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7 Joint Design


7
7.1 Basic Principles

Joint Design

Joints for adhesive bonding should be designed particularly for the use of adhesives. The habit of beginning with a design used

for another method of fastening and modifying it slightly for adhesive bonding is a poor one, often leading to disastrous results. The aim of joint design is to obtain maximum strength for a given area of a bond. In designing joints specically for adhesive bonding, the basic characteristics of adhesives must dictate the design of joints. Adhesive bonds act over areas and not a single point. For this reason, the joint should be designed with the objective of minimizing concentration of stress. The selection of joint design is inuenced by limitations in production facilities, production costs, and the desired nal appearance of the part. The strength of an adhesive joint is determined primarily by (1) the mechanical properties of the adherend and the adhesive, (2) the residual internal stresses, (3) the degree of true interfacial contact, and (4) the joint geometry. Each of these factors has a strong inuence on joint performance. The design engineer must be concerned with the elimination of stress concentrations, which reduce the strength and useful life of the joint. Localized stresses are not always apparent and may occur as a result of differential thermal expansion of the adhesive and adherends. Another cause is shrinkage of adhesive during cure, when volatiles are given off. These volatiles may become entrapped. Internal stresses decrease as adhesive thickness decreases, reducing the tendency to trap volatiles. Air can also become entrapped at the interface if the adhesive has too high a viscosity, does not ow easily as it undergoes curing, or if it does not wet the substrate.[1]

7.2 Types of Stress
Figure 7.1 shows ve types of stress found in adhesive joints. Any combination of these stresses may be encountered in an adhesive application. These stresses are described in the sections below.

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(c) Shear (a) Compression (b) Tension

(d) Peel

(e) Cleavage

Figure 7.1 Types of stresses in adhesive joints. Adapted from Shields.[2]

7.2.1 Compression When loaded in pure compression, a joint is less likely to fail than when loaded in any other manner, but compression-loaded joints are limited in application. 7.2.2 Shear This type of loading imposes an even stress across the whole bonded area, utilizing the joint area to the best advantage and providing an economical joint that is most resistant to joint failure. Whenever possible, most of the load should be transmitted through the joint as a shear load.[2] 7.2.3 Tension The strengths of joints loaded in tension or shear are comparable. As in shear, the stress is evenly distributed over the joint area, but it is not always possible to be sure that other stresses are not present. If the applied load is offset to any degree, the advantage of an evenly distributed stress is lost and the joint is more likely to undergo failure. The adherends should be thick with this type of joint and not likely to deect to any appreciable degree under the applied load. Such a situation will result in non-uniform stress.[2] Tensile stress develops when forces acting perpendicular to the plane of the joint are distributed uniformly over the entire area of the bond. The types of stress likely to result when other than completely axial loads are applied are cleavage and peel. As adhesives generally have poor resistance to cleavage and peel, joints designed to load the adhesive in tension should have physical restraints to ensure axial loading.[3]

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One or both of the adherends must be exible in this type of loading. A very high stress is applied to the boundary line of the joint, and unless the joint is wide or the load is small, failure of the bond will occur. This type of loading is to be avoided if possible.[2] 7.2.5 Cleavage Cleavage is somewhat similar to peel and occurs when forces at one end of a rigid bonded assembly act to split the adherends apart.[2] It may be considered as a situation in which an offset tensile force or a moment has been applied. The stress is not evenly distributed (as is the case with tension) but is concentrated on one side of the joint. A sufciently large area is needed to accommodate this stress, resulting in a more costly joint.[2]

7.3 Methods of Improving Joint Efciency
As mentioned earlier, joints should be specically designed for adhesive bonding. Figure 7.2 illustrates various types of adhesive joints used for at adherends. Adhesive bonds designed to follow the following general principles will result in maximum effectiveness:[4,5] The bonded area should be as large as possible within the allowable geometry and weight constraints. A maximum percentage of the bonded area should contribute to the strength of the joint. The adhesive should be stressed in the direction of its maximum strength. Stress should be minimized in the direction in which the adhesive is weakest. Thermosetting adhesives, such as epoxies, are relatively rigid and exhibit high tensile and shear strength under both dynamic loading and static loading. Such adhesives also have good fatigue resistance. However, rigid brittle adhesives are not recommended for bonds stressed in peel or cleavage. Elastomeric adhesives, on the other hand, have low tensile or shear strength, but these adhesives develop high peel or cleavage strength. Adhesives that possess high tensile and shear strength over short periods of static stress give poor results over longer periods or under vibrating stresses.[6]

