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Studies on metal matrix composite railroad wheels
C0mpw.m & Slructures Vol. 47, No. 2. pp. 2.59-263, Printed in Great Britain.
STUDIES
1993
0
004%7949/93 S6.a) + 0.00 1993 Peigamon Pms Ltd
ON METAL MATRIX COMPOSITE RAILROAD WHEELS T. C. RAMESH and N. GANESAN
Machine Dynamics Laboratory, Department of Applied Mechanics, Indian Institute of Technology,
Madras-600 036, India (Received
22
Jtznuary
1992)
Abstract-The use of metal matrix composites in railway wheei applications is explored in this paper. Static and dynamic analysis of railroad wheels made of metal matrix composites are made using the finite element method. The deflections, stresses, factor of safety and natural frequencies of these wheels are compared with those in steel as well as graphite/epoxy wheels.
INTRODt3ClTON
THEFINFFEELEMENT
The use of composite materials in the transportation industry has been increasing at a rapid rate due to their now well known properties such as high specific strength and specific stiffness. The composites which have been quite widely used are polymer-based (fibre reinforced plastics, FRP). Another type of composite which is rapidly developing is the metal matrix composite (MMC). MMCs are metals reinforced with continuous or discontinuous fibres, whiskers or particulates. The matrix can be any suitable metal or alloy, but current technology is confined mainly to aluminium, magnesium, titanium or some super-alloy [l J. The density of most MMCs is about one third that of steel and they have stiffnesses from one third to more than two times that of steel. Therefore, the specific strength and stiffnesses are high and MMCs offer potentially large weight reductions-an important factor in the transportation industry. Thus MMCs offer the same features as FRP but with added advantages such as higher tem~rature operating limits and better corrosion resistance. Recently the authors studied the feasibility of using an FRP material for railway wheels using the finite element method [2]. The present work studies the use of MMCs for railway wheels from the static and dynamic points of view. The stresses, deflections, factors of safety and natural frequencies of railway wheels made of MMC are presented and these quantities are compared with those in wheels made of steel and FRP. The semi-analytical finite element method is made use of, where the wheel is treated as an axisymmetric body. Fourier expansions are used in the circumferential direction. Conical shell elements are used in the modelling of the wheel. In the case of composite materials, the use of a thick shell theory is appropriate and hence the shell element used is based on the Reissner-Naghdi shell theory [3].
A thick conical shell finite element based on the Reissner-Naghdi shell theory incorporating the effects of transverse normal strain and shear defo~ation has been developed by Sivadas and Ganesan (41. The element is shown in Fig. 1. To accommodate the transverse normal strain, the displacements in slant length (s), circumferential (0) and radial [from the reference surface (z)] coordinate directions are represented by
where 8, and fiz represent the rotations of the tangents to the reference surface of the shell about the s and 8 coordinates, and & and fi4 are terms contributing to a non-vanishing transverse normal strain. The general strain-displacement relations in linear elasticity in an orthogonal shell coordinate system are given by Kraus [3]. Particula~zing the straindisplacement relations for conical coordinates, slant length (s), circumferential (0) and radial coordinates from the reference surface (z) (Fig. l), and substituting the displa~ment functions [eqn (I)], we get
cc4 =
1 au, U, . ;ds+Tsmq+T
1 1 + (z/R*)
2
AC&CI!Sinp,+.& r
259
f%
r
+“2” R2 >
2%
1
260
T. C. RAMESHand N.
GANESAN
Fig. 1. Conical shell element (thick). G=
B3+ ZA au0
v2
i au, -UOsinrp
1
__ K rae
YsO=ds+Zc?r+l+(z/R,)
r
>
>I ab ysz=ds +zg +g$f +/?, fz
1 ah k4 ----sincp r ( i-ae
1 yeZ=
i aw,
zap,
ANALYSIS
z*ap,
--+;de+zr,-$+s2, ( r ae
l+(z/R,)
2
>
(2) where r is the reference surface radius at any point, R, = r/cos cp. A three-noded isoparametric semi-analytical finite element with seven degrees of freedom per node is now derived to solve the problem. This type of element has been discussed by Weaver and Johnston [5]. The displacement parameters associated with the element are given by
uo, 2
00.2
wo,*
.”
