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Effect of member length on the parameter estimates of joints
PERGAMON
Computers and Structures 68 (1998) 381±391
Eect of member length on the parameter estimates of joints K. Lee a, *, E. Nikolaidis b a
Department of Mechanical Engineering, Korea University of Technology and Education, Gajeon-Ri 307, Byungchon-Myon, ChonanShi, Chungnam 333-860, Korea b Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0203, U.S.A. Received 2 April 1997; accepted 5 January 1998
Abstract Concept models of joints are used in the early stages of structural design because they are more economical and practical than detailed models. The concept joint models usually use torsional springs. The magnitudes of torsional spring constants and the positions and orientations of torsional springs can be used as parameters in order to account for the ¯exibility, location of rotation centers, and coupling eect of joints, respectively. These parameters are usually identi®ed using the results of experiments performed on a substructure that contains a joint itself and the attached members. However, the estimates of parameters may change with the length of the attached members. This is due to the fact that the magnitude of the shear deformation of the cross sections of the joint members changes with the length of members. The eect of member length on the parameters of the concept joint models is studied using system identi®cation as well as decomposition of the total deformation of the substructure into contributions from a joint itself and the attached members. The study is applied to a T-shaped joint made of simple box beams and to a joint in a real automotive body structure. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Automotive joints; Concept models; Joint members; Joint ¯exibility; System identi®cation; Decomposition of deformation
1. Notation
xi, yi, zi
xL, yL, zL xP, yP, zP f
Local coordinates which de®ne the direction of the ith torsional spring, ki. The zi-axis is de®ned to be normal to the plane in which the spring is located. Local coordinates of unconstrained member of a joint. The xL-axis is de®ned to coincide with shear center of the member. Local coordinates of B-pillar. The xP-axis is de®ned to coincide with shear center of Bpillar. Rotation of joint members.
* Author to whom correspondence should be addressed.
Parameters of concept joint models k Joint stiness. r Oset of rotation center in the direction of the corresponding joint member. h Oset of rotation center in the plane which is normal to the corresponding joint member. Angle between z1-axis and xP-axis. yz1xP
Subscripts F/A Fore/aft or inplane motion of unconstrained member of a joint. I/O Inboard/outboard or out-of-plane motion of unconstrained member of a joint. T Twist of unconstrained member of a joint.
0045-7949/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 0 6 6 - 2
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2. Introduction A joint in a structure is de®ned as a region of the structure that contains the intersection or junction of two or more load-carrying members. Joints usually have complicated geometric shapes, and thus are dicult to model. It is widely accepted that the behavior of the whole structure can be signi®cantly aected by the way joints are modeled. Borowski et al. [1] used springs in their joint models in order to account for the ¯exibility of joints. Chon [2] showed that the springs in joint models can be neglected when their stiness is very large or very small compared to the rest of the car body. In this case, the joint model can be simpli®ed by making it a rigid connection or a hinge, respectively. Sunami et al. [3, 4] studied the behavior of joints that are isolated from the whole structure. Rao et al. [5] showed that the branches of joints rotate about positions which are dierent from the geometric centers of the joints. Gangadharan [6] presented a joint model that considers the eect of coupling between rotations in dierent planes. Yamazaki and Inoue [7] considered the orientations of torsional springs in order to account for the coupling eect. Ferri [8] studied the nonlinear behavior of joints. There are also concerns about how to design joints in structures. Sakurai and Kamada [9] used bulkheads in order to increase the joint stiness eectively. Nordmark et al. [10] studied the design of joints made of aluminum. They compared the performance of joining methods such as resistance spot welding, adhesive bonding, riveting and mechanical clinching. There are two kinds of joint models which are called `detailed models' and `concept models', respectively. The detailed models consist of many ®nite elements and thus accurately describe the behavior of joints. However, these models are very complicated so that they are expensive to analyze. Large structures, such as cars or airplanes, have many joints. Thus, if all the joints are considered using detailed models in the analysis of the whole structure, the computational cost will be too high. In many applications, concept models are more economical and practical. The concept models usually use torsional springs to account for the ¯exibility of joints. A concept model of a vehicle represents its components in a simpli®ed manner using elements like beams, plates, and springs. Its main advantage is that it is de®ned in terms of a few engineering parameters (for example, cross sectional properties of beams and spring stinesses), whereas a detailed ®nite element model needs a very large number of parameters, which describe the detailed geometry of the vehicle components. The concept model allows designers to perform many iterations at a low cost in the early design
stages of a vehicle. Typically, in design of a new car, the concept model of a surrogate car (usually a previous year model) is used to investigate the eect of proposed design modi®cations. For example, the eect of increasing the stiness of a joint on a car body can be estimated by increasing the spring stinesses representing that joint in the concept model. The concept model of a car can also be used to ®nd optimum targets for the body components (for example, targets for the spring stinesses of some joints). These targets are then used in the detailed design of the components. Finally, a database can be built, which contains joints with dierent dimensions and types of construction and the values of the engineering parameters of the concept models of these joints. Then, the most appropriate joint can be found by analyzing the concept model of the car together with the concept model of each joint in the database. Two important issues can be faced when developing the concept models of joints: 1. What is a good set of engineering parameters for a concept joint model? 2. How are these parameters estimated? The authors and other researchers have proposed alternative concept models of joints and a procedure for estimating the parameters of these models [6, 11, 12]. The concept models, which in many cases are as accurate as the detailed models, consist of many parameters. Thus, in a large structure which contains many joints, all modeling parameters can not be estimated at the same time. For this reason, a substructure is isolated from the whole structure. Each substructure consists of a single joint and its connecting members. The joint parameters are estimated using the results of experiments performed on the substructure. The parameter values of the concept models are determined in a way that the analysis results, using the concept joint models, match the experimental results. This procedure is repeated for all the other joints. This paper shows that if the parameters of the concept joint model are estimated using the results of experiments performed on a substructure containing a joint, some parameter estimates will depend on the length of the members attached to the joint. This is true despite the fact that the deformation of the attached beams is subtracted from the overall substructure deformation. The reason is that the joint stiness depends on the amount of shear deformation of the cross sections of the attached members, which changes signi®cantly with the member length. This paper also shows that the estimates of joint parameters converge to constant values as the length of the attached members exceeds a threshold. It is important to consider these constant values of the joint parameters in the
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analysis of the whole structure because they represent the behavior of the joint when it is a part of the whole structure. The eect of member length on the parameters of the concept joint models will be studied using system identi®cation as well as decomposition of the deformation of a joint model. In the former approach, the parameters are estimated in a way that the dierence between experimental results and analysis results using the concept joint models is minimized. In the latter approach, the total deformation of the substructure is decomposed into contributions from the joint itself and the branches, respectively. This study will be applied to a T-shaped joint made of simple box beams and to a joint in a real automotive body structure. 3. Parameterization of concept joint model 3.1. Behavior of joints under loads It is widely accepted that `the ¯exibility of joints' may dominate the behavior of the overall structure. Chang [13] showed that the static deformation of an automotive body structure might have an error of up to 50% when joints were assumed to be rigid. Chang used torsional springs to account for the ¯exibility of joints. According to Chang's joint model, when a load was applied to a structural member which was connected to a joint, the member was assumed to rotate about the geometric center of the joint. However, the member actually rotates about a point that is dierent from the geometric center [5]. The distance between the geometric center and the rotation center is called `the oset of rotation center'. Another characteristic of joint behavior is `the coupling' [7]. When a structural branch is subjected to a moment, it rotates not only in the plane of the applied moment but also in planes which are normal to the plane of the applied moment. This coupling between rotations in dierent planes is due to the fact that joints and their connecting branches are not geometrically symmetric. 3.2. Concept joint model The behavior of a three-branch joint can be described using a concept joint model as in Fig. 1. As shown in the ®gure, the xL-axis is de®ned to coincide with the shear axis of the unconstrained branch before any load is applied. The orientations of the yL and zLaxes are arbitrary. A moment about the zL-axis is applied to the free end of the unconstrained branch while the ends of the other two branches are constrained. The behavior of this joint can be described
Fig. 1. The behavior of a joint which is described using a concept model.
