The Effect of Rivet Arrangement on the Strengths of Lap Joints and Lap Joint Design Methods

Abstract

To address the impact of rivet arrangement on the strengths of riveted lap joints, the failure modes and failure mechanisms of riveted lap joints were first studied using finite element analysis software. Next, the effects of the number of rivets, rivet rows, rivet arrangement, and row spacing on the lap joint strength were studied using the peak load as the evaluation index. Then, we proposed the concept of line load density to solve the problem that a varying rivet spacing and rivet edge distance will change the width of the sheet and thus the maximum load capacity, which is used as an index to study the effect of rivet spacing and rivet edge distance on the lap strength. Finally, a spring–mass model was developed to study the forces present in multi-row riveting. The model could accurately calculate the force–displacement curves during tensioning. The results show that when multiple rivet rows are used, higher stress concentrations cause the plate to first fracture at an outer rivet row with more rivets; therefore, the rivets should be arranged such that there are more rivets in the middle and fewer rivets on both sides. When the total rivet strength is greater than the remaining strength of the plate, the numbers of rivets and rivet rows have limited effects on the lap joint strength; however, this primarily affects the damaged form of the lap joint member. When the rivet spacing is less than 5d, the lap strength increases with increases in the rivet spacing, and when the rivet spacing is greater than 5d, the lap strength does not change significantly with increases in the rivet spacing. When the rivet edge distance is less than 3d, the lap joint strength increases with increases in the rivet edge distance, and when the rivet edge distance is greater than 3d, it has a limited effect on the lap joint strength. The rivet row spacing has no significant effect on the lap joint strength. The results of this study are valuable for improving the strengths of riveted structures in aircraft.

1. Introduction

To meet the demand for lightweight aircraft, aluminum and titanium alloys as well as fiber-reinforced composites, are commonly used in aircraft manufacturing. Aircraft structures are usually connected by riveted, glued, and glued–riveted joints [1,2]. Rivets are commonly used in aircraft manufacturing because they have high connection strengths, are simple to install, are inexpensive, and produce tight connections [3,4]. Therefore, it is necessary to analyze and evaluate the mechanical properties of riveted joints. A significant amount of research has been conducted to analyze the factors that influence the strengths of riveted joints. Hamel et al. [5], C.Y. Kim [6], and Tenorio et al. [7] investigated the effects of process parameters on joint strength by using finite element simulations. They found that riveting tool geometry and process parameters affect the riveting strength. Zhang et al. [8] investigated the mechanical properties of aluminum alloy rivets and found that the rivet size significantly affects riveted joints. S.P. Li et al. [9] deal with the effects of rivet-hole tolerance in CFRP/Al single-lap blind riveted joints by combining experimental tests and numerical simulation approaches. Aman et al. [10] et al. studied the effect of some controllable process parameters in riveting (i.e., the sequence of the riveting, distance between the rivets (pitch), and the gap between the sheets) on the quality of riveted lap joints and the formed rivets. A good combination of riveting process parameters minimizes the residual stress in sheets and rivets, bulging and material growth in sheets, and reduces the chances of post-riveting clearance in a riveted lap joint.

Li et al. [11,12] investigated the effect of the edge distance on the strengths of riveted joints, finding that the edge distance significantly affects the dynamic fatigue strength and the static properties of riveted joints. In addition, the shear and peel strengths of the lap specimens increased as the edge distance increased.

Since the riveted and bolted lap structures have similar force characteristics, relatively few domestic and foreign scholars have analyzed the load on the riveted construction. Therefore, the load distribution of the riveted lap structure is studied by studying the load distribution of the bolt lap structure. A simplified mechanical model for bolted joints has been developed to study the load distributions in bolted structures. It is assumed that a bolt transmits a load by shear deformation during plate tension or compression, and each bolt is treated as a uniform load [13]. However, the overall strength of a joint can be significantly reduced by the “short plate effect” of individual plates; therefore, it is important to quantify this difference when using bolted connections. McCarthy et al. [14,15] investigated the effects of fit clearance differences on the stress conditions in bolted members when a load was applied. They measured the magnitude of the load transferred by each bolt, and their results have formed a basis for some scholars to study rivet transfer loads. Kradinov et al. [16] obtained analytical solutions for the bolt transfer loads at different positions in a plane state by solving the static equilibrium equation and the displacement coordination equation. Gray et al. [17] experimentally investigated the effects of the joint thickness, the laminate taper, and missing fasteners on the load distributions in single-lap multi-bolt joints; they found that missing fasteners can lead to a significant decrease in load-carrying capacity.

In summary, most of the analyses of the factors that influence the strengths of riveted joints performed by scholars have focused on the rivet size and the plate thickness. However, the effect of the riveted joint arrangement on the mechanical properties is also significant. Changing the rivet arrangement can effectively improve the mechanical properties of riveted joints [11,12]; however, little research regarding this topic has been conducted. In addition, analyses of the load distributions in riveted structures have primarily focused on the uniformity of the rivet transfer loads; in addition, the established mechanical models are all one-dimensional models, which cannot effectively explain the influences of various factors on the strengths of riveted lap joints.