B 3.3
The geometry and finite element discretizations used for the wheels are shown in Figs 2 and 3. The dimensions of the wheel have been taken from [6]. The axle is considered as rigid and hence in the finite element model, node 1 is clamped. The materials used in the study are boron/aluminium and silicon carbide/aluminium (Sic/Al). The properties of these materials can be found in [7, 81. RESULTS AND DISCUSSION
Static analysis
Railway wheels are subjected to vertical and lateral loads. The stresses, deflections and factor of safety of the wheels are determined. The factor of safety (FS) is obtained from the Tsai-Hill failure criterion which is stated as (from Jones [9])
B4.317
where the subscripts 1 and 2 denote the directions parallel and perpendicular to the fibre direction,
r- .500
r-500
Flange
where subscripts 1, 2, 3 denote the nodes of the element. Using conventional finite element procedures, one can get the stiffness and mass matrices for the element.
-
400
.300
.200 .
2 s
100
100
Wheel
I Fig. 2. Cross-section
Wheel
II
of wheels.
-.-.-.Fig. 3. Discretizations
-.-.-.using conical shell element
Metal matrix composite railroad wheels
261
30 Msterirl : Boron/Al
Material : Boron/Al 20 T B U 5w
10 o-
z 5 -10 VI -20 ,111
100
150
200 250 300 Radius (mm)
350
-30
400
100
I 150
I 200
I 250
I 300
I 350
I 400
Radius (mm)
Fig. 4. Variation of stresses due to lateral load (wheel I).
Fig. 6. Variation of stresses due to lateral load (wheel II).
respectively, X and Y denote the strengths along and perpendicular to the fibre direction and S denotes the shear strength.
Figures 6 and 7 show the variation of stresses in wheel II due to the lateral load. Here too, similar results as with wheel I are observed. The magnitude of the radial and tangential stresses are higher than those in wheel I.
Wheels subjected to a lateral load A lateral load of 10,000 kgf is assumed to act on the wheel. The stresses are maximum in the plane of loading and these are represented in the various figures. The notations for subscripts and superscripts are as follows. The superscripts f and c denote the flange and counterflange side of the wheels and the subscripts r and t denote radial and tangential stresses, respectively. Figures 4 and 5 show the variation of stresses in wheel I due to the lateral load. The pattern of variation of stresses for the two materials are almost the same. The tangential stresses are nearly constant throughout the web of the wheel. The maximum value of the tangential stresses differs slightly in the case of the two materials; boron/aluminium having the lower value.
Wheels subjected to a vertical load A vertical load of 18,000 kgf is assumed to act on the wheel. The variation of stresses in the two wheels due to the vertical load are shown in Figs 8-l 1. The stresses are predominantly compressive and the maximum radial stresses on the flange and counterflange side occur at the rim and hub fillets in the case of wheel I and the hub and rim fillets, respectively, in the case of wheel II. A comparison of the deflections and factor of safety of FRP and steel wheels is now presented. As seen from the discussion on stresses, the lateral load gives rise to higher stresses. Hence, the deflections and factor of safety are presented for the wheels subjected to lateral load. These are given in Table I.
25
30 Material : Sic/Al 20 r? k
lo
c 5Om o2 f -10 v1 -20 -
05 100
150
200 250 300 Radius (mm)
350
400
Fig. 5. Variation of stresses due to lateral load (wheel I).
-30350 100
150
300 200 250 Radius (mm)
400
Fig. 7. Variation of stresses due to lateral load (wheel II).
T. C.
262
Material
-lOI 100
is0
200
250
RAMFSH and
N.
GANESAN
: Boron/Al
300
350
400
-10
’ 100
I
I
I
I
I
1
150
200
250
300
350
400
Radius (mm)
Radius (mm) Fig. 8. Variation of stresses due to vertical load (wheel I).
Material
I 1
150
I
I 250
200
Radius
I 300
: BoronlAl
Material
Fig. IO. Variation of stresses due to vertical load (wheel II).
