Strength of Mortise and Tenon Joints

As can be seen from the previous considerations, covered mortise and tenon joints maintain elasticity as long as the glue-line joining the wooden elements is not damaged. Its strength is calculated in the same way as for bridle joints, by entering the relevant dimensions and the number of glue-lines. However, it should be noted that the damage of glue-line will cause displacement of external load on the stresses between the mortise and tenon. In these conditions, the following has to be established (Fig. 6.53):

shearing strength of the tenon:

s kd

Lmax t

ngh

bending strength of the tenon:

M ld

r = J0 y *kd –

shearing strength of part of the mortise:

where

T, M cutting force and bending moment read from the graphs N, T, M, n number of tenons,

Jx moment of inertia of the n-tenons’ cross section

p distance of extreme fibres from the neutral axis,

g thickness of the tenon,

b length of the tenon,

h height of the tenon,

e depth of the tenon,

kd shearing strength of the wood across the fibres,

kgd bending strength of the wood.

Mortise and tenon joints can be treated as an adhesive connection only when the elements of the mortise and tenon are in contact with each other only along the wide planes, in which a rectangular glue-line has formed (Fig. 6.54).

In this situation, the strength of a connection depends only on tangential stresses in the glue-line. For this case, the centre of bending O of the joint is located in the geometric centre of the tenon and as long as the rotation angle a meets the con­ditions of inequality:

where

t fit between the mortise and tenon, l length of the tenon,

the strength of the joint will be specified by the equation:

SL = s2 + SLax – 2s1s2max cos У, (6.178)

where

tangential stresses caused by external moment M,

_ M

S2max=n JA(g2 + n2)dgdn;

tangential stresses caused by axial forces T, N,
where

c distance from the centre of torsion, h height of the tenon, n number of the glue-lines.

When the acceptable values of the moment bending the connection M, exceed, causing that the angle a satisfies the inequality:

some elements of the surface of the mortise and tenon begin to put pressure on each other, causing a resistance moment Mq, which reduces the external moment value M to MT, causing tangential stresses in the glue-line MT = M – Mq.

Assuming that the length of the surface of compressions represents about 10 % of half of the length of the tenon 0.5 l (Fig. 6.55), we shall obtain the relationship:

Mq = 0.05ldr[(l-2t)-0.05l], (6.182)

where

a shearing strength of the wood,

8 thickness of the tenon,

through which the stresses causing shearing of the glue-line can be easily determined:

S2max = 6(M-MQ)(h2 + l2) °’5/(nhl). (6.183)

This equation shows that the value of tangential stresses forming in the glue-line, which can cause damage to the connection, depends mainly on the strength of wood for compression. Only a portion of the external load is transferred by the glue-line to the upper limit of the wood’s strength. The rest of the part is moved by a pair of internal forces Q. The damage of wood in places of mutual pressure consequently causes an increase in the angle a and angles of shear strain of the glue-line; therefore, the increase of tangential stresses is caused by an uncompensated external moment M.

A perfectly adjusted height h of the mortise and tenon causes their exact adja­cency with narrow planes on the length l of the joint’s edge (Fig. 6.56). Loading the joint with the bending moment M and force T causes contact pressure q4 and q3 in adjacent surfaces, of resultants Q4 and Q3. Due to the lack of symmetry in the distribution of these pressures, the location of the bending centre will also change.

In drawing up balance equations for a 2-D system of forces and taking into account the similarity of triangles of pressures, the position of the centre of bending is determined by solving the system of equations:

2T = 8[q3z — q4(l – z)]

3(M + Tz) = d[q4(l — z2) + q3 z2]
q4 =l-—z-q3,

and equating the quadratic equation to zero:

-4Tz2 + z{3Tl – 4T – 6M) – 4(-4T){3Ml + 2TI) — 0, (6.185)

thus, we obtain a new location of the centre of bending at a distance z equal to:

Z1,2 = z8t((3Tl – 4T – 6M)2±{T2 [9l2 + 8(l + 2)] + 4M[3T(7l + 4) + 9M]}°’^>.

