MAE 577 PROJECT 2 FEM ANALYSIS OF POWER SCREW

MAE 577 PROJECT 2

FEM ANALYSIS OF POWER SCREW

By,

SUMIT TRIPATHI

Project Statement

MAE 477/577

Feb. 2, 2009

Project #2

Finite Element Problem

The figure below shows the upper head of a long power screw. The loading on the screw by the

downward force F keeps the screw in tension through out its life time but varies considerable during the screw’s

operation. The force F is supported by a thrust bearing and the flange surface that rests on it as shown

schematically below. The driving torque T for the screw is provided through the keyway in the upper shaft. It is

this torque that allows the screw to raise and lower its load.

In the figure, D = 1.75 inches, the base diameter of the thread run-out groove is d = 1.25 and the groove

radius r = 1/8. The power screw has an outer diameter of 1 5/8 with a thread depth of 1/8. The thrust flange has

an outer diameter of 2.5 and is 3/8 thick. The screw is made of heat treated steel with an ultimate strength of

148 ksi and yield strength of 112 ksi. All critical surfaces have a ground finish. In its operation, the screw raises

its load with an axial force F of 15,000 lbf and a required torque T of 3000 in-lbf. For movement downward, the

axial force reduces to 2000 lbf and the torque is -300 in-lbf (i.e. the torque reverses direction). These situations

define the extremes of the expected loading on the screw.

The problem of concern at this point in the design stage is the possibility of fatigue failure at the thread

run-out groove. The infinite life safety factor for the groove is to be determined under the loading conditions

described above. You are to address this problem using finite elements but also giving serious consideration to

traditional stress calculations based on stress concentration factors. Two stress concentration factor models have

been suggested for the groove. Of course, one considers a simple groove – while the other assumes that a

shoulder fillet model is more appropriate (since the thread isn’t equivalent to a solid shaft).

Part A

Consider ProMechanica finite element modeling to help resolve the stress concentration issue for this

application. Consider (a) tension only, (b) torsion only, and (c) the combined loading situation for lifting

described above. If tradional stress concentration calculations are to be used in this design process, which model

is most conservative? Based on your ProMechanica calculations, can you conclude which model is most

appropriate – simple groove or shoulder fillet? Describe your geometric modeling and your calculation process

and provide an organized set of data to support your conclusion.

Part B

Determine the infinite life safety factor for the run-out groove considering the repeated raising and

lowering of the load as described above. Clearly state your best estimate of the stresses resulting from the

specified torques and axial forces and use these stresses in an appropriate fatigue analysis under combined

stresses (be sure to reference an appropriate machine design text for fatigue failure theory).

Part C

Many months after delivery of the screw assemblies began, one was returned with an apparent crack at

the fillet lying under the bearing at the base of the thrust flange. The fillet radius here is 3/32. The screw was

returned from an amusement park ride where up to eight people are supported from free fall by the screw shaft.

Perform a finite element analysis to check the conditions at this point. Is the crack a fluke, might an overload be

expected? Or is there a design problem that needs further attention? Describe your analysis and explain your

conclusions. You may make comparisons to your fatigue analysis in Part B above – but a full fatigue analysis is

not necessary.

In Your Report…

Carefully describe your solutions to Parts A, B and C above. Present a few images to show details of

your finite element models and sample results – include a few “stress labels” to indicate maximum stresses on

your figures. Describe your process of adjusting mesh sizes and your efforts to be sure that you achieved a fine

enough mesh to get an accurate result.

It is recommended that you work with a partner on this project.

Model Assumption

We have assumed that the groove starts from the base of thread, since the groove cannot end at the thread edge

(provided that the thread is through-out the length of the power screw). With this assumption we have calculated base

diameter of the thread run out groove to be d=1.125 in, instead of 1.25 in.

in

sidebothondepthgrooveThesidebothondepthThreadThepowerscrewofDiaOuterThed

125.1

)(8

2)(

8

2)(

8

51

We have neglected any strength contribution due to threads. Hence the diameter of shaft is taken as base diameter of

thread. Hence diameter of power shaft for FEM analysis = 1.375 in

in

sidebothondepthThreadThepowerscrewofDiaOuterTheD

375.1

)(8

2)(

8

51

Material : Steel

poison's ratio =0.27

Young's modulus =2.9e+07psi

Yield strength: 112 ksi

Ultimate Strength: 148 ksi

Model Design and Constraint Set:

The FEM Analysis involves generation of hundreds of small element and calculations involved for those elements, which

costs lot of computation time. To avoid unnecessary computation time we have to carefully select our model design.

