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Calculation of joint spring rates using finite element formulation
Computers & Smcrurer Vol. 33, No. 4. pp. 977-981, Printed in Great Britain.
CALCULATION FINITE
0045.7949/89 $3.00 + 0.00 Q 1989 Pergamon Press plc
1989
OF JOINT SPRING RATES USING ELEMENT FORMULATION MOHAMEDE. M. EL-SAYED
Department of Mechanical and Aerospace Engineering, University of Missouri. Columbia, MO 65211, U.S.A.
(Received 31 October 1988)
Abstract-A method for calculating the torsional spring rates of structural joints using finite element formulation is presented. The response of the actual joint, as predicted from the experimental results or a more detailed shell finite element model, is used to calculate the torsional spring rates for any number of joint members. These joint rates are typically used in the analysis of automotive structures. The method is developed into a computer program that does not require a finite element model for the joint. The formulation of the method, the flow chart of the program and an example of a three-member joint are provided.
FORMULATION
INTRODUCTION Literature on structural joints shows the importance of joint flexibilities to the total response of the structure. Chang [l] presented sensitivity results which compared system stiffness to joint flexibilities of a passenger compartment. He found that the system stiffness can change up to 50% if flexible connections are used instead of rigid connections. The discussion included a sensitivity study of individual joint stiffness and efficiency calculations using moment-stiffness relationships. Some research efforts have been made for determination of the joint stiffness. Borowski et al. [2] incorporated linear springs into a finite element beam model of an automotive frame. The ratios of beam stiffness at a joint were used to set spring rates from an arbitrary initial value. Ratios were then scaled until the beam-spring model matched the frame test results. Rao et af. [3] discussed static and dynamic test procedures which determined joint stiffness based on an instant center approximation. They developed a joint data-bank, which consisted of measured joint rates from five different vehicles. Results were presented which correlated vehicle dynamic and acoustic response to the stiffness of two body joints. In the following, a method for calculating the joint torsional stiffness using finite element formulation is presented. The response of the actual joint, as predicted from the experimental results or a more detailed shell finite element model, is used to calculate the torsional spring rates for any number of joint members. A computer program, which does not require a finite element model, is developed. The program inputs are the coordinates of the two endpoints of each member, the cross-section properties, loads and deflection data.
In this section the approach used to calculate the joint torsional spring rates is explained. Finite element analysis is used to calculate the rates of the torsional springs to be added to the beam representation of the joint, to match the experimental or detailed analysis model. The following assumptions .# are made: (1) the springs have only three rotational degrees of freedom, which are independent of each other; (2) deflections are small and materials are linearly elastic; and (3) each joint member is represented by one beam element and three scalar joint rates at the connection point. The analysis is applicable to any number of members connected at one point to form a joint. For simplicity we consider the joint in Fig. 1 with three connected members.
2
/ Fig. I. Joint with three connected members.
978
E.R
MOHAMED
The numbers inside the squares are the element numbers, and those inside the circles are the local node numbers. The local coordinate system of each element is defined as shown in Fig. 2. The relation between the load vector, the stiffness matrix and the displacement vector of any of the elemenents in its local axes, neglecting the axial deflection, can be written as
Let element q, in its locat coordinate system shown in Fig. 2, be loaded by &, 3: and ri;, and the displacements measured for each load case separately be $,,, & and 6,Y. By constraining node @ for elements @ and q, the degrees of freedom of these nodes are eliminated and the remaining degrees of
The elements of the stiffness matrix are given by [4]. The nonzero elements are K,, =Ke6=
-Kbl=
-K,6=12EL;L3
fi?? =K7,=
-K,2=
-K2,=
K33 =Ke8=
-Ks3=
-K,g=GJjL
12EI,/L”
KM = K,, = 4EI,/L KS5 = K,o,o = 4EI:lL
K42=Kz.,= -K7,=
(2) --G7=
-Ksz=Kzy=
I(,, = K,, = K,,, = K,,, = -K,,
-Kg?=
= - K6,,, = -K,,
-KT9=
-6E&[L”
= - KG5= 6EfJL=
Kq4 = Ke9 = 2Ef,./L K t05= EC,,, = 2EI:lL,
where E is the elasticity modulus, G is the shear modulus. Is is the moment of inertia around the y-axis, I: is the moment of inertia around the z-axis, J is the polar moment of inertia and L is the member length. In Fig. 1, let element q be loaded at node 0 while elements q and a are fixed at their Iocal node 0.
