ME 751 - Group 3 - Project 2

ME 751 Project: Nonlinear analysis of a car jack A. Bayat, S. Schwade

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ME 751 – Project II

NONLINEAR ANALYSIS OF A

CAR JACK

Alireza Bayat: 150% contribution

Steve Schwade: 150 % contribution


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Introduction The objective of this project is to determine the load bearing capability, the failure mode and

failure location of a car jack under a compressive load P. Both geometric and material

nonlinearities are to be considered in the analysis.

Figure 1: Car jack

In order to find the load carrying capacity of the jack, an analytical (static equilibrium) study

was conducted to find the members carrying the highest percentage of the external load P.

Buckling theory was also used to find the critical load for each member. The maximum load

for each member was then determined by requiring that the stress be less than the yield stress

of the material or that the load be less than the critical buckling load for that member.

In all calculations and both analytical and FE analysis are done with a half model of the car

jack.

h

P


3

Static Analysis

Figure 2: Geometry of the structure with the required angles and lengths

In order to find the portion of the external load in each member the geometry much first be

determined. Trigonometric identities were use in the triangles that are formed by the

members to find the geometric properties. The detailed calculations are listed in appendix A.

Given:


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And the required angles and lengths are:

Table 1: Required angles and length for study

Calculated Dimensions

h 6 in

ᶲ 26.1˚

ᶱ 129.83˚

ᵞ 15.07˚

ᵝ 105.98˚

ᵅ 23.33˚


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Analytical study

Considering equilibrium equations in individual members, loads in different members were

calculated (Appendix B).

Figure 3: required unknown loads for Analytical and FE analysis

0

1000

2000

3000

4000

5000

6000

7000

8000

0 1000 2000 3000 4000 5000 6000

Load

s in

me

mb

ers

(lb

)

P (lb)

Loads vs P for DEF link

F

HD

CE


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Buckling

Considering buckling in members the critical loads in individual members were calculated and

the critical member DH is determined.

Critical load in DH =

1880 lbs

Critical P = 2800 lbs

Figure 4: Buckling model in DH member


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FE Analysis Two different approaches were studied in order to find the critical loads in the members; 1)

Individual members and 2) Whole structure study.

Individual members study

Based on the calculated loads from the theoretical study three critical members (DEF, DH,

and CE) were identified to be analyzed using a nonlinear finite element analysis – including

linear and nonlinear1 material properties.

Figure 5: Critical members in members

1 See Appendix C for nonlinear material properties.


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Boundary Conditions

In order to study contacts in pin areas in members different models were tried. A model with

1 pin, 2pin and without pins (Fig6, 7). The models with pins consider contact which should

yield a more accurate result; unfortunately it led to a non-convergent simulation in the

models with nonlinear material properties.

Figure 6: DEF - with 1 pin and contact in joint D

Figure 7: DEF - with 2 pins and contacts in joint E and F


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Chosen Boundary conditions for DEF, DH and CE

Comparing stress results for linear solution from the different models (1pin, 2 pin and without pins)

the maximum stress in the models with pins were identical to models without pins. So the results

from the model with bearing loads only would be reasonable enough for study.

Figure 8: DEF - Boundary conditions with no pins and bearing loads

Figure 9: DH - Boundary conditions with no pins and bearing loads


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Figure 10: CE - Boundary conditions with no pins and bearing loads

Linear material-Linear FEA results for DEF

Figure 11: DEF - Von-Mises stress result

Figure 12: DEF - Deformation results


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Linear Material-Nonlinear solution for DEF

Figure 13: DEF - Von-Mises stress results

Figure 14: DEF - Deformation results


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Nonlinear Material FEA results

Figure 15: DEF - Nonlinear Material Von-Mises stress results

Figure 16: DEF - Nonlinear Material deformation results


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FE analyses for links DH and CE were conducted as well, the results are shown in Table 1 for 3

cases:

1) Linear Material- Linear Solution (LM-LS)

2) Linear Material- Nonlinear Solution (LM-NLS)

3) Nonlinear Material (NLM)

Table 2: FEA results for different solutions for three critical members DEF, DH and CE2

P = 2000 lbs.

Link

Max Stress (Psi) Max Deformation (10^-4 in)

LM-LS LM-NLS NLM LM-LS LM-NLS NLM

DEF 37959 38171 32601 145.54 156.47 320.0

DH 29427 29434 24580 3.1427 1.459 2.940

CE 6300 6300 5779 3.3165 3.3116 6.341

The critical member in the jack is DEF as expected because of the moment in the link. So based

on trial and error – FE analyses was performed with varying loads. Failure was found to occur at

a load of 1,500 pounds.

