V. 2D Curved Beam Maximum Stress

Using plate/shell elements, find maximum bending stress due to a force couple on a curved cantilever beam.

Reference

Timoshenko, S., Strength of Materials, Part I, 3rd Edition, Van Nostrand Co., 1956.

Problem

Find the maximum bending stress

E = 3,000.0 ksi.

Poisson’s ratio = 0.3.

t = 1.0 in.

P = 100 lbs

Cantilevered, curved plate with coupling load at free end

Comparison

Table 1. Comparison of results
Result Type	Theory	STAAD.Pro	Difference	Comments
Inside stress (psi)	655.0	636.02	2.9%	The result from the Beam theory is compared with the result from a Finite Element Model output, hence the difference in results.
Outside stress (psi)	555.0	556.73	negligible

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 Curved Beam Maximum Stress.STD is typically installed with the program.

STAAD SPACE :CURVED BEAM WITH PLATE ELEMENTS
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
*  REFERENCE: TIMOSHENKO, S., "STENGTH OF MATERIALS, PART I, ELEMENTARY
*             THEORY AND PROBLEMS", 3RD EDITION, D. VAN NOSTRAND CO.,
*             INC., NEW YORK, 1956.
*
UNIT INCHES POUND
JOINT COORDINATES
1 4.5 0 0; 2 0 4.5 0; 3 0.781 4.432 0; 4 1.539 4.229 0; 5 2.25 3.897 0;
6 2.893 3.447 0; 7 3.447 2.893 0; 8 3.897 2.25 0; 9 4.229 1.539 0;
10 4.432 0.781 0; 11 4.25 0 0; 12 0 4.25 0; 13 0.738 4.185 0;
14 1.454 3.994 0; 15 2.125 3.681 0; 16 2.732 3.256 0; 17 3.256 2.732 0;
18 3.681 2.125 0; 19 3.994 1.454 0; 20 4.185 0.738 0; 21 4 0 0;
22 0 4 0; 23 0.695 3.939 0; 24 1.368 3.759 0; 25 2 3.464 0;
26 2.571 3.064 0; 27 3.064 2.571 0; 28 3.464 2 0; 29 3.759 1.368 0;
30 3.939 0.695 0; 31 3.75 0 0; 32 0 3.75 0; 33 0.651 3.693 0;
34 1.283 3.524 0; 35 1.875 3.248 0; 36 2.41 2.873 0; 37 2.873 2.41 0;
38 3.248 1.875 0; 39 3.524 1.283 0; 40 3.693 0.651 0; 41 3.5 0 0;
42 0 3.5 0; 43 0.608 3.447 0; 44 1.197 3.289 0; 45 1.75 3.031 0;
46 2.25 2.681 0; 47 2.681 2.25 0; 48 3.031 1.75 0; 49 3.289 1.197 0;
50 3.447 0.608 0;
ELEMENT INCIDENCES SHELL
1 2 3 13 12; 2 3 4 14 13; 3 4 5 15 14; 4 5 6 16 15; 5 6 7 17 16;
6 7 8 18 17; 7 8 9 19 18; 8 9 10 20 19; 9 10 1 11 20; 10 12 13 23 22;
11 13 14 24 23; 12 14 15 25 24; 13 15 16 26 25; 14 16 17 27 26;
15 17 18 28 27; 16 18 19 29 28; 17 19 20 30 29; 18 20 11 21 30;
19 22 23 33 32; 20 23 24 34 33; 21 24 25 35 34; 22 25 26 36 35;
23 26 27 37 36; 24 27 28 38 37; 25 28 29 39 38; 26 29 30 40 39;
27 30 21 31 40; 28 32 33 43 42; 29 33 34 44 43; 30 34 35 45 44;
31 35 36 46 45; 32 36 37 47 46; 33 37 38 48 47; 34 38 39 49 48;
35 39 40 50 49; 36 40 31 41 50;
ELEMENT PROPERTY
1 TO 36 THICKNESS 1
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 11 21 31 41 FIXED
LOAD 1 100 IN-IB
JOINT LOAD
2 FX 100
42 FX -100
PERFORM ANALYSIS
PRINT SUPPORT REACTION
PRINT ELEMENT JOINT STRESSES LIST 9 36
FINISH