In this paper, the effects of the number of rivets, rivet rows, rivet arrangement, row spacing, edge distance, and rivet spacing on the lap joint strength were explored through experiments and simulations. The mechanical properties of the lap joint, such as the load capacity and stress distribution under different rivet arrangements, were analyzed. The optimum rivet arrangement was obtained. Finally, the spring–mass model can be established to accurately calculate the force–displacement curve of the lap structure during the stretching process. The research results have important practical applications for the design and manufacture of high-strength and high-reliability lap joint structures.

2. Material and Methods

2.1. Experimental Studies

2.1.1. Influencing Factors

The stress of the rivet joint with an equal riveting distance was investigated in this chapter. In Figure 1 the stress condition of the lap plate was shown, including the single-rivet small section joint and the double-row rivet small section joint. When a riveted plate is subjected to tension, the plate is also subjected to tension, the rivet is subjected to shear, and the plate is subjected to both shear and squeeze forces at the rivets. To prevent the plate from being badly pulled, squeezed, or sheared, it is necessary to set the breaking tension, breaking squeezing force, and breaking shear force of the plate equally. The expressions for the breaking tensile force, P₁, the damage squeeze force, P₂, and the damage shear force, P₃, of the plate are given in Equations (1)–(3), respectively. In Equations (1)–(3), t, c, δ, and d are the rivet spacing, the edge distance, the plate thickness, and the rivet diameter, respectively, while σ_b, τ_b, and σ₁ represent the tensile strength, the shear strength, and the extrusion strength of the plate, respectively.

$$P_1=(t-d)\delta · {\sigma }_b,$$

(1)

$$P_2=d · \delta · {\sigma }_1,$$

(2)

$$P_3=2(c-\frac{d}{2}) · \delta · {\tau }_b.$$

(3)

When the damage squeeze force, P₂, at the rivet hole is equal to the damage shear force, P₃, at the edge of the plate, then Equation (4) applies:

$$\begin{gathered} d · \delta · {\sigma }_1=2(c-\frac{d}{2}) · \delta · {\tau }_b, \\ c=\frac{d}{2}(1+\frac{{\sigma }_1}{{\tau }_b}). \end{gathered}$$

(4)

For the m-row rivets, when the breaking tension, P₁, of the plate is equal to the damage squeeze force of the plate, then Equation (5) applies:

$$\begin{gathered} (t-d)\delta · {\sigma }_b=m · d · \delta · {\sigma }_1, \\ t=d(1+1.8m). \end{gathered}$$

(5)

The lap strength of a riveted component is closely related to the rivet spacing and the edge distance. In addition, the lap strength is related to the number of rivets, the number of rivet rows, the rivet arrangement, and the row spacing. Therefore, experiments must be conducted to study these parameters.

2.1.2. Experimental Parameters and Design

The test plate was 2 mm thick and was composed of the 2024-T4 aluminum alloy. The rivets, which had diameters of 4 mm [18], were round-head rivets composed of the LY10 aluminum alloy. The two plates overlapped and were connected by rivets. The plate and rivet dimensions are shown in Figure 2 in which (a) depicts the specimen model and (b) shows the rivet model. In the specimen model, c is the edge distance, a is the row spacing, t is the rivet spacing, W is the plate width, and L is the lap length. The perpendicular edge distance is considered the same as the parallel direction in this model.

The rivet arrangement labeling format is shown in Figure 3. As an example, for specimen 5-3-122-8-12-10, the “5” indicates that there are five rivets, the “3” means that there are three rows of rivets, the”122” shows that, from left to right, the three rows of rivets contained one, two, and two rivets, the “8” denotes the edge distance, the “12” represents the rivet spacing, and the “10” denotes the row spacing. The unit of edge distance, rivet spacing, and row spacing is mm.

To analyze the effects of the number of rivets, the number of rows, the rivet arrangement, the row spacing, the edge distance, and the rivet spacing on the lap joint strength, 41 sets of riveted lap joint tests were conducted. Each set was tested three times. The experimental design scheme is shown in

Table 1. Experimental design scheme.