Table 3: FE analyses results for P = 3000 lbs. that failure occurred

P = 3000 lbs.

Link

Max Stress (Psi) Max Deformation (10^-4 in)

LM-LS LM-NLS NLM LM-LS LM-NLS NLM

DEF 56939 57429 44714 218.27 243.8 519.34

2 LM: Linear material properties used in simulation, NLM: Nonlinear mater properties used in nonlinear simulation,

LS: Linear simulation used, NLS: Nonlinear simulation used


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Complete Structure Analysis

In order to validate our static analysis and boundary condition assumptions a FEA of the complete structure was needed to establish the nonlinear member interactions after deformation.

Figure 17: nonlinear structure analysis Von-Mises Stress Distribution3

The simulation was conducted with nonlinear material properties, revolute joints, and frictional body connections (with µ=0.7). Maximum stress occurs in the member DEF which verifies the outcome of our static equilibrium analysis.

3 Deformation show with 0.5 times auto scale (9.3 times magnification)


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Figure 18: DEF Isolated – Von-Mises Stress Distribution4

Member DEF experiences both displacement and rotation when the entire structure is simulated which invalidates the boundary conditions that were used in the individual member study. This rotation results in a slightly higher equivalent stress distribution and maximum stress (35.3 ksi vs 32.6 for the individual member study).

4 Deformation show with 0.5 times auto scale (9.3 times magnification)


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Empirical Results For h = 6in the jack was setup under an Instron machine and compressive load was

continuously applied on the jack till failure occurred (fig 19). The associated graphs for 3

tested jacks are shown in fig 20.

The average maximum load the jack held was 4387 lbs.

Figure 19: Experimental setup for h = 6 in.


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Figure 20: Experimental results for 3 tested jacks.

Conclusion The static and FEA results predict a similar outcome – failure due to yielding with a load of 3,000 lbs.

This is further confirmed by the experimental failure test data. All three of the jacks that were tested

displayed a nearly linear force vs. displacement curve from 500 to 3,000 pounds, as the load exceeded

3,000 pounds the structure began to deform plastically and it was this plastic deformation that eventually

resulted in the ultimate failure.

Table 4: Maximum load via different approaches [lbs]

Theory FEA individual

members model

FEA Complete

structure model

Experiment

Maximum

carrying load

2800 3000 3000 4387

The difference between experiments with theory can be justified as:

1. Maximum loads estimated in the theory and FEA is based on the safety factor of 1, while safety

factor in the real jack is definitely higher than 1.

2. Maximum stress failure in theory and FEA is based on maximum yielding stress failure, while

the maximum carrying load from the experiments are the failure in plastic region.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1 1.2

Forc

e [

kip

]

Total Displacement [in]

Jack 1

Jack 2

Jack 3


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Feasibility Study Ultimate failure was caused by out of plane buckling that the 2D simulations cannot account for. To

better predict the causes of this failure, and locate areas for improvement, a complete three dimensional

model would need to be simulated.

The two dimensional model identifies that a significant portion of the member DEF is under a

compression stress which is further compounded by the axial force added by member CE. In order to

withstand a greater load without yielding this behavior would need to be addressed.


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Appendix

Appendix A: Geometry

▲BDC ( )( )

,

,

▲ACD ( )( )

√

( )( ) ,

( )( )

▲ADH

( )( ) ( ),

( )( )

Table 5: Coordinates of the points (Xi and Yi)5

Point X (in) Y (in)

A 0 0

B AB*cos AB*sin

C AC*cos AC*sin

D AD*cos( ) AD*sin( )

E XD+ DE*cos YD + DE*sin

F XD + DF*cos YD + DF*sin

G

H AH 0

5 Note: the height of the jack h is the y value of point F + pins to top and bottom surfaces.

θ

D

A

B

C

F

E

β

α

γ

G

H


20


21


22


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Appendix B: Forces in Members

Member HD

( ) ( ) ,

or ( )

,

( )

,

From Free body GFAH as shown

∑ , ( ) ( ) Let HY be the Y component of force H:

( ) ( )

Member GF(Free body GF)

∑ , ( ) ( )

Member DEF (Free body DEF as shown)

( ) ( ) , ∑ ,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

,

,

D

A

B

C

F

E

β

α

γ

G

H

H

Ax

Ay x

y

P

E

1

C

Dx D

E

Dy

Fy

F Fx


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Appendix C: Nonlinear Material Properties

Table 6: Nonlinear Stress & Strain

Figure 21: Nonlinear Stress - Strain Curve

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ME 751 - Group 3 - Project 2

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