a new six-axis load cell - part i design
TRANSCRIPT
A New Six-axis Load Cell.Part I: Design
G. Mastinu & M. Gobbi & G. Previati
Received: 3 December 2009 /Accepted: 24 March 2010 /Published online: 13 May 2010# Society for Experimental Mechanics 2010
Abstract A new high precision six-axis load cell is presentedin two companion papers. The first paper (this one) is focusedon the presentation of the load cell, the conceptual design, themodelling and the embodiment design. The second paperrefers to the error analysis, the construction and theexperimental assessment of the performances. The new loadcell is able to measure the three components of both a forceand a moment acting on the load cell itself. The sensingstructural element of the six-axis load cell is basically a threespoke structure. Strain gauges are conveniently located onhighly stressed areas. The sensing structural element isconstrained to the frame of the load cell by means of specialjoints conceived to avoid friction. The mechanical behaviourof the load cell is described by means of analytical equationsthat allow a quick preliminary design focused on the giventechnical specifications. A finite element model has been usedto asses the mechanical behaviour of the load cell.
Keywords Six-axis load cell . Sliding spherical joint .
Analytical modelling . Calibration
NotationF vector of forces and moments acting at the load
cell centreΕb vector of the strains "i 1¼1;...;6ð ÞCb load cell calibration matrixT vector of reaction forces at the three joints Ni,
Ti i¼1;2;3ð ÞΔV vector of the output voltages ΔVi i¼1;���;6ð Þ at the
Wheatstone bridges
K gauge factorV excitation voltage at the Wheatstone bridgesL spoke lengthMi calibration matrixMtst calibration matrix (statically determined structure)Me experimental calibration matrixMthp calibration matrix (quasi-statically determined
structure)ka joint axial stiffnesskrx/kry joint radial stiffnesseskt joint torsional stiffnesskbt/kb joint bending stiffnesseskx/ky/kz load cell interface stiffnessesS vector of the internal forces and moments acting
at the spoke tipsδi displacements and rotations at a spoke tipRt(α) rotation matrixA(ω) Inertance—transfer function acceleration/forceh distance between a strain gauge and the spoke tip
(see Fig. 4)hz distance between a strain gauge and the three
spoke structure centre (see Fig. 4)ν vector of the bending moments acting on the six
strain gauge bridgesνtst vector of the bending moments—statically
determined structureνthp vector of the bending moments—quasi-statically
determined structureχ load cell sensitivity matrixχtst statically determined structure sensitivity matrixχtrj rigid joints model sensitivity matrixχtrs rigid spokes model sensitivity matrixδν error on the strain (bending) measurementsδχ error on the sensitivity matrixNote: bold symbol refers to a vector or a matrix
G. Mastinu :M. Gobbi (*) :G. PreviatiDepartment of Mechanical Engineering,Politecnico di Milano,Via La Masa, 1,20156 Milan, Italye-mail: [email protected]
Experimental Mechanics (2011) 51:373–388DOI 10.1007/s11340-010-9355-1
Introduction
In experimental testing activities the use of multi axis loadcells is widespread, e. g. balances for wind tunnels [1–4],structures [5–19],.... Many commercially available loadcells are able to measure the steady or dynamic forces andmoments acting on a structure [1, 5–10, 12–25]. Generallysuch measuring devices are quite expensive and their usageis restricted to advanced research and development activi-ties [1, 3, 4]. The high cost is often due to the relativelycomplex hardware, to time consuming calibration, and tospecial materials used for the manufacturing.
The new six-axis load cell presented in this paper is arelatively simple device, and is basically an analog deviceable to measure the three components of a force and thethree components of a moment acting on the load cell itself.
With respect to other similar six-axis load cells, thesensing element of the six-axis load cell is—in principle—astatically determined structure (actually, as it will beexplained later, it is a quasi-statically determined structure).Such a sensing element is shaped as a three spoke structure,constrained to the external body of the load cell (frame) bymeans of three special joints positioned at the tip of eachspoke. A proper design of the joints has been conceived inorder to make friction vanish. Patents are pending on theload cell [25–28].