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Plain butt unsatisfactory

Single lap (plain lap) good, practical

Bevelled lap good, usually practical, difficult to mate Scarf butt very good, usually practical, requires mating

Joggle lap good, practical

Single strap fair, sometimes desirable

Double strap good, sometimes desirable

Recessed double strap good, expensive machining

Bevelled double strap very good, difficult production

Step lap (half lap) good, requires machining

Double lap good, when applicable

Double butt lap good, requires machining

Tongue and groove excellent, requires machining

Figure 7.2 Types of joints used in adhesive bonding at adherends.[3,4,5,6]

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The types of loads and joints that concentrate stresses in small areas or on edges should be avoided. Joints that stress the adhesive in shear are preferable because adhesives generally show considerable strength under this type of stress. Sudden applications of load, such as during impact, require the use of elastic or resilient adhesives to absorb the shock. Brittle adhesives will ordinarily fail under such conditions.[6]

7.4 Joint Design Criteria
The bonded area should be large enough to resist the greatest force that the joint will be subjected to in service. The calculation of stress in the adhesive joint is not a reliable way of determining the exact dimensions required. It is relatively difcult to decide on an allowable stress. The strength of the bond is affected by environmental conditions, age, temperature of cure, composition and size of adherends, and the thickness of the adhesive layer.[2] The stress in the adhesive is ordinarily a combination of various stresses. The relative exibility of the adhesive to that of the adherends has a pronounced effect on the stress distribution. Figure 7.3 is a typical example of a simple lap joint under tensile loading. Figure 7.3c shows that most of the stress is concentrated at the ends of the lap. The greater part of the lap (adjacent to the center) carries a comparatively low stress. Therefore, if the overlap length is doubled, the load-carrying capability of the joint is increased by a relatively low percentage. The greatest gain in strength is obtained by increasing the joint width.[2] The single lap joint shown in Figure 7.4 is typical of most adhesive joints. Increasing the width of the joint results in a proportionate increase in strength, while increasing the overlap length (L) beyond a certain limit has very little effect, as seen in Figure 7.5.[2] In addition to overlap length and width, the strength of the lap joint is dependent on the yield strength of the adherend. The modulus and thickness of the adherend determine its yield strength, which should not exceed the joint strength. The yield strength of thin metal adherends can be exceeded where an adhesive with a high tensile strength is employed with a relatively small joint overlap. Figures 7.5 and 7.6 show the relationship between shear strength, adherend thickness, and overlap length.[2]

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(a)

(b)

(c)

Figure 7.3 Tensile force on lap joint showing (a) unloaded joint, (b) joint under stress, and (c) stress distribution in adhesive.[2]

L Overlap length

Jo

int

wid

th

Adhesive layer t = Adherend thickness

Figure 7.4 Single lap joint.[2]

The fall-off in the effective load-bearing capacity of the overlap joint is usually expressed as a correlation between shear strength and the L/t ratio, as seen in Figure 7.6, but sometimes the ratios t/L or t1/2/L are used. The t/L ratio is often called the “joint factor.”[2] Many variables that have signicant effects on the strength of an adhesive are related by the L/t curve. Some of these are adherend modulus, adhesive, test temperature, bondline thickness, and joint conguration. The L/t curve is generally used for each variable that may enter the design, and data are presented to the

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15,000 Width (overlap length constant at 1 cm) Failure load (N) 10,000

165

5,000

Length (overlap width constant at 1 cm)

0 0 1 2 Length or Width (cm) 3 4

Figure 7.5 Effect of overlap and width on the strength of a typical joint.[2]

12,000

Mean failure stress (N/cm2)

9,000

6,000

3,000

0 0 5 10 Overlap length (L) Adherend thickness (t ) 15 20

Figure 7.6 Correlation between shear strength and L/t ratio.[2]

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designer as families of L/t curves.[7] One variable commonly plotted is load or stress against L/t.

