V. Twist in a Tapered Tube

To find the twist at the free end of a hollow tapered shaft of uniform thickness.

Reference

Hand calculation using the following reference:

Mechanics of Materials, F.P.Beer & E.R.Johnston, 1981, McGraw-Hill Review Problem 3.120, Page 149.

Problem

The beam in the following figure which has the following geometric, load, and section properties: L = 2 m, outside diameter at fixed end = 80 mm, outside diameter at free end = 40 mm, uniform wall thickness of 10 mm, , I_Z = 5,000 cm⁴ , E = 200 KN/mm² , T = 2.0 KNm, and Poisson's ratio = 0.3.

Cantilevered member with a tapered, hollow shaft cross section

Theoretical Solution

According to the reference, the twist at the free end is

Φ_{A} = (\frac{T L}{4 π G t}) (\frac{C_{A} + C_{B}}{C_{A}^{2} C_{B}^{2}})

where

C_A	=	centerline radius at free end = 40 mm/2 - 10 mm/2 = 15 mm
C_B	=	center line radius at fixed end = 80 mm/2 - 10 mm/2 = 35 mm
G	=	$\frac{E}{2 (1 + ν)} = \frac{200 k N / {m m}^{2}}{2 (1 + 0.3)} = 76.9$

Φ_{A} = (\frac{2, 000 k N \cdot m \times 2, 000 m}{4 π \times 76.9 k N / {m m}^{2} \times 10 m m}) [\frac{15 m m + 35 m m}{{(15 m m)}^{2} {(35 m m)}^{2}}] = 0.0751 rad.

Comparison

Table 1. Comparison of results
Result Type	Theory	STAAD.Pro	Difference
Twist at free end (radians)	0.0751	0.0751	none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\01 Beams\Twist in a Tapered Tube.STDis typically installed with the program.

STAAD SPACE TORSION ON CONICAL SHAFT
START JOB INFORMATION
ENGINEER DATE 18-Sep-18
END JOB INFORMATION
*
* REFERENCE : MECHANICS OF MATERIALS, F.P.BEER &amp; E.R.JOHNSTON
* 1981, MCGRAW-HILL
*
* REVIEW PROBLEM 3.120, PAGE 149. TWIST AT FREE END SHOULD BE
* ABOUT 0.0751 RADIAN
*
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 2 0;
MEMBER INCIDENCES
1 1 2;
UNIT MMS KN
MEMBER PROPERTY AMERICAN
1 PRIS ROUND STA 80 END 40 THI 10
UNIT METER KN
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 2e+08
POISSON 0.3
END DEFINE MATERIAL
UNIT MMS KN
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED
UNIT METER KN
LOAD 1
JOINT LOAD
2 MY 2
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS
FINISH

STAAD Output

   JOINT DISPLACEMENT (CM   RADIANS)    STRUCTURE TYPE = SPACE
   ------------------
 JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
      1    1     0.0000    0.0000    0.0000    0.0000    0.0000    0.0000
      2    1     0.0000    0.0000    0.0000    0.0000    0.0751    0.0000