Rivet Number Effect	Rivet Spacing Effect	Row Spacing Effect
1-1-1-16-0-0	8-4-2222-8-8-10	6-3-222-8-12-8
2-1-2-10-16-0	8-4-2222-8-10-10	6-3-222-8-12-10
3-2-21-10-16-10	8-4-2222-8-12-10	6-3-222-8-12-12
4-2-22-10-16-10	8-4-2222-8-14-10	6-3-222-8-12-14
5-2-23-8-10-10	8-4-2222-8-16-10	6-3-222-8-12-16
6-2-33-8-10-10	8-4-2222-8-18-10	6-3-222-8-12-18
7-3-331-8-10-10	8-4-2222-8-20-10	6-3-222-8-12-20
8-3-332-8-10-10	8-4-2222-8-22-10
9-3-333-8-10-10	8-4-2222-8-24-10
Rivet Arrangement Effect	Rivet Edge Distance Effect	Row Number Effect
6-3-132-8-10-10	8-4-2222-4-12-10	3-1-3-8-10-10
6-3-213-8-10-10	8-4-2222-6-12-10	6-2-33-8-10-10
6-3-123-8-10-10	8-4-2222-8-12-10	9-3-333-8-10-10
6-3-333-8-10-10	8-4-2222-10-12-10	12-4-3333-8-10-10
6-3-323-8-10-10	8-4-2222-12-12-10
6-2-33-8-10-10	8-4-2222-14-12-10

.

Sun and Khalee [19] investigated the increases in the dynamic strengths of joints with increases in the loading speed during tensile tests. They found that the effect of the loading speed on the peak load was more significant when the loading speed was in the 0.5–2 or 10–30 mm/min ranges, and it was less significant when the loading speed was in the 2–10 mm/min range. Therefore, the tensile tests in this study were performed using a CMT4304 electronic universal testing machine at a loading speed of 3 mm/min. To eliminate the eccentric effect of the lapped specimens, an eccentric collet was used for the tests.

2.2. Simulation Analyses

2.2.1. Material Model

The simulation process considered the strain-hardening effect on the rivets and plates. Plastic strains were generated in both the rivet and the plate during the tensile tests; therefore, an isotropic elastic–plastic material model was used. The power-hardening model can be used to characterize its material properties [20].

The power-hardening model can be expressed by Equation (6):

$${\sigma }_y=\left[1+{\left(\frac{\dot{\epsilon }}{C}\right)}^{\frac{1}{p}}\right]k{\left({\epsilon }_{yp}+{\epsilon }_p\right)}^n,$$

(6)

where σ_y represents the yield stress, C and p are the Cowper–Symonds constants for the aluminum alloy, equal to 6500 s⁻¹ and 4, respectively, k and n are the strength coefficient and the hardening index, respectively, ε_yp is the elastic yield strain, and ε_p is the effective plastic strain.

The plates were tested, and plate simulations were performed to obtain the plate parameters, and the material parameters were obtained using the continuous response surface method. The plate was 2 mm thick, and the finite element model is shown in Figure 4. The material parameters of the 2024-T4 and LY10 aluminum alloys are shown in

Table 2. Power-hardening model parameters.

Material	k (MPa)	n	PR	εmax
2024-T4	618.6	0.23	0.33	0.28
LY10 [21]	515	0.2	0.33	0.2421

, where k and n are the same as in Equation (6). PR is Poisson’s ratio, and ε_max indicates the failure strain.

2.2.2. Parameter Settings

LS-DYNA was primarily used for the stress analysis of the structures subjected to various impact loadings, which was used for the numerical investigations of riveted joints under dynamic loadings in many research studies, such as Refs [22,23]. This paper established a tensile test model in the impact dynamics software LS-DYNA. The rivets and plates were made of hexahedral units. The cell size was 0.5 mm at the rivet and plate lap and 1 mm throughout the remainder of the grid. The model had a total of 42,591 nodes and 38,752 cells. The upper plate and the lower plate, the upper plate and the rivet, and the lower plate and the rivet all adopted the AUTOMATIC_SURFACE_T O_SURFACE contact method, and the friction coefficient was set at 0.18 [19]. BOUNDARY_SPC_SET was used to constrain all the degrees of freedom for 35 mm on one side of the substrate, as well as to constrain the degrees of freedom in the Y- and Z-directions for 35 mm on the other side. A velocity load of 3 mm/min in the X-direction was also applied using BOUNDARY_PRESCRIBED_MOTION_SET. A maximum principal strain random failure model was used to simulate the tensile fracture process. Based on Ref [24], the maximum principal strain random failure criterion is suitable for the large deformation failure behavior of homogeneous, isotropic metal materials with no obvious stress concentration and defects. The theoretical explanation described that when the maximum principal strain in the material reaches a certain degree, the material will be damaged and eventually fail. When applying the maximum principal strain random failure criterion, each point in the material’s principal strain must be calculated first. The principal strain refers to the strain value in the three principal stress directions, which the eigenvalue decomposition of the stress tensor can obtain. Then, determine whether the material will fail according to the size of the maximum principal strain. In LS-DYNA, MAT_ADD_EROSION was used to define the failure modes of the rivets and plates. To simulate the quasi-static stretching process, the simulation was solved using an implicit solver.

2.2.3. Simulation Model Verification

To verify the validity of the simulation model, a finite element model for one rivet lap specimen was established, as shown in Figure 5. The simulation results were compared with the experimental results, and the results are shown in Figure 6 The simulation curve agrees well with the experimental curve; the relative error was within 3%, indicating that the established model has high credibility.