7.5 Typical Joint Designs
The ideal adhesive-bonded joint is one in which, under all practical loading conditions, the adhesive is stressed in the direction in which it most resists failure. Figure 7.2 shows a number of types of joints used in bonding at adherends. These will be discussed briey.[3] Butt joints. These joints are not able to withstand bending forces because under such forces the adhesive would undergo cleavage stress. If the adherends are too thick to design simple overlap joints, modied butt joints can be designed. Such joints reduce the cleavage effect caused by side loading. Tongue-and-groove joints are self-aligning and provide a reservoir for the adhesive. Scarf butt joints keep the axis of loading in line with the joint and require no extensive machining.[3] Lap joints. These are the most commonly used adhesive joints. They are simple to make, can be used with thin adherends, and stress the adhesive in its strongest direction. The simple lap joint, however, is offset and the shear forces are not in line, as seen in Figure 7.3.[3] It can be seen in this stress distribution curve that most of the stress (cleavage stress) is concentrated at the ends of the lap. The greater part of the overlap (adjacent to the center) carries a comparatively low stress. If the overlap length is increased by 100%, the load-carrying capability is increased by a much lower percentage. The most effective way to increase the bond strength is to increase the joint width.[2] Modications of lap-joint designs that improve efciency include:[3] Redesigning the joint to bring the load on the adherends in line. Making the adherends more rigid (thicker) near the bond area (Figure 7.7). Making the edges of the bonded area more exible for better conformance, thereby minimizing peel. Modications of lap joints are shown in Figure 7.2. Joggle lap joints. This is the easiest design for aligning loads. This joint can be made by simply bending the adherends. It also provides a surface to which it is easy to apply pressure.[3]

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Lap (cm) 3,000 1 2 3 4 5 6 7

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0.081 in. (2.1 mm) 0.064 in. (1.6 mm) 0.051 in. (1.3 mm)

2,000
Failure load (lb)

0.045 in. (1.1 mm) 0.032 in. (0.8 mm) 1,000

Adherend thickness (Alclad) 0 0 1 Lap (inches) 2 3

Figure 7.7 Interrelation of failure loads, depth of lap, and adherend thickness for lap joints with a specic adhesive and adherend.[3,8]

Double lap joints. These joints have a balanced construction that is subjected to bending only if loads in the double side are not balanced.[3] Beveled lap joints. These joints are also more efcient than plain lap joints. The beveled edges allow conformance of the adherends during loading, with a resultant reduction of cleavage stress at the ends of the joint.[3] Strap joints. These joints keep the operating loads aligned and are generally used where overlap joints are impractical because of adherend thickness. As in the case of the lap joint, the single strap is subject to cleavage stress under bending forces. The double strap joint is superior when bending stresses are involved. The beveled double strap and recessed double strap are the best joint designs to resist bending forces. However, both these joints require expensive machining.[3]

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7.6 Peeling of Adhesive Joints
When thin members are bonded to thicker sheets, operating loads generally tend to peel the thin member from its base, as shown in Figure 7.8.[3] Riveting may provide extra strength at the ends of the bond, but the use of rivets may result in stress concentrations. Beading the end of the joint is helpful, but not always feasible. An increase in peel strength will result from increasing the width of the end of the joint. Finally, increasing the stiffness of the adherends is often quite effective. The stiffer the adherends, the smaller the deection of the joint for a given force, and the smaller the peel stresses.[9]

7.7 Stiffening Joints
In many cases, thin sheets of adherend are rigidized by bonding stiffening members to the sheet. When such sheets are exed, the bonded joints are subjected to cleavage stress. Figure 7.9 illustrates design methods used for reducing cleavage stress on stiffening joints.
Peel

Rivet

(a) Bead end Increase width

(b)

(c)

(d)

Increase stiffness

(e)

Figure 7.8 Designs that minimize peel.[9]

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Original design

Increased bond area

Increased flange flexibility

Increased sheet stiffness

Figure 7.9 Methods of minimizing peel for stiffening sections (ange joints).[9]

7.8 Cylindrical Joints
Several recommended designs for rod and tube joints are shown in Figures 7.10 and 7.7. These designs are preferable to simple butt joints because of (1) their resistance to bending forces and subsequent cleavage and (2) their increase in bonding area. In the case of tubular forms (Figure 7.11), the bonding area is small unless the tube walls are very heavy. Most of these joints require a machining operation.[3]

7.9 Angle and Corner Joints
Angle and corner joints for at adherends are illustrated in Figures 7.12 and 7.13. In both cases, the butt joints are susceptible to cleavage under bending stress. Among the angle joints (Figure 7.12), the dado joint is probably the best, provided that the reduction in section required for the recess is acceptable. This design is less subject to cleavage stress than the

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Butt

Socket

Landed socket

Tapered socket

Figure 7.10 Straight joints for solid bars.[9]

Butt

Taper

V-Groove

Half lap

Landed taper
[9]

Landed lap

Figure 7.11 Straight joints for tubular forms.

