Fatigue failure analysis of a leg press exercise machine

\ PERGAMON

Engineering Failure Analysis 5 "0888# 0Ð01

Fatigue failure analysis of a leg press exercise machine P[J[ Vernon\ T[J[ Mackin University of Illinois at Urbana!Champaign\ Department of Mechanical and Industrial Engineering\ 0195 West Green St[\ Urbana\ IL 50790\ U[S[A[ Received 10 August 0887^ accepted 8 September 0887

Abstract The following paper is an engineering failure analysis of an adjustment pin used in a leg press exercise machine[ The pin is used to allow adjustment of the machine for people of di}erent heights[ It was modeled as a cantilever beam subjected to varying forces depending on the weight stack setting[ The fracture occurred at a point of localized stress concentration[ The analysis shows an elastic stress concentration of 0[56 which would cause plastic deformation and a fatigue stress concentration of 0[27 that led to the eventual fatigue failure[ Based on actual user data\ a block loading analysis was used to calculate a pin lifetime of approxi! mately one year[ The pin actually failed after 001 years of use[ Design changes are recommended to reduce the size of the stress concentration and subsequently increase the life of the adjustment pin[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords] Fatigue^ Fatigue design^ Sports equipment failures

0[ Background 0[0[ Machine description The following engineering analysis concerns the fatigue failure of an adjustment pin used within a piece of weight training equipment[ The particular piece of _tness equipment is called a leg press:squat machine[ It is speci_cally designed to exercise the quadricep muscles of the upper leg\ Fig[ 0[ The machine consists of a moveable sled mounted to a set of linear bearings that travel along a pair of hardened steel shafts[ The two steel shafts are mounted to a steel support frame and are inclined at an angle of 09> to horizontal[ The user lies on the moveable sled with his:her shoulders squarely positioned against a set of padded shoulder stops and places his:her feet against

 Corresponding author[ 0249Ð5296:87:, ! see front matter Þ 0887 Elsevier Science Ltd[ All rights reserved PII] S 0 2 4 9 Ð 5 2 9 6 " 8 7 # 9 9 9 2 9 Ð 1

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P[J[ Vernon\ T[J[ Mackin:Engineering Failure Analysis 5 "0888# 0Ð01

Fig[ 0[ Person performing a leg press[

Fig[ 1[ Diagram of adjustment pin[

a _xed footpad[ To perform the exercise\ the user pushes against the footpad and shoulder stops using the upper leg muscles[ The sled rolls along the shafts until the person|s legs are fully extended[ The user then slowly allows the sled to return to the original position[ This constitutes one exercise repetition[ An adjustable stack of cast iron weights provides the resistance force[ The weight stack is connected to the sled by a steel cable[ The steel cable travels through a series of pulleys and is connected to the sled by a telescoping box beam[ To accommodate people of di}erent heights\ the distance between the shoulder stops and the footpad must be adjustable[ This is accomplished by the telescoping box beam connected in series between the cable and the bottom of the sled[ The inner tube is connected to the cable while the outer tube is _rmly welded to the sled[ The load is transferred from the inner beam to the outer one through a spring!loaded steel pin\ Fig[ 1[ The pin is _rmly attached to the outer box tube and locks into a series of holes drilled in the inner tube[ Adjustments are made by disengaging the spring loaded pin from one hole in the inner box tube and engaging it into another[ By engaging the pin into di}erent holes\ the sled can be moved a

