Single and Double Lapped Joints

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Maximum Clamping Force in Single and Double Lapped Joints H. SALIM, A. EL-DIN EL-SISI, H. EL-EMAM and H. EL-DIN SALLAM

ABSTRACT In this paper, progressive damage modeling was used in order to identify the maximum allowable clamping force in single (3DS) and double (3DD) bolted lapped joints. A constitutive model based on continuum damage mechanics was implemented and used in joint simulation. The model results were compared with experimental results and good agreement was evaluated. A 3D finite element model consists of a composite laminate plate, steel plates, steel washers, and steel bolts was adopted for the modeling of the single and double bolted joint.

The effect of bolt clearance, friction coefficient, and joint type on the maximum clamping force was studied and analyzed. It was found that using the composite plate between two steel plates (double lap) increased joint clamping force capacity significantly. This leads to enhancement in the effect of clamping force and decreases the compression damage in the thickness direction of the composite plate. According to this study it is recommended to use washer bolt hole clearance less than or equal to the composite plate bolt hole clearance as bigger washer clearance can cause early damage due to bolt tightening. It was found that, the beak point of the stress destruction under the washer or the steel plate is shifted due to the softening behavior under the washer head. The clamping stiffness in the case of single lapped joint is about double the case of double lap.

INTRODUCTION Bolted joints for its advantages, such as the ease of assembly and disassembly, are often preferred in many composite joining applications. Bolted joints are considered to be the weakest link in a structure, as drilling bolt holes creates high stress concentration [1]. _____________ Hani A. Salim, Alaa A. Elsisi, and Hesham M0 El-Emam, University of Missouri., Columbia, Missouri, 65211, U.S.A. Hossam El-Din M. Sallam, Jazan University, Kingdom of Saudi Arabia; on sabbatical leave from Materials Engineering Department, Zagazig University, Zagazig, 44519, Egypt.

The design and analysis of fiber reinforced polymeric composite bolted joints involves high degree of complexity and requires a special attention because of the anisotropic, inhomogeneous and visco-elastic properties. Joint geometry, stacking sequence, fiber orientation and bolt pre-load are some of the critical factors to be considered for a reliable joint design [2].

Experimental works studied the effect of clamping force on stiffness and the strength of composite single and double lapped joints [3–7]. Experimental results show that tightening the bolt increased the initial bearing stress by 22% and the maximum bearing stress by 105% [3–6]. Significant increase in strain was observed for the dowel pin joints when compared to the finger tightened bolted joints [3–6]. The increase in clamping pressure increased the post-peak stiffness, whereas the initial stiffness and the bolt-hole elongation decreased significantly [3–6]. Increase in clamping pressure increased the ultimate bearing strength to saturation, whereas the delamination bearing strength increased progressively [4–7], where the clamping pressure suppressed the delamination and the inter-laminar cracks. The failure mode changed from catastrophic fracture to a progressive failure as the clamping pressure was increased [4–7]. The effect of lateral supports on the failure in composite bolted joints experimentally studied [5–8]. It was found that the clamp load increased with the applied tensile load; Poisson's ratio contributed for the initial increase in the clamping force until the bearing failure, and the lateral constraints preventing the material damage increased the clamping force after the bearing failure [5–8]. Effects of clamp-up pressure on the net tension failure of bolt-filled laminated composite plates were studied experimentally [6–9]. For hole filled sample, higher the clamping pressure lower was the tensile strength of the bolt filled-hole laminate. Unlike bolt filled-hole laminate, the tensile strength of the bolted composite joints increased with clamping pressure [6–9].

Several works were adopted to analyze the progressive failure of composite materials [8–23]. The simplest technique to model degradation of stiffness is by applying an instantaneous degradation factor if the failure criteria are met, this technique is called the ply discount technique [8–23], while simple to implement, the sudden, complete failure of the elements does not compare well with experimental observations [11]. These models were followed a group of models at which damage state variables associated with the residual stiffness of a lamina were used [13–16].The stiffness terms of an element were degraded according to these internal damage state variables upon satisfying the corresponding failure criterion. The aforementioned methods of progressive damage modeling have led to the development of more phenomenal and physically-based approaches including the damage mechanics approach [17–23].

PROGRESSIVE DAMAGE MODEL A damage mechanics model (CDMM) was adopted with only three independent damage variables, i.e., di were used to describe the energy dissipation, for which each variable takes two values, one for tension mode and the other for compression mode.

