V. Curved Beam

To find the out-of-plane deflection and stress in a Circular cantilever member with a concentrated load at the free end.

Reference

Hand calculation using the following reference:

Timoshenko, S., Strength of Materials, Part I, D. Van Nostrand, 3rd Edition., 1955.

Problem

Calculate the displacement at the free end and the bending stress at the fixed end due to a concentrated load producing out–of–plane bending.

E = 30×(10)6 psi P = 50 lb r = 100 in.
Figure 1. Curved beam

Comparison

Table 1. Comparison of results
Result Theory STAAD.Pro Difference Comments
Maximum deflection at free end (in.) 2.648 2.64658 negligible
Principle stress (psi) 6,366.0 6,638.1 4.3% The result from a classical theory of a beam curved in plan is compared with the result from a piecewise linear set of beams, which closely resembles the behavior of a curved beam, but not exactly. Hence the difference in results. The difference may be further reduced to a certain level by discretizing the curved beam in more smaller subdivisions.

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro 2024\Samples \Verification Models\01 Beams\Curved Beam.STDis typically installed with the program.

STAAD SPACE :A CURVED BEAM
START JOB INFORMATION
ENGINEER DATE 18-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, 1955.
*
INPUT WIDTH 72
UNIT INCHES POUND
JOINT COORDINATES
1 100 0 0; 2 99.619 0 -8.716; 3 98.481 0 -17.365; 4 96.593 0 -25.882;
5 93.969 0 -34.202; 6 90.631 0 -42.262; 7 86.603 0 -50;
8 81.915 0 -57.358; 9 76.604 0 -64.279; 10 70.711 0 -70.711;
11 64.279 0 -76.604; 12 57.358 0 -81.915; 13 50 0 -86.603;
14 42.262 0 -90.631; 15 34.202 0 -93.969; 16 25.882 0 -96.593;
17 17.365 0 -98.481; 18 8.716 0 -99.619; 19 0 0 -100;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 5 6; 6 6 7; 7 7 8; 8 8 9; 9 9 10;
10 10 11; 11 11 12; 12 12 13; 13 13 14; 14 14 15; 15 15 16; 16 16 17;
17 17 18; 18 18 19;
MEMBER PROPERTY AMERICAN
1 TO 18 PRIS AX 3.14 IX 1.57 IY 0.7854 IZ 0.7854 YD 2 ZD 2
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
19 FIXED
LOAD 1 POINT LOAD APPLIED AT THE FREE END
JOINT LOAD
1 FY 50
PERFORM ANALYSIS
PRINT MEMBER STRESSES LIST 18
PRINT JOINT DISPLACEMENTS LIST 1
FINISH

STAAD Output

  ALL UNITS ARE POUN/SQ INCH
 MEMB   LD  SECT    AXIAL    BEND-Y     BEND-Z   COMBINED  SHEAR-Y  SHEAR-Z
     18    1   .0      0.0        0.0    6082.7    6082.7     23.9      0.0
              1.0      0.0 C      0.0    6638.1    6638.1     23.9      0.0 
   ************** END OF LATEST ANALYSIS RESULT **************
    39. PRINT JOINT DISPLACEMENTS LIST 1
  JOINT    DISPLACE LIST     1        
      :A CURVED BEAM                                           -- PAGE NO.    4 
   JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = SPACE
   ------------------
 JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
      1    1    0.00000   2.64658   0.00000  -0.02438   0.00000   0.01076