The six-axis load cell was especially developed for theindoor characterisation of vehicle suspension systems; see[29] for further details. The construction principle of thepresented new six axis load cell has been used extensivelyreferring to measuring hubs (a special class of six-axis loadcells used to measure the forces and moments acting on awheel [11, 25–28, 30–37]).
In the following sections, the load cell concept design [25–28], the mathematical modelling, and the embodiment designwill be presented. Another paper [38], companion of thisone, will present the error analysis, the construction and theexperimental assessment of the performance of the load cell.
Load Cell Design Concept
The sensing element of the six-axis load cell is a threespoke structure [25–28], Fig. 1. The three spoke structure,is constrained to the outer (frame) of the load cell by meansof three special joints positioned at the tip of each spoke.Each joint allows 4 degrees of freedom, i.e. three rotationsand a translation (sliding) along the axis of the spoke. Sucha joint will be named hereafter ‘sliding spherical joint’. Thethree spoke structure constrained by this type of joints (1, 2,3 in Fig. 1) is statically determined. The proof can beobtained simply by inspection of Fig. 1, in which the twoconstraints (for ex. 1, 3) at two adjacent spokes are
equivalent to the virtual rotational joint (1+3). This virtualjoint makes the structure a simply supported beam which isclearly a statically determined structure. The exploitation ofa statically determined structure prevents the negativeeffects due to the non-uniform distribution of the temper-ature on the load cell and the negative effects due to thecompliance of constraints.
When the load cell is loaded, the three spoke structure inFig. 1 is subject to two forces at each spoke tip, a total of 6forces are acting at the three spoke tips. Namely, there are thethree forces T1 T2 T3 and other three forces N1 N2 N3 normalto them. By sensing the bending moments at each spoke root(or acting at a known position along the spokes) the sixforces Ti Ni at the spoke tips are obtained. Given such sixforces, the three forces and the three moments acting at thecentre of the load cell (F) can be computed by solving theequilibrium equations (see next Sect. Mathematical Model).
A special design [25, 28] of the sliding spherical jointshas been performed in order to make friction vanish. After acomprehensive analysis, the sliding spherical joint at thespoke tip has been designed as a circular spoked structure(Fig. 2) (this circular spoked structure has not to beconfused with the other three-spoke sensing structure).Some stiffnesses have been associated to the degrees offreedom of the sliding spherical joint, so actually we havean elastic sliding spherical joint. By employing these elasticjoints, the whole load cell structure is no longer staticallydetermined. Anyway, it can be considered quasi-staticallydetermined as the three sliding spherical joints are actuallysprings with weak stiffness associated to the three rotationsand to the sliding motion along the axis of the spoke.Figure 2 shows the geometry of the developed elasticsliding spherical joint. The central hole is fitted and fixed toeach spoke tip while the outer ring is constrained to theload cell frame.
y
Fx
Fz
23
11+3
y x
z
x3
z3
z2x2
x1
z1
T1
T2
T3
N1
N2
N3
Px
Pyy
Fx
Fz
23
11+3
y x
z
x3
z3
z2x2
x1
z1
T1
T2
T3
N1
N2
N3
Px
Pz
Py
Fig. 1 Concept design of the six-axis load cell suitable to measure the 3forces (Fx, Fy, Fz) and 3 moments (Px, Py, Pz) acting at the centre of theload cell. The concept design refers to a statically determined structure
374 Exp Mech (2011) 51:373–388
As the target is a quasi-statically determined structure,the design of the joint stiffnesses is a fundamental task. Thespoked structure in Fig. 2 has low axial, low torsional andlow bending stiffness compared to radial stiffness. Thedesign of the circular spoked structure (elastic slidingspherical joint) has been successfully performed based onanalytical modelling (see Sect. Mathematical Model, thesame model can be applied to design the elastic slidingspherical joint) and on finite element analysis (Fig. 3). Theaxial stiffness (associated to the sliding motion) is set lessthan 1% of the radial stiffness.