(6.186)

In the provided situation, tangential stresses in the glue-line T2max will appear once the elastic strains of wood e caused by mutual pressures of joint elements increase. The larger these strains are, the higher the value of tangential stresses in the glue-line. The value of moment MT, causing tangential stresses in a glue-line, will therefore be the difference of the acting external moment M and moment of the pair of forces Q3 and Q4.

Mt — M-Mq,

where

Mq = (S/3)[q4(l – z)2 + q3z2] – Tz. (6.188)

The similarity of the triangles of pressures (Fig. 6.56) shows further that:

q4

q4x — , xl-z,

q-z (6.189)

q3x = — xz. z

Therefore, in order to calculate the value of tangential stresses in the glue-line of shape-adhesive mortise and tenon joints, the size of deformations should be determined caused by compressing wood in places of mutual pressures of elements of the joint. The stresses associated with these deformations are generally expressed by Hooke’s law a = Ee.

For load distributed increasingly linearly (Fig. 6.57), we can provide the general equation:

from which we will eventually obtain for q4:

and for q3:

q3 = Eez(x2;Z – x1,z) fX2Z* d*.

*1 z z

and hence, the maximum tangential stresses in the glue-line caused by the reduced moment MT for the range of elastic deformations of wood can be written as follows:

where

Taking into account the largest resultant vector of tangential stresses (Fig. 6.58), maximum stresses shall be written in the form of the equation:

in which = T/nhl.

When the external load M exceeds the value, at which moment Mq causes to exceed the range of elastic strains of wood, the load of the glue-line will suddenly increase. It will begin to shift the bending moment MT dependent only on the strength of wood to compression a. In this case, the equation describing the maximum tangential stress caused by this moment shall take the form:

In the place of the glue-line, in which the value of the vector of tangential stresses is the largest, processes of decohesion are initiated which reduce the strength of the joint and therefore the strength of the whole furniture construction.

Another commonly practised way of gluing is three-sided contact of elements of the tenon to elements of the mortise (Fig. 6.59). In this case, the pressures will also occur in the lower part of vertical planes of contact. By writing an equation of balance for such a system and entering geometric relationships between the graphs of pressures, the position of the centre of bending of the tenon can be determined by solving the system of equations:

from which we obtain:

If in this case we assume that the state of tangential stresses in the glue-line will depend on the values of the external moment M and the moment Mq caused by pressures on the surface of the wood, then the equation of maximum tangential stresses shall obtain the form:

in which

Tz – Nh. (6.201)

In the scope of elastic deformations of wood, the glue-line will transfer smaller external moments than loads; thus, the strength of the structural node will be much higher than the simple strength of glue-line subject to shearing. Upon crossing the
border of elastic deformations for wood and achieving strength of wood to com­pression, damage to the joint can occur unexpectedly due to the sudden increase of the moment of twisting the glue-line. In this situation, the greatest tangential stresses caused by this moment can be written as follows:

————————– :——– • (6.202)

I-(i-z)f-h/2 (g2 + П )d^dn

In both of the cases discussed above, the maximum tangential stresses generating the process of decohesion of the glue-line occur in one, strictly defined place, dependent on the method of applying external load and the geometry of the mortise and tenon (Fig. 6.60). The value of this stress is expressed by the equation:

SLax = s2 + SLax – 2s1s2max cos (90° + ^rtgy + arctgjQ ; (6.203)

where

Si = (T2 + N2 )05/nhl. (6.204)

Influence of Wood Species and Shape of Glue-Line on the Strength of Mortise and Tenon Joints

Despite the new technologies of wood-based materials available for multidimen­sional building constructions, glue-lines still play a dominant role (Pellicane 1994).

