10.1111/j.1460-2695.2007.01149.x

Torsion response of a cracked stainless steel shaft

C. J. LISSENDEN1, S. P. TISSOT1, M. W. TRETHEWEY2 and K. P. MAYNARD3

1Department of Engineering Science and Mechanics, 2Department of Mechanical Engineering, 3Applied Research Laboratory, Penn State, UniversityPark, PA 16802, USA

Received in final form 26 December 2006

A B S T R A C T Pump shafts used for power generation are susceptible to fatigue cracking while oftenin near-continuous operation. Technology based on torsional vibration is under develop-ment for condition-based assessment of shaft health. The focus of this paper is on therelationship between a crack, which propagated due to bending loads, and the torsionalstiffness of the shaft. An analytical method to determine the compliance associated witha crack has been implemented for a semi-elliptical surface crack. A 3-D finite elementmodel of a shaft section with a crack has also been used to predict the effect of a crack onstiffness. Fatigue cracks were seeded in shafts on a three-point bend apparatus. A benchtoptest rig was constructed to evaluate the torsional natural frequencies of a cracked shaftsystem. Quasistatic torsional stiffness tests indicated that crack closure has an effect on theresults. A torsional finite element model of the benchtop test rig indicates that the firsttorsional natural frequency is reduced by the propagation of a crack. The reduction intorsional stiffness of the shaft inferred from the natural frequency results is in reasonableagreement with the quasistatic results and the model predictions.

Keywords semi-elliptical surface crack, torsional stiffness, natural frequency, condition-based monitoring.

I N T R O D U C T I O N

The growth of fatigue cracks in rotating components inthe power generation industry, especially pump shafts, is asignificant economic and safety concern.1 Fatigue crackstypically initiate at stress concentrators due to some com-bination of thermal, bending and torsional loads. In manycases the equipment is required to operate continuouslyfor extended periods of time, highlighting the impor-tance of on-line structural health monitoring. The mea-surement of vibrational response features can be used tomonitor structural damage, such as that due to a fatiguecrack because crack growth decreases structural stiffness,which changes the vibrational characteristics. The workpresented herein was undertaken in response to the occur-rence of fatigue cracks in reactor coolant pump shafts fornuclear power plants. In this case, the crack apparently ini-tiates at a pinhole due to thermal loading and then prop-agates due to bending stresses. These cracks have beenobserved to propagate as either a semi-elliptical surface

Correspondence: C. J. Lissenden. E-mail: lissenden@psu.edu

crack or a circumferential crack. Shaft deflection-basedvibration monitoring systems were ineffective in provid-ing an early warning to the plant operators. Recent workhas shown the potential of torsional vibration monitoringto detect structural damage.2,3

Laboratory tests have shown the ability to detect tor-sional natural frequency changes as successively deeperseeded faults were placed in shafts and a model bladeddisk assembly.3 However, effective prognostics require thechange in torsional natural frequency to be related to thechange in the shafts structural state (i.e., fatigue cracksize). If the crack geometry can be related to the changein torsional stiffness, the shafts health can be directly as-sessed from the observed torsional vibration signature fea-tures. Previous work4,5 has examined the effects of tor-sional stiffness for cuts with straight fronts, but cuts arenot necessarily representative of the effects that a fatiguecrack has on a shafts torsional stiffness. The difference isprimarily due to the surface contact associated with crackclosure present for a fatigue crack, but not for a cut. Theeffect of fatigue cracks on torsional stiffness has not beenthoroughly investigated until very recently by Vaziri andNayeb-Hashemi,6,7 who consider circumferential cracks.

734 c 2007 The Authors. Journal compilation c 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 734747

TORSION RESPONSE OF A CRACKED STAINLESS STEEL SHAFT 735

Here, motivated by cracks observed in reactor coolantpump shafts, we investigate the effect of semi-ellipticalsurface cracks on torsional stiffness. The evaluation uti-lizes analytical modelling, finite element analysis (FEA),and a series of seeded fault experiments with both cuts andfatigue cracks. Torsional stiffness is determined by apply-ing a quasistatic torque to a (damaged) shaft and measuringthe angle of twist. The slope of the angle of twisttorquecurve is the torsional stiffness. This can be done in thelaboratory or simulated using FEA. Torsional stiffness canalso be inferred from dynamic testing through a combi-nation of experiments and modelling as will be describedbelow.