: Sic/Al
I 350
Material
I
-10
400
100
: Sic/AI
I
I
I
I
I
150
200
250
300
350
(mm)
Radius
1 400
(mm)
Fig. 9. Variation of stresses due to vertical load (wheel I).
Fig. 11. Variation of stresses due to vertical load (wheel II).
From Table 1, it can be seen that the maximum deflection of MMC wheels is much lower than in the case of the FRP wheel. Thus, the increase in web thickness suggested in [lo] in order to reduce the deflections may not be needed in the case of MMC wheels.
steel and grap~te/epoxy wheels (from [ 1I]) are also listed in the tables. The fundamental frequency of the MMC wheels are higher than those of steel and graphite/epoxy wheels. The difference between the fundamental frequency of the two MMCs is not large.
Dynamic analysis: evaluation of natural frequencies
The natural frequencies of the wheels are determined using standard eigenvalue solvers. The frequencies corresponding to zero nodal diameters for the first few circumferential modes are presented in Tables 2 and 3. The corresponding frequencies for
CONCLUSIONS
The use of metal matrix composites for railway wheels has been investigated from the static strength point of view. The natural frequencies of these wheels have also been found and compared with steel and
Table 1. Deflection and factor of safety for different materials Wheel I Material Steel [lo] Graphite/epoxy [lo] Boron/Al SiClAI
Wheel II
Deflection (mm)
FS
Deflection (mm)
FS
0.593 3.510 0.960 0.895
1.91 2.52 3.84 1.95
0.677 3.590
1.089
1.85 2.06 3.25
1.037
1.93
Metal matrix composite railroad wheels
263
Table 2. Frequencies in hertz for wheel I steel [ 111 Graphite/epoxy [ 1I] Boron/Al Sic/Al
ha
Jil
419.2 384.1 660.7 586.4
280.1 321.3 405.7 391.9
j-20 490.4 387.1 629.6 608.9
h 1253.9 751.1 1640.3 1527.6
ha 2233.9 1288.2 2893.2 2712.6
Table 3. Frequencies in hertz for wheel II Steel [1l] Graphite/epoxy [l 1] Boron/Al Sic/Al
f&l
flo
fm
401.5 397.2 616.8 545.8
310.1 374.3 448.9 428.8
474.9 427.6 624.2 601.8
graphite/epoxy wheels. As metal matrix composites are lightweight and have strengths comparable to that of steel, they could be an exciting prospect in railway wheels in particular and in the transportation industry in general. Other aspects such as thermal strength, wear and fatigue properties, etc. will certainly need to be investigated before deciding on their suitability.
REFERENCES
S. M. Lee (Ed.), Reference Book for Composites Technology, Vol. 1. Edited by Technomic, Lancaster (1989). N. Ganesan and T. C. Ramesh, Stress analysis of composite railroad wheels. Comput. Struct. 40, 805-813 (1991). H. Kraus, Thin Eiustic Shells. John Wiley, New York (1967).
h 1123.2 701.8 1526.0 1381.0
fuJ 2003.0 1152.3 2712.1 2451.0
4. K. R. Sivadas and N. Ganesan, Vibration analysis of thick composite clamped conical shells of varying thickness. J. Sound Vibr. 152, 27-37 (1992). 5. W. Weaver Jr and P. R. Johnston, Finite Elements for Structural Analysis. Prentice-Hall, Englewood Cliffs, NJ (1984). 6. K. Nishioka and Y. Morita, The strength of railroad wheels (3rd report. Stress analysis on wheels subjected to lateral and vertical forces.) Bull. LYME 14, 1l-19 (1971). 7. W. S. Johnson (Ed.), Metal Matrix Composites, Testing, Analysis and Failure Modes, pp. 19-42. ASTM STP 1032 (1989). 8. G. J. Dvorak and C. Zweben, Metal Mutrix Composites-A Short Course (Seminar Notes). Technomic, Lancaster (1987). 9. R. M. Jones, Mechanics of Composite Materials. McGraw-Hill, New York (1975). 10. N. Ganesan and T. C. Ramesh, Comparison of shell theories in the analysis of railroad wheels. Composite Srruct. 21, 75-84 (1992). Il. N. Ganesan and T. C. Ramesh, Free vibration analysis of composite railway wheels. J. Sound Vibr. 153, 113-124 (1992).