using three torsional springs. The magnitudes of the spring constants account for the joint ¯exibility. These springs are located at hinges H1, H2, and H3 and lie in planes P1, P2, and P3, respectively, in Fig. 1. Three hinges, H1, H2, and H3, account for the location of rotation centers. The orientations of spring planes, P1, P2, and P3, account for the coupling between rotations in dierent planes. Note that the planes, P1, P2, and P3, can have arbitrary orientations with respect to the xLyL, yLzL, and zLxL-planes. 3.3. Parameters of the concept joint model As described in the previous section, three dierent types of parameters are used to represent the behavior of joints [11]; the magnitudes of three torsional spring constants and the positions and orientations of three torsional springs. In this study, translational springs are not considered. The parameters of a joint model are the magnitudes of three torsional spring constants, three coordinates for the location of each spring, and three angles between the orientation of each spring and the three coordinate axes. Consequently, there are 21 parameters for each joint. However, all three angles between the orientation of each spring and the three coordinate axes are not independent. The sum of squares of the three direction cosines of the angles between the orientation of each spring and the three coordinate axes is equal to 1.0. Due to the above constraint, the magnitude of one angle is automatically
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determined if those of the other two angles are known. Thus, the total number of parameters is reduced from 21 to 18. The number of parameters of the concept model is reduced further using the following assumptions [11]. For each spring in the concept joint model, three coordinates are needed to de®ne the position of each of three hinges H1 to H3 in Fig. 1. The hinge H3, which is mostly related with twist of the joint branch, is assumed to lie in a plane normal to the shear axis of the beam passing through O. Thus, the coordinate of H3 in the xL-direction is zero. The remaining hinges H1 and H2 are assumed to lie on the shear axis of the beam so that their coordinates in the yL and zL-directions are zero. Thus, only four parameters are needed to de®ne the positions of the three hinges (one for H1, one for H2, and two for H3) instead of nine parameters. This reduces the number of parameters of the concept joint model from 18 to 13.
4. Eect of member length on concept joint models
In Eq. (1), fB is rotation due to bending deformation of the vertical branch, fT is rotation due to torsion deformation of the horizontal branch, and fS is rotation due to the deformation of the joint itself. It is impractical to measure fS directly. In experiments, f is measured and the parameters of the concept joint models are estimated from f. The magnitude of fS will change with the length of the constrained member, which is the horizontal branch for the joint in Fig. 2. The change in the magnitude of fS is caused by the change in the magnitude of the shear deformation of the horizontal branch. The change in the magnitude of fS results in changes in the parameter estimates of the concept joint model. Thus, the eect of member length must be considered in order to use concept models of joints in the analysis of the whole structure. In this study, the parameters of the concept joint model will be estimated for dierent values of the length of the members using system identi®cation as well as decomposition of the total deformation of the joint using Eq. (1).
The parameters of the concept joint model are usually identi®ed from a substructure that consists of a joint and its connecting members. The response of the substructure can be decomposed into that of the joint itself and those of its connecting members. When a force in the z-direction is applied to the free end of a vertical branch of a T-shaped joint in Fig. 2, the rotation about the y-axis, f, at the free end of the vertical branch can be decomposed as follows;
4.1. System identi®cation
f fB fT fS :
WSOS um ÿ up T Ce ÿ1 um ÿ up :
1
System identi®cation can be used to estimate the parameters of concept joint models. In this technique, the parameters are estimated in a way that minimizes the weighted sum of squares (WSOS), which is a measure of the dierence between the experimental results and the analysis results using the concept joint models [14]; 2
In Eq. (2), um and up are vectors of the measured responses and the predicted responses obtained using the concept models of joints, respectively. Matrix [Ce] is the covariance matrix of measurement errors, which is introduced in order to weigh the measurements according to the con®dence in each measurement. The parameters are estimated using the following iterative procedure [12]; bi1 bi ZT Ce ÿ1 Zÿ1 ZT Ce ÿ1 um ÿ up :
3
In Eq. (3), bi + 1 and bi are the updated and the previous vectors of joint parameter estimates, respectively. Matrix [Z] is the matrix of sensitivity derivatives whose jth column is the derivative of the response vector with respect to the jth parameter, @u/@bj, and is obtained by the Direct method [15]; K@u=@bj @f=@bj ÿ @K=@bj u:
Fig. 2. A T-shaped joint made of simple box beams.
4
In Eq. (4), [K] is a stiness matrix and f is a force vector. This study requires repeated experiments on dierent substructures containing the same joint. The
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Fig. 3. Sectional deformation of horizontal member of a T-shaped joint.
members attached to the joint have dierent lengths in each substructure. Because such experiments are expensive, very detailed ®nite element models of the joint and its connecting members are generated and analyzed. The analytical results are considered as measurements. However, the procedure in this study can be applied to real experimental measurements. In the numerical experiments of this study, the measured responses are the static deformations of the substructure that contains a joint and its connecting members. There are no measurement errors in numerical experiments. Therefore, the covariance matrix of measurement errors, [Ce], is used in Eq. (3) in order to nondimensionalize all the measurements in this study. The diagonal elements of [Ce] are variances of measurement errors. The o-diagonal elements correspond to the correlation between measurement errors. For the purpose of nondimensionalization, the standard deviation of each displacement is assumed to be 10.0% of the measured displacements. It is also assumed that there is no correlation between measurement errors. Consequently, all the o-diagonal elements of [Ce] are zero. 4.2. Decomposition of deformation Decomposition of deformation can be also used to estimate the modeling parameters of joints [4]. In this method, the deformation of a joint, fS, is obtained by decomposing the deformation of the cross section, of the constrained member of a joint, into one due to rigid body rotation and the other due to shear deformation. When a force in the z-direction is applied at the end of the vertical branch of the joint in Fig. 2, the sectional deformation of the horizontal branch is illustrated in Fig. 3. The sectional deformation can be decomposed into the rigid body rotation and the shear deformation, which is shown in Fig. 3(a and b), respectively. Fig. 3(c) shows their summation. Each of the 4 cross-sectional sides of the horizontal branch rotates by the same angle, fT, in Fig. 3(a). The two adjacent cross sectional sides of the horizontal branch rotate by the same angle, fS, but in opposite directions
in Fig. 3(b). The two angles fT and fS correspond to the rigid body rotation and the shear deformation, respectively. The two angles fT and fS are due to twist rotation of the horizontal member and the joint ¯exibility, respectively. When a numerical experiment is performed on detailed joint models, the values of f1 and f2 in Fig. 3(c) can be found. The shear deformation and the rigid body rotation can be easily calculated using f1 and f2, which are obtained in the analysis of detailed models; fS 0:5 f1 f2 ;
5
fT 0:5 f1 ÿ f2 :
6
The joint stiness in the direction of out-of-plane or inboard/outboard, kI/O, can be found as follows; kI=O M=fS :
7
In the above equation, M is a moment in the out-ofplane direction and is approximately equal to the magnitude of the applied force multiplied by the length of the vertical branch. The decomposition of rotation can be used only for joints whose members have simple cross sectional shapes. Another limitation of this method is that it can only identify the joint stiness in the out-of-plane direction. 5. Example of a simple box beam joint 5.1. Joint description The joint in Fig. 2 which was constructed by connecting two simple box beams was considered. This joint and its connecting branches are symmetric in the xy and xz-planes. Thus, no coupling eect needs to be considered. In other words, the three spring planes, which are denoted as P1, P2 and P3 in Fig. 1, coincide with the three orthogonal coordinate planes, xLyL, xLzL, and yLzL-planes, respectively. All shear axes of the three connecting branches intersect at the same point. Thus, the rotation center of the spring that is mostly related to twist deformation, denoted as H3 in Fig. 1, coincides with the geometric center of the joint.