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Butt

Right-angle butt

Dado

Double right-angle butt

Figure 7.12 Angle joints.[9]

right-angle butt joints (also called “L” angle joints), and is easier to form. The double right-angle butt joint is also called the “T” angle joint. Corner joints (Figure 7.13) for at adherends are best designed to use xtures. For solid rods and tubular forms, xtures are always required.[9]

7.10 Joints for Plastics and Elastomers
7.10.1 Flexible Materials Thin or exible polymeric substrates may be joined using a simple or a modied lap joint. The double strap joint shown in Figure 7.2 is best, but time-consuming to form. The strap material should be fabricated from the same material as the parts to be joined. If this is not possible, it should have approximately equivalent strength, exibility, and thickness. The adhesive should have the same degree of exibility as the adherends. If the sections to be bonded are relatively thick, a scarf joint, also shown in Figure 7.2, is acceptable. The length of the scarf should be at least four times the thickness, as shown in Figure 7.14.[3]
Right-angle corner plates

Butt

Corner lap

Recessed

Figure 7.13 Corner joints.[9]

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Figure 7.14 Recommended scarf joint conguration for exible plastics and elastomers.[3]

Figure 7.15 shows several types of joints for rubber under tension. The horizontal white lines are equidistant when the joints are unstressed. It is obvious that the scarf joint is least subject to stress concentration with materials of equal modulus, and the double scarf joint is the best for materials of unequal modulus. These designs offer the best resistance to peel and, all other factors being equal, represent the best choices.[9] When bonding elastic material, forces on the elastomer during cure should be carefully controlled, as too much pressure will cause residual stresses at the bond interface. Stress concentrations may also be minimized in rubberto-metal joints by elimination of sharp corners and by the use of metal adherends sufciently thick to prevent peel stresses that may arise with thinner-gauge metals. As with all joint designs, polymeric joints should avoid peel stresses. Figure 7.16 illustrates methods of bonding exible substrates so that the adhesive will be stressed in its strongest direction.[9]

Straight lap

Double lap (a) Equal modulus

Beveled double lap

Scarf

Scarf

Double scarf

(b) Unequal modulus

Figure 7.15 Joints for rubber under stress.[9]

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Shear Peel Peel Good Poor Poor

173

Peel Poor

Shear Good

Figure 7.16 Joints for exible materials.[9]

7.10.2 Rigid Plastics In the case of rigid plastics, the greatest problems are in reinforced plastics, which are often anisotropic, having directional strength properties. Joints made from such substrates should be designed to stress both the adhesive and the adherend in the direction of greatest strength. Laminates should be stressed parallel to the laminations to prevent delamination of the substrate.[3] Single and joggle lap joints (Figure 7.2) are more likely to cause delamination than scarf or beveled lap joints. Strap joints may be used to support bending loads.[3]

7.11 Stress Analysis of Adhesive Joints
This is an extremely complicated subject and will only be touched upon here. The ultimate objective is to develop a design method for bonded construction based on the principles of mechanics and rational engineering design so that joint behavior can be predicted. 7.11.1 Theoretical Analysis of Stresses and Strains Most theoretical analyses have been carried out on single or double lap joints, which are the primary types of joints used for determining the strength of adhesives. Properly designed joints stress the adhesive in shear. Adhesives are especially weak in peel, and are also weak under tensile loads applied normal to the plane of the joint. The earliest theoretical lap-joint work involved simplifying assumptions that: (1) the joint was a