STUDIES
1993
0
004%7949/93 S6.a) + 0.00 1993 Peigamon Pms Ltd
ON METAL MATRIX COMPOSITE RAILROAD WHEELS T. C. RAMESH and N. GANESAN
Machine Dynamics Laboratory, Department of Applied Mechanics, Indian Institute of Technology,
Madras-600 036, India (Received
22
Jtznuary
1992)
Abstract-The use of metal matrix composites in railway wheei applications is explored in this paper. Static and dynamic analysis of railroad wheels made of metal matrix composites are made using the finite element method. The deflections, stresses, factor of safety and natural frequencies of these wheels are compared with those in steel as well as graphite/epoxy wheels.
INTRODt3ClTON
THEFINFFEELEMENT
The use of composite materials in the transportation industry has been increasing at a rapid rate due to their now well known properties such as high specific strength and specific stiffness. The composites which have been quite widely used are polymer-based (fibre reinforced plastics, FRP). Another type of composite which is rapidly developing is the metal matrix composite (MMC). MMCs are metals reinforced with continuous or discontinuous fibres, whiskers or particulates. The matrix can be any suitable metal or alloy, but current technology is confined mainly to aluminium, magnesium, titanium or some super-alloy [l J. The density of most MMCs is about one third that of steel and they have stiffnesses from one third to more than two times that of steel. Therefore, the specific strength and stiffnesses are high and MMCs offer potentially large weight reductions-an important factor in the transportation industry. Thus MMCs offer the same features as FRP but with added advantages such as higher tem~rature operating limits and better corrosion resistance. Recently the authors studied the feasibility of using an FRP material for railway wheels using the finite element method [2]. The present work studies the use of MMCs for railway wheels from the static and dynamic points of view. The stresses, deflections, factors of safety and natural frequencies of railway wheels made of MMC are presented and these quantities are compared with those in wheels made of steel and FRP. The semi-analytical finite element method is made use of, where the wheel is treated as an axisymmetric body. Fourier expansions are used in the circumferential direction. Conical shell elements are used in the modelling of the wheel. In the case of composite materials, the use of a thick shell theory is appropriate and hence the shell element used is based on the Reissner-Naghdi shell theory [3].
A thick conical shell finite element based on the Reissner-Naghdi shell theory incorporating the effects of transverse normal strain and shear defo~ation has been developed by Sivadas and Ganesan (41. The element is shown in Fig. 1. To accommodate the transverse normal strain, the displacements in slant length (s), circumferential (0) and radial [from the reference surface (z)] coordinate directions are represented by
where 8, and fiz represent the rotations of the tangents to the reference surface of the shell about the s and 8 coordinates, and & and fi4 are terms contributing to a non-vanishing transverse normal strain. The general strain-displacement relations in linear elasticity in an orthogonal shell coordinate system are given by Kraus [3]. Particula~zing the straindisplacement relations for conical coordinates, slant length (s), circumferential (0) and radial coordinates from the reference surface (z) (Fig. l), and substituting the displa~ment functions [eqn (I)], we get
cc4 =
1 au, U, . ;ds+Tsmq+T
1 1 + (z/R*)
2
AC&CI!Sinp,+.& r
259
f%
r
+“2” R2 >
2%
1
260
T. C. RAMESHand N.
GANESAN
Fig. 1. Conical shell element (thick). G=
B3+ ZA au0
v2
i au, -UOsinrp
1
__ K rae
YsO=ds+Zc?r+l+(z/R,)
r
>
>I ab ysz=ds +zg +g$f +/?, fz
1 ah k4 ----sincp r ( i-ae
1 yeZ=
i aw,
zap,
ANALYSIS
z*ap,
--+;de+zr,-$+s2, ( r ae
l+(z/R,)
2
>
(2) where r is the reference surface radius at any point, R, = r/cos cp. A three-noded isoparametric semi-analytical finite element with seven degrees of freedom per node is now derived to solve the problem. This type of element has been discussed by Weaver and Johnston [5]. The displacement parameters associated with the element are given by
uo, 2
00.2
wo,*
.”