P[J[ Vernon\ T[J[ Mackin:Engineering Failure Analysis 5 "0888# 0Ð01

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Fig[ 2[ Diagram of leg press machine[

total of 7 23 in[ to adjust for di}erent height people[ Figure 2 shows a diagram of the major components of the machine[ The resistance force from the stack of weights is transferred through the set of cables[ These cables run through a series of pulleys that magnify the e}ective resistance force supplied by the weight stack[ One end of a cable is attached to the inner tube of the telescoping box beam[ The adjustment pin in question prevents the inner and outer tubes from moving relative to each other[ Consequently\ the pin transfers the e}ective force supplied by the weight stack to the sled[ The pin experiences a complete loading cycle with each exercise repetition[ The pin is loaded in a cantilevered manner from zero load to a maximum load and then back to zero during each lifting repetition[ The weight stack is composed of a series of cast iron plates and can be adjusted to provide a resistance force from 04 to 174 lb in 04 lb increments\ Fig[ 3[ From the wear marks along the center of the weight stack\ it is evident that the most use was in the range of 34Ð049 lb[ 0[1[ Pin failure The pin was in service for approximately 001 years before failure occurred[ At that time\ the machine was disassembled to reveal the fractured adjustment pin\ Fig[ 4[ Upon close inspection of the pin\ two distinct regions on the fracture surface were seen[ One region is indicative of fatigue crack growth and the other of fast fracture[ Once the fatigue crack reached a critical size of approximately 2 mm\ the pin failed completely due to fast fracture\ Fig[ 5[ A micrograph of the fracture surface clearly shows the transition line from the fatigue driven crack growth to fast fracture as well as a micrograph of the transition region[ It was at this transition line that the crack reached a critical length and the pin failed completely due to fast fracture\ Fig[ 5[ In Fig[ 5\ the upper portion of the micrograph shows a relatively smooth "smeared# surface consistent with fatigue crack growth[ The surface smearing was caused by crack closure and surface contact[ The lower portion of the picture in Fig[ 5 shows a region of ductile tearing with less smearing evident at the point of crack instability and the region of fast fracture[ The transition line from fatigue

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P[J[ Vernon\ T[J[ Mackin:Engineering Failure Analysis 5 "0888# 0Ð01

Fig[ 3[ Picture of weight stack[

Fig[ 4[ Diagram of pin loading and failure[

crack growth "upper portion# to fast fracture "lower portion# is very distinct and was measured to be about 2 mm from the top surface of the pin[ 0[2[ Material properties To calculate the fatigue lifetime of the adjustment pin\ an estimate of the ultimate tensile strength is needed[ With most common metals\ the tensile strength can be directly correlated to the surface hardness[ For this particular case\ a Rockwell B hardness test was performed on a cylindrical section of the pin[ The test measured a surface hardness of 89 HRB[ Since the test was performed on a circular cross section 7[9 mm diameter\ a corresponding circular correction factor "CF# of 3 HRB was added to the measured hardness\ resulting in a surface hardness of 83 HRB ð0Ł[ From the hardness\ the approximate tensile strength of the steel can be calculated using the following relationship ð1Ł]

P[J[ Vernon\ T[J[ Mackin:Engineering Failure Analysis 5 "0888# 0Ð01

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Fig[ 5[ SEM picture of the transition region from fatigue to fast fracture[

sult  2[34×"HRB¦CF#[

"0#

According to this relationship the approximate ultimate tensile strength of the steel was calculated to be 213 MPa[ This relatively low tensile strength indicates that the pin was probably made of a low carbon steel such as AISI 0994 or 0909 low carbon steel[

1[ Fatigue failure 1[0[ Beam stresses To determine the cyclic stresses experienced by the pin during the exercise repetitions\ the pin was modeled as a circular cantilever beam of 7[9 mm diameter and a length of 04 mm[ From this model\ the maximum bend stresses along the top edge of the pin were calculated using the following standard beam ~exure equation\ sbend 

My \ I

"1#

where M  moment created by the cantilevered load\ y  pin radius and I  _rst moment of inertia[ From examining the wear patterns of the fractured pin\ it was evident that the force from the weight stack was applied about 09[4 mm from the site of failure\ Fig[ 4[ Using a 09[4 mm moment arm and a pin diameter of 7[9 mm\ the bend stress can be calculated for any weight stack setting[ Since the amount of weight was adjustable to account for people of varying strengths\ the bending stresses experienced by the pin depends on the particular machine user[ To correctly analyze the entire loading spectrum\ a block loading scenario must be employed and a typical distribution of users must be determined[