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simple overlap type; (2) both adherends were made of the same metal and had the same geometry; (3) the adherends and adhesive behaved elastically; (4) bending or peeling stresses were not involved; (5) thermal expansion or residual stresses were ignored; and (6) the deections were small.[10] Recent theoretical studies have become much more complex. New computerassisted techniques permit the use of nite-element matrix-theory type approaches. The effects of important variables are being determined by parametric studies. More complex joints are also being studied. New adherend materials, including advanced lamentary composites, are also being evaluated. The elastic, low-deection, constant temperature behavior of scarf and stepped-lap joints has been replaced by elastic-plastic, large-deection behavior, combined with thermal expansion differences, or curing shrinkage-induced residual stresses. 7.11.2 Experimental Analyses Typically, the yardstick for qualitatively measuring the internal resistance of an adhesive bond to an external load has been the determination of the strain distribution in the adhesive and adherends. This is a difcult task. Even in simple lap joints, the actual stress–strain distributions under load are extremely complex combinations of shear and tensile stresses, and are very prone to disturbance by non-uniform material characteristics, stress concentrations or localized partial failures, creep and plastic yielding, etc. It is extremely difcult to accurately measure the strains in adhesive joints with such small glue line thicknesses and such relatively inaccessible adhesive. Extensometers, strain gauges, and photoelasticity are being used with limited success.[10] The stress distribution on the adhesive affects the ability of the joint to accommodate loads. Joint design should strive to distribute the stresses equally over the bond area in order to create uniform stress on the adhesive. Adhesive bonds subjected to tensile, compressive, or shear stress during loading experience a more uniform stress distribution than bonds exposed to cleavage or peel stress. Tensile and compressive stress are evenly distributed throughout the bond area; stress is represented by a straight line in Figure 7.17. The compressive strength of most adhesive lms is greater than their tensile strength; optimal joint design should maximize compression and minimize tensile stresses. The stress distribution of a cleavage or peel stress is concentrated at one end of the joint (Figure 7.18). Peel strength of any adhesive may be as low as 1% of its

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Stress

Tension/Compression

Bond location

Figure 7.17 Distribution of tensile and compressive stress along the joint. Stress is distributed evenly along the joint for both tensile and compressive stress, as shown by the straight line.[13]

Stress

Cleavage Peel

Bond location

Figure 7.18 Distribution of peel and cleavage stress. In both types of stress, stress is concentrated at one end of the joint.[13]

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shear strength; low-modulus elastic adhesives usually have higher peel strengths. Peel stress can be reduced through symmetrical joints, such as double lap. Joint design should ensure that peel and cleavage stress are minimized, and shear stress is maximized.[11,12] In shear stress, the ends of the bond resist a greater amount of stress than the middle of the bond (Figure 7.19). The maximum stress experienced by

Shear stress T

Tmax. Tavg. O (a)

O Adhesive stress-strain curve (Brittle)

(c) Adhesive stress-strain curve (Elastic-plastic)

Shear stress T Shear stress (b)

Shear stress T Shear stress (d)

Figure 7.19 Shear stress distribution in brittle and elastic materials. In brittle plastics, with a typical stress vs. strain curve as shown in (b), stress increases linearly from the center to the ends of the joint (a). In elastic materials, with a typical stress vs. strain curve as shown in (d), the shear stress distribution is non-linear (c), and stress is distributed over a larger area near the ends of the joints.[13]

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the ends of the joint is greater than the average stress (joint load divided by bond area); lower than average stress occurs in the middle of the joint. This stress distribution is due to the exibility of plastic materials, which tend to bend when a load is applied, increasing stress concentrations at the joint ends. Stress ratios (highest stress/average stress) of plastics (15), with relatively low elastic moduli (2,068 MPa unlled plastic), are much greater than stress ratios for steel (1.7), with a high elastic modulus (212,000 MPa). Stress concentrations can lead to joint failure at relatively low loads; however, stress concentrations can be reduced by a joint design that takes into account the elastic modulus of the adhesive, the joint overlap length, and the bond line thickness.[11,14] The elastic modulus of the adhesive inuences the stress distribution of the joint. The shear stress distribution of a more brittle, higher-modulus adhesive, with a stress–strain curve as shown in Figure 7.19, shows a linear increase in stress from the center to the ends of the joint (Figure 7.19a). A more elastic adhesive with a higher elongation and stress–strain behavior as shown in Figure 7.19d exhibits a non-linear stress distribution (Figure 7.19c). The exibility of the more elastic adhesive allows the joint to more readily accommodate motion of the adherends; stress is then distributed over a larger area, and the stress ratio (highest stress/average stress) is reduced. Substitution of a lower-modulus (1.4 MPa) adhesive for a higher-modulus (2,068 MPa) adhesive can reduce the stress ratio from 22.4 to 1.2. Although a lower modulus adhesive may, for some applications, produce a stronger joint, it may not be able to accommodate structural loads without excessive deformation. Due to the greater area under the stress distribution curve, the more elastic adhesive experiences a higher average stress than a brittle adhesive of the same strength. For two adhesives of the same strength and elongation, the higher-modulus, brittle adhesive would tolerate higher loads. Brittle adhesives, however, are more sensitive to crack propagation and generally have lower fatigue life than more elastic adhesives.[11,14] Although bonds with larger areas generally have higher strength, bond width is a more important design parameter than bond length or overlap. Bond strength increases slightly with overlap length (Figure 7.20) up to a point, then remains constant. Due to the shear stress concentration at the ends of the joint, however, shear strength is directly proportional to bond width (Figure 7.20). A 2-cm wide joint is twice as strong as a 1 cm (0.4 in.) wide bond, but a 2-cm long joint is not twice as strong as a 1-cm joint. A short, wide bond area (Figure 7.21b) is stronger than a long, narrow bond

