA 1.2e-05
END DEFINE MATERIAL
CONSTANTS
MATERIAL ALUMINUM ALL
ELEMENT PROPERTY
1 TO 56 THICKNESS 0.1
SUPPORTS
16 29 30 41 42 51 52 59 60 65 66 69 70 INC REFJT 15 FIXED BUT FX MX MZ
2 TO 14 FIXED BUT FX FZ MY MZ
17 31 43 53 61 67 FIXED BUT FY MY MZ
1 FIXED BUT MY MZ
15 71 FIXED
LOAD 1 UNIFORM BENDING TEMPERATURE
TEMPERATURE LOAD
1 TO 56 TEMP 0 450
PERFORM ANALYSIS PRINT STATICS CHECK
PRINT JOINT DISPLACEMENTS LIST 2 TO 14
PRINT ELEMENT JOINT STRESSES LIST 1 8 14 55
FINISHOutput
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