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K. Lee, E. Nikolaidis / Computers and Structures 68 (1998) 381±391 Table 1 Change in osets of rotation centers due to change in the length of horizontal member of a T-shaped joint LR$ (mm) 440.0 450.0 833.3 1140.0 1560.0 1980.0 2400.0
LR/LR0% (%) 18.3 18.8 34.7 47.5 65.0 82.5 100.0
rF/A (mm)
rI/O (mm)
70.13 69.98 68.70 68.45 68.44 68.34 68.46
121.5 121.3 114.6 112.4 111.1 110.4 110.0
$ LR: Length of horizontal member. % LR0: Maximum LR (LR0 = 2400mm).
Fig. 4. Change in joint stiness due to change in the length of horizontal member of a T-shaped joint.
The total number of parameters in the concept joint model is reduced to the following 5 parameters for the joint in Fig. 2; 1. kF/A, kI/O, and kT: a spring which is related to inplane or fore/aft bending deformation, a spring which is related to out-of-plane or inboard/outboard bending deformation, and a spring which is related to twist deformation, respectively. 2. rF/A and rI/O: the distances between the geometric center of a joint and the location of springs, kF/A and kI/O, respectively, in the direction of the vertical branch. These distances are called the osets of rotation centers. The distance between the geometric center and the location of spring, kT, is assumed to be zero. The geometric properties of this joint are as follows; Lp=1200, lf=150, lw=100, tf=tw=1.0 (mm) for the vertical branch and lf=150, lw=110, tf=tw=1.0 (mm) for the horizontal branch. Seven values are used for the length of the horizontal branch, LR; 440, 450, 883.3, 1140, 1560, 1980, 2400 (mm). Loads were applied at the end of the vertical branch to take measurements in the directions of correspond-
ing loads. The applied loads were forces in the y and z-directions and a twisting moment about the x-axis. All the measurements were taken at 18 dierent positions along the vertical branch. 5.2. Eect of member length on parameter estimates 5.2.1. System identi®cation The system identi®cation technique was repeatedly applied to the T-shaped joint in Fig. 2 as the length of the horizontal member was increased. The change in the joint stiness is depicted in Fig. 4. In this ®gure, joint stiness is nondimensionalized using the value of stiness corresponding to LR=2400 mm. According to the ®gure, the joint stiness which corresponded to inplane bending and twist, kF/A and kT, respectively, was not signi®cantly changed. However, the joint stiness which corresponded to out-of-plane bending, kI/O, was signi®cantly aected by the change in the length of the horizontal member. The signi®cant decrease in kI/O was due to the increase in shear deformation of the horizontal branch. According to Fig. 4, the change in the joint stiness becomes smaller as the length of the horizontal member becomes larger. When the length of the horizontal branch reaches 2400 mm, the change in the joint stiness is negligible. There is a threshold for the value of the member length, beyond which the joint stiness does not change.
Table 2 Change in I/O stiness due to change in the length of horizontal member of a T-shaped joint LR$ (mm)
440.0
450.0
833.3
1140.0
1560.0
1980.0
2400.0
kI/O (107 Nmm/rad) from System Identi®cation kI/O (107 Nmm/rad) from Deformation Decomposition
119.0
116.6
23.5
10.0
4.5
2.8
2.1
112.4
108.3
24.0
10.2
4.6
2.8
2.1
$ LR: Length of horizontal member.
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The osets of the rotation centers in the concept joint model are shown in Table 1 for the dierent values of the member lengths. It is observed that the osets of the rotation centers do not seem to be signi®cantly aected by the member length change. 5.3. Eect of member length on parameter estimates 5.3.1. Decomposition of deformation The modeling parameters of joints can also be estimated by decomposing their cross sectional deformation. Eqs. (5) and (7) were used for this purpose. The out-of-plane bending stiness of a T-shaped joint in Fig. 2, kI/O, was estimated using decomposition of deformation. The estimation procedure was repeated for the dierent values of the length of the horizontal branch. Table 2 summarizes the results using decomposition of deformation as well as system identi®cation. The approximations using decomposition of deformation agree well with the results from system identi®cation. When the length of the horizontal branch was 2400 mm, the rotation of the vertical branch about the y-axis was decomposed using Eq. (1) and is illustrated in Fig. 5. The eect of shear deformation, fS, is the most dominant. However, the two other eects, fT and fB, cannot be neglected. If the rotations due to the bending of the vertical member and the twist of the horizontal member, fB and fT, respectively, are neglected, the estimate of joint stiness can be erroneous. 6. Example of a joint in a passenger car body structure 6.1. Joint description The joint in Fig. 6 was considered, which was isolated from a real car body structure. The horizontal and vertical members in this joint are called rocker and B-pillar, respectively. This joint has a complicated shape and does not have any geometrical symmetry. Thus, all 13 parameters of the concept joint model must be considered to be independent. Because the geometry of the B-pillar was complicated, it was dicult to describe it using a detailed model and to perform numerical experiments. In order to avoid these problems, a large portion of the B-pillar was cut o and a straight beam was attached as shown in Fig. 6. In order to simulate measurements that were needed in system identi®cation, loads were applied at the attached straight beam and displacements and rotations were calculated using ®nite element analysis at several positions along this beam. Because of geometrical non-symmetry, the coupling eect could not be
Fig. 5. Decomposition of rotation of vertical member of a Tshaped joint.
neglected in this joint. Thus, in numerical experiments, de¯ections not only in the load direction but also in the other directions were measured. All the measurements were considered simultaneously. 6.2. Eect of member length on parameter estimates 6.2.1. System identi®cation The system identi®cation technique was repeatedly applied to the real automotive body joint in Fig. 6 as the length of the horizontal member was increased. Four dierent values of the rocker length were considered; LR=474.8, 757.6, 930.9, 1138.3 mm. The results of system identi®cation are summarized in Table 3 for the four values of rocker length. The notations in Table 3 are as follows; 1. xP, yP, zP: local coordinates of the B-pillar. The xPaxis is selected to coincide with the shear axis of the B-pillar (Fig. 6). 2. k1, k2, k3: magnitudes of three spring constants. 3. r1, r2: osets of rotation centers in xP-direction for springs, k1 and k2, respectively. 4. hyP , hzP: osets of rotation center in yP and zP-directions, respectively, for spring, k3. 5. xi, yi, zi (i = 1 0 3): coordinates which de®ne the direction of the ith spring, ki. The plane in which the ith spring is located is selected to be normal to the zi-axis. 6. yz1xP: angle between the z1-axis and the xP-axis. According to Table 3, the estimates of parameters that de®ne the positions and directions of springs changed considerably as the rocker length changed. Note that in the case of the T-shaped joint made of simple box beams, there was no considerable change in the pos-
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Fig. 6. A joint in an automotive body structure. Table 3 Parameter estimates of an automotive body joint for dierent values of rocker length (13-parameter model) LR=474.8 k1 k2 k3 yz1xP yz1yP yz1zP yz2xP yz2yP yz2zP yz3xP yz3yP yz3zP r1 r2 hyt hzt
LR=757.6 8
3.8010 10 2.5851 107 5.8473 107 107.32 129.11 44.23 102.76 12.78 90.76 9.04 82.23 94.60 145.24 92.62 ÿ128.84 41.06
LR=930.0 8
2.8333 10 1.2672 107 5.5219 107 103.23 134.82 47.83 107.48 17.49 89.53 24.04 66.39 94.28 102.81 89.83 ÿ177.59 43.30
Units: LR, ri, hi (mm), kt(N mm/rad), yl (deg).