My approach:

First we are going to study the actual design with minimal mesh size so that we will have fair idea about the location of

stress concentrations.

The following figure displays that the shaft flange, which is supported on the bearing is constrained at bottom surface of

it in all 6 degrees of freedom. The tensile load of 15000lbf is applied at the bottom surface of the shaft. The FEM analysis

accomplished by Pro Mechanica shows that the stress generation in all the volume that above top of the shaft flange is

insignificant. Hence we will cut off this volume in subsequent study.

The motor is attached to top shaft with key. My assumption is that the motion is constrained my motor (only for

analysis, because for every action of motor there will be equal and opposite reaction by shaft). Without loss of

generality we can constrain the top of the flange in modifies model to study stress generation in groove.

Simple Groove model geometry:

Fig: 1

Simple Groove ,Axial Loading

Loading Description:

Neutral Axis of Shaft is aligned in yy world

coordinate.

All of the 6 degrees of freedom of Top face are

constrained .

Axial load of 15000lbf in negative yy direction.

Mesh Geometry:

Mesh refinement is implemented by changing the allowable angles between the edges, number of nodes on the edges

and maximum edge turn angle.

H1

Points: 73 Edges: 324

Faces: 439 Elements: 187

H2



H3



FEM solutions might be less reliable, if the elements

involved in FEM computations are too stiff, i.e. the

angle between the edges of the element. This happens

because of difficulty in fitting the polynomial over sharp

corners of elements.

My Approach in Mechanica:

Increase the minimum allowable angle between edges

so that better polynomial curve fitting is achieved. This

process in effect increases the number of elements to

mesh volume. We did this in conjunction with reduction

in aspect ratio and increment of manual nodes.

Why not reduce only Aspect ratio?

Reducing only maximum allowed aspect ratio and

leaving the minimum allowable angle b/w edges as

default (5 Degree), will not serve the purpose. Still there

may be some adjacent elements having maximum

aspect ratio reached and leaving the angle between

the edges somewhat steep.

Initial Mesh H1: Default setting in pro Mechanica, i.e.

Allowable angles:

Edge Max=175 degree, Edge min=5degree

Face max=175 degree Face min=5degree

Max Aspect Ratio=30 Max Edge Turn=95 degree

Second Mesh H2:

Allowable angles:




Final Mesh H3: Default Setting In pro Mechanica i.e.

Allowable angles:




For H3 Manual Nodes were also generated around the groove region to facilitate smooth generation of mesh.

Location of Nodes can is visible in above figure.

Number of Nodes : 20 at each edge of groove.

In subsequent analysis mesh generation will remain almost same, therefore the details will not be repeated except the

number elements for H-Convergence in each mesh refinements.

The stress under consideration is max_yy, since only load that is acting is in axial direction. The maximum stress

generated will be in yy ( the axis of shaft) direction, other stress will be insignificant. Considering Von Mises stress in this

case might not be desirable since it takes all three principal stresses into consideration.

P_Convergence:

We can see in above figure that polynomial convergence in case of finest mesh (H3) is very precise.

Region Of maximum stress concentration for the Finest Mesh (the region in solid, i.e. Groove)

Please note that the region of maximum Stress

concentration is groove/fillet only for all the

cases of loading for Part A.

Stress Distribution: max_stress_yy:-

Maximum Stress

Nominal Stress=2^

4

d

Pnom

Where, P= Applied axial load, d= Dia of minimum cross section

In our case P=15000 lbf,

in

sidebothondepthgroovesidebothondepthThreadpowerscrewofDiaOuterd

125.1

)(8

2)(

8

2)(

8

51

Hence,

psinom 24.150902^125.1

150004

H – Convergence:

The exact solution based on FEM solutions of the 3 finest meshes is carried out by Lagrangian Interpolating polynomial,

which gives as exact solution as

21

21

2

2)^(

nn

nn

n

nexact

, Where ‘n’ is subscript for the finest mesh and ‘n-1’ and ‘n-2’ are previous two

meshes.:

%Convergence= 100X( exact - fem )/ exact

Stress concentration factor based on FEM computation Stress concentration

factor based on” R. E.

