Assuming that we are dealing with a 2-D element placed in a Cartesian coordinate system XOY, in which the Z-axis is perpendicular to the other two, strains e and у can be written in a general manner by equations (Ashkenazi et al. 1958;
Ambarcumian 1967; Lekhnickij 1977; Leontiev 1952; Mitinskij 1948; Norris 1942; Nowacki 1970; Rabinowicz 1946; Timoshenko and Goodier 1951):
£x a11 rx E a12ry E a13sxy
£y = a21 rx + a22ry + a23sxy • (6.124)
yxy = a31 rx + a32ry + a33sxy
In this case, the equations consist of nine coefficients of susceptibility atj. The value of these coefficients can be determined experimentally by analysing the character of deformation of a 2-D element in a Cartesian coordinate plane
(Fig. 6.46).
To present the relations between stresses and strains in an element stretched biaxially and sheared in the XY plane, the principle of superposition can be applied, by summing up the deformations of a rectangular element shown in Fig. 6.46. Under the influence of stresses <rx, the element shall sustain extension £x, = rx/Ex in the direction of the X-axis, while narrowing ey, = vxy£x towards the Y-axis and shear strain y,,, = ixxy ■ ex in the XY plane. Due to stresses oy, the appropriate elongation in the direction of the Y-axis will amount to ey = ry/Ey, while narrowing towards the X sx» = vyx ■ sy and the shear strain y,,, = iy xy ■ £y. The effect of tangential stresses Txy will cause shear strain y, = sxyGxy, elongation in the direction of X equal to £x« = ixy xyxy and elongation in the direction of Y, respectively,
syn = ixyy • y. By summing up the appropriate strains of the element, caused by the joint effect of stresses ax, ay, Txy, we shall obtain:
According to Rabinowicz (1946), the coefficients of susceptibility atj should be called technical coefficients, while the above equation can be written in the form:
|
1 |
V |
№xy, x " |
_ |
|||
|
sx |
Ex |
Ey |
Gxy |
rx |
||
|
sy |
mxy |
Exy, y |
Gy |
|||
|
Ex |
Ey |
Gxy |
||||
|
yxy |
E |
1 x Ey |
1 Gxy |
sxy |
Taking into account the symmetry of matrix of deformations atj = a,;, we shall obtain obvious dependencies:
1yxy E ’
Ey
which enable us to determine normal strains ex ey in the main directions of the plane’s axis and shear strain yxy in the plane XOY in the following form:
Functions of strains and stresses written in the form of constitutive equations express the relationships between the coordinates of tensors of strains and stresses. However, it is necessary to determine the coordinates of the tensor of susceptibility in any directions of the global plane of coordinates. This is connected with determining the state of stresses or strains in selected directions of anisotropy of the tested body. Therefore, let us consider the method of transforming elastic properties of wood after entering a new plane of coordinates.
The next equation allows us to calculate the values of normal and shear strains of wood in any direction tilted to the main axes of the plane of coordinates at an angle ф.
we obtain:
a’n = — = an cos4 u + a22 sin4 u + (2aj2 + 053) sin2 ucos2 u + 2aj3 sin u cos2 u + 2a23 sin3 cos u,
a’22 = — = a\ sin4 u + 022 cos4 +(2aj2 + 033) sin2 u cos2 u — 2aj3 sin3 cos u — 2a23 sin u cos3 u, 33 = – y – = (an — 2aj2 + a22)4 sin2 ucos2 u + [(a23 — aj3)4 sin u cos u + a33] (cos2 u — sin2 u)
(6.131)
and
Hence, we ultimately obtain equations describing modules of wood elasticity in any direction ф, in the form of:
g ____________________________ ExEyOxy_________________________
x Gxy(Ey cos4 u + Ex sin4 u + Ex(—2Gxyvyx + Ey) sin2 u cos2 u + 2ExEy (lxyx cos2 u + 1xyy sin2 u) sin u cos u
E’ = __________________________ ExEyGxy__________________________
y Gxy(Ey sin4 u + Ex cos4 u + Ex(—2Gxyvxy + Ey) sin2 u cos2 u — 2ExEy (ixyx sin2 u + ixy, y cos2 u) sin u cos u
ExEyGxy
4Gxy(Ey — Ex(2vyx — 1)) sin2 u cos2 u + ExEy{(lxyy — 1) sin u cos / + 1(cos2 u — sin2 u) (6.134)
In order to determine the deformation of an anisotropic body, the following equation can be used:
|
ex |
a11 |
a12 |
a13 |
X‘ |
|||
|
ey |
= |
a21 |
a22 |
a23 |
ry |
(6.135) |
|
|
.Ciy. |
a31 |
a33 |
a33 |
or in the case of a uniaxial state of stresses, the equation:
eX = et
in which the values of factors Ex, Ey, Gxy, vyx, fixy, x, fixy, y should be determined in the primary XOY system.