Load Cell Modeling
Mathematical Model
In this section the mechanical and mathematical modellingof the six-axis load cell is presented. Three models havebeen derived and used. At first, a simple model, represent-ing the statically determined structure (shown in Fig. 1), isintroduced. Then, a more refined analytical model referringto the quasi-statically determined structure, is derivedwhich takes into account the actual stiffnesses of the elasticsliding spherical joints (Fig. 3) and the compliance of thethree spokes pertaining to the sensing structure in (Fig. 4).Finally a finite element model is presented.
Let us consider:
– the vector of the forces and of the moments acting atthe load cell F ¼ Fx Fy Fz Fx Fy Fz½ � (seecoordinate system in Fig. 1),
– the vector of the strains measured at the locations shownin Fig. 4 Es¼ "1a "1b "2a "2b "3a "3b "4a "4b "5a "5b "6a "6b½ �T .
The following relationship holds
F ¼ F Esð Þ ð1Þ Fig. 4 Location of the 12 strain gauges
Fig. 3 FEM analysis of the elastic sliding spherical joint. Axial (a),torsional (b) and bending (c) deformations related to the axial,torsional and bending stiffnesses respectively
Fig. 2 Circular spoked (steel) structure as an elastic spherical slidingjoint
Exp Mech (2011) 51:373–388 375
The above equation becomes F ¼ C � ES if a linearrelationship between F and ΕS holds. As stated in theprevious section, half bridge connections of the straingauges have been used for measuring the two bendingmoments. Therefore the vector of the bending strainssignals reduces to Eb ¼ "1 "2 "3 "4 "5 "6½ �T . Therelationship between F and Εb becomes:
F ¼ Cb � Eb ð2Þ
where Cb is a 6×6 constant square matrix referred to as theload cell stiffness matrix.
By considering the ideal statically determined structure(Fig. 1), the vector Εb is directly related to the vector of the6 reaction forces at the three joints T¼ N1 T1 N2 T2 N3 T3½ �T(Fig. 1) (l denotes the length of a single spoke)
N1
T1N2
T2N3
T3
26666664
37777775¼
0 �1=3 0 2=3l 0 0�2=3 0 0 0 �1=3l 00 �1=3 0 �1=3l 0 �1=l
ffiffiffi3
p1=3 0 1=
ffiffiffi3
p0 �1=3l 0
0 �1=3 0 �1=3l 0 1=lffiffiffi3
p1=3 0 �1=
ffiffiffi3
p0 �1=3l 0
26666664
37777775�
Fx
Fy
Fz
Px
Py
Pz
26666664
37777775¼ Ast � F ð3Þ
From the vector of the reaction forces, the strains can bederived by the following equation
Eb ¼ h
E
1=Wxx 0 0 0 0 00 1=Wyy 0 0 0 00 0 1=Wxx 0 0 00 0 0 1=Wyy 0 00 0 0 0 1=Wxx 00 0 0 0 0 1=Wyy
26666664
37777775T
ð4Þwhere E is the elastic modulus of the material, h theapplication arm (see Fig. 4) and W the spoke sectionmodulus. A well-known relationship between the vector ofthe bridges output voltages ΔV and the vector Εb exists (k
and V are respectively the gauge factor and the bridgeexcitation voltage)
ΔV ¼ ΔV1ΔV2ΔV3ΔV4ΔV5ΔV6½ �T ¼ V
2k � Eb ð5Þ
The theoretical calibration matrix Mtst is computed asfollows
ΔV ¼ V2 k � Eb � Ast
� � � F ¼ inv M tstð Þ � FF ¼ M tst �ΔV
ð6Þ
For square section spokes, the matrix Eb can be replacedby the scalar expression h/EW. In general, the matrix Mtst
has the expression
Mtst ¼ 2
V
1
k
E
h�
0 0 0 3Wyy
2 0 3Wyy
2Wxx 0 Wxx 0 Wxx 0
0ffiffiffi3
pWyy 0
ffiffi3
pWyy
2 0ffiffi3
pWyy
2
�WxxL 0 WxxL2 0 WxxL
2 00 �3LWyy 0 �3LWyy 0 �3LWyy
0 0ffiffi3
pLWxx2 0 � ffiffi
3p
LWxx2 0
266666664
377777775
The actual load cell has elastic sliding spherical joints atthe spoke tips (Figs. 3, 8 and 9), so, in order to compute thecalibration matrix, the analytical mathematical model of thequasi-statically determined structure of the load cell hasbeen derived. A single spoke tip pertaining to the sensingstructure in Fig. 4 is shown Fig. 5. The actual stiffnesses atthe spoke tip (axial ‘ka’, radial ‘krx–kry’, torsional ‘kt’ andbending ‘kbt–kb’ stiffnesses) are taken into account. Themodel accounts for the compliance of each spoke of thesensing structure, as shown in Fig. 6.