B 3.3
The geometry and finite element discretizations used for the wheels are shown in Figs 2 and 3. The dimensions of the wheel have been taken from [6]. The axle is considered as rigid and hence in the finite element model, node 1 is clamped. The materials used in the study are boron/aluminium and silicon carbide/aluminium (Sic/Al). The properties of these materials can be found in [7, 81. RESULTS AND DISCUSSION
Static analysis
Railway wheels are subjected to vertical and lateral loads. The stresses, deflections and factor of safety of the wheels are determined. The factor of safety (FS) is obtained from the Tsai-Hill failure criterion which is stated as (from Jones [9])
B4.317
where the subscripts 1 and 2 denote the directions parallel and perpendicular to the fibre direction,
r- .500
r-500
Flange
where subscripts 1, 2, 3 denote the nodes of the element. Using conventional finite element procedures, one can get the stiffness and mass matrices for the element.
-
400
.300
.200 .
2 s
100
100
Wheel
I Fig. 2. Cross-section
Wheel
II
of wheels.
-.-.-.Fig. 3. Discretizations
-.-.-.using conical shell element
Metal matrix composite railroad wheels
261
30 Msterirl : Boron/Al
Material : Boron/Al 20 T B U 5w
10 o-
z 5 -10 VI -20 ,111
100
150
200 250 300 Radius (mm)
350
-30
400
100
I 150
I 200
I 250
I 300
I 350
I 400
Radius (mm)
Fig. 4. Variation of stresses due to lateral load (wheel I).
Fig. 6. Variation of stresses due to lateral load (wheel II).
respectively, X and Y denote the strengths along and perpendicular to the fibre direction and S denotes the shear strength.
Figures 6 and 7 show the variation of stresses in wheel II due to the lateral load. Here too, similar results as with wheel I are observed. The magnitude of the radial and tangential stresses are higher than those in wheel I.
Wheels subjected to a lateral load A lateral load of 10,000 kgf is assumed to act on the wheel. The stresses are maximum in the plane of loading and these are represented in the various figures. The notations for subscripts and superscripts are as follows. The superscripts f and c denote the flange and counterflange side of the wheels and the subscripts r and t denote radial and tangential stresses, respectively. Figures 4 and 5 show the variation of stresses in wheel I due to the lateral load. The pattern of variation of stresses for the two materials are almost the same. The tangential stresses are nearly constant throughout the web of the wheel. The maximum value of the tangential stresses differs slightly in the case of the two materials; boron/aluminium having the lower value.
Wheels subjected to a vertical load A vertical load of 18,000 kgf is assumed to act on the wheel. The variation of stresses in the two wheels due to the vertical load are shown in Figs 8-l 1. The stresses are predominantly compressive and the maximum radial stresses on the flange and counterflange side occur at the rim and hub fillets in the case of wheel I and the hub and rim fillets, respectively, in the case of wheel II. A comparison of the deflections and factor of safety of FRP and steel wheels is now presented. As seen from the discussion on stresses, the lateral load gives rise to higher stresses. Hence, the deflections and factor of safety are presented for the wheels subjected to lateral load. These are given in Table I.
25
30 Material : Sic/Al 20 r? k
lo
c 5Om o2 f -10 v1 -20 -
05 100
150
200 250 300 Radius (mm)
350
400
Fig. 5. Variation of stresses due to lateral load (wheel I).
-30350 100
150
300 200 250 Radius (mm)
400
Fig. 7. Variation of stresses due to lateral load (wheel II).
T. C.
262
Material
-lOI 100
is0
200
250
RAMFSH and
N.
GANESAN
: Boron/Al
300
350
400
-10
’ 100
I
I
I
I
I
1
150
200
250
300
350
400
Radius (mm)
Radius (mm) Fig. 8. Variation of stresses due to vertical load (wheel I).
Material
I 1
150
I
I 250
200
Radius
I 300
: BoronlAl
Material
Fig. IO. Variation of stresses due to vertical load (wheel II).
: Sic/Al
I 350
Material
I
-10
400
100
: Sic/AI
I
I
I
I
I
150
200
250
300
350
(mm)
Radius
1 400
(mm)
Fig. 9. Variation of stresses due to vertical load (wheel I).