Peterson Stress

concentration Hand

Book” , Chart 2.19

Simple Groove Maximum Stress Convergence, Axial Loading

No of

Elements

Max Stress in axial

direction(yy) psi

%Convergence

187.00 36882.84 5.34

1197.00 36089.57 3.08

2846.00 35632.57 1.77

Exact Solution

35011.49 Nominal

Stress 15090.24

2.3201

11.0125.1

125.0

d

r

Ref fig:1

22.1125.1

375.1

d

D

From chart 2.19

Kt = 2.3

100

Sex

SfeSex

Stress Nominal

Stress MaximumFactorion Concentrat Stress

Simple Groove ,Torsion Loading


Neutral Axis of Shaft is aligned in yy world coordinate.


constrained .

Torsion load of 3000 lbf-in in tangential direction as

shown.

Force applied is force per unit area

psihdd

T27.577

75.16875.26875.0

3000

)2/(2)2/(

For torsion load maximum principal stress magnitude is considered, since principal stresses are the eigenvalues of the

stress matrix it will include shear forces in all directions for each Mesh Element.

Mesh Geometry:

Mesh geometry remains almost same as previous case.

P_Convergence:

Again, it is clear that polynomial convergence in case of finest mesh (H3) is very precise.

Stress Distribution: max_stress_prin

J

cTnom

Where, T= 3000 lbf-in,

c= distance of extreme fiber of minimum cross section from neutral axis = 1.125/2 =d/2

32

4^dJ

in

sidebothondepthgroovesidebothondepthThreadpowerscrewofDiaOuterd

125.1

)(8

2)(

8

2)(

8

51

Hence,

psid

T

J

cTnom 84.10730

3^125.1

300016

3^

16

Maximum Stress

H – Convergence:



21

21

2

2)^(

nnn

nnn

exact

, Where ‘n’ is the finest mesh and ‘n-1’ and ‘n-2’ are previous two meshes.:


factor based on” R. E.

Peterson Stress

concentration Hand

Book” , Chart 2.47

Simple Groove Maximum Stress Convergence,Torsional Loading

No of

Elements

Max Principal Stress

Magnitude

%Convergence

147 16784.64 3.76

1210 16258.63 0.51

1752 16187.53 0.07

Exact Solution

16176.42 Nominal

Stress 10730.84

1.5075

11.0125.1

125.0

d

rRef

fig:1

22.1125.1

375.1

d

D

From Chart 2.47

Kts = 1.55

Clearly Kts is smaller

than Kt(2.3), which

implies that stress

concentration is higher

in tensile loading in our

case.

100

Sex

SfeSex

Stress Nominal


Simple Groove ,Combined Loading


Please refer previous definitions of axial and

torsion loading, both are acting simultaneously.

Von mises stress is considered here, as it combines all the shear and tensile stress with the calculation of principal stress

, the eigen values of matrix of shear and normal stresses

2

2)^32(2)^32(2)^21(

vm , as we can see that Vm stress average out the principal stresses,

it is considered to be the best stress distribution in case of combined loading.

Please refer to H- Convergence table for number of elements. The initial mesh geometry is almost same.

P_Convergence: (Von mises stress is considered over here)

Stress Distribution: max_stress_vm

Maximum Stress

H – Convergence:


as stated earlier.

Stress concentration factor based on FEM computation

Simple Groove Maximum Stress Convergence, Combined Loading

No of

Elements

Max Von Mises stress

psi

%Convergence

147 45375.37 6.37

1210 43046.08 0.91

1752.00 42668.54 0.03

2846.00 42657.56 0.00

Exact Solution

42657.23 Nominal

Stress 20663.03

2.0644

100

Sex

SfeSex

Stress Nominal


Shoulder Fillet model geometry:

Fig: 1

Shoulder Fillet ,Axial Loading


Neutral Axis of Shaft is aligned in yy world

coordinate.


constrained .

Axial load of 15000lbf in negative yy direction.

P_Convergence:

We can see in above figure that polynomial convergence in case of finest mesh (H3) is very precise.