By considering
– the vector S of the 18 internal forces and momentsacting at the 3 spoke at the central node of the threespoke sensing structure as shown in Fig. 5 (localcoordinate system xi; yi; zi i¼1;2;3ð Þ in Fig. 1),
Si i¼1;2;3ð Þ ¼ SxiSyiSziMxiMyiMzi
h i
S ¼ S1S2S3½ �T
376 Exp Mech (2011) 51:373–388
– the 3 vectors δi of the displacements “δ” androtations “φ” of each spoke tip root (local coordinatesystem)
di i¼1;2;3ð Þ ¼ dxi dyi dzi ϕxi ϕyi ϕzi
h iT– the rotation matrix Rt að Þ a ¼ �2=3pð Þ defined as
follows
Rt að Þ ¼ R að Þ 00 R að Þ
� �
R að Þ ¼cos að Þ 0 sin að Þ
0 1 0� sin að Þ 0 cos að Þ
24
35
In order to compute the 18 components of the vectorS, a 18 equations system has been written in the followingform
F ¼ I � S1 þ Rt 2=3pð Þ � S2 þ Rt �2=3pð Þ � S3
d1 ¼ Rt 2=3pð Þ � d2d1 ¼ Rt �2=3pð Þ � d3
8><>: ð7Þ
The first equation refers to the force equilibrium at thecentral node (roots of the three spokes), while the other 2equations refer to the displacement consistency at thecentral node between the spokes 1–2 and 1–3 respectively.The displacements and rotations of each spoke arecomputed as follows. Figure 6 shows the displacementof a spoke root in the x local direction when a force �Sxacts.
The total displacement dx Sx� �
depends on the radialand bending compliances of the elastic sliding sphericaljoint (respectively dx krxð Þ and dx kbtð Þ), additionally
dx Sx� �
depends on the compliance of the spokedx Jy� �� �
dx Sx� � ¼ dx krxð Þ þ dx Jy
� �þ dx kbtð Þ
dx Sx� � ¼ Sx
krxþ Sxl3
3EJyþ Sxl2
kbtð8Þ
The force Sx produces a rotation of the spoke tip aroundthe y local axis, due to the bending compliance of thesliding spherical joint (kbt) and due to the spoke bendingcompliance respectively
8y Sx� � ¼ � Sxl
kbt� Sxl2
2EJyð9Þ
A similar approach has been followed to derive all thedisplacements and rotations resulting at each spoke root( Sx ! dx Sx
� �=ϕy Sx� �
; Sy ! dy Sy
=ϕx Sy
; Sz ! dz Sz� �
;
Mx ! dy Mx
� �=ϕx Mx
� �;My ! dx My
=ϕy My
;Mz !