freedom for the whole system are those of node 0, element a and joint node @ of all the elements. By adding the stiffness contribution of each element at node @ to element IJ in its local coordinate, the system of equations for each load case without considering the spring rates can be written as
979
Calculation of joint spring rates where R represents stiffness coefficients that resulted from the contribution of all elements. We note that the deflection along the local x-axis is neglected. Now, let S,, $ and Sz be the torsional spring stiffness contribution per member at node 0 in the x, y and .z directions respectively. If element q is loaded in the x direction with TVthen S, will be the resisting torsional spring stiffness, whereas S, is the spring stiffness when loading with pz and Sr is the spring stiffness when loading is FV. We now consider loading fX and adding the scalar spring stiffness S, to the stiffness matrix in the proper location [5, 61. For this loading case, the system of equations becomes
0.
K,,
Kn
K,,
KM
K,,
KM
Ku
4s
KM
K,,o
0.
K2,
K22
K2,
K24
K25
f&6
K27
K20
K29
K21o
fY
KN
K,2
f&3
KM
K,s
&6
47
&8
K,,
f&o
0.
f&i
&2
Ku
KM
&s
fG6
Ka
KG
&9
ho
4,
K,,
KS,
Ks4
4,
Ks6
4,
KS,
K59
Go
0.
=
0.
K6,
K62
&
KM
K6s
%6
%7
%B
%9
%,o
0.
Ku
fG2
Kn
KM
Kn
jE76
ET,
$8
x,9
%,o
0.
Ku
Kg2
Ku
Ku
Ka,
&6
%,
kt88
&9
ho
0.
KS,
K92
K93
K94
K9,
%6
IE,,
%s
%9
%,o
0.
K 101
Ko2
Ko,
K,,
K,os
R,,
%o,
%o,
%B
%o,o
where e, is the measured torsional displacement node 0 of element q due to load Fr and
Now
Fig. 2. Local coordinate system.
at
&* = &R + s,.
(5)
s, %: = fix,.
(6)
let
The system of eqn (4) can be written as
0. -f&k= Ku% +K,,&, +Ge,, +W:, f&$,.,+ 0. -
K23&
=
K22&,
+
Kd?,,
+
K254
+
K26&
+
K27A2
f, - KM& = KX 6!.,+
fG24
+
f@,.,
+
K&4,
+
K,66y2
+
K+Z2 + Kd,
0. - Ka 4.x= KN 6.n +
42
+
&A
0. - Ku& = KY& +
424,
+
he,.,
0. - KM & = Ku a,., +
K62 &,
+
&a
0. - Kn&=
K2, h?., +
KU&~+ &A,+
K,,$
4
f,., + + e,,., +
&J
01, +
&6
6?., +
+
Ks66p2
+
Kd2
+ Kd,,
K6s 01, +
Ks6 6,,
+
K6, a,,
+
f&R,
+ K,@:, + Kwe,., + Kd:,
+ K&,.,+
K&+
+ Kw,8.1.2 + K&
0. -
K93 6, =
+
0. -
K103&=
6,,
40,4,
+
K92 4 +K,o,%
+
&de,, +&we,,
+
K2Bex2
K9, e:, +
K,oA,
K96 By2 + +
K,o6&
+
K29Q
+
K2,,&
+
K390n
+
Km,O:2
Ku a,, + Ka 8x, + Ke $, + K4,,,O:>
0. - KS,& = Ku% + Ksd:, + Ku@,, + Kd:, &,
+
K,,e,v,+ K,,,e:,
+ K59O,v,+ KS100r2
Km ox2+ KG9e,., + K6,of& K,d’x2 +K,,O,.,+
+ K.dx2 + K&,
K976, + Kg8Ox2+ +
K,o,6r2
+
K,,oO:2 + Ks10er2+ Rx,
K99e,,, + K,,, ei2
K,o&
+ K,,&,,+
K,o,oO;,.