LR=1138.3 8
2.6346 10 0.9799 107 4.7175 107 100.89 136.56 48.61 109.54 19.59 89.22 37.42 52.88 94.03 69.04 32.41 ÿ215.29 46.50
2.4465 108 0.8889 107 4.2472 107 99.16 138.12 49.59 110.16 20.18 89.09 42.40 47.86 93.88 53.72 5.24 ÿ232.10 47.65
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Table 4 Parameter estimates of an automotive body joint for dierent values of rocker length (5-parameter model) LR=474.8 kF/A kI/O kT rF/A rI/O
LR=757.6 8
7.0279 10 2.6596 107 5.2971 107 165.63 104.11
LR=930.0 8
6.0530 10 1.3386 107 4.4552 107 147.83 92.74
LR=1138.3 8
5.7584 10 1.0259 107 4.0138 107 141.21 90.68
5.5330 108 0.9203 107 3.8131 107 138.82 90.63
Units: LR, ri (mm), ki (N mm/rad).
itions of springs as the horizontal member length changed. For the T-shaped joint, the directions of springs were not considered because there was no coupling eect. The inplane stiness and its oset of rotation center (kF/A and rF/A), the out-of-plane stiness and its oset of rotation center (kI/O and rI/O), and the twist stiness (kT) are considered as the most important parameters in the design of joints in automotive body structures. For this purpose, the parameters of this joint were identi®ed assuming that this joint did not have a coupling eect and that the osets of rotation centers in the yP and zP-directions were zero. The results are summarized in Table 4. The change in the magnitudes of the osets of the rotation centers in this joint (Table 4) was more signi®cant than in the T-shaped joint made of simple box beams (Table 1). The change in joint stiness is also illustrated in Fig. 7, where joint stiness is normalized using the value of stiness corresponding to LR=1138.3 mm. The change in the outof-plane stiness, kI/O, was the most signi®cant. The value of this parameter corresponding to the shortest
Fig. 7. Change in joint stiness due to change in the rocker length of an automotive body joint. No coupling eect was considered. (5-parameter model).
rocker is almost three times larger than the stiness corresponding to the longest rocker. The signi®cant change in the out-of-plane stiness can be explained using Fig. 8. Using the results in Table 3, the directions of the three springs were generated in Fig. 8 for the cases where LR=474.8 and 1138.3 mm. When the rocker length was the shortest (LR=474.8 mm), the spring axis z1 almost lay in the yPzP-plane, the spring axis z2 was almost parallel to the yP-axis, and the spring axis z3 was almost parallel to the xP-axis. When the rocker length was increased to its maximum value (LR=1138.3 mm), the directions of the spring axes z1 and z2 did not change signi®cantly. However, the direction of spring axis z3 changed signi®cantly. The z3-axis almost coincided with the xP-axis when LR was 474.8 mm. When LR was increased to 1138.3 mm, the z3±axis rotated in the xPyP-plane from the xP-axis towards the yP-axis. In other words this spring was mostly related to twist ¯exibility about the xP-axis when the rocker was the shortest (LR=474.8 mm). As the rocker length was increased (LR=1138.3 mm), the orientation of the plane of spring k3 changed so that the B-pillar could rotate about both the xP and the yP-axes. Thus, the increase in out-of-plane bending ¯exibility with the increase in rocker length can be explained by the change in the direction of spring k3.
Fig. 8. Change in the orientation of spring directions due to change in the rocker length of an automotive body joint.
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For the example of the real car joint, the threshold value for rocker length above which the joint stiness becomes insensitive to the change in rocker length cannot be observed in Tables 3 and 4, nor in Fig. 7. The threshold cannot be observed because the member length can be increased only up to the adjacent joint in the car body structure. If the estimates of parameters of concept joint models are to be used in the analysis of overall structure, the parameters must be identi®ed from a substructure, where the length of attached members is as long as possible and also any other joints are not included in the same substructure. Otherwise, the response of the overall structure may not be accurately predicted. In order to illustrate the importance of the eect of the member length on the parameter estimates and thus on the prediction of structural responses of the overall car body, two dierent substructures which contain the same joint were considered. The only dierence in the two substructures was that they had dierent rocker lengths; LR=474.8 and 1138.3 mm. The estimation results are summarized in Table 3. Each set of parameter estimates was used to predict the displacements of the B-pillar under a force in the I/O direction for the substructure whose rocker length was LR=1138.3 mm. The predicted displacements are shown in Fig. 9. The predicted displacements using parameters whose estimates were found from a substructure with rocker length LR=1138.3 mm almost coincided with the displacements obtained from the analysis of the detailed models. However, when the parameter estimates, which were found from a substructure with rocker length LR=474.8 mm, were used in the analysis of a substructure with rocker length
Fig. 9. Predicted displacements in I/O direction under a force of ÿ196 (N) in the same direction. The displacements along B-pillar were predicted for the substructure with LR = 1138.3 mm.
LR=1138.3 mm, the results were signi®cantly dierent from the analysis of the detailed models. Thus, in the analysis of the whole structure, it is not reasonable to use the values of parameters that are estimated from a substructure with short members, because this leads to a serious underestimation of some displacements of the overall car body. It is also sensible to consider the values of the joint parameters that correspond to long members when comparing design alternatives or trying to assess the eect of design modi®cations on the performance of a joint. Because the cross section of a horizontal member is complicated, the decomposition of deformation can not be used in the estimation of parameters of this joint.
7. Summary and conclusions The eect of member length on the parameter estimates of concept joint models was investigated in this study. A T-shaped joint made of simple box beams and a joint isolated from a real automotive body structure were considered for examples. The following are the main observations: 1. The ¯exibility of concept joint models increased as the length of attached members increased. The increase in the ¯exibility was primarily due to the increase in shear deformation of cross section. Thus, for a joint that is located in a plane, the increase in the out-of-plane ¯exibility was the most signi®cant. 2. In a T-shaped joint made of simple box beams, the change in ¯exibility became insensitive when the length of the attached members increased above a speci®c threshold. In a real automotive body joint, the threshold could not be observed. This is because another joint would be included in the substructure before the threshold was reached. 3. As the length of joint members increased, the changes in osets of rotation centers were more signi®cant in a real car joint than in a T-shaped joint. 4. In order to account for the coupling eect, the orientations of the springs were considered. The coupling eect did not exist in a T-shaped joint made of simple box beams, which was geometrically symmetric. The eect was signi®cant in a real car joint that was not geometrically symmetric. As the member length increased, the directions of springs in a real car joint changed in a way that the out-ofplane ¯exibility increased signi®cantly. Based on the above observations, the following conclusions can be made for the application of the concept joint models to the analysis of the overall structures:
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1. When the member length is above a certain threshold, the parameter estimates become insensitive to the change in the member length. Thus, the parameter estimates should be obtained from an isolated joint or a substructure whose members are longer than the threshold if the concept joint model is to be used in the analysis of overall structure. 2. The threshold value for the member length may not exist when the size or geometry of the overall structure limits the length of an attached member. In this case, joint parameters should be identi®ed using a substructure with the longest possible attached members. Otherwise, the application of concept joint models in the analysis and design of the overall structure can yield erroneous results.
Acknowledgements The work in this paper was supported in part by Non-directed Research Fund, Korea Research Foundation, and by a research grant from Ford Motor Company. References [1] Borowski VJ, Steury RL, Lubkin JL. Finite element dynamic analysis of an automobile frame. SAE Automobile Engineering Meeting. Detroit, Michigan. SAE Paper No.730506, 1973. [2] Chon CT. Sensitivity of total strain energy of a vehicle structure to local joint stiness. AIAA Journal 1987;25:1391±5. [3] Sunami Y, Yugawa T, Yoshida Y. Analysis of joint rigidity in-plane bending of plane-joint structures. JSAE Review 1987;9:44±51.