ϕz Mz
� �see equation (10)). All the parameters pertaining
to each of the three spokes (e.g. l, Jy,...) are assumed tobe equal. The following relationship between the vectorof the displacements and rotations δi (at the spoke root)
yi
zi
krx kbt kb
kakry
zi
xi
ka
Mzi
Szi
Sxi
Mxi
Syi
Myi
Mzi
Szi
Fig. 5 Model of the six-axis load cell (quasi-statically determinedstructure)—only a single spoke of the sensing structure in Figs. 1 andin 4 is shown
Fig. 6 The displacement in the x direction of a spoke root, due to aforce −Sx
Exp Mech (2011) 51:373–388 377
and the vector of the internal forces and moments Siholds
di ¼ Δ � Si;where
Δ ¼
1krx
þ l3
3EJyþ l2
kbt0 0 0 � l2
2EJy� l
kbt0
0 1kry
þ l3
3EJxþ l2
kb0 l2
2EJxþ l
kb0 0
0 0 1kaþ l
EA 0 0 0
0 lkbþ l2
2EJx0 1
kbþ l
EJx0 0
� lkbt
� l2
2EJy0 0 0 1
kbtþ l
EJy0
0 0 0 0 0 1ktþ l
GJp
26666666664
37777777775
ð10Þ
Where JP accounts for the torsion cross section coeffi-cient of non circular sections. The system in equation (7)can be rewritten in the following compact form
I Rt 2=3pð Þ Rt �2=3pð ÞΔ �Rt 2=3pð Þ �Δ 0Δ 0 �Rt �2=3pð Þ �Δ
24
35 � S ¼
F00
24
35ð11Þ
The above system has been solved symbolically, the 18components of internal forces and moments S, have beenfound as function of the vector F of force and momentacting at the load cell.
S ¼ S Fð Þ
The above system equation reads
Sx1 ;My1¼ f Fx; Ty� �
Sy1;Mx1 ¼ f Fy; Tx
� �Sx2�3
;My2�3
¼ f Fx;Fz; Ty� �
Sy2�3;Mx2�3
¼ f Fy; Tx; Tz� �
Sz1 ¼ f Fzð Þ;Mz1 ¼ f Tzð ÞSz2�3
¼ f Fx;Fzð ÞMz2�3
¼ f Tx; Tzð Þ
ð12Þ
The analytical expressions of equation system (12) arereported in Appendix.
As the vector of the internal forces and moments S hasbeen computed, it is possible to compute the outputdifferential voltages ΔV of the Wheatstone bridges (see(5)). Let us consider the single spoke shown in Fig. 10. Theoutput voltage (strain gauges iA� iB i¼1;::;6ð Þ, correspondingstrains "iA � "iB i¼1;::;6ð Þ respectively) can be computed bythe following equation
ΔVi ¼ V
4k "iA � "iBð Þ ð13Þ
The voltages are function of the system parameters (jointstiffnesses, three spoke structure geometry, strain gauges
characteristics,...) and of the vector F of the force and of themoment acting at the load cell centre
ΔVi ¼ ΔVi Sð Þ ¼ MΔVi½ �F
ΔV ¼
MΔV1
MΔV2
MΔV3
MΔV4
MΔV5
MΔV6
26666664
37777775� F ð14Þ
Fig. 7 One spoke of the load cell model with rigid ring
378 Exp Mech (2011) 51:373–388
By simply inverting the terms in equation (14), thecalibration matrix Mthp can be derived:
ΔV ¼M̂ � F ) F ¼ Mthp �ΔV ð15ÞThe three spokes are connected to the central part of the
structure (see Figs. 5 and 7). The central part of the threespoke structure has a very low compliance and it can beconsidered rigid with respect to the spokes. The very lowcompliance of this component of the structure can be takeninto account as follows:
& a rigid ring is considered to model the central part of thestructure (Fig. 7)
& the three beams representing the spokes of the structureare connected to the rigid ring
& the rigid ring is modelled as a rigid beam connectingeach spoke to the centre of structure. Thus each spoke iscomposed by a deformable and by a non-deformablepart.
& the deformable length of each spoke is calculated as thedifference between the spoke length (l) and the radius ofthe rigid disk (r)
& the equations referring to the displacement consistencyof the structure have been modified to include the rigidpart of each spoke.
The described procedure is depicted in Fig. 8. Theresulting system has been solved symbolically and theexpression of the internal forces and moments S have beenreported in Appendix.
Load Cell Construction
Load Cell Technical Specifications
The technical specifications of the six-axis load cell werederived in order to perform experimental tests on a vehiclesuspension system [11]. The design forces and moments arerespectively (Fig. 1):
– 10 kN (Fy along load cell axis), 5 kN (Fx, Fz)– 0.5 kNm (Px, Py, Pz)
Additionally, the first natural frequency of the load cellhas to be over 500 Hz in order to perform high frequencymeasurement of the forces acting on the suspension systemunder test. The natural frequency has been computed byconsidering the load cell constrained as in an experimentaltest. That is, the load cell is connected to a very siff surface
R.R.