(7)
MOHAMED E. M. EL-SAYED
980 Rewriting eqn (7) in a matrix form
By solving eqn (8), using Gauss elimination, we obtain O,, and R1?, and using eqn (6) we can calculate s,. Similarly, we can write the following set of equations for loading with p:=,: 0. - K,,&
Fz- K22d; 0. - K,2$z 0. - Kd2& 0. - K,2&
(9)
0. - Kb2& 0. - K,& 0. - K,& 0. - K& 3. - Klo2$
and
s, or2= I?,,.
(10)
Solving eqn (9), and using eqn (IO), we can calculate S,.. For loading condition &. the system of equations is 0. -K,,&
Fl 0. 0. 0. 0. 0. 0. 0. 3. and
K,, d, K,,c( KA18, K,,8,, K,,i, K,,i, K,, 8, Kg&,
K,o,&
=
981
Calculation of joint spring rates Table 1 Properties, loading data and talc. stiffness J (cm4)
I,, (cm? 1,: (cm4) i;. (N) S,. (cm) S, (N/cm)
Member
q
30.40 36.50 486.00 1334.00 0.049 2.35 x IO’
Solving eqns (1 1), and using eqn (12) we can calculate S,. It is clear that the same calculations can be applied to the other joint members for any number of these members.
Member
q
2.24 5.74 15.80 934.00 0.102 1.33 x 10’
Member
q
7.62 9.21 177.00 1401.00 0.401 0.33 x 10’
ALGORITHM
The flow chart in Fig. 3 shows the calculation steps for all the joint rates. Some modifications are necessary when only some selective rates are required. EXAMPLE
-Read number of memben ti -Read geomeby and loading da& of each member.
-calculate rrtifncJS and ormVa.don of ellch member . Set Ue member counter J= I
Fig. 3. Flow chart.
The developed program was used to calculate the torsional spring stiffness S, for each member of the joint shown in Fig. 4. The cross-sectional properties, loading data and the calculated spring rates are given in Table 1. The joint material properties are E = 0.2 x 10’ N/cm and G = 7.81 x lo6 N/cm. CONCLUSIONS
A method of calculating the joint torsional stiffness using finite element formulation is developed, using the response of the actual joint, as predicted from the experimental results or a more detailed shell finite element model, to calculate the torsional spring rates for any number of joint members. This method is developed into a computer program that does not require a finite element model. By using the super-element technique the program could be modified to use more than one beam element to model a joint member. The calculated scalar spring rates could also be grouped into one spring stiffness and assigned to one member of the joint. Because the axial springs of the joint are usually considered to be rigid the calculation of these rates was not discussed. The stiffness of any axial spring, however, could be calculated using the same formulation. REFERENCES
Fig. 4. Joint.
1. D. C. Chang, Effects of flexible connections on body structural response. SAE Trans. Paper 740041 (1974). 2. V. J. Borowski, R. L. Steury and J. L. Lubkin. Finite element dynamic analysis of an automobile frame. SAE Trans. Paper 730506 (1973). 3. M. K. Rao, M. P. Zebrowski and H. C. Crabb, Automotive body joint analysis for improved vehicle response. In Proc. Int. Symp. Automotive Technology and Automarion, Vol. 2, pp. 953-973 (1983). 4. J. S. Przemieniecki, Theory of Matrix Srructurnl Analysis. McGraw-Hill, New York (1968). 5. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1981). 6. R. H. MacNeal, The NASTRAN Theoretical Manual, Level 15.5. MacNeal-Schwendler, Los Angeles (1972).