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P[J[ Vernon\ T[J[ Mackin:Engineering Failure Analysis 5 "0888# 0Ð01

Fig[ 6[ Elastic stress concentration for a round shaft in bending[ "Reprinted with permission from Machine Design\ 0840[ A Penton publication[#

1[1[ Elastic stress concentration As seen in Fig[ 4\ the fracture of the pin occurred at a point where the diameter changed abruptly[ The abrupt change in part diameter created an elastic stress concentration at the shoulder root radius[ The root radius at the shoulder was measured to be 9[7 mm "210 in[#[ The elastic stress concentration factor depends on two di}erent ratios of part dimensions[ The _rst ratio is the root radius divided by the smaller part diameter and the second is the ratio of the larger diameter to the smaller diameter[ The elastic stress concentration factor "Kt# can be read from standard charts developed from experimental data\ Fig[ 6 ð2Ł[ For this particular application\ the ratio of the diameters is D:d  0[5 and the ratio of the root radius to the smaller part diameter "r:d# is 9[0[ Using Fig[ 6\ the corresponding elastic stress concentration factor "Kt# is approximately 0[56[ 1[2[ Plastic yielding As an initial check of the system design\ it is prudent to determine the maximum static stress on the pin at the maximum selectable load[ The maximum setting of the weight stack was 174 lb[ This setting produced a force of 0157 N on the pin[ Using this value and equation "1# produces a bending stress of 154 MPa[ As was stated in the previous section\ an elastic stress concentration

P[J[ Vernon\ T[J[ Mackin:Engineering Failure Analysis 5 "0888# 0Ð01

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factor must be included to account for the stress riser caused by the abrupt change in pin diameters[ Multiplying the bend stress by the elastic stress concentration factor "Kt  0[56# produces a maximum bending stress of approximately 332 MPa[ This value is greater than the ultimate stress of 213 MPa as calculated from the hardness test and equation "0#[ Even though the maximum bend stress is greater than the ultimate stress\ the pin will not necessarily fail under this loading scenario[ As the material along the top surface of the pin is plastically deformed\ the stress level remains relatively constant and additional loads are redistributed toward the center of the pin[ After the pin has undergone plastic deformation\ the elastic beam equation for the bend stress is no longer valid[ Once this happens\ an inelastic bending analysis must be performed[ The important point to note from this analysis is that under the maximum loading scenario\ the pin will be plastically deformed[ This fact alone indicates that the pin was poorly designed[ 1[3[ Fatigue stress concentration Close examination of the fracture surfaces has indicated a fatigue failure[ The elastic stress concentration factor previously determined is measured under conditions of elastic stress and cannot be directly applied in fatigue calculations[ Experiments have shown that the fatigue stress concentration factor may di}er signi_cantly from the elastic stress concentration factor[ From the microscopic viewpoint\ the steep gradients of an elastic stress _eld may not be su.cient to shuttle dislocations and nucleate a fatigue crack[ Hence\ considerable experimental evidence has shown that the fatigue stress concentration "Kf # is less than or equal to the elastic stress concentration "Kt #[ The elastic concentration factor can be converted to a fatigue stress concentration factor by the following formula ð3Ł\ q