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[4] Sunami Y, Yugawa T, Yoshida Y. Analysis of joint rigidity of the automotive body structureÐout-of-plane bending of plane-joint structures. JSAE Review 1990;11:345±8. [5] Rao MK, Zebrowski MP, Crabb HC. Automotive body joint analysis for improved vehicle response. Proceedings of International Symposium on Automotive Technology and Automation 1983;2:953±73. [6] Gangadharan SN. Statistical system identi®cation of structures with ¯exible joints. Ph.D. Dissertation. Virginia Polytechnic Institute and State University, 1990. [7] Yamazaki I, Inoue T. An application of structural-acoustic coupling analysis to boom noise. SAE Paper No. 891996, 1989. [8] Ferri AA. Analysis of ¯exible structures with nonlinear joints. Proceedings of the International Modal Analysis Conference 1989;IMAC892:962±8. [9] Sakurai T, Kamada Y. Structural joint stiness of automotive body. SAE International Congress and Exposition. Detroit, Michigan. SAE Paper No. 880550, 1988. [10] Nordmark GE, Miller JM, Banthia VK. Joint design for aluminum automotive structures. SAE Paper No. 930492, 1993. [11] Nikolaidis E, Lee K. A 3-D joint model for automotive structures. SAE Transactions, 101. In: Section 6. SAE Paper No. 921088, 1992:251±63. [12] Lee K, Nikolaidis E. Identi®cation of ¯exible joints in vehicle structures. Transaction of the Canadian Society of Mechanical Engineers 1993;17(4B):851±63. [13] Chang DC. Eects of ¯exible connections on body structural response. SAE Transactions 1974;83:233±44. [14] Allen JJ, Martinez DR. Automating the identi®cation of structural model parameters. In: 30-th AIAA/ASME/ ASCE/AHS Structures, Structural Dynamics, and Materials Conference. Mobile, Alabama, 1989;89-1242CP:728±36. [15] Haftka RT, Kamat MP. Elements of structural optimization. Netherlands: Martinus Nijho Publisher, 1985:156±61.
Computers and Structures 68 (1998) 381±391
Eect of member length on the parameter estimates of joints K. Lee a, *, E. Nikolaidis b a
Department of Mechanical Engineering, Korea University of Technology and Education, Gajeon-Ri 307, Byungchon-Myon, ChonanShi, Chungnam 333-860, Korea b Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0203, U.S.A. Received 2 April 1997; accepted 5 January 1998
Abstract Concept models of joints are used in the early stages of structural design because they are more economical and practical than detailed models. The concept joint models usually use torsional springs. The magnitudes of torsional spring constants and the positions and orientations of torsional springs can be used as parameters in order to account for the ¯exibility, location of rotation centers, and coupling eect of joints, respectively. These parameters are usually identi®ed using the results of experiments performed on a substructure that contains a joint itself and the attached members. However, the estimates of parameters may change with the length of the attached members. This is due to the fact that the magnitude of the shear deformation of the cross sections of the joint members changes with the length of members. The eect of member length on the parameters of the concept joint models is studied using system identi®cation as well as decomposition of the total deformation of the substructure into contributions from a joint itself and the attached members. The study is applied to a T-shaped joint made of simple box beams and to a joint in a real automotive body structure. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Automotive joints; Concept models; Joint members; Joint ¯exibility; System identi®cation; Decomposition of deformation
1. Notation
xi, yi, zi
xL, yL, zL xP, yP, zP f
Local coordinates which de®ne the direction of the ith torsional spring, ki. The zi-axis is de®ned to be normal to the plane in which the spring is located. Local coordinates of unconstrained member of a joint. The xL-axis is de®ned to coincide with shear center of the member. Local coordinates of B-pillar. The xP-axis is de®ned to coincide with shear center of Bpillar. Rotation of joint members.
* Author to whom correspondence should be addressed.
Parameters of concept joint models k Joint stiness. r Oset of rotation center in the direction of the corresponding joint member. h Oset of rotation center in the plane which is normal to the corresponding joint member. Angle between z1-axis and xP-axis. yz1xP
Subscripts F/A Fore/aft or inplane motion of unconstrained member of a joint. I/O Inboard/outboard or out-of-plane motion of unconstrained member of a joint. T Twist of unconstrained member of a joint.
0045-7949/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 0 6 6 - 2
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2. Introduction A joint in a structure is de®ned as a region of the structure that contains the intersection or junction of two or more load-carrying members. Joints usually have complicated geometric shapes, and thus are dicult to model. It is widely accepted that the behavior of the whole structure can be signi®cantly aected by the way joints are modeled. Borowski et al. [1] used springs in their joint models in order to account for the ¯exibility of joints. Chon [2] showed that the springs in joint models can be neglected when their stiness is very large or very small compared to the rest of the car body. In this case, the joint model can be simpli®ed by making it a rigid connection or a hinge, respectively. Sunami et al. [3, 4] studied the behavior of joints that are isolated from the whole structure. Rao et al. [5] showed that the branches of joints rotate about positions which are dierent from the geometric centers of the joints. Gangadharan [6] presented a joint model that considers the eect of coupling between rotations in dierent planes. Yamazaki and Inoue [7] considered the orientations of torsional springs in order to account for the coupling eect. Ferri [8] studied the nonlinear behavior of joints. There are also concerns about how to design joints in structures. Sakurai and Kamada [9] used bulkheads in order to increase the joint stiness eectively. Nordmark et al. [10] studied the design of joints made of aluminum. They compared the performance of joining methods such as resistance spot welding, adhesive bonding, riveting and mechanical clinching. There are two kinds of joint models which are called `detailed models' and `concept models', respectively. The detailed models consist of many ®nite elements and thus accurately describe the behavior of joints. However, these models are very complicated so that they are expensive to analyze. Large structures, such as cars or airplanes, have many joints. Thus, if all the joints are considered using detailed models in the analysis of the whole structure, the computational cost will be too high. In many applications, concept models are more economical and practical. The concept models usually use torsional springs to account for the ¯exibility of joints. A concept model of a vehicle represents its components in a simpli®ed manner using elements like beams, plates, and springs. Its main advantage is that it is de®ned in terms of a few engineering parameters (for example, cross sectional properties of beams and spring stinesses), whereas a detailed ®nite element model needs a very large number of parameters, which describe the detailed geometry of the vehicle components. The concept model allows designers to perform many iterations at a low cost in the early design
stages of a vehicle. Typically, in design of a new car, the concept model of a surrogate car (usually a previous year model) is used to investigate the eect of proposed design modi®cations. For example, the eect of increasing the stiness of a joint on a car body can be estimated by increasing the spring stinesses representing that joint in the concept model. The concept model of a car can also be used to ®nd optimum targets for the body components (for example, targets for the spring stinesses of some joints). These targets are then used in the detailed design of the components. Finally, a database can be built, which contains joints with dierent dimensions and types of construction and the values of the engineering parameters of the concept models of these joints. Then, the most appropriate joint can be found by analyzing the concept model of the car together with the concept model of each joint in the database. Two important issues can be faced when developing the concept models of joints: 1. What is a good set of engineering parameters for a concept joint model? 2. How are these parameters estimated? The authors and other researchers have proposed alternative concept models of joints and a procedure for estimating the parameters of these models [6, 11, 12]. The concept models, which in many cases are as accurate as the detailed models, consist of many parameters. Thus, in a large structure which contains many joints, all modeling parameters can not be estimated at the same time. For this reason, a substructure is isolated from the whole structure. Each substructure consists of a single joint and its connecting members. The joint parameters are estimated using the results of experiments performed on the substructure. The parameter values of the concept models are determined in a way that the analysis results, using the concept joint models, match the experimental results. This procedure is repeated for all the other joints. This paper shows that if the parameters of the concept joint model are estimated using the results of experiments performed on a substructure containing a joint, some parameter estimates will depend on the length of the members attached to the joint. This is true despite the fact that the deformation of the attached beams is subtracted from the overall substructure deformation. The reason is that the joint stiness depends on the amount of shear deformation of the cross sections of the attached members, which changes signi®cantly with the member length. This paper also shows that the estimates of joint parameters converge to constant values as the length of the attached members exceeds a threshold. It is important to consider these constant values of the joint parameters in the
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analysis of the whole structure because they represent the behavior of the joint when it is a part of the whole structure. The eect of member length on the parameters of the concept joint models will be studied using system identi®cation as well as decomposition of the deformation of a joint model. In the former approach, the parameters are estimated in a way that the dierence between experimental results and analysis results using the concept joint models is minimized. In the latter approach, the total deformation of the substructure is decomposed into contributions from the joint itself and the branches, respectively. This study will be applied to a T-shaped joint made of simple box beams and to a joint in a real automotive body structure. 3. Parameterization of concept joint model 3.1. Behavior of joints under loads It is widely accepted that `the ¯exibility of joints' may dominate the behavior of the overall structure. Chang [13] showed that the static deformation of an automotive body structure might have an error of up to 50% when joints were assumed to be rigid. Chang used torsional springs to account for the ¯exibility of joints. According to Chang's joint model, when a load was applied to a structural member which was connected to a joint, the member was assumed to rotate about the geometric center of the joint. However, the member actually rotates about a point that is dierent from the geometric center [5]. The distance between the geometric center and the rotation center is called `the oset of rotation center'. Another characteristic of joint behavior is `the coupling' [7]. When a structural branch is subjected to a moment, it rotates not only in the plane of the applied moment but also in planes which are normal to the plane of the applied moment. This coupling between rotations in dierent planes is due to the fact that joints and their connecting branches are not geometrically symmetric. 3.2. Concept joint model The behavior of a three-branch joint can be described using a concept joint model as in Fig. 1. As shown in the ®gure, the xL-axis is de®ned to coincide with the shear axis of the unconstrained branch before any load is applied. The orientations of the yL and zLaxes are arbitrary. A moment about the zL-axis is applied to the free end of the unconstrained branch while the ends of the other two branches are constrained. The behavior of this joint can be described
Fig. 1. The behavior of a joint which is described using a concept model.