CALCULATION FINITE
0045.7949/89 $3.00 + 0.00 Q 1989 Pergamon Press plc
1989
OF JOINT SPRING RATES USING ELEMENT FORMULATION MOHAMEDE. M. EL-SAYED
Department of Mechanical and Aerospace Engineering, University of Missouri. Columbia, MO 65211, U.S.A.
(Received 31 October 1988)
Abstract-A method for calculating the torsional spring rates of structural joints using finite element formulation is presented. The response of the actual joint, as predicted from the experimental results or a more detailed shell finite element model, is used to calculate the torsional spring rates for any number of joint members. These joint rates are typically used in the analysis of automotive structures. The method is developed into a computer program that does not require a finite element model for the joint. The formulation of the method, the flow chart of the program and an example of a three-member joint are provided.
FORMULATION
INTRODUCTION Literature on structural joints shows the importance of joint flexibilities to the total response of the structure. Chang [l] presented sensitivity results which compared system stiffness to joint flexibilities of a passenger compartment. He found that the system stiffness can change up to 50% if flexible connections are used instead of rigid connections. The discussion included a sensitivity study of individual joint stiffness and efficiency calculations using moment-stiffness relationships. Some research efforts have been made for determination of the joint stiffness. Borowski et al. [2] incorporated linear springs into a finite element beam model of an automotive frame. The ratios of beam stiffness at a joint were used to set spring rates from an arbitrary initial value. Ratios were then scaled until the beam-spring model matched the frame test results. Rao et af. [3] discussed static and dynamic test procedures which determined joint stiffness based on an instant center approximation. They developed a joint data-bank, which consisted of measured joint rates from five different vehicles. Results were presented which correlated vehicle dynamic and acoustic response to the stiffness of two body joints. In the following, a method for calculating the joint torsional stiffness using finite element formulation is presented. The response of the actual joint, as predicted from the experimental results or a more detailed shell finite element model, is used to calculate the torsional spring rates for any number of joint members. A computer program, which does not require a finite element model, is developed. The program inputs are the coordinates of the two endpoints of each member, the cross-section properties, loads and deflection data.
In this section the approach used to calculate the joint torsional spring rates is explained. Finite element analysis is used to calculate the rates of the torsional springs to be added to the beam representation of the joint, to match the experimental or detailed analysis model. The following assumptions .# are made: (1) the springs have only three rotational degrees of freedom, which are independent of each other; (2) deflections are small and materials are linearly elastic; and (3) each joint member is represented by one beam element and three scalar joint rates at the connection point. The analysis is applicable to any number of members connected at one point to form a joint. For simplicity we consider the joint in Fig. 1 with three connected members.
2
/ Fig. I. Joint with three connected members.
978
E.R
MOHAMED
The numbers inside the squares are the element numbers, and those inside the circles are the local node numbers. The local coordinate system of each element is defined as shown in Fig. 2. The relation between the load vector, the stiffness matrix and the displacement vector of any of the elemenents in its local axes, neglecting the axial deflection, can be written as
Let element q, in its locat coordinate system shown in Fig. 2, be loaded by &, 3: and ri;, and the displacements measured for each load case separately be $,,, & and 6,Y. By constraining node @ for elements @ and q, the degrees of freedom of these nodes are eliminated and the remaining degrees of
The elements of the stiffness matrix are given by [4]. The nonzero elements are K,, =Ke6=
-Kbl=
-K,6=12EL;L3
fi?? =K7,=
-K,2=
-K2,=
K33 =Ke8=
-Ks3=
-K,g=GJjL
12EI,/L”
KM = K,, = 4EI,/L KS5 = K,o,o = 4EI:lL
K42=Kz.,= -K7,=
(2) --G7=
-Ksz=Kzy=
I(,, = K,, = K,,, = K,,, = -K,,
-Kg?=
= - K6,,, = -K,,
-KT9=
-6E&[L”
= - KG5= 6EfJL=
Kq4 = Ke9 = 2Ef,./L K t05= EC,,, = 2EI:lL,
where E is the elasticity modulus, G is the shear modulus. Is is the moment of inertia around the y-axis, I: is the moment of inertia around the z-axis, J is the polar moment of inertia and L is the member length. In Fig. 1, let element q be loaded at node 0 while elements q and a are fixed at their Iocal node 0.
