Optimal Condition for Best Quality Joint in SPR Processing

1-     Introduction

In spite of the long history of riveting in joining of metals, coming back to the Bronze Age, its applications have been limited by the availability of wieldable metals results in the usage of the less expensive, and easy automation welding techniques [1]. Since 1975 [2], a developed method of riveting named as self-piercing riveting (SPR) started to find its way in industrial application due to its ability to join dissimilar materials, aluminium sheets, and pre-coated panels. The SPR process is used to attach two or more similar or dissimilar sheet materials by driving a rivet to the top (or top and middle) sheet(s) and  partially piercing to the bottom sheet creates an interlock between the sheets by flaring the rivet skirt, directed by a suitable die, without piercing the bottom sheet [3, 2]. The SPR processing is a cold process which unlike the conventional riveting process does not require a pre-drilled hole results in the same joining speed with that of the spot resistance welding [4]. As observed in Fig. 1a the riveting performance depends on several parameters, including the stack thickness, load of the process, die shape, die geometry, and geometry, shape, and hardness of the rivet. Besides, the primary important parameter to have a good joint is its interlock which can be define by sectioning the joint. A schematic of a joint section is illustrated in Fig. 1b. While deformed rivet diameter ( $D_{\mathit{dr}}$

) and rivet flaring ( $X$

) represent the interlocking, bottom thickness ( $t_b$

) has a strong effect on the strength of the joint. The procedure that is used by the automotive industry to evaluate the quality of a SPRed joint involves measurement of the rivet head height ( $h$

) above (or below) the surface of the top sheet and the rivet flaring ( $X$

) [5, 6]. Preferably, $h$

should be around zero ( $\pm 0.05\mathit{mm}$

), and as long as no crack appears in the rivet, the greater the interlock distance the better quality of the joints. After achieving a satisfactory joint it is important to check the strength of the joint using shear and peel tests. No need to mention that the higher strength joints are preferable for industrial usages.

To select the best condition for SPR processing to achieve all of the goals including a good interlock, sufficient $t_b$

, around-zero $h$

, and high peel and shear strengths, several experiments are needed. Therefore, comparing joints, ranking them, and choosing the best joint is one of the important stages in SPR processing.  Generally these stages (comparing, ranking, and choosing) are significant in any of the material and process selection issues [7, 8].  In the cases like the SPR process in which more than of a single definite criterion for selecting is involved and a large number of selection criteria should be taken into account, the need of a simple, methodical, and rational scientific way to guide designers in taking a proper material selection decision is undeniable.  For a specified engineering module dependent to several criteria, the proposed multi-criteria decision making (MCDM) [9] approach is a powerful tool to rank the alternatives. Simply, in the selection procedure, the multi-criteria decision making (MCDM) methods have been used commonly for the properties that can be represented by numerical values [10]. In 1990, Saaty [11] introduced a new methods of MCDM named as Analytic Hierarchy Process (AHP) that has been frequently used by decision makers facing a complex problem with multiple conflicting and subjective criteria due to its high potential and its simplicity.

In the present study a systematic assessment model is proposed to determine the optimal condition in order to obtain the best quality joint in the SPR processing with the aim of the AHP method.

2-     Theoretical basis of the AHP 

In a general sense, the AHP is a methodology for “relative measurement” [12]. The use of pair-wise comparisons between the quantities instead of direct allocation of their weights is the essence of the relative measurement and the AHP as well [13].  The AHP is based on three steps: problem modelling, comparative judgment of the alternatives and the criteria, and synthesis of the priorities [12, 13, 14].

2.1. Problem modelling

To facilitate a complex decision problem, AHP has the advantage of permitting a structure of the criteria as a hierarchy. At first, AHP overwhelm a complex multi-criteria decision-making problem into a hierarchy of interconnected decision elements such as criteria, decision and alternatives. The hierarchical structure of the objectives, criteria and alternatives produced in this step, similar to a family tree, provides the decision makers a better focus on a criteria and sub-criteria once assigning the weights. A hierarchy is compounded by a goal, a set of alternatives ( $X=\left\{x_1, x_2, \dots ., x_n\right\}$

), a set of criteria, and a relation on the goal, the criteria and the alternatives.

2.2. Pair-wise comparisons and Judgement scales

After the decomposition of the problem and the construction of the hierarchy, prioritization starts with the purpose of determination of the relative importance of the criteria within each level. The pair-wise judgment starts from the second level and finishes in the lowermost level. The criteria are compared pairwise in each level according to their altitudes and based on the specified criteria in the higher level. The decision maker does not need to deliver a numerical judgement; as an alternative a relative verbal appreciation, more acquainted in everyday lives, is satisfactory [13]. Therefore, the decision maker can declare opinions on pairs using linguistic terms associated to real numbers. In Saaty’s scale [11, 12]  the verbal statements are converted into integers from one to nine. Table 1 displays the match of Saaty’s scale to verbal judgments used in the weighting of two elements.