3. Results

3.1. Single-Lap Shear Failure Behavior Analysis

3.1.1. Analysis of the Test Results

The test results show that two basic types of failure modes occurred in the joint: rivet shear damage and plate fracture. Figure 7 shows the different failure modes observed.

During the stretching process, the rivet-hole wall in the upper plate produces plastic deformation due to rivet extrusion and the end of the plate warps and deforms. At this time, the shear force is mainly transmitted by the self-locking structure, the upper plate rivet-hole wall, and the bottom of the lower plate riveting point. When the strength of a transfer link is smaller than that of other links competing with it, the shear transfer connection becomes the weak point in the specimen and will thus be the first to break. This occurrence is referred to as a “strength competition”. When the strengths of the rivet-hole wall in the upper plate and the bottom of the rivet point in the lower plate are greater than that of the self-locking structure, that is, the residual strength of the plate is greater than the total strength of the rivets, then rivet shear failure occurs. On the contrary, when the remaining strength in the plate is less than the total strength of the rivets, then the plate fractures. The fracture location is related to the arrangement of the rivets. Fractures occur at the load end when each end contains the same number of rivets; this occurs primarily because the load is first transferred to the rivets at the load end, thereby causing the stress concentration to be relatively high. Fractures occur on the side with more rivets when the number of rivets in each row varies; this occurs primarily because each rivet reduces the effective cross-sectional area of the plate. When there are many rivets, the effective cross-sectional area of the plate is reduced, and more stress will be generated during a tensile process; therefore, this region fractures first.

3.1.2. Analysis of the Simulation Results

Stress analyses were performed for the single-lap joints labeled 6-3-231-8-10-10 and 6-3-222-10-16-10. The positions of rows 1 through 12 were defined, and their finite element models are shown in Figure 8 and Figure 9. Figure 8a shows the front view of the “222” arrangement, with rows 1 through 3 defining the three rows of rivets in the upper plate. Figure 8b shows the back view of the “222” arrangement, with rows 4 through 6 defining the three rows of rivets in the lower plate. Figure 9 is constructed similarly to Figure 8. The displacement–stress curves of the units around rows 1 through 12 are shown in Figure 10, where (a) shows the displacement–stress curve for each rivet in the “222” arrangement and (b) shows the displacement–stress curve for each rivet in the “231” arrangement. In this simulation process, the failure determination criteria were obtained by the fitting method based on the previous single-riveting LS-OPT in Chapter 2.2.1. The stress values were usually taken as the average stress of the element between the row 1 rivet and row 2 rivet, where the von Mises model was used to describe the stress. The simulated fracture location is consistent with the location from the test results. Figure 10a shows that the stress in row 3 was greater than the stresses in rows 1 and 2, and the stress in row 4 was greater than the stresses in rows 5 and 6. For the front and back of the specimen, the stress in the outer row (either the row closer to the load end or the fixed end) was always greater than the stress in the inner row. The stress increased with the proximity to the outer row. The stress in row 3 was greater than the stress in row 4, causing a fracture to occur at the load end. Figure 10b shows that the stress in row 7 was greater than the stresses in rows 8 and 9, and the stress in row 12 was greater than the stresses in rows 10 and 11. The stress in the outer row was always greater than the stress in the inner row. The stress in row 7 was greater than the stress in row 12, thus causing a fracture to occur at the fixed end. The simulation results show that when there was an equal number of rivets in each row, the stresses in the outer rows of both the upper and lower plates were always greater than the stresses in the inner rows. Since the load was first transferred to the rivets at the load end, resulting in greater stress at the load end than at the fixed end, fractures occurred at the outside of the load end. When there were different numbers of rivets in each row, the outer row—which had more rivets—fractured first because the stress there was greater than at other locations. The cloud diagram shows that, as the load increased, the stress in the outer row increased first and yielded. The stress concentration area then expanded to form an “M”-type stress concentration area around the rivet, and the stress was greatest between the two rivets. When the principal strain reached the failure strain, a fracture occurred, first between the two rivets, then extended outward until the entire fracture had been completed.

3.2. Rivet Arrangement Trend Study

3.2.1. Line Load Density

Different rivet spacings and edge distances lead to changes in the plate width. However, the maximum load-carrying capacity is equal to the product of the remaining strength limit of the plate and the cross-sectional area; therefore, different rivet spacings and edge distances cause variations in the peak load. To eliminate the effect of the plate width on the peak load, this paper proposes using the line load density to measure the effects of rivet spacing and edge distance on the lap strength.

The lap strength recovery rate can be expressed by Equation (7):

$$\eta =\frac{F}{F_1}=\frac{F}{{\sigma }_{b}S}=\frac{F}{{\sigma }_{b}W\delta },$$

(7)

where η represents the strength recovery rate, F is the peak load of the lap specimen, F₁ is the peak load of an intact plate of equal width, and σ_b is the strength limit of the plate. S and δ are the plate’s cross-sectional area and thickness, respectively, and W is the plate width.

