Orthotropic Material Laws

Technical Article

Orthotropic material laws are used wherever materials are arranged according to their loading. Examples include fiber-reinforced plastics, trapezoidal sheets, reinforced concrete or timber.

Hooke's law with $\mathrm u\;=\;\frac{\mathrm E}{2\;\mathrm G}\;\;-\;1\;\leq\;0.5$ and $-1\;<\;\mathrm u\;<\;0.5$ is generally no more valid for orthotropic materials.

Figure 01 - Transversal Strain

The following material parameters refer to two-dimensional stiffnesses and, unless otherwise stated, to the material timber. A local axis system is the basis as shown in Figure 01.

  • Ex = stiffness in local x-direction of the surface
  • Ey = stiffness in local y-direction of the surface
  • Gxz = shear stiffness in local x-direction of the surface (thickness direction of the plate)
  • Gyz = shear stiffness in local y-direction of the surface (thickness direction of the plate)
  • Gxy = shear stiffness in-plane area
  • νxy = transversal strain in x-direction
  • νyx = transversal strain in y-direction

The stresses in Figure 02 are related to the stiffnesses mentioned here.

Figure 02 - Shear Stresses

The material law is subject to the following rules.

Equation 1:
$\frac{{\mathrm u}_\mathrm{yx}}{{\mathrm E}_\mathrm y}\;=\;\frac{{\mathrm u}_\mathrm{xy}}{{\mathrm E}_\mathrm x}$

Equation 2:
${\mathrm u}_\mathrm{xy}\;\geqslant\;{\mathrm u}_\mathrm{yx}$

Equation 3:
${\mathrm u}_\mathrm{xy}\;=\;-\frac{{\mathrm\varepsilon}_\mathrm y}{{\mathrm\varepsilon}_\mathrm x}\;=\;\frac{{\mathrm E}_\mathrm x}{{\mathrm E}_\mathrm y}\;\cdot\;{\mathrm u}_\mathrm{yx}$

Equation 4 (stiffnesses in-plane area): 
$\begin{bmatrix}{\mathrm\varepsilon}_\mathrm x\\{\mathrm\varepsilon}_\mathrm y\\{\mathrm\gamma}_\mathrm{xy}\end{bmatrix}\;=\;\begin{bmatrix}\frac1{{\mathrm E}_\mathrm x}&-\;\frac{{\mathrm u}_\mathrm{xy}}{{\mathrm E}_\mathrm x}&0\\-\;\frac{{\mathrm u}_\mathrm{xy}}{{\mathrm E}_\mathrm x}&\frac1{{\mathrm E}_\mathrm y}&0\\0&0&{\mathrm G}_\mathrm{xy}\end{bmatrix}\;\cdot\;\begin{bmatrix}{\mathrm\sigma}_\mathrm x\\{\mathrm\sigma}_\mathrm y\\{\mathrm\tau}_\mathrm{xy}\end{bmatrix}$

The ratio of the strains in the equations mentioned above underlines the relations in Figure 01.

The stiffnesses in-plane area are calculated as follows.

Equation 5:
${\mathrm d}_\mathrm i=\begin{bmatrix}{\mathrm d}_{11,\mathrm i}&{\mathrm d}_{12,\mathrm i}&0\\&{\mathrm d}_{22,\mathrm i}&0\\\mathrm{sym}&&{\mathrm d}_{33,\mathrm i}\end{bmatrix}=\begin{bmatrix}\frac{{\mathrm E}_{\mathrm x,\mathrm i}}{1\;-\;\mathrm u_{\mathrm{xy},\mathrm i}^2\;\frac{{\mathrm E}_{\mathrm y,\mathrm i}}{{\mathrm E}_{\mathrm x,\mathrm i}}}&\frac{{\mathrm u}_{\mathrm{xy},\mathrm i}\;{\mathrm E}_{\mathrm y,\mathrm i}}{1\;-\;\mathrm u_{\mathrm{xy},\mathrm i}^2\;\frac{{\mathrm E}_{\mathrm y,\mathrm i}}{{\mathrm E}_{\mathrm x,\mathrm i}}}&0\\&\frac{{\mathrm E}_{\mathrm y,\mathrm i}}{1\;-\;\mathrm u_{\mathrm{xy},\mathrm i}^2\;\frac{{\mathrm E}_{\mathrm y,\mathrm i}}{{\mathrm E}_{\mathrm x,\mathrm i}}}&0\\\mathrm{sym}&&{\mathrm G}_{\mathrm{xy},\mathrm i}\end{bmatrix}$

Transversal Strain v

As explained in Figure 01, modified deformations and stresses in this direction result from the smoother material behavior in the respective direction.