Table 1. Pair comparison evaluation scale (Saaty’s scale) [11, 12]

The pair-wise comparisons are recorded in a positive reciprocal matrix:

$A=\left[\begin{array}{cc}\begin{array}{cc}1& a_{12}\\ a_{21}& 1\end{array}& \begin{array}{cc}\dots & a_{1n}\\ \dots & a_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ a_{n1}& a_{n2}\end{array}& \begin{array}{cc}⋮& ⋮\\ \dots & 1\end{array}\end{array}\right]$

         (1)

where $a_{\mathit{ij}}$

is the comparison between elements $i$

and $j$

.

According to Saaty’s theory, each entries of Matrix A (1) is supposed to be the ratio between two quantities (weights)

$a_{\mathit{ij}=\frac{w_i}{w_j} \forall i,\mathit{ j}}$

.          (2)

Considering Eq. (1), condition of multiplicative reciprocity is satisfied as

$a_{\mathit{ij}}=\frac{1}{a_{\mathit{ji}}} \forall i,\mathit{ j}$

.          (3)

Inserting Eq. (3) into Matrix A (1)

$A=\left[\begin{array}{cc}\begin{array}{cc}1& a_{12}\\ \frac{1}{a_{12}}& 1\end{array}& \begin{array}{cc}\dots & a_{1n}\\ \dots & a_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ \frac{1}{a_{1n}}& \frac{1}{a_{2n}}\end{array}& \begin{array}{cc}⋮& ⋮\\ \dots & 1\end{array}\end{array}\right]$

.        (4)

Therefore, the number of performed pair-wise comparisons is $\frac{n(n–1)}{2}$

[15] assuming n is the number of criteria involved in the evaluation.

Once a pairwise comparison matrix is completed, there are many methods to derive the priority vector w ( $\mathit{w}={\left\{w_1, w_2,\dots ,w_n\right\}}^T$

) such as eigenvector [11, 14], and geometric mean [16] methods. In the current study we used geometric mean method due to three main reasons. First, to avoid “rank reversal” problem for scale inversion with the eigenvalue method [13]. Generally rank reversal can be expressed by changing the priorities by adding a new alternative. For further details, the authors reference readers to [16, 12]. Second, contrary to the eigenvector method, the weights can be expressed as analytic functions of the entries of the matrix [12]. Third, the final weights achieved from the whole hierarchy can be uttered as analytic expressions of the entries of all the matrices involving in the hierarchy. According to this method each component of w is obtained by

$\mathit{w}=\frac{\sqrt[n]{\prod _{j=1}^n{a}_{\mathit{ij}}}}{\sum _{i=1}^n\sqrt[n]{\prod _{j=1}^n{a}_{\mathit{ij}}}}$

.         (5)

To complete the analysis, the described procedure is repeated for all subsystems in the hierarchy. In order to synthesize the various priority vectors, these vectors are weighted with the global priority of the parent criteria and synthesized.

2.3. Consistency

Priorities are acceptable only if obtained from consistent or near consistent matrices. Therefore, the last step in the HP is the consistency check. For a perfectly consistent matrix, the transitivity rule is essential for all comparisons:

$a_{\mathit{ij}}=a_{\mathit{ik}}.a_{\mathit{jk}} \forall i,\mathit{ j},\mathit{ k}$

.         (6)

Among several methods proposed to check the consistency (for more detail please refer to [9, 12, 13]), Consistency Index (CI) [17], and Geometric Consistency Index (GCI) [18] have attracted more attention. In spite of the popularity of the CI, it has been criticised due to the permission of inconsistent matrices [19] or rejection of rational matrices [20]. Therefore, GCI was selected for consistency check in this research. GCI was defined as [21]

$\mathit{GCI}\left(A\right)=\frac{2}{(n–1)(n–2)}\sum _{i<j}{\log }^2{e}_{\mathit{ij}}$

       (7)

where $e_{\mathit{ij}}=a_{\mathit{ij}}{w}_j/w_i$

. Once GCI is calculated, the consistency index can be calculated using [21]

$\mathit{CR}=\frac{\mathit{GCI}}{k\left(n\right)},$

          (8)

in which, $k\left(n\right)$

is a parameter depends on n and can be find in table 2. According to AHP, the consistency is acceptable as long as CR is less than 0.1 (CR=10%). To satisfy this condition, GCI should be less than 0.3147 for $n=3$

, 0.3256 for $n=4$

, and 0.370 for $n>4$

[13].

Table 2. Values of $k\left(n\right)$

for $n=3,\dots , 16$

[21]

References