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.

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.

E = 10×(10)⁶ lb/in²

υ = 0.3

α = 12 (10)^-6 in/in °F

ΔT = T_top – T_bottom = 4500

t = 0.1 inches

a = 3.0 inches

Simply-supported, equilateral triangle with thermal load

Finite element model of a simply-supported, equilateral triangle

Theoretical Solution

From page 96 of the reference:

M_{x} = - \frac{α Δ T E t^{2}}{24} (1 + \frac{3 x^{'}}{a})

= - \frac{12 {(10)}^{- 6} (450.0) 10 {(10)}^{6} {(0.1)}^{2}}{24} (1 + \frac{3 x^{'}}{3}) = - 22.5 (1 + x^{'})

M_{y} = - \frac{α Δ T E t^{2}}{24} (1 - \frac{3 x^{'}}{a}) = - 22.5 (1 - x^{'})

M_{x y} = \frac{α Δ T E t^{2} y}{8 a}

= \frac{12 {(10)}^{- 6} (450.0) 10 {(10)}^{6} {(0.1)}^{2}}{8 (3)} y = 22.5 y

w = \frac{α Δ T (1 + ν)}{4 a t} ({x^{'}}^{3} - 3 y^{2} x^{'} - a {x^{'}}^{2} - a y^{2} + \frac{4 a^{3}}{27})

= 0.00585 ({x^{'}}^{3} - 3 y^{2} x^{'} - 3 {x^{'}}^{2} - 3 y^{2} + 4)

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 = -22.5x

My = -22.5 (2 - x)

Mxy = 22.5y

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 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\2D Triangular Surface with Thermal Load.STD is typically installed with the program.

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
FINISH

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