Fig. 8 Structure with central rigid ring (R.R.). Bold line rigid parts ofthe spokes. Continuous line non-deformed structure. Dotted lineQualitatively deformation of the spoke
Table 1 Comparison between the analytical model and the corresponding FEM model
Half bridge n° Load case
Fy [1000 N] Px [1000 Nm] Pz [1000 Nm]
Analyticalmodel
FEMmodel
Difference % Analyticalmodel
FEMmodel
Difference % Analyticalmodel
FEMmodel
Difference %
1 42.9 44.4 3.25 2172 2166 0.25 – – –
3 42.9 44.4 3.25 −1086 −1083 0.25 −1881 −1876 0.25
5 42.9 44.4 3.25 −1086 −1083 0.25 1881 1876 0.25
Half bridge n° Load case
Fx [1000 N] Fz [1000 N] Py [1000 Nm]
Analyticalmodel
FEMmodel
Difference % Analyticalmodel
FEMmodel
Difference % Analyticalmodel
FEMmodel
Difference %
2 79.5 76.1 4.45 – – – 1103 1132 2.65
4 −39.8 −38.1 4.45 −68.8 −65.9 4.45 1103 1132 2.65
6 −39.8 −38.1 4.45 68.8 65.9 4.45 1103 1132 2.65
Exp Mech (2011) 51:373–388 379
by means of the three connecting holes on the base of theload cell (see Fig. 10).
FEM Model of the Load Cell
Given Fx, Fy, Fz, Px, Py, Pz the reaction forces at thejoints can be computed by equation (3). Thus a pre-liminary design of both the joints and the three spokestructure can be performed. On the basis of the pre-liminary design, a FEM model of the load cell has beenrealized. The FEM model (Fig. 8) takes into account allthe mechanical components needed to realize the loadcell.
This FEM model has been used to optimize the design interms of mechanical strength and stiffness of the compo-nents. The sensing structure has been optimized in order toget the maximum level of stress in the zones of applicationof the strain gauges.
The comparison between the analytical model ofthe load cell and the corresponding FEM model isreported in Table 1. The comparison shows the valuesof the deformations read by the half bridges on eachspoke. The half bridges are numbered as shown inFig. 4. The computations have been performed for thesix pure loads (three forces and three moments, seeFig. 1 for symbols). Being that the system is linear, anyother load case is a linear combination of the consideredload cases. The table is divided into two parts. In thefirst part the odd half bridges are considered. These halfbridges read only the deformation due to Fy, Px and Pz.In the second part the even half bridges. These half bridgesread only the deformation due to Fx, Fz and Py. Thecomparison shows a satisfying agreement between the twomodels.
Embodiment Design
The drawing of the prototype six-axis load cell is shown inFig. 9.
Particular attention has been devoted in designingthe reference vertical axis of the load cell, definedby pins (see Fig. 10). The sliding spherical joint (Figs. 2and 3) has been constrained to the load cell structureby means of ring nuts. Each spoke has been connected tothe corresponding sliding spherical joint by means of anut which compresses the annular inner part of thecorresponding sliding spherical joint and realizes theconstraint.
Fig. 9 FEM model of the load cell
z
x
y
1
Fig. 10 Mechanical drawingof the six-axis load cell(dimensions are in mm)
380 Exp Mech (2011) 51:373–388
A flange provided with 4 threaded holes aligned alongthe x and z load cell axes is the connection interface withthe tested structure. The forces and moments measuredby the load cell refer to the centre of the cell (i.e. centreof the three spoke structure).
The overall dimensions of the assembled load cell areabout 150� 175� 100mm. The mass is about 5 kg.
The embodiment design considered the sequence that isneeded for assembling the load cell. Such sequence iscrucial for aligning the vertical axis to the centre line of theload cell and for suppressing internal pre-loads, dangerousfor structural integrity.