Since the strength limits and thicknesses of the plates were held constant, the lap strength was measured using the line load density, D = F/W, when studying the effects of the rivet spacing and the edge distance on the lap strength. The proposed concept can greatly reduce the workload of studying rivet spacing and rivet edge distance.

3.2.2. Number of Rivets

A sample with plate dimensions of 130 mm × 36 mm × 2 mm and from one to nine rivets was tested. The results show that when there were four rivets or fewer, the shear damage occurred in the rivets. When there were more than four rivets, a fracture occurred on the plate. The effect of the number of rivets on the peak load is shown in Figure 11. For specimens with fewer than eight rivets, the effect of the number of rivets on the peak load was significant and increased with the number of rivets. For the specimens with eight or nine rivets, the effect of the number of rivets on the peak load was limited. When the total rivet strength was less than the remaining strength of the plate, rivet shear damage occurred. The peak load also varied linearly with the number of rivets; in other words, the total rivet strength was equal to the product of the number of rivets and the strengths of the individual rivets. When the total strength of the rivets was greater than the remaining strength of the plate, the plate fractured. At this point, increasing the number of rivets could not significantly increase the strength of the lap joint. A comparison of the nine-rivet and seven-rivet specimens indicates that when the total rivet strength was greater than the remaining strength of the plate, an increase in the number of rivets may have even caused a decrease in the static strength. Therefore, when the total rivet strength is greater than the remaining strength of the plate, the number of rivets is not crucial to the lap joint strength but rather primarily determines the form of the damage that occurs in the lap joint structure.

3.2.3. Number of Rivet Rows

Tests were performed for specimens with an edge distance of 8 mm, a rivet spacing of 10 mm, a row spacing of 10 mm, plate dimensions of 130 mm × 36 mm × 2 mm, and from one to four rivet rows. The effect of the number of rivet rows on the peak load is shown in Figure 12. The results show that when there was a single row of rivets, shear damage occurred in the rivet; however, when there was more than one rivet row, fracture occurred in the plate. When the total rivet strength was greater than the remaining strength of the plate, increasing the number of rivet rows did not significantly increase the lap strength. The specimen with three rows of rivets had the greatest lap strength, but its peak load was only 0.68 kN larger than that of the specimen with two rows of rivets. There was no improvement in the lap strength when there were four rivet rows than when there were three. From the lap length analysis perspective, increasing the lap length did not significantly increase the lap strength. Therefore, when the total rivet strength is greater than the remaining strength of the plate, the number of rivet rows is not critical to the lap strength. Improving the strength of a lap joint requires changing the rivet arrangement.

3.2.4. Rivet Arrangement

Tests were conducted for specimens with an edge distance of 8 mm, a rivet spacing of 10 mm, a row spacing of 10 mm, plate dimensions of 130 mm × 36 mm × 2 mm, and different rivet arrangements. The results show that plate fractures occurred in all the specimens. The effect of the rivet arrangement on the peak load is shown in Figure 13. A comparison of the 1-3-2, 2-1-3, and 1-2-3 arrangements shows that when the total number of rivets was the same between specimens, and there were more than two rows, the specimen with more rivets in the middle row had a higher static strength. This result primarily occurred because when there were many outer rivets and few middle rivets, the stress concentrations around the outer rivets were superimposed, causing larger stress concentrations and a smaller static strength in the member. A comparison of the 1-3-2 and 3-3-3 arrangements shows that the number of rivets was inversely proportional to the peak load when the lap areas were equal; therefore, an increase in the number of rivets does not necessarily increase the lap strength. Multiple rows of rivets should be arranged with more rivets in the middle and fewer on the sides. A comparison of the 3-3-3 and 3-2-3 arrangements shows that there was no significant difference between the static strengths of the staggered and parallel arrangements.

3.2.5. Rivet Row Spacing

Tests were conducted for specimen 6-3-222-8-12-a using different row spacings, where a = 2d, 2.5d, 3d, 3.5d, 4d, 4.5d, and 5d. The plate had dimensions of 130 mm × 28 mm × 2 mm. The effects of the row spacing and the lap length on the peak load are shown in Figure 14. The results show that the row spacing and the lap length have limited effects on the peak load.

3.2.6. Rivet Spacing

To eliminate the effect of the plate width on the lap strength, the lap strengths of the specimens were measured using the line load density, F/W. Tests were performed for specimen 8-4-2222-8-t-10 using different riveting spacings, where t = 2d, 2.5d, 3d, 3.5d, 4d, 4.5d, 5d, 5.5d, and 6d. The plate had dimensions of 130 mm × W mm × 2 mm, where W = 24, 26, 28, 30, 32, 34, 36, 38, and 40. The effects of the rivet spacing on the line load density and the maximum stress are shown in Figure 15. The results show that the line load density increased with increases in the rivet spacing. However, when the rivet spacing was greater than 5d, the line load density did not change significantly with further increases in the rivet spacing. Therefore, the maximum stress decreases as the rivet spacing increases and the degree of reduction decreases when t > 5d.