Ratio of the strains:

Equation 6:
${\mathrm\nu}_\mathrm{xy}\;=\;-\frac{{\mathrm\varepsilon}_\mathrm y}{{\mathrm\varepsilon}_\mathrm x}\;\rightarrow\;{\mathrm\varepsilon}_\mathrm y\;=\;-{\mathrm\nu}_\mathrm x\;\cdot\;{\mathrm\varepsilon}_\mathrm x$

Equation 7:
${\mathrm\nu}_\mathrm{yx}\;=\;-\frac{{\mathrm\varepsilon}_\mathrm x}{{\mathrm\varepsilon}_\mathrm y}\;\rightarrow\;{\mathrm\varepsilon}_\mathrm x\;=\;-{\mathrm\nu}_\mathrm y\;\cdot\;{\mathrm\varepsilon}_\mathrm y$

For ${\mathrm\sigma}_\mathrm x\; eq\;0,\;{\mathrm\sigma}_\mathrm y\;=\;0,\;{\mathrm\sigma}_\mathrm{xy}\;=\;0$, the following equations are given with Hooke's law.

Equation 8:
${\mathrm\varepsilon}_\mathrm x\;=\;\frac{{\mathrm\sigma}_\mathrm x}{{\mathrm E}_\mathrm x}$

Equation 9:
${\mathrm\varepsilon}_\mathrm y\;=\;\frac{{\mathrm\sigma}_\mathrm y}{{\mathrm E}_\mathrm y}$

Equation 10:
${\mathrm\sigma}_\mathrm y\;=\;{\mathrm\varepsilon}_\mathrm y\;\cdot\;{\mathrm E}_\mathrm y\;\mathrm{with}\;{\mathrm E}_\mathrm y\;=\;\frac{{\mathrm\nu}_\mathrm{yx}\;\cdot\;{\mathrm E}_\mathrm x}{{\mathrm\nu}_\mathrm{xy}}\;\hspace{5mm}\rightarrow\;\hspace{5mm}{\mathrm\sigma}_\mathrm y\;=\;{\mathrm\varepsilon}_\mathrm y\;\cdot\;\frac{{\mathrm\nu}_\mathrm{yx}\;\cdot\;{\mathrm E}_\mathrm x}{{\mathrm\nu}_\mathrm{xy}}$

Equation 11:
$\rightarrow\;{\mathrm\sigma}_\mathrm y\;=\;{\mathrm\varepsilon}_\mathrm y\;\cdot\;\frac{{\mathrm\nu}_\mathrm{yx}\;\cdot\;{\displaystyle\frac{{\mathrm\sigma}_\mathrm x}{{\mathrm\varepsilon}_\mathrm x}}}{{\mathrm\nu}_\mathrm{xy}}$

Equation 12:
$\rightarrow\;{\mathrm\varepsilon}_\mathrm y\;=\;\frac{{\mathrm\varepsilon}_\mathrm y\;\cdot\;{\displaystyle\frac{{\mathrm\nu}_\mathrm{yx}\;\cdot\;{\displaystyle\frac{{\mathrm\sigma}_\mathrm x}{{\mathrm\varepsilon}_\mathrm x}}}{{\mathrm\nu}_\mathrm{xy}}}}{{\mathrm E}_\mathrm y}\;\mathrm{with}\;{\mathrm\nu}_\mathrm{xy}\;=\;-\frac{{\mathrm\varepsilon}_\mathrm y}{{\mathrm\varepsilon}_\mathrm x}$