Innovative wooden structures, such as the Norwegian Viking Ship stadium in Hamar (Aasheim 1993) or the German pedestrian overpass over the River Altmuhl in Essing (BrUninghoff 1993), show how great a load they bear and how safely they should be designed. Solutions to improve the behaviour of loaded elements and constructional joints made of wood are constantly sought, and methods of con­trolling stresses are being developed. One of the oldest and most commonly used joints of elements of wooden structures is shape-adhesive joints and among them mortise and tenon joints. Nakai and Takemura (1995) conducted studies on the torsion stiffness of mortise and tenon joints, proving that the mathematical formulas developed by them correctly describe the stiffness of tenons of rectangular and elliptical cross sections. In subsequent works Nakai and Takemura (1996a, b), by analysing the distribution of tangential stresses using the method of electrical strain gauges, numerically verified the value of tangential stresses and indicated the necessity to avoid loads, which could cause cracks in the tenon. These tests, however, did not take into account the presence of glue-lines and their impact on stiffness and strength of the joint. A few other works were devoted to the issue of the distribution of tangential stresses in wooden glue-lines of shape-adhesive joints (Haberzak 1975; Matsui 1990a, b, 1991; Pellicane et al. 1994; Smardzewski 1996, 1998a, b). The character of the distribution of stresses was analysed in tests of stretching, bending and twisting loads. Mathematical models describing these distributions were also developed. Hill and Eckelman (1973) dealt with the deformation and bending stresses in mortise and tenon joints. Eckelman (1970) and Smardzewski (1990) also developed computer programs for analysing the stiffness and strength of furniture with susceptible joints, including mortise and tenon joints. These studies, however, did not take up the subject matter of mutual stresses of the joint’s elements, including the mortise and tenon, nor the impact of the species of wood and type of adhesive used on this phenomenon.

It is well known that mathematical modelling is a rational alternative to costly and time-consuming laboratory tests. This chapter presents the possibilities of using numerical methods for the analysis of contact stresses in mortise and tenon joints. In particular, the size of normal stresses was described in places of mutual pressures of the tenon and mortise and the impact of these stresses on the value of changes in the species of wood and type of adhesive used, or no adhesive in the joint.

Due to the common use of mortise and tenon joints in the construction of chairs, for examination, a chair with a muntin was chosen, establishing the connection of a horizontal strip and rear leg as the structural node (Fig. 6.61). From the practice of a university laboratory, tests and validation of furniture, it results that the most common cause of damage to a chair construction is its improper use and loading with forces that significantly exceed the mass of one user.

By taking into account the symmetry of a chair as a 3-D structure, one side frame was selected for strength analysis, loading the front edge of the seat with a con­centrated force of 800 N, corresponding to the impact of a user weighing approximately 160 kg (Fig. 6.62). On the basis of the distribution of internal forces (Fig. 6.63), the appropriate bending moments, cutting forces and normal forces were transferred onto the chosen structural node (Fig. 6.64).

Fig. 6.62 Static scheme of the construction

The numerical model of the mortise and tenon joints was developed in the environment of the program Algor® (Fig. 6.65). To this end, for modelling the wooden parts of the joint, 20-node, orthotropic solid elements were used, while for modelling glue-lines, isotropic elements were used. The fit between tenon and mortise is equal 0.1 mm (Fig. 6.61). Inside the gap between the side surface of the mortise and tenon, a glue-line was formed. Additionally, in order to trace the process of pressure of the tenon on the mortise, in all gaps, perpendicular to the surface of the tenon and mortise, gap-type stress elements were distributed with a

Fig. 6.64 The balance of internal forces acting on the joint of the rear leg with the bar

stiffness of 16,000 N/mm. The developed solid model was supported and loaded with external forces as in Fig. 6.66, which correspond to the load of the node by internal forces from Fig. 6.64.

In the selection of wood species for comparative material, it was decided to perform numerical calculations for representatives of deciduous wood and conif­erous wood. In order to do this, four of the most popular wood species in the production of furniture were selected: beech, ash, pine and alder (Table 6.4), assuming their elastic properties on the basis of the studies of Hearmon (1948) and Bodig and Goodman (1973).

In industrial practice, for the assembly of furniture, polyvinyl chloride adhesive (PVC) is commonly used, while urea-formaldehyde adhesive is less frequently used. In determining the module of linear elasticity of these adhesives, both own research and data from the literature were used. By indicating the stiffness of layer-glued beam elements in the bending test, Dzi^gielewski and Wilczynski (1990) established the value of the linear elasticity module of PVC glue at the level of 2845-33,450 MPa. According to Wilczynski (1988), the linear elasticity module

Fig. 6.65 Mesh of finite elements of the bridle mortise and tenon joints

Fig. 6.66 Scheme of loading the joint with external forces

of PVC glue, established in the torsion test of prismatic glued samples, amounts to 358 MPa. A similar value to PVC glue, equal to 465.74 MPa, was also obtained by Smardzewski (1998a), in conducting tests on samples in the shape of a paddle, subjected to tension. Clada (1965) dealt with determining the stiffness of urea – formaldehyde glue, according to whom the value of Young’s modulus varies from 4940 to 5200 MPa. Based on this, the following values of the linear elasticity module of glue were selected in MPa: 33,450, 4940, 465, 100, 50, 10 and 0.

