 # 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 θx in degrees at the free end.
• T1 = 3,000 N·mm
• T2 = 2,000N·mm
• T3 = 800N·mm ## Theoretical Solution

Moment of Inertia:

$I p 1 = π ( 80 m m / 2 ) 4 2 = 4.021 ( 10 ) 6 m m 4$
$I p 2 = π ( 60 m m / 2 ) 4 2 = 1.272 ( 10 ) 6 m m 4$
$I p 2 = π ( 40 m m / 2 ) 4 2 = 0.251 ( 10 ) 6 m m 4$

Angle of twist is given by:

$θ = ∑ i L i T i G I p$
$= 5 , 800 N-mm ( 500 m m ) 4.021 ( 10 ) 6 m m 4 80 G P a + 2 , 800 N-mm ( 500 m m ) 1.272 ( 10 ) 6 m m 4 80 G P a + , 800 N-mm ( 500 m m ) 0.251 ( 10 ) 6 m m 4 80 G P a$
$= 0.0090 + 0.0138 + 0.0199 = 0.0427$

which is equal to 2.446°.

## Comparison

Table 1. Comparison of results
Angle of twist (rad.) 0.0427 0.0427 none

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
2 MX 3000
3 MX 2000
4 MX 800
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 4
FINISH


   JOINT DISPLACEMENT (CM   RADIANS)    STRUCTURE TYPE = SPACE