Conclusions
This paper has been devoted to the design of a new six axisload cell. The sensing element of the load cell is a quasi-statically determined three spoke structure constrained bymeans of elastic sliding spherical joints. The design of such
joints has been performed in order to avoid friction. Twelveresistive strain gauges have been used to measure bendingdeformations at the three spokes of the sensing structure.The vector of the external forces and moments applied atthe load cell is linearly related to six voltage output signalsvia a square calibration matrix. Different mathematicalmodels have been derived to design the load cell and totune its performances. Particular attention has been devotedto the derivation of an analytical model of the quasi-statically determined structure of the load cell. This hasenabled the preliminary design of the load cell. A finiteelement model has been derived to assess the stress-strainbehaviour of the load cell. A comparison between theanalytical model and the finite element model has beenperformed with satisfactory results. The six axis load cell isrelatively simple to be manufactured and the embodimentdesign has been presented. The error analysis, the con-struction and the experimental assessment of the perform-ances of the load cell are presented in another paper, whichis the companion paper of this one.
Appendix
Analytical expressions of the 18 internal forces andmoments acting at the 3 spoke tips (rigid disk consideredin the model):
Sx1
� 1=3 1=3L3
EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ Lð Þ2
kbt
!L
EJyþ kbt�1
� �Ty �1=2
L2
EJy� RL
EJyþ �R� L
kbt
� �
þ2=3Fx 1=3L3
EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ Lð Þ2
kbt
!L
EJyþ kbt�1
� �
� L
EAþ ka�1
� �þ 1=3 �1=2
L2
EJy� RL
EJyþ �R� L
kbt
� �3
Ty� 1=3 �1=2L2
EJy� RL
EJyþ �R� L
kbt
� �L
EJyþ kbt�1
� �Ty
L
EAþ ka�1
� �
0BBBBBBBBBB@
1CCCCCCCCCCA
� �1=2L2
EJy� RL
EJyþ �R� L
kbt
� �2
þ 1=3L3
EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ lð Þ2
kbt
!L
EJyþ kbt�1
� �þ L
EJyþ kbt�1
� �L
EAþ ka�1
� � !�1
1=3L3
EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ Lð Þ2
kbt
!�1
Exp Mech (2011) 51:373–388 381
Sy1
1=3 � L2
EJx� 2
RL
EJx� 2
Rþ L
kb
� �Tx
L
JpGþ kt�1
� �� 1=3Fy 1=2
L2
EJxþ RL
EJxþ Rþ L
kb
� �2
þ1=3 1=3L3
EJxþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!Fy
L
JpGþ Kt�1
� �
þ1=3L
EJxþ kb�1
� �1=3
L3
EJxþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!Fy
0BBBBBBBBBBB@
1CCCCCCCCCCCA
1=3L3
EJxþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!L
JpGþ kt�1
� �þ L
EJxþ kb�1
� ��
1=3L3
EJxþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!� 1=2
L2
EJxþ RL
EJxþ Rþ L
kb
� �2
0BBBBBB@
1CCCCCCA
�1
Sz1
2=3FZ � �1=2L2
EJy� RL
EJyþ �R� L
kbt
� �2
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EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ Lð Þ2
kbt
!L
EJyþ kbt�1
� � !
� �1=2L2
EJy� RL
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EJyþ krx�1 þ Rþ Lð Þ2
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� �þ L
EJyþ kbt�1
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Mx1
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EJxþ RL Rþ Lð Þ
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kb
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� �
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EJxþ RL Rþ Lð Þ
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� 1=2L2
EJxþ RL
EJxþ Rþ L
kb
� �� 1=3
L
EJxþ kb�1
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L3
EJxþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!