Fig. 11. Variation of stresses due to vertical load (wheel II).
From Table 1, it can be seen that the maximum deflection of MMC wheels is much lower than in the case of the FRP wheel. Thus, the increase in web thickness suggested in [lo] in order to reduce the deflections may not be needed in the case of MMC wheels.
steel and grap~te/epoxy wheels (from [ 1I]) are also listed in the tables. The fundamental frequency of the MMC wheels are higher than those of steel and graphite/epoxy wheels. The difference between the fundamental frequency of the two MMCs is not large.
Dynamic analysis: evaluation of natural frequencies
The natural frequencies of the wheels are determined using standard eigenvalue solvers. The frequencies corresponding to zero nodal diameters for the first few circumferential modes are presented in Tables 2 and 3. The corresponding frequencies for
CONCLUSIONS
The use of metal matrix composites for railway wheels has been investigated from the static strength point of view. The natural frequencies of these wheels have also been found and compared with steel and
Table 1. Deflection and factor of safety for different materials Wheel I Material Steel [lo] Graphite/epoxy [lo] Boron/Al SiClAI
Wheel II
Deflection (mm)
FS
Deflection (mm)
FS
0.593 3.510 0.960 0.895
1.91 2.52 3.84 1.95
0.677 3.590
1.089
1.85 2.06 3.25
1.037
1.93
Metal matrix composite railroad wheels
263
Table 2. Frequencies in hertz for wheel I steel [ 111 Graphite/epoxy [ 1I] Boron/Al Sic/Al
ha
Jil
419.2 384.1 660.7 586.4
280.1 321.3 405.7 391.9
j-20 490.4 387.1 629.6 608.9
h 1253.9 751.1 1640.3 1527.6
ha 2233.9 1288.2 2893.2 2712.6
Table 3. Frequencies in hertz for wheel II Steel [1l] Graphite/epoxy [l 1] Boron/Al Sic/Al
f&l
flo
fm
401.5 397.2 616.8 545.8
310.1 374.3 448.9 428.8
474.9 427.6 624.2 601.8
graphite/epoxy wheels. As metal matrix composites are lightweight and have strengths comparable to that of steel, they could be an exciting prospect in railway wheels in particular and in the transportation industry in general. Other aspects such as thermal strength, wear and fatigue properties, etc. will certainly need to be investigated before deciding on their suitability.
REFERENCES
S. M. Lee (Ed.), Reference Book for Composites Technology, Vol. 1. Edited by Technomic, Lancaster (1989). N. Ganesan and T. C. Ramesh, Stress analysis of composite railroad wheels. Comput. Struct. 40, 805-813 (1991). H. Kraus, Thin Eiustic Shells. John Wiley, New York (1967).
h 1123.2 701.8 1526.0 1381.0
fuJ 2003.0 1152.3 2712.1 2451.0
4. K. R. Sivadas and N. Ganesan, Vibration analysis of thick composite clamped conical shells of varying thickness. J. Sound Vibr. 152, 27-37 (1992). 5. W. Weaver Jr and P. R. Johnston, Finite Elements for Structural Analysis. Prentice-Hall, Englewood Cliffs, NJ (1984). 6. K. Nishioka and Y. Morita, The strength of railroad wheels (3rd report. Stress analysis on wheels subjected to lateral and vertical forces.) Bull. LYME 14, 1l-19 (1971). 7. W. S. Johnson (Ed.), Metal Matrix Composites, Testing, Analysis and Failure Modes, pp. 19-42. ASTM STP 1032 (1989). 8. G. J. Dvorak and C. Zweben, Metal Mutrix Composites-A Short Course (Seminar Notes). Technomic, Lancaster (1987). 9. R. M. Jones, Mechanics of Composite Materials. McGraw-Hill, New York (1975). 10. N. Ganesan and T. C. Ramesh, Comparison of shell theories in the analysis of railroad wheels. Composite Srruct. 21, 75-84 (1992). Il. N. Ganesan and T. C. Ramesh, Free vibration analysis of composite railway wheels. J. Sound Vibr. 153, 113-124 (1992).