A Sample of Calculation of Anisotropy Loaded Beams

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And then :

(1.13)

Deformation we can to describe by equation:

(1.14)

Geometrical relationship between deformation, dislocations and curvature as well as location of considered fiber describes equation:

(1.15)

Bending moment can be described by equation:

(1.16)

(1.17)

This equations of binding large dislocation with dimensions of section of beam, curvature and physical properties.

2. Application of the Method

In the next part of this article the derived equations were applied to calculations of composite lift spring. Internal burdens of any element of spring we can count use well-known conditions. Sum of projections of strengths in studied section Figure 2:

(2.1)

(2.2)

and from this:

(2.3)

(2.4)

Sum of moments:

(2.5)

Equation 2.3, 2.4, 2.5 and 1.16, 1.17 give possibility to count radial dislocations (U) and circumferential (V) of spring axis in dependence from external loads Figure 2.

Figure 2. Computational diagram of leaf spring

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Relationships between radial dislocations (U) and circumferential (V) and vertical dislocations (f) and horizontal (f_p) are was possible to get from equations:

Horizontal dislocations - f_p

(2.6)

Vertical dislocations – f

(2.7)

If we'll put to equation of inertia moment of spring section (1.7):

(2.8)

and we’ll accept, that:

(2.9)

We will get possibility of calculation of leaf spring at any shape.

For spring at solid widths - and variable thickness - to have substituted (2.8) and (2.6) to the equation (1.2) we receive maximum squeezing tensions:

(2.10)

3. Experimental Research

The equations presented above were used to calculate the composite curved leaf spring. In the further step the results of calculations were compared with the FEM calculations. The leaf spring has a regular width and a changing thickness. The scheme of leaf spring was shown on the Figure 3. It has the three-ply construction. The core was made from glass epoxy composite reinforced by textile. External layers have regular thickness, they are reinforced by continuous glass fibers. The width of layers is equal to the width of leaf spring.

Figure 3. Geometrical dimensions of composite leaf spring

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The width of composite leaf spring was 80 mm and its length was 1110 mm. The changes of the thickness in the function of his length were introduced in Table 1. Signs used in the table can be found in Figure 3.

Table 1. The radial thickness of composite leaf spring in the function of the his length

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Tables index

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Using the Algor system the calculations of composite leaf spring were conducted. The results of FEM were compared with the calculations made using the new author's analytic method and with the results of laboratory research. Leaf springs were tested on the static load. The perpendicular and horizontal dislocation of center leaf was measured on various loading. The experimental data was compared with the results of calculations obtained using the theory of small deflections (TMU), the method FEM (MES) and the theory of large deflections (TDU). In Figure 4 the horizontal dislocation of the central part of one composite leaf spring was shown. Experimental results were marked by symbol “EXP”.

Figure 4. The perpendicular dislocation of central part of composite leaf spring

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Similarly, the horizontal dislocation of central part of leaf spring was put together in Figure 5.

Figure 5. The horizontal dislocation of central part of composite leaf spring

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In case of the perpendicular dislocation, exactitudes of particular computational methods are similar. The dislocation calculated by the FEM method and by the method derived in the support of the theory of large dislocation (TDU) are approximate to measured. The horizontal displacements calculated by TDU better describes the actual situation. In Figure 4 and Figure 5 the line marked EXP measurement data indicated a line described TDU symbol calculation results obtained using the derived equations. It should be noted much less time-consuming the TDU method in comparison to the method of MES. The methodology described above, has been used to calculate stress in the selected cross-section spring. In Figure 6 the normal tension in external layers of spring was shown.

Figure 6. Normal tension exterior cladding of leaf spring

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