For simply adjusting the damage law, a damage activation functions (DAF) based on the maximum strain criteria was used. For comparison and for solution time consideration Ansys material degradation model (PDM) which depends on the ply

discount technique was used for comparisons and for parametric study [8,22]. In Ansys PDM, if the failure criterion is met, an instantaneous degradation factor (km) is applied to the material properties, inn this study km was selected equal to 5%.

Stress Strain Relation Six damage variables were adopted to describe the degradation of material properties di, where i= {1, 2, 3, 4, 5, 6}. Among the six variables, three independent variables are found, as the damage for shear stiffness component are taken from the normal damage component, Eq. 2: d = 1 − (1 − d )(1 − d ) d = 1 − (1 − d )(1 − d ) Eq. 2 d = 1 − (1 − d )(1 − d ) The damage tensor d relates the nominal stress σ to the effective stress σ, i.e. σ = (I − I: d): σ where I is the identity tensor. The stress-strain relation is derived from Matzenmiller [12] complementary free energy density of a damaged orthotropic lamina ψ, Eq. 3; σ υ σ σ Eq. 3 + + σ σ + ψ= 2(1 − d )E 2(1 − d )E E 2(1 − d )G Where E , G , and υ are the Young’s modulus, shear modulus, and Poisson’s ratio for the undamaged elastic material, respectively. The partial derivative of Eq. 3 gives the corresponding two-dimensional (2D) compliance tensor.; From Eqs. 3 an expression for ψ that can be used for the 3D was driven which result in the compliance tensor in Eq. 4 , [21]; υ υ 1 1 1 1 E E (1 − d )E 2 2 2 υ 1 1 1 1 υ E 2 2 2 (1 − d )E E 1 1 1 1 υ υ 2 2 2 (1 − d )E E E ε= 1 1 1 1 1 1 (1 − d 2 2 )G 2 2 2 1 1 1 1 1 1 (1 − d 2 )G 2 2 2 2 1 1 1 1 1 1 (1 − d )G 2 2 2 2 2

.σ

Eq. 4

The stiffness tensor is the relation between the effective stress σ and the strain ε, and can be obtained from the inverse of the 3D compliance tensor.

Damage Activation Functions Damage Activation Function is the function that defines the elastic domain [36], it can be defined as; Eq. 6 g =g −γ ≤0

g is the positive loading function (normal) that depends on the stress states, and γ is the updated damage threshold function, where m refers to the failure mode. The aim of this paper is to develop the simplest 3D CDM model and compare it with other simple models like PDM and SPDM, and thus a simple loading function was used to describe the damage initiation process based on the maximum strain criteria [24-27].

Table I describes the proposed criteria; it can be observed that shear strength was neglected, since the degradation of shear stiffness components was considered to be a function of normal stress damage variable d₁, d₂ ,d₃. This means that the activation of damage in normal stress components would also activate the damage in the shear stress components.

TABLE I. LOADING FUNCTIONS g Tension ε For ε > 0 g = ϵ ε Matrix Direction 2 For ε > 0 g = ϵ ε Matrix Direction 3 For ε > 0 g = ϵ

Damage direction Fiber Direction 1

Compression ε − For ε < 0 ϵ ε − For ε < 0 ϵ ε − For ε < 0 ϵ

For example, the fiber tension failure strain is calculated from ϵ

=

. The

damage development in the material initiates when the value of g exceeds the initial damage threshold of γ = 1. Further damage growth occurs when the value of g in the current stress state exceeds the value of γ in the previous loading history[17].

Damage Evolution Laws The evolution of the damage threshold values γ is expressed by the three Kuhn– Tucker conditions, i.e., γ ≥ 0 , g ≤ 0 , and g . γ = 0 [36, 24], where γ = , in which T is time. Neglecting viscous effects, the damage activation functions of Eq. 6 always have to be non-positive [16,17]. While g is negative, the material response is elastic. When the strain state activates the criterion g = 0 , it is necessary to evaluate the gradient g [16,17]. If the gradient is not positive, the state is one of unloading or neutral loading. If the gradient g is positive, there is damage evolution, and the consistency condition has to be satisfied, i.e., g = g − γ = 0.