Equation 13:
$\rightarrow\;{\mathrm\varepsilon}_\mathrm y\;=\;\frac{{\mathrm\varepsilon}_\mathrm y\;\cdot\;{\mathrm\nu}_\mathrm{yx}\;\cdot\;{\mathrm\sigma}_\mathrm x}{{\mathrm\varepsilon}_\mathrm x\;\cdot\;\left(-{\displaystyle\frac{{\mathrm\varepsilon}_\mathrm y}{{\mathrm\varepsilon}_\mathrm x}}\right)\;\cdot\;{\mathrm E}_\mathrm y}\;=\;-\frac{{\mathrm\nu}_\mathrm{yx}\;\cdot\;{\mathrm\sigma}_\mathrm x}{{\mathrm E}_\mathrm y}$

Stiffness Matrix

Caculation of the global stiffness matrix of the plate.

Equation 14:
$\mathrm D\;=\;\begin{bmatrix}{\mathrm D}_{11}&{\mathrm D}_{12}&0&0&0&0&0&0\\&{\mathrm D}_{22}&0&0&0&0&0&0\\&&{\mathrm D}_{33}&0&0&0&0&0\\&&&{\mathrm D}_{44}&0&0&0&0\\&&&&{\mathrm D}_{55}&0&0&0\\&\mathrm{sym}&&&&{\mathrm D}_{66}&0&0\\&&&&&&{\mathrm D}_{77}&0\\&&&&&&&{\mathrm D}_{88}\end{bmatrix}$

Bending components:

Equation 15:
${\mathrm D}_{11}\;=\;\frac{\mathrm d^3}{12}\;\cdot\;{\mathrm d}_{11}\;\widehat{=\;}\frac{{\mathrm E}_\mathrm x\;\mathrm d^3}{12\;\left(1\;-\;{\mathrm\nu}_\mathrm{xy}\;{\mathrm\nu}_\mathrm{yx}\right)}$

Equation 16:
${\mathrm D}_{12}\;=\;\frac{\mathrm d^3}{12}\;\cdot\;{\mathrm d}_{12}\;\widehat{=\;}\mathrm{sgn}\left({\mathrm\nu}_\mathrm{xy}\right)\;\sqrt{{\mathrm\nu}_\mathrm{xy}\;{\mathrm\nu}_\mathrm{yx}\;{\mathrm D}_{11}\;{\mathrm D}_{22}}$

Equation 17:
${\mathrm D}_{22}\;=\;\frac{\mathrm d^3}{12}\;\cdot\;{\mathrm d}_{22}\;\widehat{=\;}\frac{{\mathrm E}_\mathrm y\;\mathrm d^3}{12\;\left(1\;-\;{\mathrm\nu}_\mathrm{xy}\;{\mathrm\nu}_\mathrm{yx}\right)}$

Equation 18:
${\mathrm D}_{33}\;=\;\frac{\mathrm d^3}{12}\;\cdot\;{\mathrm d}_{33}\;\widehat{=\;}{\mathrm G}_\mathrm{xy}\;\frac{\mathrm d^3}{12}$

Membrane Components:

Equation 19:
${\mathrm D}_{66}\;=\;\mathrm d\;\cdot\;{\mathrm d}_{11}\;\widehat{=\;}\frac{{\mathrm E}_\mathrm x\;\mathrm d}{1\;-\;{\mathrm u}_\mathrm{xy}\;{\mathrm u}_\mathrm{yx}}$

Equation 20:
${\mathrm D}_{67}\;=\;\mathrm d\;\cdot\;{\mathrm d}_{12}\;\widehat{=\;}\mathrm{sgn}\left({\mathrm\nu}_\mathrm{xy}\right)\;\sqrt{{\mathrm\nu}_\mathrm{xy}\;{\mathrm\nu}_\mathrm{yx}\;{\mathrm D}_{66}\;{\mathrm D}_{77}}$

Equation 21:
${\mathrm D}_{77}\;=\;\mathrm d\;\cdot\;{\mathrm d}_{22}\;\widehat{=\;}\frac{{\mathrm E}_\mathrm y\;\mathrm d}{1\;-\;{\mathrm u}_\mathrm{xy}\;{\mathrm u}_\mathrm{yx}}$