using three torsional springs. The magnitudes of the spring constants account for the joint ¯exibility. These springs are located at hinges H1, H2, and H3 and lie in planes P1, P2, and P3, respectively, in Fig. 1. Three hinges, H1, H2, and H3, account for the location of rotation centers. The orientations of spring planes, P1, P2, and P3, account for the coupling between rotations in dierent planes. Note that the planes, P1, P2, and P3, can have arbitrary orientations with respect to the xLyL, yLzL, and zLxL-planes. 3.3. Parameters of the concept joint model As described in the previous section, three dierent types of parameters are used to represent the behavior of joints [11]; the magnitudes of three torsional spring constants and the positions and orientations of three torsional springs. In this study, translational springs are not considered. The parameters of a joint model are the magnitudes of three torsional spring constants, three coordinates for the location of each spring, and three angles between the orientation of each spring and the three coordinate axes. Consequently, there are 21 parameters for each joint. However, all three angles between the orientation of each spring and the three coordinate axes are not independent. The sum of squares of the three direction cosines of the angles between the orientation of each spring and the three coordinate axes is equal to 1.0. Due to the above constraint, the magnitude of one angle is automatically
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determined if those of the other two angles are known. Thus, the total number of parameters is reduced from 21 to 18. The number of parameters of the concept model is reduced further using the following assumptions [11]. For each spring in the concept joint model, three coordinates are needed to de®ne the position of each of three hinges H1 to H3 in Fig. 1. The hinge H3, which is mostly related with twist of the joint branch, is assumed to lie in a plane normal to the shear axis of the beam passing through O. Thus, the coordinate of H3 in the xL-direction is zero. The remaining hinges H1 and H2 are assumed to lie on the shear axis of the beam so that their coordinates in the yL and zL-directions are zero. Thus, only four parameters are needed to de®ne the positions of the three hinges (one for H1, one for H2, and two for H3) instead of nine parameters. This reduces the number of parameters of the concept joint model from 18 to 13.
4. Eect of member length on concept joint models
In Eq. (1), fB is rotation due to bending deformation of the vertical branch, fT is rotation due to torsion deformation of the horizontal branch, and fS is rotation due to the deformation of the joint itself. It is impractical to measure fS directly. In experiments, f is measured and the parameters of the concept joint models are estimated from f. The magnitude of fS will change with the length of the constrained member, which is the horizontal branch for the joint in Fig. 2. The change in the magnitude of fS is caused by the change in the magnitude of the shear deformation of the horizontal branch. The change in the magnitude of fS results in changes in the parameter estimates of the concept joint model. Thus, the eect of member length must be considered in order to use concept models of joints in the analysis of the whole structure. In this study, the parameters of the concept joint model will be estimated for dierent values of the length of the members using system identi®cation as well as decomposition of the total deformation of the joint using Eq. (1).
The parameters of the concept joint model are usually identi®ed from a substructure that consists of a joint and its connecting members. The response of the substructure can be decomposed into that of the joint itself and those of its connecting members. When a force in the z-direction is applied to the free end of a vertical branch of a T-shaped joint in Fig. 2, the rotation about the y-axis, f, at the free end of the vertical branch can be decomposed as follows;
4.1. System identi®cation
f fB fT fS :
WSOS um ÿ up T Ce ÿ1 um ÿ up :
1
System identi®cation can be used to estimate the parameters of concept joint models. In this technique, the parameters are estimated in a way that minimizes the weighted sum of squares (WSOS), which is a measure of the dierence between the experimental results and the analysis results using the concept joint models [14]; 2
In Eq. (2), um and up are vectors of the measured responses and the predicted responses obtained using the concept models of joints, respectively. Matrix [Ce] is the covariance matrix of measurement errors, which is introduced in order to weigh the measurements according to the con®dence in each measurement. The parameters are estimated using the following iterative procedure [12]; bi1 bi ZT Ce ÿ1 Zÿ1 ZT Ce ÿ1 um ÿ up :
3
In Eq. (3), bi + 1 and bi are the updated and the previous vectors of joint parameter estimates, respectively. Matrix [Z] is the matrix of sensitivity derivatives whose jth column is the derivative of the response vector with respect to the jth parameter, @u/@bj, and is obtained by the Direct method [15]; K@u=@bj @f=@bj ÿ @K=@bj u:
Fig. 2. A T-shaped joint made of simple box beams.
4
In Eq. (4), [K] is a stiness matrix and f is a force vector. This study requires repeated experiments on dierent substructures containing the same joint. The
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385
Fig. 3. Sectional deformation of horizontal member of a T-shaped joint.
members attached to the joint have dierent lengths in each substructure. Because such experiments are expensive, very detailed ®nite element models of the joint and its connecting members are generated and analyzed. The analytical results are considered as measurements. However, the procedure in this study can be applied to real experimental measurements. In the numerical experiments of this study, the measured responses are the static deformations of the substructure that contains a joint and its connecting members. There are no measurement errors in numerical experiments. Therefore, the covariance matrix of measurement errors, [Ce], is used in Eq. (3) in order to nondimensionalize all the measurements in this study. The diagonal elements of [Ce] are variances of measurement errors. The o-diagonal elements correspond to the correlation between measurement errors. For the purpose of nondimensionalization, the standard deviation of each displacement is assumed to be 10.0% of the measured displacements. It is also assumed that there is no correlation between measurement errors. Consequently, all the o-diagonal elements of [Ce] are zero. 4.2. Decomposition of deformation Decomposition of deformation can be also used to estimate the modeling parameters of joints [4]. In this method, the deformation of a joint, fS, is obtained by decomposing the deformation of the cross section, of the constrained member of a joint, into one due to rigid body rotation and the other due to shear deformation. When a force in the z-direction is applied at the end of the vertical branch of the joint in Fig. 2, the sectional deformation of the horizontal branch is illustrated in Fig. 3. The sectional deformation can be decomposed into the rigid body rotation and the shear deformation, which is shown in Fig. 3(a and b), respectively. Fig. 3(c) shows their summation. Each of the 4 cross-sectional sides of the horizontal branch rotates by the same angle, fT, in Fig. 3(a). The two adjacent cross sectional sides of the horizontal branch rotate by the same angle, fS, but in opposite directions
in Fig. 3(b). The two angles fT and fS correspond to the rigid body rotation and the shear deformation, respectively. The two angles fT and fS are due to twist rotation of the horizontal member and the joint ¯exibility, respectively. When a numerical experiment is performed on detailed joint models, the values of f1 and f2 in Fig. 3(c) can be found. The shear deformation and the rigid body rotation can be easily calculated using f1 and f2, which are obtained in the analysis of detailed models; fS 0:5 f1 f2 ;
5
fT 0:5 f1 ÿ f2 :
6
The joint stiness in the direction of out-of-plane or inboard/outboard, kI/O, can be found as follows; kI=O M=fS :
7
In the above equation, M is a moment in the out-ofplane direction and is approximately equal to the magnitude of the applied force multiplied by the length of the vertical branch. The decomposition of rotation can be used only for joints whose members have simple cross sectional shapes. Another limitation of this method is that it can only identify the joint stiness in the out-of-plane direction. 5. Example of a simple box beam joint 5.1. Joint description The joint in Fig. 2 which was constructed by connecting two simple box beams was considered. This joint and its connecting branches are symmetric in the xy and xz-planes. Thus, no coupling eect needs to be considered. In other words, the three spring planes, which are denoted as P1, P2 and P3 in Fig. 1, coincide with the three orthogonal coordinate planes, xLyL, xLzL, and yLzL-planes, respectively. All shear axes of the three connecting branches intersect at the same point. Thus, the rotation center of the spring that is mostly related to twist deformation, denoted as H3 in Fig. 1, coincides with the geometric center of the joint.