Under tension loading the produced cracks do not affect the compression response; the compression elastic domains are unaffected by γ , where m = { t , t , t }. The following expressions can be used to represent the compression damage thresholds: Eq. 7 and m = { c , c , c } γ = max{1, max{g }} where τ = 0: T, On the other hand, under compression loading the produced cracks completely affect the tension response; the tension elastic domains are affected by γ , where m = { c , c , c }. The following expressions can be used to represent the tension damage thresholds: γ = max{1, max{g , g }} Eq. 8 where τ = 0: T , m = { t , t , t } and n = { c , c , c }

According to Bazant’s crack band theory [19,20], the damage energy dissipated per unit volume G for uniaxial or shear stress condition is related to the critical strain energy release rate G along with the characteristic length LC of the finite element as follows: where, m is the failure mode number.

G = Eq. 9 This condition was used to adjust the laws for the damage variables or the damage evolution laws. In this paper, an analytical procedure was performed to identify a suitable degradation law. First, a two-coefficient material law is proposed for the stress-strain relation σ (ε ). To find these coefficients, the following two equations were used: G σ (ϵ ) = S ; G = σ (ε ) ∂ε = Eq. 10 L Where, i is the direction corresponding to the mode m. By solving the produced material law, σ (ϵ ), with the proposed damage threshold function γ , another function in terms of d and γ instead of σ and ε can be found, where σ = (I − I: d): σ. The proposed material laws are listed in Table II along with the corresponding damage activation laws. For the exponential damage law case, Camanho and Matthews [2] found the same damage evolution law. The positive α adjustment parameter can be defined as following: α =

where, i is the direction corresponding to the

Eq. 11

mode m Therefore, the maximum size for the finite element for each damage law m is L < [25]. If a FEM contains elements with LC larger than , the parameter α

will be negative and the mesh becomes unfeasible. This can be avoided by reducing the corresponding strength form S <

[19]., and in this case the damage

variable will take its maximum value of 1 if γ > 1 [15].

TABLE II. MATERIAL DEGRADATION AND DAMAGE EVOLUTION LAW Degradation Law Material Law Damage Law α 1 Linear d = (1 + )(1 − ) σ = B − Aε 2 γ A & B are material law constants, α is the adjustment parameter of damage law.

Ansys user Material Subroutine (USERMAT) The user material routine, USERMAT, is an ANSYS user-programmable feature [22]. Its function is to allow users to write their own material constitutive equations within a newly developed general material framework. This subroutine is called at all material integration points of these elements during the solution phase. The input parameters for the USERMAT subroutine are defined in the modeling stage before the solution. In the field of damage mechanics and composite materials modeling, there is very limited research that uses ANSYS User Material Subroutine in the modeling.

Berbero [26] developed a group of 2D models based on damage mechanics to

simulate the matrix failures in composites; all the implementations were done using USERMAT. Elisa [8] used USERMAT in the damage model of a plate with a hole.

NUMERICAL MODELLING In this section the problem under investigation will be discussed. The finite element modeling, i.e. element type, material type, contact details will be described in details.

Bolted Joint Model Description and Boundary Condition In order to study the single or double lapped joint in the three dimensional (3D) space, a series of finite element models were developed. Figure 1 shows the joints under investigation. Each 3D model consists of one plates (PL₁) in the case of single lap joint (3DS) and three plates (PL₂, PL₁ & PL₂) for the case of the double lap joint (3DD), figure 2 shows the finite element mesh of 3DD and 3DS models.

PL₁

W

PL₂

PL₂

Bolt Head

L

Washer

PL₁

Shank

Hole

3DS

3DD

Figure 1. Specimen components and geometry.

PL1 is always a composite laminate while PL2 is a metal plate according to the case study. The plates and the bolts were modeled by using 3D twenty nodes elements SOLID186 [22]. SOLID186 can handle the layered medium so that layered element option was activated.

δ

δ

Figure 2. Finite element models for 3DD and 3DS.

For the case of composite plate composite material model was selected i.e (Ansys Model and User Subroutine, in which elastic orthotropic damage model was used while for the case of steel, aluminum and titanium the material properties can found in table III. TABLE III. METALLIC MATERIAL PROPERTIES. E1 Item GPa Steel Plates, Nuts and Washers 200 Aluminum Alloy Plates 60 Titanium Bolts 110

ν 0.3 0.29 0.29

The laminated plate that was used in this study is composed of plies made from T300/1034-C carbon/epoxy composite (Table IV) with a stacking sequence of [0/(±45)3/903]s,as listed in table table IV. TABLE IV: PROPERTIES OF T300/1034-C CARBON/EPOXY LAMINATE [9,15].