Kf−0 Kt−0

"2#

where q is the notch sensitivity parameter of the given material[ The notch sensitivity is dependent upon the notch root radius and the materials ultimate tensile strength[ The notch sensitivity was determined from a plot of notch root radius versus notch sensitivity for steels\ Fig[ 7 ð3Ł[ In this case\ the notch root radius was 9[7 mm "210 in[# and the approximate ultimate tensile strength of the steel was 213 MPa as calculated from equation "0#[ Using these values and Fig[ 7 produces a notch sensitivity value of q  9[47[ Using equation "2# and the value for q\ the fatigue stress concentration factor "Kf# was calculated to equal 0[27[ The fatigue stress concentration factor was incorporated into the analysis by multiplying the bending stress calculated in equation "1# by Kf [ 1[4[ Machine usage To make an accurate prediction of the life of the adjustment pin\ it is crucial to determine the amount of use the machine typically sees[ The repetitions at each weight level were recorded over a period of three weeks[ Figure 8 shows a graph of the number of repetitions at each weight level during the three week period[ The majority of the use was between 34 and 024 lb with only a small amount of use at the very high weight levels[ We will consider the three week time as one period in the block loading scenario[

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Fig[ 7[ Notch sensitivity for various materials[ "Metal Fatigue\ George Sines\ editor\ Þ 0848[ Reprinted by permission of Prof[ George Sines[#

Fig[ 8[ Machine usage over a three week period[

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Table 0 Bend stress incurred in the adjustment pin at each weight stack setting Weight stack "lb#

Fatigue bend stress "MPa#

04 29 34 59 64 89 094 019 024 049 054 079 084 109 114 139 144 169 174

08 27 47 66 85 004 024 043 062 081 101 120 149 158 178 297 216 235 255

1[5[ Bending stresses From the previous section\ we see that individuals use varying weight stack settings[ Each weight stack setting produces a unique bending stress in the adjustment pin[ For each weight stack setting\ the bend stress was calculated using equation "1# and multiplied by the fatigue stress concentration at the pin shoulder[ The corrected bending stresses for each weight stack setting are summarized in Table 0[ 1[6[ Full reversed loading The expected number of cycles to failure for each bending stress were estimated using Basquin|s law ð4Ł\ where\ sar  A"Nf#b[

"3#

However\ Basquin|s law is only valid for the case of fully reversed loading[ In the case of the adjustment pin\ it was being loaded from zero to various values[ Therefore\ the cyclic bending stresses were converted to equivalent fully reversed stresses[ This was done using the Goodman mean stress correction ð5Ł\

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Table 1 Equivalent reversed stresses and cycles to failure for each weight stack setting Weight stack setting "lb#

Fully reversed equivalent stress "MPa#

Cycles to failure

04 29 34 59 64 89 094 019 024 049 054 079 084 109 114 139 144 169 174

12 36 69 82 005 039 052 075 109 122 145 168 292 215 238 262 285 308 331

0\226\123\908\652 5\354\249\263 174\676\843 20\148\007 4\506\027 0\270\636 311\026 040\023 50\966 16\047 02\936 5570 2598 1930 0199 620 347 184 084

0

1

sm sa  0− \ sar sult

"4#

where 0 "5# sa  "smax−smin#\ 1 0 "6# sm  "smax¦smin#[ 1 The ultimate stress of the pin used in Goodman|s equation "4# was estimated to be 213 MPa based on a Rockwell hardness test and equation "0#[ The smin was zero for all weight levels while smax was taken from Table 0 for each weight level[ The Goodman equation "4# was used to calculate an equivalent fully reversed stress for each weight level[ These new equivalent stresses take into account the e}ect of the mean stress on the fatigue life[ Once the equivalent fully reversed stresses were calculated\ each was substituted into Basquin|s law "4# to determine the number of cycles to failure at each weight level[ Using the following constants for AISI 0994 steel ð4Ł] A 3 767 MPa\ b 3−9[02 and equation "4#\ the number of cycles to failure were calculated at each weight level[ Table 1 summarizes the equivalent fully reversed stresses using the Goodman equation and the corresponding cycles to failure according to Basquin|s law for each weight setting[