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K. Lee, E. Nikolaidis / Computers and Structures 68 (1998) 381±391 Table 1 Change in osets of rotation centers due to change in the length of horizontal member of a T-shaped joint LR$ (mm) 440.0 450.0 833.3 1140.0 1560.0 1980.0 2400.0
LR/LR0% (%) 18.3 18.8 34.7 47.5 65.0 82.5 100.0
rF/A (mm)
rI/O (mm)
70.13 69.98 68.70 68.45 68.44 68.34 68.46
121.5 121.3 114.6 112.4 111.1 110.4 110.0
$ LR: Length of horizontal member. % LR0: Maximum LR (LR0 = 2400mm).
Fig. 4. Change in joint stiness due to change in the length of horizontal member of a T-shaped joint.
The total number of parameters in the concept joint model is reduced to the following 5 parameters for the joint in Fig. 2; 1. kF/A, kI/O, and kT: a spring which is related to inplane or fore/aft bending deformation, a spring which is related to out-of-plane or inboard/outboard bending deformation, and a spring which is related to twist deformation, respectively. 2. rF/A and rI/O: the distances between the geometric center of a joint and the location of springs, kF/A and kI/O, respectively, in the direction of the vertical branch. These distances are called the osets of rotation centers. The distance between the geometric center and the location of spring, kT, is assumed to be zero. The geometric properties of this joint are as follows; Lp=1200, lf=150, lw=100, tf=tw=1.0 (mm) for the vertical branch and lf=150, lw=110, tf=tw=1.0 (mm) for the horizontal branch. Seven values are used for the length of the horizontal branch, LR; 440, 450, 883.3, 1140, 1560, 1980, 2400 (mm). Loads were applied at the end of the vertical branch to take measurements in the directions of correspond-
ing loads. The applied loads were forces in the y and z-directions and a twisting moment about the x-axis. All the measurements were taken at 18 dierent positions along the vertical branch. 5.2. Eect of member length on parameter estimates 5.2.1. System identi®cation The system identi®cation technique was repeatedly applied to the T-shaped joint in Fig. 2 as the length of the horizontal member was increased. The change in the joint stiness is depicted in Fig. 4. In this ®gure, joint stiness is nondimensionalized using the value of stiness corresponding to LR=2400 mm. According to the ®gure, the joint stiness which corresponded to inplane bending and twist, kF/A and kT, respectively, was not signi®cantly changed. However, the joint stiness which corresponded to out-of-plane bending, kI/O, was signi®cantly aected by the change in the length of the horizontal member. The signi®cant decrease in kI/O was due to the increase in shear deformation of the horizontal branch. According to Fig. 4, the change in the joint stiness becomes smaller as the length of the horizontal member becomes larger. When the length of the horizontal branch reaches 2400 mm, the change in the joint stiness is negligible. There is a threshold for the value of the member length, beyond which the joint stiness does not change.
Table 2 Change in I/O stiness due to change in the length of horizontal member of a T-shaped joint LR$ (mm)
440.0
450.0
833.3
1140.0
1560.0
1980.0
2400.0
kI/O (107 Nmm/rad) from System Identi®cation kI/O (107 Nmm/rad) from Deformation Decomposition
119.0
116.6
23.5
10.0
4.5
2.8
2.1
112.4
108.3
24.0
10.2
4.6
2.8
2.1
$ LR: Length of horizontal member.
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The osets of the rotation centers in the concept joint model are shown in Table 1 for the dierent values of the member lengths. It is observed that the osets of the rotation centers do not seem to be signi®cantly aected by the member length change. 5.3. Eect of member length on parameter estimates 5.3.1. Decomposition of deformation The modeling parameters of joints can also be estimated by decomposing their cross sectional deformation. Eqs. (5) and (7) were used for this purpose. The out-of-plane bending stiness of a T-shaped joint in Fig. 2, kI/O, was estimated using decomposition of deformation. The estimation procedure was repeated for the dierent values of the length of the horizontal branch. Table 2 summarizes the results using decomposition of deformation as well as system identi®cation. The approximations using decomposition of deformation agree well with the results from system identi®cation. When the length of the horizontal branch was 2400 mm, the rotation of the vertical branch about the y-axis was decomposed using Eq. (1) and is illustrated in Fig. 5. The eect of shear deformation, fS, is the most dominant. However, the two other eects, fT and fB, cannot be neglected. If the rotations due to the bending of the vertical member and the twist of the horizontal member, fB and fT, respectively, are neglected, the estimate of joint stiness can be erroneous. 6. Example of a joint in a passenger car body structure 6.1. Joint description The joint in Fig. 6 was considered, which was isolated from a real car body structure. The horizontal and vertical members in this joint are called rocker and B-pillar, respectively. This joint has a complicated shape and does not have any geometrical symmetry. Thus, all 13 parameters of the concept joint model must be considered to be independent. Because the geometry of the B-pillar was complicated, it was dicult to describe it using a detailed model and to perform numerical experiments. In order to avoid these problems, a large portion of the B-pillar was cut o and a straight beam was attached as shown in Fig. 6. In order to simulate measurements that were needed in system identi®cation, loads were applied at the attached straight beam and displacements and rotations were calculated using ®nite element analysis at several positions along this beam. Because of geometrical non-symmetry, the coupling eect could not be
Fig. 5. Decomposition of rotation of vertical member of a Tshaped joint.
neglected in this joint. Thus, in numerical experiments, de¯ections not only in the load direction but also in the other directions were measured. All the measurements were considered simultaneously. 6.2. Eect of member length on parameter estimates 6.2.1. System identi®cation The system identi®cation technique was repeatedly applied to the real automotive body joint in Fig. 6 as the length of the horizontal member was increased. Four dierent values of the rocker length were considered; LR=474.8, 757.6, 930.9, 1138.3 mm. The results of system identi®cation are summarized in Table 3 for the four values of rocker length. The notations in Table 3 are as follows; 1. xP, yP, zP: local coordinates of the B-pillar. The xPaxis is selected to coincide with the shear axis of the B-pillar (Fig. 6). 2. k1, k2, k3: magnitudes of three spring constants. 3. r1, r2: osets of rotation centers in xP-direction for springs, k1 and k2, respectively. 4. hyP , hzP: osets of rotation center in yP and zP-directions, respectively, for spring, k3. 5. xi, yi, zi (i = 1 0 3): coordinates which de®ne the direction of the ith spring, ki. The plane in which the ith spring is located is selected to be normal to the zi-axis. 6. yz1xP: angle between the z1-axis and the xP-axis. According to Table 3, the estimates of parameters that de®ne the positions and directions of springs changed considerably as the rocker length changed. Note that in the case of the T-shaped joint made of simple box beams, there was no considerable change in the pos-
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Fig. 6. A joint in an automotive body structure. Table 3 Parameter estimates of an automotive body joint for dierent values of rocker length (13-parameter model) LR=474.8 k1 k2 k3 yz1xP yz1yP yz1zP yz2xP yz2yP yz2zP yz3xP yz3yP yz3zP r1 r2 hyt hzt
LR=757.6 8
3.8010 10 2.5851 107 5.8473 107 107.32 129.11 44.23 102.76 12.78 90.76 9.04 82.23 94.60 145.24 92.62 ÿ128.84 41.06
LR=930.0 8
2.8333 10 1.2672 107 5.5219 107 103.23 134.82 47.83 107.48 17.49 89.53 24.04 66.39 94.28 102.81 89.83 ÿ177.59 43.30
Units: LR, ri, hi (mm), kt(N mm/rad), yl (deg).
