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
Plate Corner Stress at Node 41 (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 2023\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