P[J[ Vernon\ T[J[ Mackin:Engineering Failure Analysis 5 "0888# 0Ð01

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As expected\ Table 1 shows that the higher weight levels require considerably fewer cycles to failure than the lower weight levels[ 1[7[ Cumulative damage The _nal step in determining the total number of cycles to failure was to combine the various weight levels and the user distribution for the three week period using Miner|s rule for block loading[ According to Palmgren and Miner\ failure occurs when the cumulative damage caused by each loading cycle equals one[ The general form of the PalmgrenÐMiner rule ð5Ł is given by\ k

ni  0\ il Ni s

"7#

where k  number of stress levels in the block loading spectrum\ ni  number of cycles at each stress level in the block loading spectrum and Ni  number of cycles to failure at each stress level[ Using equation "7# and the values from Table 1 the cumulative damage incurred by one three week loading block was 9[946[ Therefore\ a total of 06[6 loading blocks would be required to cause failure of the adjustment pin[ This translates into approximately 42 weeks of typical use before failure[ According to the purchase records of the machine the actual failure occurred after 001 years of operation[ It should be remembered that fatigue calculations are only an estimate and the calculated lifetimes are very sensitive to small changes in geometry that a}ect stress levels[ This analysis ignores the presence of plastic deformation that occurs at the higher stress levels[ In addition\ the standard notch sensitivity formulas are derived from data where the notch depth does not exceed four times the notch radius[ So\ according to R[E[ Peterson\ {{This means that caution should be exercised in application to cases of deep sharp notches or very small _llets on stepped shafts [ [ [||[ However\ the analysis does show the pin design to be inadequate[

2[ Design implications The analysis con_rms that the pin failure was due to a poor design and lack of engineering[ The stress calculations indicate that the pin plastically deformed under the maximum load[ The fatigue analysis determined that the pin should fail after only approximately one year of typical service[ This is clearly unacceptable and the design should be altered to take the maximum load and fatigue loading into account[ One way to improve the design to avoid fatigue failure would be to redesign the machine to add support at the end of the pin[ This would reduce the bending stresses by half since the pin could be modeled as a simply supported beam instead of a cantilever beam[ Another method of changing the design would be to increase the pin diameter and:or use a stronger grade of steel[ However\ this would probably increase material and manufacturing costs[ It has also been shown that the addition of a relief groove behind the shoulder causes a reduction in the stress concentration\ Figure 09[ The addition of relief grooves does require extra machining and may not be possible due to the geometry of the machine[ Since the localized stress concentration is dependent on the shoulder root

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Fig[ 09[ Examples of relief grooves behind the shoulder to reduce the stress concentration at the radius\ r[ "Peterson|s Stress Concentration Factors\ Walter D[ Pilkey\ Þ 0886[ Reprinted by permission of John Wiley + Sons[#

radius\ increasing the radius causes a decrease in both the elastic and fatigue stress concentrations[ Reducing the stress concentration decreases the e}ective stress at the shoulder\ avoiding plastic deformation and increasing the fatigue lifetime[ References ð0Ł ð1Ł ð2Ł ð3Ł ð4Ł ð5Ł

Newby JR[ ASM handbook\ mechanical testing\ Vol[ 7\ 0874[ pp[ 71[ Callister WD[ Materials science and engineering[ New York] John Wiley + Sons\ 0883[ pp[ 021[ Peterson RE[ Design factors for stress concentration[ Machine Design 0840^4"12#]048Ð050[ Shigley JE\ Mischke CR[ Mechanical engineering design[ St[ Louis] McGraw!Hill Inc[\ 0878[ Lampman SR[ ASM handbook\ fatigue and fracture\ Vol[ 08\ 0885[ pp[ 122\ 858[ Hertzberg RW[ Deformation and fracture mechanics of engineering materials[ New York] John Wiley + Sons\ 0885[ pp[ 429Ð21[ ð6Ł Pilkey W[ Peterson|s stress concentration factors[ New York] John Wiley and Sons\ 0886[ pp[ 035Ð6[ ð7Ł Peterson RE\ Sines G[ editors[ Metal fatigue[ New York] McGraw!Hill Book Company\ Inc[\ 0848[ pp[ 185Ð6[