The Manchester Metropolitan University School of Engineering

6E5Z2001 Advanced Mechanical Science

Laboratory: Principal stress determination
OBJECTIVES
To determine the principal stresses at the surface of an aluminium tube subjected to a torsional load by processing measured data from s strain rosette. To validate the theoretical solutions and finite element predictions using test data.

EXPERIMENTAL METHOD (See Appendix for other information)
1. Set and zero the Wheatstone bridge circuit for all channels. 2. Load the torsion rig with a set of masses, recording the strain gauge readings at each load level. 3. Take necessary measurements (e.g., the loading arm length). (Data is provided on the last page)

THEORY
(a) From a selected set of experimental readings, draw a Mohr’s strain circle and determine the principal strains, the principal directions and the maximum shear strain. Calculate the corresponding principal stresses and the maximum shear stress by applying the generalised Hooke’s law. ? (b) Using the torsion theory, determine the shear stress acting at the outer radius of the cross section and then the principal stresses.

DELIVERABLES (with marking scheme)
A FULL report structure should be adopted and the following aspects should be covered. (a) Cover page, summary, contents page, introduction, apparatus list, experimental procedure, and references, etc. [10%]

(b) Experimental data processing to capture the proportional behaviour between the weight applied (W) and the strain (?) for each strain gauge (?i = gi x W where i = A,B,C and gi is the gradient of the ‘best-fit’ line) [15%] (c) Select a weight value W’ and estimate the corresponding strain values eA’, eB’ and eC’ using the results in (b). [5%] (d) Construct a strain Mohr’s circle using eA’, eB’ and eC’. Graphically determine the principal strains e1 , e2 and the 1st principal direction relative to the shaft axis (longitudinal) direction ?1. Also determine the maximum shear strain, ?max. Calculate e1 , e2, ?1 and ?max using the formula method to verify the solution from the Mohr’s circle. [35%] (e) Calculate the principal stresses s1exp , s2exp and the maximum shear stress tmax_exp using the stress-strain transformation formulae given below. [5%]

where These complete the experimental data processing and solution. (f) Calculate the theoretical shear stress tth acting at the outer radius of the shaft with an outer diameter D and inner diameter d due to torque T’ using the formula below. [5%] where (g) For the stress state of a point P on the surface, tth tth
and T’ = arm length x W’

tth

draw a Mohr’s stress circle. Determine the principal stresses s1th , s2th and the 1st principal direction ?1_th relative to the shaft axis direction as well as the maximum shear stress tmax_th. [5%] (h) Compare the experimental result, the theoretical solution and the FEA prediction showing the % difference for each of the four quantities. [10%] (i) Comment on the likely error sources causing the differences between the experimental, the theoretical and the FEA results. [5%] (j) Comment on the usefulness and limitations of experimental principal stress determination in practical applications. [5%]

APPENDIX Apparatus list: Torsion rig with instrumented strain rosette Strain indicator Channel switch box Masses Ruler

Pre-measured data:
Outer diameter of the shaft specimen: 22.3 mm Inner diameter: 18.9 mm Young’s modulus (E): 68 GN/m2 Poisson’s ratio (?): 0.3

Rosette gauge arrangement:

Gauge C (green)

Gauge B (blue)

Gauge A (red) Torque
45o 45o

??
Shaft fixed end

Note: ? depends on the specific rig and is listed below. Rig No. ??? o ) 1 31 2 34 3 25 5 32 6 28

Some torsion lab images:

Figure 1: A torsion rig and a blue multi-channel strain indicator

Half-bridge connection is used for each channel

Figure 2: A yellow strain indicator and a channel switch box