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Width Shear strength Overlap

Overlap

Figure 7.20 Dependence of shear strength on bond width and bond overlap. Strength increases linearly with increasing joint width. The increase in strength with increasing overlap is gradual; after a particular overlap is reached, there is no further increase in joint strength.[13]
W 2

W

L (a) Inferior joint: Long, narrow bond area

L 2 (b) Superior joint: Short, wide bond area

Figure 7.21 Dependence of shear strength on bond geometry. A long, narrow bond area (a) produces a lower strength bond than a short bond area with greater width (b).[13]

area (Figure 7.21a). Bending and differential shear stress concentrations are reduced with shorter bond overlaps; decreases in overlap length from 2.54 to 0.32 cm can result in reduction of the stress ratio (highest stress/ average stress) from 22.5 to 3.78.[11,14,15] A thicker bond line can reduce shear stress concentration by spreading the strain over a larger dimension, resulting in less strain on the adhesive. An increase in bond line gap from 0.0025 to 0.10 cm can decrease the stress ratio from 18.2 to 3.06.[11,14] The most common method for reducing stress concentration in lap joints is by tapering the adherends, in a tapered lap joint (Figure 7.22a). Stress at the joint ends is reduced, allowing for a more uniform stress distribution. Modeling studies indicate that both adhesive peel and shear stresses

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(a)

(b)

Figure 7.22 Joints used to reduce stress in adhesive bonding. (a) A tapered lap joint reduces stress at the joint ends. (b) A step lap joint avoids a large change in stress concentration from the middle to the ends of the joint when long overlap lengths are necessary.[13]

decrease with a decrease in taper angle, with the optimum angle being the smallest angle that can be economically machined and assembled. A step lap joint (Figure 7.22b) can be used to avoid a large change in stress concentration from the ends of the joint to the center when long overlap lengths are necessary.[11,14] 7.11.3 Failure Analyses The function of a structural adhesive joint is to transmit an external load to the structural member. If the joint fails to function as it is intended, it will undergo damage or failure. The damage could be actual fracture of the structure, excessive elastic deformation, or excessive inelastic ow. The criteria for what constitutes structural failure depend on the performance requirements of the joint. The fundamental problem in the mechanics of adhesives and joints is to obtain some relationship between the loads applied to the joint and a parameter that will adequately describe the criteria for structural failure. The most common criterion for such failure of lap-type joints is actual fracture of the joint. For a given combination of adherend and adhesive, the stress analyst must decide what the mode or theory of failure would be if the applied loads become large enough to cause failure. The decision as to which theory would realistically determine the mode of failure is usually based on past experience, or upon some form of experimental evidence.[10]

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The next step is to determine a relationship between the applied load and a parameter that will describe the failure of the joint. Such parameters might be stress, strain, strain energy, etc. Finally, when maximum tolerable stresses have been obtained, the allowable stress values or factors of safety are decided upon to allow for factors such as long- and short-term loading, fatigue loading, special environmental conditions, and other special considerations. This step is ordinarily based on experience, engineering judgment, and legal, government, or commercial specications.[10]