The seeded fault experimental database was developedusing the following procedure:

1 A fatigue crack is grown in a stainless steel shaft usingthree-point bending (Fig. 1).

2 The crack is characterized visually and by ultrasonicinspection.

3 Torsional stiffness (rigidity) is measured by applying atorque to the shaft and measuring the angle of twist.

4 The shaft is placed in a torsional vibration test stand andthe line shaft torsional natural frequency measured.

5 Steps 14 are repeated multiple times to acquire datafor progressively deeper cracks.

6 At completion of the sequence, the shaft was destruc-tively sectioned to measure the beach marks on the frac-ture surface.

For reference purposes, shafts with open cuts are alsoexamined. The same test sequence is used with the excep-tion that a sequence of straight front cuts is used insteadof fatigue cracks.

Analytical and finite element models of the cracked shaftare created. Both static and dynamic results are comparedwith the experimental results.

In order to determine how well various methods com-pare, the torsional stiffness will be determined using fourdifferent methods:

1 Quasistatic torsion experiments on cracked shafts,2 Dynamic tests on a benchtop test rig to determine

the torsional natural frequency and iteratively back-calculate the torsional spring constant representative ofthe crack using a finite element model to match theexperimental frequency data,

3 3-D finite element model of the cracked shaft,4 Inverse of the compliance obtained from an analytical

model.

Comparison of the model and experimental results pro-vides insight into the torsional stiffness degradation asso-ciated with fatigue cracks and cuts in shafts.

M O D E L S

Models are used to quantify the additional torsional shaftcompliance (flexibility) due to the presence of a crack andto predict the effect of a crack on the natural frequenciesand mode shapes of a line shaft system. The additionalshaft compliance due to a crack is modelled both analyt-ically and with finite elements, however neither of theseaccount for the frictional effects of a closed crack.

Cracked shaft compliance

Analysis of pump shaft systems, or the table-top systembuilt for this study, is readily performed using a line shaftmodel where the shaft is represented by beam elements.A crack in the shaft creates additional compliance that canbe represented by a discrete spring. The stiffness of thespring can be determined by (i) analytical methods, (ii)numerical methods or (iii) experimental methods. Eachof these is considered, starting with an analytical method.

The general theoretical framework applied byPapadopoulos4,5,8 starts with the Paris equation fordescribing the additional generalized displacement (threetranslations and three rotations), associated with a crack,

u j = Pj

Ac

d A j = 1, 2, . . . , 6, (1)

where Ac is the crack surface area, Pj are generalized forces(three force components and three moment components),and

= 1E

[ 6j=1

KI j

(

h

)]2+

[6

j=1KII j

(

h

)]2

+ m[

6j=1

KIII j

(

h

)]2 (2)is the strain energy density function, with K Ij(/h),K IIj(/h) and K IIIj(/h) being the stress intensity factorsassociated with generalized force Pj, /h is the crack depthto section depth ratio, and

E ={

E for plane stressE

1 for plane strain

m = E

2G, (3)

where E is Youngs modulus, is Poissons ratio, and Gis the shear modulus. The theorem of Castigliano can beemployed to determine the 6 6 compliance matrix as-sociated with the crack,

Si j = ui Pj

= 2

Pi Pj

Ac

d A. (4)

Papadopoulos5 has implemented this analysis for an opencrack having a straight crack front using the geometry

c 2007 The Authors. Journal compilation c 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 734747

736 C. J. L I SSENDEN et al.

a

0.25 mmmachined notch with maximum

root radius of 0.076 mm

P

Ls

76 mm 76 mm

2Ra

Fig. 1 Three-point bend specimen geometry.

shown in Fig. 2(a). The geometric variables are:

h = 2R2 z2a = R2 z2 R + ab =

R2 [R a]2 , (5)

which represent the section height, crack depth and halfcrack width, respectively, at the integration point. For thespecial case of a crack in a shaft subjected solely to a torque,T , the only non-zero stress intensity factors and geometryfactors are

KII6(

h

)= 2Tz

R4

FII(

h

)KIII6

(

h

)= 2T

R2 z2

R4

FIII(

h

)(6)

and

FII(

h

)= 1.122 0.561

(h

) + 0.085 ( h )2 + 0.180 ( h )31 ( h )