3.2.7. Rivet Edge Distance

Tests were performed for specimen 8-4-2222-c-12-10 using different edge distances, where c = d, 1.5d, 2d, 2.5d, 3d, and 3.5d. The plate had dimensions of 130 mm × W₁ mm × 2 mm, where W₁ = 20, 24, 28, 32, 36, and 40. The effects of the edge distance on the line load density and the maximum stress are shown in Figure 16. The results show that the line load density increased with increases in the edge distance; however, when the edge distance was greater than 3d, there was no significant change in the line load density with further increases in the edge distance. Therefore, the maximum stress decreases as the rivet edge distance increases and the degree of reduction decreases when c > 3d.

4. Mechanical Model and Numerical Analysis

4.1. Stress Distributions around the Rivet Holes

While symmetry is present in single-riveted structures, multi-riveted structures are antisymmetric because each rivet hole is subjected to a different load. The proportion of the load applied to each specific rivet hole is closely related to the material properties, the number of rivets, and the lap joint format. The rivets can be divided into three categories according to the forces applied at different locations: first rivet, end rivet, and middle rivet. The first rivet is closest to the external load and is subject to the rivet transfer load, the bypass load, and the interference force. The end rivet is furthest from the external load and is subject only to the rivet transfer load. A middle rivet is located between the first and end rivets and is simultaneously subject to the rivet transfer load, the bypass load, and the interference force. In addition, the farther away a middle rivet is from the first rivet, the smaller the bypass load and interference force. After the rivet transfer load is applied to the rivet–hole structure, it releases the interference strain due to peri-hole interference extrusion deformation, then extrusion of the rivet hole occurs. The bypass load can be considered stretching with an orifice plate. The force on the rivet hole can be considered a superposition of the rivet transfer load, F_r, the bypass load, F_a, and the interference force, F_int, as shown in Figure 17.

Therefore, the riveted structural stress can be expressed by Equation (8):

$$\sigma ={\sigma }_r+{\sigma }_a+{\sigma }_{\mathrm{int}}$$

(8)

where σ_r represents the rivet transfer load stress, σ_a is the bypass load stress, and σ_int is the residual stress around the hole.

4.2. Spring–Mass Model

The stiffness of the linear elastic is fixed in this model. If plasticity is to be considered, a nonlinear spring with stiffness varying with strain (deformation) must be used. The spring–mass model is usually used to discuss rivet joints [25,26].

The study indicated that the stress in a riveted structure is primarily determined by the rivet transfer load and the bypass load, while the interference force has a smaller effect [27]. The force of the lap plate is shown in Figure 18 where F_t represents the tensile load, Fri is the rivet transfer load, and Fai is the bypass load.

The rivet transfer load can be calculated using Equation (9):

$$F_t={\displaystyle {\sum }_i^N{F}_r{}^i}$$

(9)

where N is the number of rivet lines.

According to its definition, the bypass load can be expressed by Equation (10):

$$F_a{}^i{=F}_a{}^{i+1}{+F}_r{}^{i+1}$$

(10)

The bypass load calculation is based on the rivet transfer load; therefore, calculating the rivet transfer load is crucial for analyzing the load distributions of the riveted members.

Equations (9) and (10) show that the algebraic sum of all the rivet transfer loads is equal to the plate tensile load, F_t. The bypass load that is transferred backward around the previous rivet hole is equal to the sum of the external loads of all the subsequent rivets; therefore, no load is transferred backward around the last hole. There are significant differences between the bypass loads at different locations, which result in differences in the stress distributions at different locations in riveted lap joints. The bypass load causes the outer-row stress to be consistently larger than the inner-row stress.

During a load analysis of a lap joint, the spring–mass model can be used to fit the load transfer process when the member is loaded, as shown in Figure 19. In Figure 19a shows the simplified model, in which W is the plate width, a is the row spacing, r is the rivet radius, t₁ and t₂ are the thicknesses of the upper and lower plates, respectively, and Figure 19b depicts the spring–mass model, where KA and KC represent the spring stiffnesses of the upper and lower plates, respectively, and KB is the spring stiffness of the rivet.