Stress Distribution: max_stress_yy

2^

4

d

Pnom

Where, P= Applied axial load, d= Dia of minimum cross section

In our case P=15000 lbf,

in

sidebothondepthThreadpowerscrewofDiaOuterd

375.1

)(8

2)(

8

51

Hence,

psinom 60.101012^375.1

150004

Maximum Stress

H – Convergence:



factor based on” Richard

M. Phelan fig-6-14

Shoulder Fillet, Maximum Stress Convergence,Axial Loading

No of

Elements

Max Stress in axial

direction(yy)

%Convergence

206 17656.85 10.96

1034 19585.74 1.23

2219 19802.49 0.14

Exact Solution

19829.93 Nominal

Stress 10101.60

1.9630

09.0375.1

125.0

d

rRef

fig:1

27.1375.1

75.1

d

D

Kt = 1.8

Clearly Kt in case of

shoulder fillet is much

smaller than simple

groobe(2.3), the stress

concentration is less for

fillet, which implies that

fillet design is better.

100

Sex

SfeSex

Stress Nominal


Shoulder Fillet ,Torsion Loading


Neutral Axis of Shaft is aligned in yy world coordinate.


constrained .

Torsion load of 3000 lbf-in in tangential direction as

shown.

Force applied is force per unit area

psihdd

T27.577

75.16875.26875.0

3000

)2/(2)2/(

P_Convergence:

Again, it is clear that polynomial convergence in case of finest mesh (H3) is very precise.

Stress Distribution: max_stress_prin

J

cTnom

Where, T= 3000 lbf-in,

c= distance of extreme fiber of minimum cross section from neutral axis = 1.375/2 =d/2

32

4^dJ

in

sidebothondepthThreadpowerscrewofDiaOuterd

375.1

)(8

2)(

8

51

Hence,

psid

T

J

cTnom 37.5877

3^375.1

300016

3^

16

Maximum Stress

H – Convergence:



21

21

2

2)^(

nnn

nnn

exact

, Where ‘n’ is the finest mesh and ‘n-1’ and ‘n-2’ are previous two meshes.:


factor based on

Richard M. Phelan fig

– 6-17

Shoulder Fillet, Maximum Stress Convergence,Torsional Loading

No of

Elements

Max Principal Stress

Magnitude

%Convergence

206 9319.64 3.41

1034 9137.89 1.39

2602 9063.65 0.57

Exact Solution

9012.38 Nominal

Stress 5877.37

1.5334

11.0125.1

125.0

d

rRef

fig:1

22.1125.1

375.1

d

D

Kts = 1.45

100

Sex

SfeSex

Stress Nominal


Shoulder Fillet ,Combined Loading


Please refer previous definitions of axial and

torsion loading, both are acting simultaneously.

Please refer to H- Convergence table for number of elements. The initial mesh geometry is almost same.

P_Convergence: (Von mises stress is considered here, as it combines all the shear and tensile stress with the calculation

of principal stress , the eigen values of matrix of shear and normal stresses )

Stress Distribution: max_stress_vm

Maximum Stress

H – Convergence:


Stress concentration factor based on FEM computation

Shoulder Fillet Maximum Stress Convergence,Combined Loading

No of

Elements

Maximum Von Mises Stress

psi

%Convergence

206 23451.72 4.45

1034 24508.23 0.15

1394.00 24542.68 0.00

Exact Solution

24543.84 Nominal

Stress 12800.00

1.9175

100

Sex

SfeSex

Stress Nominal


Conclusion:

Refinement of mesh by increasing minimum allowable angle between edges gives more accurate result as

compared to reduction of only aspect ratio.

Defining manual nodes at appropriate locations smooths out mesh, giving more reliable results.

Stress concentration in tensile loading is higher than torsion loading.

Stress concentration for shoulder fillet is much less than simple groove.

Shoulder fillet model is more appropriate for the type of loading stated in the problem.