7.11.4 Methods of Stress Analysis Theory of Volkersen: In 1938, Volkersen analyzed the distribution of shearing stresses in the adhesive layers of a lap joint. Volkersen’s model is useful only with very stiff adhesives, which do not bend on loading the joint. A dimensionless stress concentration factor is found to depend on the geometry and the physical parameters of the joint. By introducing further simplications, certain reasonable geometric conditions, and identical adherends, a simple formula is obtained:[11] Δ = GL2/Etd where G is the shear modulus of the adhesive, E is the Young’s modulus of the adherend, d is the thickness of the adherend, L is the length of the overlap, and t is the thickness of the adherend. DeBruyne has suggested that, when all other variables are kept constant, the quantity √t/L with dimension (length)–1/2, the “joint factor” derived from the above equation, is useful in correlating joint strengths.[11] Volkersen’s theory predicts that the shear stresses in the adhesive layer reach a maximum at each end of the overlap, when the bonded plates are in pure tension. Photoelastic analyses of these composite structures show that stresses are uniform in the central part of the model adhesive, but high near the edges of the steel plate used in the analysis (Figure 7.2). Stress distributions at the end were found to be independent of the length of the overlap, when its length was at least three times the thickness of the adhesive layer.[11] Theory of Goland and Reissner: In Volkersen’s theory, the so-called “tearing” or “peeling” stresses were ignored. Goland and Reissner[14] took the bending deformation of the adherends into account, as well as the transverse strains in the adhesive and the associated tearing stresses. These researchers

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showed that the maximum tearing and shear stresses reach asymptotic values for large overlap lengths. Provided the system remains linearly elastic, the joint strength reaches a limiting value with increasing overlap length. In actual practice, however, a limiting strength is obtained because the adherends are loaded to their ultimate strength.[12]

References
1. Design considerations. Adhesives in Modern Manufacturing (E.J. Bruno, ed.), Society of Manufacturing Engineers (SME), St. Louis, MO, 1970. 2. Shields, J., Adhesives Handbook, 3rd ed., Butterworth, London, 1984. 3. Petrie, E.M., Plastics and elastomers as adhesives. Handbook of Plastics and Elastomers (C.A. Harper, ed.), 4th ed., McGraw-Hill, New York, 2002. 4. Kubo, J.T., Joints. Advanced Composites Design Guide, Vol. 1—Design, 3rd ed., 2nd Revision, September 1976. Prepared for Air Force Flight Dynamics Laboratory, WPAFB, OH, by Rockwell International Corp. 5. Sharpe, L.M., The materials, processes, and design methods for assembling with adhesives. Machine Design, 38(19): 179–200, 1966. 6. Kubo, J.T., Joints. Advanced Composites Design Guide, 3rd ed., 3rd Revision, January 1977. Prepared for Air Force Flight Dynamics Laboratory, WPAFB, OH, by Rockwell International Corp. 7. Lunsford, L.R., Stress analysis of bonded joints. Applied Polymer Symposia No. 3, Structural Adhesive Bonding, Presented at a Symposium Sponsored by Picatinny Arsenal and held at Stevens Institute of Technology, September 14–16, 1965, pp. 57–73, Wiley-Interscience, New York, 1966. 8. Perry, H.A., Room temperature setting adhesives for metals and plastics. Adhesion and Adhesion Fundamentals and Practices (J.E. Rutzler and R.L. Savage, eds.), Society of Chemical Industry, London, 1954. 9. Military Handbook MIL-HDBK-691A, Adhesives, May 17, 1965. 10. Design, Analysis and Test Methods, Vol. 4, Treatise on Adhesion and Adhesives, Structural Adhesives with Emphasis on Aerospace Applications, A Report of the Ad hoc Committee on Structural Adhesives for Aerospace Use, National Materials Advisory Board, National Research Council, Marcel Dekker, New York, 1976. 11. The Design Guide for Bonding Plastics, Loctite (LT-2197), Henkle Loctite Corporation, www.loctite.com, 2006. 12. Williams, J. and W. Scardino, Adhesives selection. Engineered Materials Handbook: Composites, Reference Book (ASM 620.1), ASM International, 1987. 13. Handbook of Plastics Joining, 1st ed., William Andrew Publishing, Norwich, NY, 1997. 14. Lewis, A., R. Gosnell, K. Berg, C.E. Chastain, and N. Berry, Introduction to adhesives technology. Adhesives Digest, Reference Book (Edition 8), Data Business Publishing, Englewood, CO, 1994. 15. Techniques: Adhesive Bonding, Solvent Bonding, and Joint Design, Supplier Technical Report (#SR-401A), Borg-Warner Chemicals, Inc., Parkersburg, WV, 1986.


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