The rivet–hole contact surface was treated as a rigid body with no mass, and the plate and rivet between the contact surfaces were replaced with a spring that had a coefficient of elasticity. Since the tensile direction of the plate and the shear deformation direction of the rivet are both horizontal, the axial deformation can be ignored in this direction. Therefore, the plate and rivet were reduced to a spring with elasticity, but only in the horizontal direction. According to Hooke’s law, the load transmitted by the spring is equal to the product of the coefficient of elasticity and the displacement difference between the two ends of the spring. Therefore, Equation (11) can be used for each mass:

$$\left\{\begin{array}{l}F{A}_i=K{A}_i(x_{Ai}-x_{Ai+1}-2{△}_i)\\ F{C}_i=K{C}_i(x_{Ci}-x_{Ci+1}-2{△}_i)\\ F{B}_i=K{B}_i(x_{Ci}-x_{Ai})\end{array}\right.,$$

(11)

where F_Ai, F_Ci, and F_Bi are the loads transmitted by each spring, K_Ai, K_Ci, and K_Bi are the spring stiffnesses of the upper plate, the lower plate, and the rivet, respectively, x_Ai and x_Ci are the displacements of each mass, and Δ is the amount of interference on one side.

The equivalent elasticity coefficient of the plate can be calculated using Equation (12):

$$\left\{\begin{array}{l}KA=\frac{E_b\times W\times t_1}{a-2r}\\ KC=\frac{E_b\times W\times t_2}{a-2r}\end{array}\right.,$$

(12)

where E_b is the modulus of elasticity of the plate, W is the plate width, a is the row spacing, r is the radius of the rivet, and t₁ and t₂ represent the thicknesses of the upper and lower plates, respectively.

The equivalent spring stiffness coefficient for the riveted plate is generally obtained from the empirical formula shown in Equation (13), which is based on the influence of the riveting method [28]:

$$\frac{1}{KB}=\frac{2(t_1+t_2)}{3G_b{A}_b}+\left[\frac{2(t_1+t_2)}{t_1{t}_2{E}_r}+\frac{t_1+t_2}{t_1{t}_2{E}_b}\right](1+3\beta ).$$

(13)

In Equation (13), A_b represents the cross-sectional area of the rivet, G_b is the shear modulus of the rivet, E_r is the modulus of elasticity of the rivet, E_b is the modulus of elasticity of the plate, and β is the coefficient of secondary bending. Since only the load bearing of the tensile load in the horizontal direction was analyzed during this study, β = 0 was applied.

After determining the equivalent spring stiffness, the equations of motion for each mass could be determined. The free body diagram and equations of motion for masses 1 and 4 are shown in Figure 20. Since different rivet spacings and edge distances can cause different stress concentrations in the specimen, the concept of a stress severity factor is proposed here. The expression for the stress severity factor, L, is given in Equation (14):

$$L=W(\beta D_1+\gamma D_2),$$

(14)

where W is the width of the plate, β and γ are the stress concentration coefficients due to different rivet spacings and edge distances, respectively, and D₁ and D₂ represent the equations for the variations in the line load density as functions of the rivet spacing and the edge distance, respectively. D₁ and D₂ can be obtained by fitting the half-life equation [29], which is expressed by Equation (15):

$$y(t)=y_0+A_1{e}^{-\frac{t}{{\tau }_1}}.$$

(15)

In Equation (15), y₀ is the offset, A₁ is the amplitude, and τ₁ is the decay time.

Figure 21 shows the fitting curves for the line load density, where (a) depicts the curve for the line load density as a function of the rivet spacing and (b) depicts the curve for the line load density as a function of the edge distance. The fitting results shown in Figure 21a,b produced Equations (16) and (17), respectively.

$$D_1=0.68-0.4e^{-\frac{2t}{3}},$$

(16)

$$D_2=0.7-0.4e^{-\frac{5c}{8}}.$$

(17)

Therefore, the stress severity factor, L, can be expressed by Equation (18):

$$L=W\left[\beta (0.68-0.4e^{-\frac{2t}{3}})+\gamma (0.7-0.4e^{-\frac{5c}{8}})\right].$$

(18)

Since the aluminum alloy is an elastic–plastic material, it is necessary to introduce a discount factor, ν, into the calculation process. Its expression, provided in Equation (19), was obtained by stretching the plate aluminum alloy and fitting the results, as shown in Figure 22:

$$\nu =\frac{F}{F_{\max }}=\frac{15.7-16e^{-\frac{5x}{11}}}{F_{\max }}=1-e^{-\frac{5x}{11}}.$$

(19)

In Equation (19), F represents the tensile load of the specimen, F_max is the maximum tensile load, and x is the tensile displacement.