Fy 1=2L2
EJxþ RL
EJxþ Rþ L
kb
� �þ 1=3Fy 1=2
L2
EJxþ RL
EJxþ Rþ L
kb
� �3
0BBBBBBBBBBBBBBBBB@
1CCCCCCCCCCCCCCCCCA
� L
EJxþ kb�1
� ��11=3
L3
EJXþ RL Rþ Lð Þ
EJXþ kry�1 þ Rþ Lð Þ2
kb
!L
JpGþ kt�1
� �
þ L
EJxþ kb�1
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L3
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EJxþ kry�1 þ Rþ Lð Þ2
kb
!� 1=2
L2
EJxþ RL
EJxþ Rþ L
kb
� �2
0BBBBBB@
1CCCCCCA
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382 Exp Mech (2011) 51:373–388
My1
�1=3Ty �1=2L2
EJy� RL
EJyþ �R� L
kbt
� �2
þ 1=3 1=3L3
EJyþ RL Rþ Lð Þ
EJyþ krx�1 Rþ Lð Þ2
kbt
!L
EJyþ kbt�1
� �Ty
þ1=3L
EJyþ kbt�1
� �Ty
L
EAþ ka�1
� �� 1=3 � L2
EJy� 2
RL
EJyþ 2
�R� L
kbt
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L
EAþ ka�1
� �
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1CCCCCA
� �1=2L2
EJy� RL
EJyþ �R� L
kbt
� �2
þ 1=3L3
EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ Lð Þ2
kbt
!L
Ejyþ kbt�1
� �þ L
EJyþ kbt�1
� �L
EAþ ka�1
� � !�1
Mz1
2=3Tz1
EJxþ kb�1
� �1=3
L3
EJxþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!� 1=2
L2
EJxþ RL
EJxþ Rþ L
kb
� �2 !
1=3L3
EJxþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!L
JpGþ kt�1
� �þ
þ L
EJxþ kb�1
� �1=3
L3
EJxþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!� 1=2
L2
EJxþ RL
EJxþ Rþ L
kb
� �2
0BBBBBB@
1CCCCCCA
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Sx2
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EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ Lð Þ2
kbt
! ffiffiffi3
p L
EJyþ kbt�1
� �Fz
L
EAþ ka�1
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EJyþ RL Rþ Lð Þ
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kbt
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EJyþ kbt�1
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EAþ ka�1
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L2
EJy� RL
EJyþ �R� L
kbt
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Ty� 1=3 1=3L3
EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ Lð Þ2
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Exp Mech (2011) 51:373–388 383
Sy2
1=3 1=3L3
EJrþ RL Rþ Lð Þ
EJxþ kry�1 þ Rþ Lð Þ2
kb
!Fy
L
JpGþ kt�1
� �þ 1=3
L
EJrþ kb�1
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L3
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EJxþ kry�1 þ Rþ Lð Þ2
kb
!
Fy� 1=3Fy 1=2L2
EJxþ RL
EJxþ Rþ L
kb
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þ 1=3 1=2L2
EJxþ RL
EJxþ Rþ L
kb
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L
JpGþ kt�1
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þ 1=3 1=2L2
EJxþ RL
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kb
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p L
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1=3L3
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kb
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L
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kb
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L
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EJxþ Rþ L
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384 Exp Mech (2011) 51:373–388
My2
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EJy� RL
EJyþ �R� L
kbt
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EJyþ RL Rþ Lð Þ
EJyþ krx�1 þ Rþ Lð Þ2
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þ 1=3L
EJyþ kbt�1
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L
EAþ ka�1
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L2
EJy� RL
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kbt
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EJyþ �R� L
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L
EAþ ka�1
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EJy� RL
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Tz
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L
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Exp Mech (2011) 51:373–388 385
Sy3
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EJxþ RL
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kb
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kb
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L
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kb
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EJxþ RL
EJxþ Rþ L
kb
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þ 1=3 1=2L2
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kb
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L
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0BBBBBBBBBBB@
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EJxþ RL Rþ Lð Þ
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kb
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ffiffiffi3
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þ 1=3 �1=2L2
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EJxþ RL Rþ Lð Þ
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kb
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þ 1=3ffiffiffi3
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L
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kb
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EJxþ kb�1
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kb
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L2
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EJxþ Rþ L
kb
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EJxþ kb�1
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EJrþ Rþ L
kb
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386 Exp Mech (2011) 51:373–388
My3
1=3 1=3L3
EJyþ RL Rþ Lð Þ
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L
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L2
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EAþ ka�1
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EJyþ �R� L
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L
EAþ ka�1
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� �1=2L2
EJy� RL
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References
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