freedom for the whole system are those of node 0, element a and joint node @ of all the elements. By adding the stiffness contribution of each element at node @ to element IJ in its local coordinate, the system of equations for each load case without considering the spring rates can be written as
979
Calculation of joint spring rates where R represents stiffness coefficients that resulted from the contribution of all elements. We note that the deflection along the local x-axis is neglected. Now, let S,, $ and Sz be the torsional spring stiffness contribution per member at node 0 in the x, y and .z directions respectively. If element q is loaded in the x direction with TVthen S, will be the resisting torsional spring stiffness, whereas S, is the spring stiffness when loading with pz and Sr is the spring stiffness when loading is FV. We now consider loading fX and adding the scalar spring stiffness S, to the stiffness matrix in the proper location [5, 61. For this loading case, the system of equations becomes
0.
K,,
Kn
K,,
KM
K,,
KM
Ku
4s
KM
K,,o
0.
K2,
K22
K2,
K24
K25
f&6
K27
K20
K29
K21o
fY
KN
K,2
f&3
KM
K,s
&6
47
&8
K,,
f&o
0.
f&i
&2
Ku
KM
&s
fG6
Ka
KG
&9
ho
4,
K,,
KS,
Ks4
4,
Ks6
4,
KS,
K59
Go
0.
=
0.
K6,
K62
&
KM
K6s
%6
%7
%B
%9
%,o
0.
Ku
fG2
Kn
KM
Kn
jE76
ET,
$8
x,9
%,o
0.
Ku
Kg2
Ku
Ku
Ka,
&6
%,
kt88
&9
ho
0.
KS,
K92
K93
K94
K9,
%6
IE,,
%s
%9
%,o
0.
K 101
Ko2
Ko,
K,,
K,os
R,,
%o,
%o,
%B
%o,o
where e, is the measured torsional displacement node 0 of element q due to load Fr and
Now
Fig. 2. Local coordinate system.
at
&* = &R + s,.
(5)
s, %: = fix,.
(6)
let
The system of eqn (4) can be written as
0. -f&k= Ku% +K,,&, +Ge,, +W:, f&$,.,+ 0. -
K23&
=
K22&,
+
Kd?,,
+
K254
+
K26&
+
K27A2
f, - KM& = KX 6!.,+
fG24
+
f@,.,
+
K&4,
+
K,66y2
+
K+Z2 + Kd,
0. - Ka 4.x= KN 6.n +
42
+
&A
0. - Ku& = KY& +
424,
+
he,.,
0. - KM & = Ku a,., +
K62 &,
+
&a
0. - Kn&=
K2, h?., +
KU&~+ &A,+
K,,$
4
f,., + + e,,., +
&J
01, +
&6
6?., +
+
Ks66p2
+
Kd2
+ Kd,,
K6s 01, +
Ks6 6,,
+
K6, a,,
+
f&R,
+ K,@:, + Kwe,., + Kd:,
+ K&,.,+
K&+
+ Kw,8.1.2 + K&
0. -
K93 6, =
+
0. -
K103&=
6,,
40,4,
+
K92 4 +K,o,%
+
&de,, +&we,,
+
K2Bex2
K9, e:, +
K,oA,
K96 By2 + +
K,o6&
+
K29Q
+
K2,,&
+
K390n
+
Km,O:2
Ku a,, + Ka 8x, + Ke $, + K4,,,O:>
0. - KS,& = Ku% + Ksd:, + Ku@,, + Kd:, &,
+
K,,e,v,+ K,,,e:,
+ K59O,v,+ KS100r2
Km ox2+ KG9e,., + K6,of& K,d’x2 +K,,O,.,+
+ K.dx2 + K&,
K976, + Kg8Ox2+ +
K,o,6r2
+
K,,oO:2 + Ks10er2+ Rx,
K99e,,, + K,,, ei2
K,o&
+ K,,&,,+
K,o,oO;,.