Therefore, the entire model can be expressed as the system of linear equations given in Equation (20):

$$Ma+\nu KX=L{F}_t.$$

(20)

Acceleration can be neglected during quasi-static stretching; therefore, Equation (20) can be replaced by Equation (21):

$$\nu KX=L{F}_t,$$

(21)

where the equivalent spring stiffness coefficient matrix, K, is provided in Equation (22):

$$\left(\begin{array}{ccccccccc}KC4+KC3+KB4& -KB4& -KC3& 0& 0& 0& 0& 0& 0\\ -KB4& KB4+KA4& 0& -KA4& 0& 0& 0& 0& 0\\ -KC3& 0& KC3+KC2+KB3& -KB3& -KC2& 0& 0& 0& 0\\ 0& -KA4& -KB3& KA4+KA3+KB3& 0& -KA3& 0& 0& 0\\ 0& 0& -KC2& 0& KC2+KC1+KB2& -KB2& -KC1& 0& 0\\ 0& 0& 0& -KA3& -KB2& KA3+KA2+KB2& 0& -KA2& 0\\ 0& 0& 0& 0& -KC1& 0& KC1+KB1& -KB1& 0\\ 0& 0& 0& 0& 0& -KA2& KB1& KA2+KB1+KA1& -KA1\\ 0& 0& 0& 0& 0& 0& 0& -KA1& KA1\end{array}\right).$$

(22)

The displacement vector, X = (x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉)^T, represents the displacement of C4, A4, C3, A3, C2, A2, C1, A1, and A0, respectively.

The load vector, F_t, can be expressed by Equation (23):

$$\left(\begin{array}{c}-2KB4{\Delta }_c{}_4\\ 2KB4{\Delta }_c{}_4\\ -2KB3{\Delta }_c{}_3\\ 2KB3{\Delta }_c{}_3\\ -2KB2{\Delta }_c{}_2\\ 2KB2{\Delta }_c{}_2\\ -2KB1{\Delta }_c{}_1\\ 2KB1{\Delta }_c{}_1\\ F_t\end{array}\right).$$

(23)

The simulation curves were obtained using the data in

Table 3. Spring–mass model input data.

Gb (GPa)	Er (GPa)	Eb (GPa)	W (mm)	t1 (mm)	t2 (mm)	r (mm)	Δ (μm)
28 [25]	69 [25]	68 [25]	2c + t	2	2	4	25

and by building finite element models for different rivet arrangements. Theoretical values were also calculated, and they were compared with the curves calculated by the finite element model, as shown in Figure 23. The theoretical curves agree well with the curves produced by the finite element simulations, which indicates the accuracy of the mechanical model developed during this study.

In this chapter, the riveted lap structure is simplified to a spring–mass point model. The forces of the whole model under tensile loading are studied by analyzing the forces at each mass point. The beneficial effects of the developed model are:

(1) The spring–mass model is established according to Hooke’s law. The variation of force with displacement during the tensile process of the riveted lap specimen is derived completely. At the same time, the equivalent spring stiffness coefficient-displacement load equation system is established.

(2) The system of equivalent spring stiffness coefficient-displacement-load equations can be used to calculate the force–displacement curves for different rivet spacing, edge distance, and row spacing under tensile loading. It can be used to study the prediction of the strength of riveted lap joints.

5. Conclusions

During this study, the effects of the number of rivets, number of rivet rows, rivet arrangement, and the rivet row spacing on the lap joint strength were first investigated with the peak load used as the evaluation index. Since changing the rivet spacing or the edge distance leads to changes in the plate width, the effects of the rivet spacing and the edge distance on the lap joint strength were also studied while using the line load density as the evaluation index. Finally, a spring–mass model was developed. The study produced four primary conclusions:

(1)

The primary failure forms in riveted lap joints are rivet shear damage and plate fracture. The failure form follows the “strength competition” principle. When the total rivet strength is less than the remaining strength of the plate, rivet shear damage occurs. At this point, the peak load is linearly related to the number of rivets. When the total rivet strength is greater than the remaining strength of the plate, the plate then fractures. In this case, the fracture area is primarily related to the rivet arrangement. When each row contains the same number of rivets, fracture occurs at the load end. When the number of rivets varies between rows, fracture occurs on the side with more rivets.

(2)

When there are more than two rivet rows, more rivets should be placed in the middle, and fewer rivets should be placed on both sides. There is no significant difference between the lap strengths of the staggered and parallel arrangements.

(3)

For a certain edge distance and row spacing arrangement, the rivet spacing is the primary factor that affects the mechanical properties of a lap joint. When the rivet spacing is less than 5d, the lap joint strength increases with increases in the rivet spacing. When the rivet spacing is greater than 5d, the lap joint strength does not change significantly. For a certain rivet spacing and row spacing arrangement, the edge distance is the primary factor that affects the mechanical properties of the lap joint. When the edge distance is less than 3d, the lap strength increases with increases in the edge distance; however, when the edge distance is greater than 3d, the lap strength does not change significantly. For a certain rivet spacing and edge distance arrangement, the effect of the row spacing on the lap strength is limited. Further analysis shows that the lap length has a limited effect on the lap strength.

(4)

A spring–mass model was developed. The model explains that the presence of a bypass load leads to significantly higher stresses around the riveted holes at the ends of a lap joint than at the intermediate locations. In addition, the force–displacement curves corresponding to different rivet spacing, edge distance, and row spacing can be calculated by the model, which reduces the test costs of studying the rivet arrangement method.