Property of wood

Species of wood

Beech

Ash

Pine

Alder

Density of wood [g/cm3]

0.75

0.67

0.55

0.38

Table 6.4 Elastic properties of chosen wood species (Hearmon 1948; Bodig and Goodman 1973)

Linear elasticity module [MPa]

II

.Сч

N

13,969

15,788

16,606

10,424

£

II

X

2284

1509

1117

809

Et = Ey

1160

799

583

355

Shear modulus [MPa]

GLT = GZY

1082

889

693

313

GLR = GZX

1645

1337

1181

632

GRT = GXY

471

471

70

144

Poisson’s ratio

ULR = UZX

0.450

0.460

0.420

0.440

ULT = UZY

0.510

0.510

0.510

0.560

URT = UXY

0.750

0.710

0.680

0.570

UTR = UYX

0.360

0.360

0.310

0.290

URL = UXZ

0.075

0.051

0.038

0.031

UTL = UYZ

0.044

0.030

0.015

0.013

The values 100, 50 and 10 MPa correspond to those types of adhesives, which have a technological application mistakes and characterised by lower mechanical prop­erties. The use of the indication 0 MPa only shows the lack of glue-line in the joint. In the course of numerical calculations, the change in the thickness of the glue-line Дg was determined, caused by pressure of the tenon and mortise and the change of normal stresses in points A and B, indicating the possibility of mutual pressure of elements of the tenon and mortise (Fig. 6.67).

The analysis of deformed meshes of the numerical model shows that the tenon sustained rotation and bending. Its upper part was subjected to sliding out and moving downwards, while the lower edge deeper with smaller movement down­wards (Fig. 6.68). The effect of mutual stress of joint elements is, therefore, a result of both bending of the tenon and twisting of the glue-line. As it is shown in Fig. 6.69, the change in the species of wood, of which the joint was made, does not determine the change of the strength of the glue-line. These changes are mainly caused by the change of the value of linear elasticity modulus of the glue-line. Adhesives characterised by a very high linear elasticity module, above 4940 MPa, are not subject to significant geometrical changes and do not change the thickness of the glue-line. The thickness of the glue-line is reduced by 10 % for glue-line of module E = 465 MPa, by 50 % for glue-line of module E =100 MPa and by 80 % for glue-line of module E = 50 MPa. For all of these glue-lines, however, it can be assumed that the stress of wooden elements in point A occurs only by compression of the glue-line. Reducing the elasticity module of glue to 10 MPa or no glue in the gap between elements results in a direct stress of the tenon and mortise.

Displacement

2,49 mm

2,10 mm I 1,85 mm I 0.89 mm I 0.57 mm

Fig. 6.69 The impact of the linear elasticity module of glue and wood species on the thickness of the glue-line in the place of the stress of mortise and tenon

Figure 6.70 shows clearly that the tenon is in a state of bending stresses, and in point B, normal stresses ayy are triggered caused by mutual pressure of the bar on the leg. When analysing this impact, it can be noticed that in the bar, of a longi­tudinal course of fibres, stresses occur which are six times greater than in the leg, where the fibres run perpendicular to the direction of the pressure. Furthermore, stresses in pinewood and alder wood are approximately 9-13 MPa higher than those in beech wood and ash wood (Fig. 6.71). The greatest stresses ayy in the leg are formed, if alder wood is used. In relation to ash wood, these stresses are higher by 1-1.5 MPa and in relation to beech wood and pinewood by 0.5-1.0 MPa. The type of adhesive used has a significant impact on the value of stresses ayy in the place of the bar’s pressure on the leg.