(7)
MOHAMED E. M. EL-SAYED
980 Rewriting eqn (7) in a matrix form
By solving eqn (8), using Gauss elimination, we obtain O,, and R1?, and using eqn (6) we can calculate s,. Similarly, we can write the following set of equations for loading with p:=,: 0. - K,,&
Fz- K22d; 0. - K,2$z 0. - Kd2& 0. - K,2&
(9)
0. - Kb2& 0. - K,& 0. - K,& 0. - K& 3. - Klo2$
and
s, or2= I?,,.
(10)
Solving eqn (9), and using eqn (IO), we can calculate S,.. For loading condition &. the system of equations is 0. -K,,&
Fl 0. 0. 0. 0. 0. 0. 0. 3. and
K,, d, K,,c( KA18, K,,8,, K,,i, K,,i, K,, 8, Kg&,
K,o,&
=
981
Calculation of joint spring rates Table 1 Properties, loading data and talc. stiffness J (cm4)
I,, (cm? 1,: (cm4) i;. (N) S,. (cm) S, (N/cm)
Member
q
30.40 36.50 486.00 1334.00 0.049 2.35 x IO’
Solving eqns (1 1), and using eqn (12) we can calculate S,. It is clear that the same calculations can be applied to the other joint members for any number of these members.
Member
q
2.24 5.74 15.80 934.00 0.102 1.33 x 10’
Member
q
7.62 9.21 177.00 1401.00 0.401 0.33 x 10’
ALGORITHM
The flow chart in Fig. 3 shows the calculation steps for all the joint rates. Some modifications are necessary when only some selective rates are required. EXAMPLE
-Read number of memben ti -Read geomeby and loading da& of each member.
-calculate rrtifncJS and ormVa.don of ellch member . Set Ue member counter J= I
Fig. 3. Flow chart.
The developed program was used to calculate the torsional spring stiffness S, for each member of the joint shown in Fig. 4. The cross-sectional properties, loading data and the calculated spring rates are given in Table 1. The joint material properties are E = 0.2 x 10’ N/cm and G = 7.81 x lo6 N/cm. CONCLUSIONS
A method of calculating the joint torsional stiffness using finite element formulation is developed, using the response of the actual joint, as predicted from the experimental results or a more detailed shell finite element model, to calculate the torsional spring rates for any number of joint members. This method is developed into a computer program that does not require a finite element model. By using the super-element technique the program could be modified to use more than one beam element to model a joint member. The calculated scalar spring rates could also be grouped into one spring stiffness and assigned to one member of the joint. Because the axial springs of the joint are usually considered to be rigid the calculation of these rates was not discussed. The stiffness of any axial spring, however, could be calculated using the same formulation. REFERENCES
Fig. 4. Joint.
1. D. C. Chang, Effects of flexible connections on body structural response. SAE Trans. Paper 740041 (1974). 2. V. J. Borowski, R. L. Steury and J. L. Lubkin. Finite element dynamic analysis of an automobile frame. SAE Trans. Paper 730506 (1973). 3. M. K. Rao, M. P. Zebrowski and H. C. Crabb, Automotive body joint analysis for improved vehicle response. In Proc. Int. Symp. Automotive Technology and Automarion, Vol. 2, pp. 953-973 (1983). 4. J. S. Przemieniecki, Theory of Matrix Srructurnl Analysis. McGraw-Hill, New York (1968). 5. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1981). 6. R. H. MacNeal, The NASTRAN Theoretical Manual, Level 15.5. MacNeal-Schwendler, Los Angeles (1972).