V. Torsion on a Stepped Cantilever

To find end rotation due to torques on a stepped cantilever shaft.

Reference

Hand calculation using the following reference:

Gere J. M., and Timoshenko, S. P., Mechanics of Materials, 2nd Edition, PWS Engineering, Page 171, Problem 3.3 - 1.

Problem

A stepped shaft is subjected to torques as shown in the figure. The material has shear modulus of elasticity G = 80 Gpa. Determine the angle of twist θ_x in degrees at the free end.

T₁ = 3,000 N·mm
T₂ = 2,000N·mm
T₃ = 800N·mm

Cantilevered member subject to torsional loads

Theoretical Solution

Moment of Inertia:

I_{p 1} = \frac{π {(80 m m / 2)}^{4}}{2} = 4.021 {(10)}^{6} {m m}^{4}

I_{p 2} = \frac{π {(60 m m / 2)}^{4}}{2} = 1.272 {(10)}^{6} {m m}^{4}

I_{p 2} = \frac{π {(40 m m / 2)}^{4}}{2} = 0.251 {(10)}^{6} {m m}^{4}

Angle of twist is given by:

θ = \sum_{i} \frac{L_{i} T_{i}}{G I_{p}}

= \frac{5, 800 N-mm (500 m m)}{4.021 {(10)}^{6} {m m}^{4} 80 G P a} + \frac{2, 800 N-mm (500 m m)}{1.272 {(10)}^{6} {m m}^{4} 80 G P a} + \frac{, 800 N-mm (500 m m)}{0.251 {(10)}^{6} {m m}^{4} 80 G P a}

= 0.0090 + 0.0138 + 0.0199 = 0.0427

which is equal to 2.446°.

Comparison

Table 1. Comparison of results
Result	Theory	STAAD.Pro	Difference
Angle of twist (rad.)	0.0427	0.0427	none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\01 Beams\Torsion on a Stepped Cantilever.STDis typically installed with the program.

STAAD SPACE :A STEPPED CANTILEVER SHAFT
START JOB INFORMATION
ENGINEER DATE 18-Sep-18
END JOB INFORMATION
*
* REFERENCE: MECHANICS OF MATERIALS, GERE AND TIMOSHENKO, 2ND EDITION
*            PROBLEM 3.3-1 PAGE 171
*
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0.5 0 0; 3 1 0 0; 4 1.5 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4;
UNIT MMS KN
MEMBER PROPERTY AMERICAN
1 TABLE ST PIPE OD 80 ID 0
2 TABLE ST PIPE OD 60 ID 0
3 TABLE ST PIPE OD 40 ID 0
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 200
POISSON 0.25
END DEFINE MATERIAL
UNIT METER KN
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED
UNIT MMS KN
LOAD 1 TORSIONAL MOMENT
JOINT LOAD
2 MX 3000
3 MX 2000
4 MX 800
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 4
FINISH

STAAD Output

   JOINT DISPLACEMENT (CM   RADIANS)    STRUCTURE TYPE = SPACE
   ------------------
 JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
      4    1     0.0000    0.0000    0.0000    0.0427    0.0000    0.0000