STAAD Output

   SUPPORT REACTIONS -UNIT POUN INCH    STRUCTURE TYPE = SPACE
   -----------------
 JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
      1    1    -15.30     61.91      0.00      0.00      0.00     -2.09
     11    1      3.80     69.53      0.00      0.00      0.00      0.72
     21    1     12.47      5.36      0.00      0.00      0.00      1.21
     31    1     11.58    -66.95      0.00      0.00      0.00      1.75
     41    1    -12.55    -69.85      0.00      0.00      0.00     -1.59
   ************** END OF LATEST ANALYSIS RESULT **************
    52. PRINT ELEMENT JOINT STRESSES LIST 9 36
  ELEMENT  JOINT    STRESSES LIST     
      :CURVED BEAM WITH PLATE ELEMENTS                         -- PAGE NO.    4
   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
      9      1         0.00        0.00        0.00        0.00        0.00
                     432.15      432.15     -433.11       -2.49        8.94
                     433.30      433.30
       TOP : SMAX=      -2.31 SMIN=    -433.30 TMAX=     215.50 ANGLE= 88.8
       BOTT: SMAX=      -2.31 SMIN=    -433.30 TMAX=     215.50 ANGLE= 88.8
         JOINT         0.00        0.00        0.00        0.00        0.00
           10        592.15      592.15     -560.35       58.12       15.37
       TOP : SMAX=      58.50 SMIN=    -560.73 TMAX=     309.62 ANGLE= 88.6
       BOTT: SMAX=      58.50 SMIN=    -560.73 TMAX=     309.62 ANGLE= 88.6
         JOINT         0.00        0.00        0.00        0.00        0.00
            1        527.90      527.90     -556.47      -63.65       11.40
       TOP : SMAX=     -63.39 SMIN=    -556.73 TMAX=     246.67 ANGLE= 88.7
       BOTT: SMAX=     -63.39 SMIN=    -556.73 TMAX=     246.67 ANGLE= 88.7
         JOINT         0.00        0.00        0.00        0.00        0.00
           11        285.05      285.05     -310.91      -64.85       13.47
       TOP : SMAX=     -64.11 SMIN=    -311.65 TMAX=     123.77 ANGLE= 86.9
       BOTT: SMAX=     -64.11 SMIN=    -311.65 TMAX=     123.77 ANGLE= 86.9
         JOINT         0.00        0.00        0.00        0.00        0.00
           20        351.99      351.99     -318.48       58.96        8.48
       TOP : SMAX=      59.15 SMIN=    -318.67 TMAX=     188.91 ANGLE= 88.7
       BOTT: SMAX=      59.15 SMIN=    -318.67 TMAX=     188.91 ANGLE= 88.7
     36      1         0.00        0.00        0.00        0.00        0.00
                     457.10      457.10      473.43       34.66        2.24
                     473.44      473.44
       TOP : SMAX=     473.44 SMIN=      34.64 TMAX=     219.40 ANGLE=  0.3
       BOTT: SMAX=     473.44 SMIN=      34.64 TMAX=     219.40 ANGLE=  0.3
         JOINT         0.00        0.00        0.00        0.00        0.00
           40        314.58      314.58      298.80      -24.94      -23.18
       TOP : SMAX=     300.45 SMIN=     -26.59 TMAX=     163.52 ANGLE= -4.1
       BOTT: SMAX=     300.45 SMIN=     -26.59 TMAX=     163.52 ANGLE= -4.1
         JOINT         0.00        0.00        0.00        0.00        0.00
           31        247.25      247.25      282.26      108.74       10.51
       TOP : SMAX=     282.90 SMIN=     108.11 TMAX=      87.39 ANGLE=  3.5
       BOTT: SMAX=     282.90 SMIN=     108.11 TMAX=      87.39 ANGLE=  3.5
         JOINT         0.00        0.00        0.00        0.00        0.00
           41        594.55      594.55      634.80       95.42       25.72
       TOP : SMAX=     636.02 SMIN=      94.20 TMAX=     270.91 ANGLE=  2.7
       BOTT: SMAX=     636.02 SMIN=      94.20 TMAX=     270.91 ANGLE=  2.7
         JOINT         0.00        0.00        0.00        0.00        0.00
           50        680.99      680.99      656.97      -43.84      -21.63
       TOP : SMAX=     657.64 SMIN=     -44.51 TMAX=     351.08 ANGLE= -1.8
       BOTT: SMAX=     657.64 SMIN=     -44.51 TMAX=     351.08 ANGLE= -1.8
      :CURVED BEAM WITH PLATE ELEMENTS                         -- PAGE NO.    5
          **** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES ****
             MAXIMUM       MINIMUM       MAXIMUM       MAXIMUM       MAXIMUM
            PRINCIPAL     PRINCIPAL       SHEAR       VONMISES       TRESCA
             STRESS        STRESS        STRESS        STRESS        STRESS
          6.576412E+02 -5.607280E+02  3.510752E+02  4.571006E+02  4.734368E+02
 PLATE NO.      36             9            36            36            36
 CASE  NO.       1             1             1             1             1