LR=1138.3 8
2.6346 10 0.9799 107 4.7175 107 100.89 136.56 48.61 109.54 19.59 89.22 37.42 52.88 94.03 69.04 32.41 ÿ215.29 46.50
2.4465 108 0.8889 107 4.2472 107 99.16 138.12 49.59 110.16 20.18 89.09 42.40 47.86 93.88 53.72 5.24 ÿ232.10 47.65
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389
Table 4 Parameter estimates of an automotive body joint for dierent values of rocker length (5-parameter model) LR=474.8 kF/A kI/O kT rF/A rI/O
LR=757.6 8
7.0279 10 2.6596 107 5.2971 107 165.63 104.11
LR=930.0 8
6.0530 10 1.3386 107 4.4552 107 147.83 92.74
LR=1138.3 8
5.7584 10 1.0259 107 4.0138 107 141.21 90.68
5.5330 108 0.9203 107 3.8131 107 138.82 90.63
Units: LR, ri (mm), ki (N mm/rad).
itions of springs as the horizontal member length changed. For the T-shaped joint, the directions of springs were not considered because there was no coupling eect. The inplane stiness and its oset of rotation center (kF/A and rF/A), the out-of-plane stiness and its oset of rotation center (kI/O and rI/O), and the twist stiness (kT) are considered as the most important parameters in the design of joints in automotive body structures. For this purpose, the parameters of this joint were identi®ed assuming that this joint did not have a coupling eect and that the osets of rotation centers in the yP and zP-directions were zero. The results are summarized in Table 4. The change in the magnitudes of the osets of the rotation centers in this joint (Table 4) was more signi®cant than in the T-shaped joint made of simple box beams (Table 1). The change in joint stiness is also illustrated in Fig. 7, where joint stiness is normalized using the value of stiness corresponding to LR=1138.3 mm. The change in the outof-plane stiness, kI/O, was the most signi®cant. The value of this parameter corresponding to the shortest
Fig. 7. Change in joint stiness due to change in the rocker length of an automotive body joint. No coupling eect was considered. (5-parameter model).
rocker is almost three times larger than the stiness corresponding to the longest rocker. The signi®cant change in the out-of-plane stiness can be explained using Fig. 8. Using the results in Table 3, the directions of the three springs were generated in Fig. 8 for the cases where LR=474.8 and 1138.3 mm. When the rocker length was the shortest (LR=474.8 mm), the spring axis z1 almost lay in the yPzP-plane, the spring axis z2 was almost parallel to the yP-axis, and the spring axis z3 was almost parallel to the xP-axis. When the rocker length was increased to its maximum value (LR=1138.3 mm), the directions of the spring axes z1 and z2 did not change signi®cantly. However, the direction of spring axis z3 changed signi®cantly. The z3-axis almost coincided with the xP-axis when LR was 474.8 mm. When LR was increased to 1138.3 mm, the z3±axis rotated in the xPyP-plane from the xP-axis towards the yP-axis. In other words this spring was mostly related to twist ¯exibility about the xP-axis when the rocker was the shortest (LR=474.8 mm). As the rocker length was increased (LR=1138.3 mm), the orientation of the plane of spring k3 changed so that the B-pillar could rotate about both the xP and the yP-axes. Thus, the increase in out-of-plane bending ¯exibility with the increase in rocker length can be explained by the change in the direction of spring k3.
Fig. 8. Change in the orientation of spring directions due to change in the rocker length of an automotive body joint.
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For the example of the real car joint, the threshold value for rocker length above which the joint stiness becomes insensitive to the change in rocker length cannot be observed in Tables 3 and 4, nor in Fig. 7. The threshold cannot be observed because the member length can be increased only up to the adjacent joint in the car body structure. If the estimates of parameters of concept joint models are to be used in the analysis of overall structure, the parameters must be identi®ed from a substructure, where the length of attached members is as long as possible and also any other joints are not included in the same substructure. Otherwise, the response of the overall structure may not be accurately predicted. In order to illustrate the importance of the eect of the member length on the parameter estimates and thus on the prediction of structural responses of the overall car body, two dierent substructures which contain the same joint were considered. The only dierence in the two substructures was that they had dierent rocker lengths; LR=474.8 and 1138.3 mm. The estimation results are summarized in Table 3. Each set of parameter estimates was used to predict the displacements of the B-pillar under a force in the I/O direction for the substructure whose rocker length was LR=1138.3 mm. The predicted displacements are shown in Fig. 9. The predicted displacements using parameters whose estimates were found from a substructure with rocker length LR=1138.3 mm almost coincided with the displacements obtained from the analysis of the detailed models. However, when the parameter estimates, which were found from a substructure with rocker length LR=474.8 mm, were used in the analysis of a substructure with rocker length
Fig. 9. Predicted displacements in I/O direction under a force of ÿ196 (N) in the same direction. The displacements along B-pillar were predicted for the substructure with LR = 1138.3 mm.
LR=1138.3 mm, the results were signi®cantly dierent from the analysis of the detailed models. Thus, in the analysis of the whole structure, it is not reasonable to use the values of parameters that are estimated from a substructure with short members, because this leads to a serious underestimation of some displacements of the overall car body. It is also sensible to consider the values of the joint parameters that correspond to long members when comparing design alternatives or trying to assess the eect of design modi®cations on the performance of a joint. Because the cross section of a horizontal member is complicated, the decomposition of deformation can not be used in the estimation of parameters of this joint.
7. Summary and conclusions The eect of member length on the parameter estimates of concept joint models was investigated in this study. A T-shaped joint made of simple box beams and a joint isolated from a real automotive body structure were considered for examples. The following are the main observations: 1. The ¯exibility of concept joint models increased as the length of attached members increased. The increase in the ¯exibility was primarily due to the increase in shear deformation of cross section. Thus, for a joint that is located in a plane, the increase in the out-of-plane ¯exibility was the most signi®cant. 2. In a T-shaped joint made of simple box beams, the change in ¯exibility became insensitive when the length of the attached members increased above a speci®c threshold. In a real automotive body joint, the threshold could not be observed. This is because another joint would be included in the substructure before the threshold was reached. 3. As the length of joint members increased, the changes in osets of rotation centers were more signi®cant in a real car joint than in a T-shaped joint. 4. In order to account for the coupling eect, the orientations of the springs were considered. The coupling eect did not exist in a T-shaped joint made of simple box beams, which was geometrically symmetric. The eect was signi®cant in a real car joint that was not geometrically symmetric. As the member length increased, the directions of springs in a real car joint changed in a way that the out-ofplane ¯exibility increased signi®cantly. Based on the above observations, the following conclusions can be made for the application of the concept joint models to the analysis of the overall structures:
K. Lee, E. Nikolaidis / Computers and Structures 68 (1998) 381±391
1. When the member length is above a certain threshold, the parameter estimates become insensitive to the change in the member length. Thus, the parameter estimates should be obtained from an isolated joint or a substructure whose members are longer than the threshold if the concept joint model is to be used in the analysis of overall structure. 2. The threshold value for the member length may not exist when the size or geometry of the overall structure limits the length of an attached member. In this case, joint parameters should be identi®ed using a substructure with the longest possible attached members. Otherwise, the application of concept joint models in the analysis and design of the overall structure can yield erroneous results.
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