Stiffness Parameters Strength Parameters Fracture Toughness

E1

146.8 GPa

St1

1730 MPa

Gt1

89.83 N/mm

E2

11.4

GPa

Sc1

1370 MPa

Gc1

78.27 N/mm

G12

6.1

GPa

St1

66.5 MPa

Gt2

0.23 N/mm

Sc2

268.2 MPa

Gc2

0.76 N/mm

Ss12

58.2 MPa

Gs12

0.46 N/mm

0.3

Two symmetric boundary conditions (BC) were used about XY and ZY planes which makes the Finite element model stable and constrained in all directions. To simulated the Calmping force effect the bolt shank cross section in XZ plane is subjected to incrementally increased displacement. The load is increase until failure and the bolt reaction is calculated δ, figure 2.

Contact Properties In order to simulate the contactgu between the joint parts, such as the interfaces between the bolts heads and the plates, the nuts and the plates, the washers and the plates, the two plates, in addition to the bolt shank and the holes, surface-to-surface contact pairs were used.

ANSYS contact pair consists of contact elements (CONTA174) and target elements (TARGE170). Coulomb friction model was used as it is supported by ANSYS [28,29]. Coefficient of friction of 0.1, 0.2, and 0.3 were considered. In the

present work, the effect of the clearance between the bolt hole and the bolt shank was taken into consideration.

VALIDATION OF FINITE ELEMENT MODEL The laminate that was considered in this section is from existing literature [8,23].

The laminate was tested in compression, and is composed of T800H/3633 composite plies (Table 5), with a length, a width, and a thickness of 118 mm, 38 mm and 1.1 mm, respectively. Figure 3 shows the dimensions of the plate with hole specimen.

Figure 3. Configuration of the open-hole test specimen.

The displacement (Î&#x201D;) load (P) result obtained by Suemasu was compared to the numerical results obtained from the three numerical models as shown in Figure 4. It can be seen that the model gives the good agreement with the experiemental results.

The failure stress of the present model (CDMM) was 21.3 kN compared to the experimentally measured value of 21.95 kN, which represents a good correlation between the present model and experiment with an error of about -2.9%.

TABLE V: PROPERTIES OF T800H/3633-C CARBON/EPOXY LAMINATE [8,23].

Stiffness Parameters

Strength Parameters

E1

148

St1

2000 MPa

E2

9.50 GPa

Sc1

1500 MPa

G12

4.50 GPa

St1

54

ν12

0.3

Sc2

150 MPa

Ss12

100 MPa

GPa

MPa

Furthermore, it can be found that the PDM underestimates the failure loads by about -16%, this due to the absence of damage evolution in the ply discount model.

The stiffness is reduced to 5% of the original stiffness once the failure criteria achieved, which makes the estimated strength of PDM is lower that the estimated

strength of CDMM. The present models do not include the effect of plasticity which might enhance the results if it were included, since the plasticity effect enhances the solution convergence [17].

25 EXP PDM CDMM

P, kN

20 15 10 5 0 0.00

0.40

0.80

Î&#x201D;, mm

1.20

1.60

Figure 4. Comparison between numerical and experimental results.

DETERMINATION OF MAXIMUM CLAMPING FORCE IN LAPPED JOINTS In order to evaluate the maximum accepted value of clamping force in 3DS and 3DD, two finite element models were created. The laminate that was considered is composed of plies made from T300/1034-C carbon/epoxy composite (Table IV) with a stacking sequence of [0/(Âą45)3/903]s for PL1 and steel for PL2. The specimen geometries are typical 3DS and 3DD joints in figure 2. In which W=25.4 mm, L=25.4 mm, s=15 mm, e=12.7 mm, d=6.35 mm, tPL1= 2.616 mm and tPL2= 2.6 mm. The bolt head diameter is 6.35 mm and the washer internal and external diameters are 6.75, 12 mm, respectively. In the case of double lapped joint tPL1 will be equivalent to half the thickness of the composite plate; however a symmetric BC will be added at the midplane of the joint.

The 3DD model can be seen in figure 2.a, this model was created to simulate the bolt tightening process in the case of double lapped joint model. The second model that found in figure 2.b, was created to simulate the bolt tightening process in the case of 3DS model. These models subjected to incremental clamping force up to failure.

RESULTS AND DISCUSSION Damage Due to Clamping Force

Figure 5 shows that damage index of compression failure mode at thickness direction d3c, it can be observed that the area under washer is approximately totally damaged. For the 3DD case the first crack occurred at 16 kN for the 3DS 4.8 kN. For this case, it is recommended to design for a value less than these values.