Figure 6.71 shows that the proportional growth of stresses occurs together with the fall of the value of the linear elasticity module of the glue-line up to 50 MPa, while these stresses do not reach the acceptable limits for particular species of wood. Below this value, stresses both in the bar and in the leg rise suddenly, reaching a maximum in the absence of a glue-line in the connection. For an element of a bar made of pinewood, the maximum stress reaches a value of 93 MPa, which is higher than the strength of this wood to compression along the fibres

Rc || = 80 MPa. For the remaining wood species, maximum contact stresses were as follows: for alder 88 MPa > Rc || =51 MPa, for ash 83 MPa » Rc | = 63 MPa and for beech 79 MPa < Rc || = 84 MPa. In the leg, the stresses were directed per­pendicular to the fibres. Also here it was noticed that the value of stresses exceeded compression strength of the wood across the fibres, which is as follows: for the alder wood 15.1 MPa > Rc^ = 2 MPa, for beech 15.05 MPa > Rc^ = 7 MPa, for pine 14.67 MPa > Rc^ = 4.4 MPa and for ash 14.61 MPa > Rc^ = 7 MPa.

Much greater stress pressure azz appeared in the same connection, and the place particularly exposed to damage is proved to be point A, in which the edge of the tenon after inserting into the mortise is pressed on the edge of the mortise (Fig. 6.72). Based on Fig. 6.73, it is clear that in the mortise, higher stresses azz are generated than in the tenon. When analysing this impact, it can be noticed that in the mortise of a longitudinal course of fibres, stresses <7zz occur which are 65-130 % greater than in the tenon, where the fibres run perpendicular to the direction of the pressure. Furthermore, stresses in beech wood are approximately 15-45 MPa higher

than in alder wood (Fig. 6.73). In relation to ash wood, these stresses are higher by 8 MPa and in relation to pinewood by 9-30 MPa. In the element of the tenon, the stresses in beech wood are approximately 17-34 MPa higher than in alder wood, 8-12 MPa than in ash wood and 12-23 MPa than in pinewood. The type of adhesive used has a significant impact on the value of stresses azz in the point B, place of the tenon pressure on the mortise. Figure 6.73 shows the nonlinear fall of values of stresses ozz, together with a decrease in the values of the linear elasticity module of the glue-line to the 50 MPa. It should be noted here that these stresses teeter on the verge of acceptable values for each particular species of wood. Below, the border value of the module of linear elasticity of adhesive equal to 50 MPa, stresses in the tenon and mortise grow suddenly, reaching a maximum in the absence of a glue-line. For an element with a mortise made of beech wood, the maximum stress reaches a value of 165 MPa, which is higher than the strength of this wood to compression along the fibres Rc ц = 84 MPa. For the remaining wood species, maximum contact stresses were as follows: for ash 157 MPa > Rc || = 63 MPa, for pine

Modulus of elasticity [MPa]

Fig. 6.73 Impact of linear elasticity module of the glue and wood species on the value of stresses o^: a tenon at the point of stress on the mortise, b mortise at the point of stress on the tenon
134 MPa > Rc || = 80 MPa and for alder 120 MPa > Rc || =51 MPa. In the tenon, the stresses were directed perpendicular to the fibres. Also here it was noticed that the value of stresses exceeded compression strength of the wood across the fibres, which is as follows: for the beech wood 100 MPa > Rc^ = 7 MPa, for ash 88 MPa > Rcu = 7 MPa, for pine 78 MPa > Rcu = 4.4 MPa and for alder 65 MPa > R^u = 2 MPa.

In the indicated places of the construction, in which glue line were used of Young’s modulus above 50 MPa, stresses are similar or slightly exceed the acceptable values for particular species of wood. However, this does not mean that the construction gets completely destroyed. These stresses are concentrated on the edges and corners and have a local character. Moreover, the concentration of stresses is eased by the compression of the glue-line. Without a doubt, the most dangerous, because damaging, are stresses resulting from the connections of weak glue-line or with a damaged glue-line. They constitute a direct cause of damage of the node in the entire construction.