Equation 22:
${\mathrm D}_{88}\;=\;\mathrm d\;\cdot\;{\mathrm d}_{33}\;\widehat{=\;}{\mathrm G}_\mathrm{xy}\;\mathrm d$

Shear Components:

Equation 23:
${\mathrm D}_{44}\;=\;\frac56\;{\mathrm G}_\mathrm{xz}\;\cdot\;\mathrm d$

Equation 24:
${\mathrm D}_{55}\;=\;\frac56\;{\mathrm G}_\mathrm{yz}\;\cdot\;\mathrm d$

A prerequisite for these equations is that the stiffness matrix is defined positive, so that all eigenvalues of the matrix are positive.

For this reason, RFEM checks, amongst others, the definition of the transversal strain according to the following equation.

Equation 25:
${\mathrm u}_\mathrm{xy}\;\leq\;0.999\;\cdot\;\sqrt{\frac{{\mathrm E}_\mathrm x}{{\mathrm E}_\mathrm y}}$

Example

With the following example (Figure 03), the orthotropic material behavior will be explained. An orthotropic material will be compared to an isotropic material. In addition, the stiffness of the orthotropic plate will be defined with the high stiffness in x-direction and also in y-direction.

Figure 03 - Structure

Structure:

  • Plate thickness 200 mm
  • Material C 24
  • Orthotropic stiffnesses
    $\begin{array}{l}{\mathrm E}_\mathrm x\;=\;1,100.0\;\mathrm{kN}/\mathrm{cm}^2\\{\mathrm E}_\mathrm y\;=\;37.0\;\mathrm{kN}/\mathrm{cm}^2\\{\mathrm G}_\mathrm y\;=\;6.9\;\mathrm{kN}/\mathrm{cm}^2\\{\mathrm G}_\mathrm x\;=\;69.0\;\mathrm{kN}/\mathrm{cm}^2\\{\mathrm G}_\mathrm{xy}\;=\;69.0\;\mathrm{kN}/\mathrm{cm}^2\\{\mathrm u}_\mathrm{xy}\;=\;2.52\\{\mathrm u}_\mathrm{yx}\;=\;0.085\end{array}$
  • Isotropic stiffnesses
    $\begin{array}{l}\mathrm E\;=\;1,100\;\mathrm{kN}/\mathrm{cm}^2\\\mathrm G\;=\;500\;\mathrm{kN}/\mathrm{cm}^2\\\mathrm u\;=\;0.1\end{array}$
  • Dimension w = 2.0 m, l = 4.0 m
  • Load 20 kN/m²
  • FE mesh size 50 cm

The structure is supported as rigidly fixed in vertical z-direction. The support conditions in x- and y-direction have been selected in such a way that no effects due to restraint occur.

The calculation is performed according to the linear static analysis with linear elastic material behavior and support conditions.

The following transversal strain results from Hooke's Law, together with the given values.

Equation 26:
$\mathrm\nu\;=\;\left(\sqrt{{\mathrm E}_\mathrm x\;\cdot\;{\mathrm E}_\mathrm y}\right)\;/\;\left(2\;\mathrm G\right)\;-\;1\;=\;6.97$

This high transversal strain is not possible with the selected material model. With the equations from [1], the values can be, however, adjusted.

Equation 27:
${\mathrm\nu}_\mathrm{xy}\;\approx\;\left(\frac{\sqrt{{\mathrm E}_\mathrm x\;{\mathrm E}_\mathrm y}}{2\;{\mathrm G}_\mathrm{xy}}\;-\;1\right)\;\cdot\;\sqrt{\frac{{\mathrm E}_\mathrm x}{{\mathrm E}_\mathrm y}}$

Equation 28:
${\mathrm\nu}_\mathrm{yx}\;\approx\;\left(\frac{\sqrt{{\mathrm E}_\mathrm x\;{\mathrm E}_\mathrm y}}{2\;{\mathrm G}_\mathrm{xy}}\;-\;1\right)\;\cdot\;\sqrt{\frac{{\mathrm E}_\mathrm y}{{\mathrm E}_\mathrm x}}$