As the analysis of the state of stresses showed in lap joints, wooden elements are subject to stretching. Therefore, in order to appoint dependencies of strains from a uniaxial stress ax and angle of cutting samples ф in the plane of orthotropy, the following equation should be used:
4 = a’n • rXi or 4 = et • rxi
in which the parameters a’n are subject to transformation. Because the wooden elements joined at lap after gluing may form different planes of anatomical build, it was decided to provide a description of normal strains of wood for the following planes:
– the radial plane LR,
– the tangential plane LT.
Normal strains e’l in the LR plane amount to:
where
Q value of the external load.
Based on the analysis of the state of tangential stresses in rectangular angular joints, it was shown that wooden elements, as a result of external forces, are in a complex state of stresses. This state causes that except to normal strains in the adherent (glued element) shear strains appear. Therefore, prior to establishing tangential stresses in the glue-line of a wooden joint, it would be necessary to determine the appropriate, due to the direction of stresses, global deformations in the selected plane of the adherent (Fig. 6.47).
Normal strains e’xi and e’yi of any adherent in the direction of X or Y, caused by stresses rX or ry for particular types of planes of wood glued together, are expressed by equations described in the previous chapter. The causes of the emergence of such strains are axial forces in structural nodes. In addition to these forces in angular joints, also bending moments appear, causing states of tangential stresses in glue-lines and shear strains of elements of joints. Load of an angular joint in the plane of a glue-line resembles a complex state of stresses in an orthotropic body, shown in Fig. 6.47. For such a case, shear strains are expressed by the equation:
cXy = a31 rX + ^32r’y + a33 V (6-140)
Coefficients a’31, a’32, a!33 allow us to determine the value of shear strains for the examined planes of gluing wood: description of the deformations y’LR
(6.141)
description of the strains y’LT
Tlt= I (—j^ ~ E“) 2 sin u cos3 u + [E + ELJ 2 sin3 u cos u + ^ £
. У VLT, L 2 , 1LT, L -2 , 1
+ 1—– — cos u +—— — sin u +——–
V Glt
mTL 1 – . 3 . ( 1 . vLt – . 3 , (1L, LT 1T, LTA -2 2
– ——- — 2sin u cos u 2sin u cos u + l —=———- d:— 2sin u cos u
jT jL/ Et El) v jL jT /
. / MlT, L • 2 1 1LT, T 2 1 / 2 • 2u)l /
+ I——- — sin u +—— — cos u———- sin u cos u I (cos u — sinuJ J оу
glt glt glt /
Г/1 2Vtl 1 A. 2 2 . ((1LT, L MlT. A. • . 1 / 2 -2 )1 0
+ Ж + ~W + Er) 4 sm u cos u + M——————————— ~^j 4 sin u cos u + —I (cos u — sin uJtlt-
(6.142)
Presented description of normal and shear strains of adherents must be used in equations describing tangential stresses in glue-line which connect these adherents. The mathematical models obtained in such a way allow us to determine the relation between the layout of wood fibres in elements of a joint and the form and values of tangential stresses in the glue-line.