A separate structural problem is to determine the impact of the shape of glue-line on its strength and form of distribution of tangential and normal stresses in the glue-line. To this end, the construction of a chair has been chosen with a bar, selecting a horizontal bar and rear leg connection as the structural node (Fig. 6.74). Due to the perpendicular position of the bar and leg in the global plane of coordinates and the resulting different longitudinal course of wood fibres, two separate local systems have been used. For a bar, it was assumed that wood fibres will run parallel to the Y-axis, and the radial-tangential plane will lie on the XZ plane of the global system of coordinates. For the leg, the fibres will be oriented in the direction of the Z – axis, and the radial-tangential plane shall lie on the XYplane. Between the tenon and the mortise around the entire perimeter, a gap with a thickness of 0.1 mm has been formed. An oval glue-line has been created in this gap.

By taking into account the symmetry of the construction, one side frame was selected for strength analysis, loading the front edge of the seat with a concentrated force of 800 N (Fig. 6.62). The value, direction and rotation of this force corre­sponded to extreme conditions of use of the furniture by a user weighing approx­imately 160 kg. By analysing the static side frame of a chair, the concentrated forces appropriate for it have been transferred onto the structural node, which correspond to internal bending moments, shearing forces and normal forces (Fig. 6.63). The numerical model of the mortise and tenon joints was developed in the environment of the program Algor® (Fig. 6.75). To this end, for modelling the wooden parts of the joint, 20-node, orthotropic solid elements were used, while for modelling glue-lines—isotropic elements with a thickness of 0.1 mm.

For beech wood (Fagus silvatica L.), from which the model of the chair frame was made from, numerical calculations were conducted. Elastic properties of wood are provided in Table 6.4, based on the studies of Hearmon (1948) and Bodig and Goodman (1973). Taking into account the different orientation of individual ana­tomic directions of wood in elements of joints, this table provides values that correspond to two local coordinate planes. Another material component, which is

widely used for furniture assembly, is polyvinyl chloride glue (PVAC), for which the value of Young’s modulus was selected equal to 465.74 MPa.

Numerical calculations included two models of joints, which differed in shape of glue-line. The first model constituted a joint, in which the glue-line in the form of two parallelepipeds of the dimensions 0.1 x 20 x 33 mm, was set up on opposite surfaces of the tenon. In Fig. 6.76, by numbers from 1 to 11 vertically and from 1 to 8 horizontally, the nodes of the mesh of finite elements have been indicated. The second model represented a joint with a glue-line in an oval shape, with a thickness of 0.1 mm, formed on the entire side of the tenon. In this case, the vertical numbers of mesh nodes, from 1 to 23, also included hardened parts of the glue-line on both parts of the tenon (Fig. 6.76). During the course of numerical calculations, the change of the shape and thickness of glue-line was determined, caused by the

Fig. 6.75 Mesh of finite elements of the mortise and tenon joints a tenon, b glue-line, c mortise

Fig. 6.76 Dimensions of the glue-line and determination of the points of measuring stresses

pressure of the tenon on the mortise, as well as change of reduced stresses, tan­gential stresses and normal stresses in the nodes of the mesh on the surface of the glue-line.

An analysis of deformed meshes shows that in connection, the tenon sustained rotation and bending. The consequence of this deformation is a visible dispropor­tion of normal and shear strains of the glue-line along its edges. In a rectangular glue-line, shear strain dominates, of which the largest are concentrated in the corners and on the edge adjacent to the base of the tenon (Fig. 6.77a). These deformations were formed as a result of bending the tenon, in which the extreme fibres were subject to elongation or shortening. In other parts of the tenon, normal strains were not significant enough to affect the deformations of the glue-line. Therefore, in this part, the glue-line sustained proportional twisting between two stiff adherents. By testing the deformation of an oval glue-line, it was noted that both normal and shear strains occur in them. Figure 6.77b shows the change in the thickness of the glue-line, caused by pressure of the mortise and tenon. At the beginning and at the end of the tenon, as a result of compression, the glue-line reduces its thickness from 0.1 to 0.087 mm or 0.099 mm, while as a result of stretching, it increases its thickness to 0.102 or 0.103 mm. Pressure of the tenon on the mortise by the glue-line significantly reduces shear strain and thus decreases the value of stresses forming.