Figure 5. Thickness damage index in the composite plate (3DS).

Figure 6 shows a comparison between the stress-strain curve of the cases of 3DD and 3DS,CDMM and PDM results were presented. The stress (Ď&#x192;b) is the average stress over the bolt shank while the strain (Îľth) is the average strain over the composite thickness. It can be observed that the through thickness stiffness of the 3DS is about twice the stiffness the 3DD. On the contrary, the failure clamping load of the 3DD model is twice the failure load of the 3DS. The main difference between the two cases is that in the case of the 3DD the load is transferred from the bolt head to the washed, then from the washer to the steel plate, finally, it distributed over the composite plate. While for the 3DS the load is transferred from the bolt head to the washed, then from the washer to the composite plate directly, with a stress value much larger that the case of the 3DD. The value of maximum stress in the 3DS case is about twice the value in the case of 3DD. It can be concluded that, for the friction type connections or combined friction bearing connection with composite plates, it is preferred to use double lapped joint, and it will be more efficient. Furthermore, it can be found that the predicted values of PDM is always less that the CDMM model due to the absence of damage evolution process as mentioned in the validation section.

0.8 0.7 Bolt Stress (σb), GPa

0.6 0.5 0.4

3DD (CDMM) 3DD (PDM) 3DS (CDMM) 3DS (PDM)

0.3 0.2 0.1 0 0

0.01

0.02 Strain (εth)

0.03

0.04

Figure 6. Bolts Stress vs. composite laminate thickness Strain (εth) .

Figure 7 shows a comparison between the elastic stress distributions in the 3DS model before damage at P=4 kN and after damage at 12.8 kN , it can be observed that after damage the beak point shifter toward the direction outside the plate due to the softening effect.

250 200

σy

σy, MPa

150 d

100 4 kN 12.8 kN

50 0 -50

0

2

4

6

8

10

12

14

d, mm Figure 7. Stress distributions over half width in 3DS model.

This is more apparent in the case of 3DD in figure 8. At the same value of clamping force i.e. 4.0 kN the stress in the case 3DS is about 3 times the case 3DD, which is the main reason of early failure in the case of 3DS.

250 4 kN 26 kN

200

Ď&#x192;y

Ď&#x192;y, MPa

150 100 50 0 -50

0

2

4

6

8

10

12

14

d, mm Figure 8. Stress distributions over the half width in (3D).

Effect Bolt Hole Clearance Figure 9 shows the maximum clamping force Pmax )several bolt hole clearance (C) values for the 3DS model. It can be found that by increasing the bolt hole clearance the failure load increases until a specific value of C at which it begins to decrease. In this study it was found that the critical value of clearance is equal to the washer bolt hole clearance. Furthermore, it can be observed that this transition behavior does not appear in the double lapped joint due to the similarity of clearance of both steel plate and composite plate.

0.7 0.6

3DD

0.5

3DS

σb, GPa

0.4 0.3 0.2 0.1 0 0

500

1000

1500

C, μm

2000

2500

Figure 9. Effect of clearance on maximum clamping force.

Effect of Friction Coefficient Figure 10 shows the maximum clamping force (Pmax) several friction coefficients (μ) and bolt hole clearance (C) values for the 3DS model. It can be found that by increasing the friction coefficient the failure load increases for all small values of clearance except the biggest value.

0.41 0.39 0.37

σb, GPa

0.35 0.33 μ=0.1 μ=0.2 μ=0.3

0.31 0.29 0.27 0.25 0

500

1000

1500

2000

C, μm Figure 10. Effect of friction coefficient on maximum clamping force (3DS).

CONCLUSIONS In this paper, the maximum clamping force that can be induced to the single and double lapped composite plates were studied. In order to do this a 3D damage mechanics constitutive model (CDMM) was presented, implemented, and validated.

The developed model results were compared with previously published experimental data. Good comparison was evaluated from the proposed model. The results were compared with Ansys built in composite damage model. It was found that failure loads obtained from Ansys material model always underestimate the values obtained from CDMM, due to the absence of damage evolution process.

A 3D finite element model for the single and double lapped bolted joints was adopted. The effect of Different parameters such as, bolt hole clearance, washer hole clearance, and friction coefficient on the maximum failure load and stress distribution.

It was found that the stiffness of the 3DS is about double the stiffness of 3DD model, on the other hand the failure load of 3DD model is double the failure load 3DS. It can be concluded that the clamping force becomes more efficient in the case of double lapped joint.

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