Equation 29:
${\mathrm\nu}_\mathrm{xy}\;=\;\left(\frac{\sqrt{1,100\;\cdot\;37}}{2\;\cdot\;69}\;-\;1\right)\;\sqrt{\frac{1,100}{37}}\;=\;2.52$

Equation 30:
${\mathrm\nu}_\mathrm{xy}\;=\;\left(\frac{\sqrt{1,100\;\cdot\;37}}{2\;\cdot\;69}\;-\;1\right)\;\sqrt{\frac{37}{1,100}}\;=\;0.085$

Figure 04 - Surfaces with Stiffnesses

Results
As expected, the largest deformations occur with the orientation of the stiffnesses in y-direction (Figure 06). The support reaction and the moment of the isotropic plate are displayed in Figure 05.

Figure 05 - Results Isotropic Material

Since the plate with the high stiffness in y-direction (Ey = 1,100 kN/cm²) has the high resistance in this direction, the support reactions are also higher there (125.4 kNm compared to 58.3 kNm).

Figure 06 - Deformations and Support Reactions of Orthotropic Plates

The resulting maximum bending moments for the orthotropic plates are equal to mx with the stiffness in x-direction and for my with a high stiffness in y-direction.

For the plate with the high stiffness in y-direction, the maximum bending moment my is almost in the center of the plate (Figure 07).

Figure 07 - Bending Moments of Orthotropic Plate mx (left) and my (right)

Variation of Transversal Strain

The transversal strain according to the strain diagrams can reach the maximum and minimum values listed in the table.

 

 Max.Min.
νxy5.447-5.447
νyx0.183-0.183

The plate introduced at the beginning with the high stiffness (Ex = 11,000) will be defined with these high transversal strains for this purpose. The other stiffnesses of the plate, however, remain unchanged.

Figure 08 shows the results of the variation of νxy = 5.44 to -5.44. 

For νxy = 5.44, the support reactions are qualitatively identical to the isotropic material behavior. The bending moment enlarges from mx = 18.1 kNm/m (isotropic plate) to mx = 34.9 kNm/m (orthotropic plate).

Compared to the orthotropic plate with common transversal strains (νxy = 2.5), the bending moment is slightly reduced. 

Figure 08 - Results of Variation According to Table (Left: νxy = 5.44, Center: vxy = 0, Right: νxy = -5.44)

With νxy = 0, the high amplitude of the support reaction at the free end of the plate is shifted to a constant value of 43 kN/m.

The moment mx increases to 38.1 kNm/m. Compared to the previous result (νxy =5.44), the influence of the transversal strain is shown here. For ν =0, no deformation or distortion due to the transversal strain is caused.

For νxy = -5.44, a post-critical failure is shown at the free plate end and the support reactions turn negative. The maximum moment occurs in the center of the plate with 59.5 kNm/m.

The plate behaves now more than an uniaxial stressed plate without the third support in its longitudinal direction.

This behavior can be explained with Figure 01 and the relation listed there.

Due to the high negative transversal strain (νxy = -5.44), the plate is completely overpressed at the free edge and can therefore not be deformed.

The influence of the orthotropy in y-direction is almost zero here (Ey ≈ 0).

Summary

With the orthotropic material model in RFEM, almost any material parameters can be defined. With the variation of the transversal strains, very different results are possible. A transversal strain after modification of the values according to [1] results in values which lie close to the solution for a single-span beam.

Equation 31:
${\mathrm M}_\mathrm y\;=\;\frac{\mathrm q\;\mathrm l^2}8\;=\;40\;\mathrm{kNm}$

Too high negative transversal strains show a modified structural system which does no more correspond to the modeling.

Literature

[1] Huber, M. T : The Theory of Crosswise Reinforced Ferroconcrete Slabs and Its Application to Various Important Constructional Problems Involving Rectangular Slabs, Der Bauingenieur 12, pages 354 - 360, and 13, pages 392 - 395. 1923

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