PRAT B

PART B:

The factor of safety is defined as:

In the case of fatigue loading, the material property which determines the life of the part is the endurance limit. This

value depends on the quality of the surface finish of the part. In our case, the part has ground finish. The material of the

part is assumed to be ductile. The following diagram summarizes the Soderberg criteria:

In the triangle, the failure line connects Se and Sy. After a value of factor of safety is chosen, a line is drawn from Se/f.s.

to Sy/f.s. This line is the locus of all combinations of steady and reversed loads which correspond to the chosen factor of

safety. The triangle is called Soderberg triangle. The equation for the factor of safety is:

For: 10X10^6 cycles

In the calculations, the symbols above are replaced by the following symbols for stress:

Se = σe (endurance limit)

Sy = σy (yield strength)

Sr = σ0 (reversed stress component)

Sav = σav (average stress component)

For the purpose of calculation of fatigue life we consider the diameter d to be 1.125 inches. Using the following

formulae for torsion and axial stress:

Tmax = 3000 lbf-in (Maximum Torque)

Tmin = -300 lbf-in (Minimum Torque)

Pmax = 15000 lbf (Maximum Axial Force)

Pmin = 2000 lbf (Minimum Axial Force)

psipsid

TKts

J

cTKts

psipsid

TKts

J

cTKts

28.166308.107355.13^125.1

)300(1655.1

3^

min16min

802.1663284.1073055.13^125.1

30001655.1

3^

max16max 1

We also need to consider the notch sensitivity. Notch sensitivity is a ratio of fatigue factor ordinate to the theoretical-

factor ordinate in the graph of notch radius to stress concentration factor. (figure 6-10, Fundamentals of Mac. Des. ,

psipsiKt

psipsiKt

67.462703.20123.22^125.1

20004min

55.3470724.150903.22^125.1

150004max

Phelan).

Hence,

Kf is the factor for axial or bending loads, and Kfs is the factor for torsion or shear. The material properties given are:

Ultimate tensile strength = 148 ksi

Yield Strength = 112 ksi

As calculate in part A:

Kt = 2.3

Kts = 1.55

The notch sensitivity factor is the measure of the sensitivity of a specimen to the presence of a notch. The value above is

obtained for annealed Steel for a notch radius of 0.125 inches (Fig. 6-11, ‘Average notch sensitivity’, Fundamentals of

Mac. Des. , Phelan)

Notch sensitivity factor q = 0.9

So, from the equations above:

Kf = 2.17

Kfs= 1.49

We take the endurance limit psie 68000 for ground finish (Fig. 6-2, ‘Endurance limit of steel as related to ultimate

strength and surface condition’, Fundamentals of Mac. Des., Phelan]

σe (endurance limit)=68000psi

σy (yield strength)=112000psi

Putting the respective values,

We get Infinite life fs = 2.36.

Part C:

My design is based on the stress concentration of thread run out groove. All the analysis done in part A reflects that the

maximum stress is in the region of groove.

There one more region, the flange fillet, which is to be addressed, since we have sudden change in cross section at

meeting section of shaft and flange. Although the fillet is provided to take care of the change in cross section, the radius

of fillet is of much importance while considering stress concentration.

Since the flange is resting on the thrust bearing , we have to constrain the bottom surface of the flange to study the

effect of stress concentration in flange fillet.

Loading : Torsion 3000lbf-in, Tensile Load: 15000lbf

Mesh Generation for Part_C

Final Mesh:

Points: 323

Edges: 1730

Faces: 2614

Elements: 1206

Maximum Stress

From above figure it is evident that Stress at outer edge of the fillet is very high.

Let us refine the mesh little further

Points: 502

Edges: 2766

Faces: 4237

Elements: 1972

Max Von Mises Stress=59009.7psi . This value is very high as comparison to max von mises stress in case of simple

groove combined loading, which was 42657 psi.

383.142657

7.59009

max

max

groovesimple

filletflange

The infinite life safety factor is calculated based on the groove design. However, the stress generation at outer periphery

of the flange fillet is 1.383 times greater.

The infinite life safety factor is 2.36, which leads to allowable maximum stress to 2.36*42657=100670.52psi. The max

stress in flange fillet is well within the range. It appears that “ filletflangemax ” might not cause the crack.

However, we have to keep in mind that the shaft is subjected to alternating loading. The F.S calculation is considered for

10X10^6 cycles. Since filletflangemax is higher flange will not sustain the loading for 10X10^6 cycles.

Hence there might be two cases:

Case 1 : Shaft running time < < 10X10^6 cycles (Very less than 10X10^6 cycles)

The crack might have appeared because of mishandling or overloading of the shaft.

Case 1 : Shaft running time close to 10X10^6 cycles

The crack might is appeared due to wrong design.

MAE 577 PROJECT 2 FEM ANALYSIS OF POWER SCREW

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