The disproportion of figural deformations in the glue-line also causes migration of its centre of rotation and uneven distribution of stresses. By assigning the complex state of stress, which is created in the glue-line, the uniaxial state has been characterised by reduced stress aR according to the equation:

ffRm“ _ "^2 ((rxx ~ ffYY)2+(ffXX – rZZ)2+(rZZ – Cyy)2+6(siy + sZy + TXz)) ;

(6.205)

where

otj normal stress for the chosen direction, otj tangential stress for the chosen plane.

Fig. 6.78 The distribution of stresses in a rectangular glue-line: a reduced stresses according to Mises, b tangential stresses txz, tx,

It was determined that for rectangular glue-line, the biggest reduced stresses according to Mises occur in the lower left corner of the glue-line, reaching a value of 112.84 MPa (Fig. 6.78a). It should be noted here that the stresses in other corners are from 1.33 to 2.62 times smaller. The components of reduced stress are appropriate normal and tangential stresses. Figure 6.78b presents the distribution of tangential stresses txz, txy. These stresses are responsible for shearing the glue-line and therefore for the strength of mortise and tenon joints. The biggest tangential stresses, like the biggest reduced stresses, occur in the left lower corner of the glue-line. Their value is 59.58 MPa. In the remaining corners, stresses constitute from 30 to 50 % of the value of maximum stresses. Taking into account the technical strength of glue-line of PVAC glue to shearing, ranging from 17 to 23 MPa, the damaging process is initiated in the corners of the bond. However, this does not mean that in these particular points of the glue-line, along with reaching the acceptable values by the tangential stresses, damage to the entire joint must occur. High concentration of reduced or tangential stresses in the corners of the glue-line results only in the decohesion in a few places on a small area. At a small distance from those corners, the level of stresses falls rapidly below the acceptable values. By calculating the average value of reduced and tangential stresses on the entire surface of the glue-line, the following values were obtained, respectively, or = 32.46 MPa and т = 16.28 MPa. Therefore, the tangential stresses, at extremely unfavourable operational load of a furniture construction, teeter on the verge of shearing strength of the adhesive.

In addition, in these glue-lines, it was also observed that the centre of rotation of a glue-line does not lie in its geometric centre. By connecting opposite sides of the glue-line by sections in point of the lowest value of reduced and tangential stresses (Fig. 6.78b), the actual position of the centre of rotation 0′ was obtained. This point shifted to the right on the horizontal axis of symmetry of the glue-line. The dis­placement of the centre of rotation is largely due to the bending of the tenon.

For oval glue-line, it was determined that the greatest reduced stresses according to Mises also occur in the lower left corner of a developed glue-line (that is under the axis of the tenon), reaching a value of 52.96 MPa (Fig. 6.79a). It should be noted here that the stresses in the corners of the opposite edge are from 7.8 to 18.3 times smaller. Figure 6.79b presents also distribution of tangential stresses txz, txy. The largest of these, with a value of 18.92 MPa, occur in the point of transition of the glue-line’s shape from the rectangular part into a round one (node of number 17). It is also interesting that the stresses at the horizontal edges of the glue-line are negligibly small.

By calculating the average value of reduced and tangential stresses on the entire surface of the glue-line, the following values were obtained: or = 14.31 MPa and т = 4.35 MPa. Tangential stresses, at a chosen unfavourable operational load of a furniture construction, constitute only 25 % of the values of acceptable stresses, corresponding to the acceptable strength of the adhesive to shearing. Slightly higher average values of reduced stresses are caused by the presence of normal stresses ozz and Oyy. During bending of the joint and twisting of the glue-line, the tenon, sustaining bending and rotation, put pressure on the lower and upper parts of the glue-line, causing very high normal stresses ozz in it. For this reason, small and safe tangential and reduced stresses were formed, contributing to a smaller stress of material, than in the rectangular glue-line.

In this way, it was shown that the shape of the glue-line clearly differentiates the mortise and tenon joints. Stress of the tenon on a mortise by the layer of glue-line changes the form and sizes of its deformations. In rectangular glue-line, only shear strains occur, which generate tangential stresses of values that exceed the ultimate strength. In oval glue-line, shear strains are clearly limited by the pressure of the tenon on the mortise, through which the level of dangerous shearing stresses sig­nificantly decreases.

Fig. 6.79 The distribution of stresses in an oval glue-line: a reduced stresses according to Mises, b tangential stresses rxz, Txy