Skip to main content
SpringerLink
Log in

Stiffness matrix of the finite element of a plate compliant in transverse shear

  • Published:
Mechanics of Composite Materials Aims and scope

A four-node rectangular finite element is elaborated for a plate calculated with account of transverse shear strains. On variational realization of the finite-element method, the stiffness matrix and the vector of equivalent nodal forces are obtained for the case of a surface load. The basic nodal kinematic parameters adopted also include the angles of transverse shear.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Reissner, “The effect of transverse shear deformation on the bending of elastic plates,” Trans. ASME, J. Appl. Mech., 12, No. 2, 69–77 (1945).

    Google Scholar 

  2. R. D. Mindlin, “Influence of rotary inertia and shear on flexural motions of elastic plates,” Trans. ASME, J. Appl. Mech., 18, 31–38 (1951).

    Google Scholar 

  3. V. V. Vasil’ev, Mechanics of Composite Structures [in Russian], Mashinostroenie, Moscow (1988).

    Google Scholar 

  4. D. Zenkert, An Introduction to Sandwich Construction, Chameleon Press, London (1995).

    Google Scholar 

  5. J. R. Vinson, The Behavior of Sandwich Structures of Isotropic and Composite Materials, Technomic, Lancaster (1999).

    Google Scholar 

  6. L. P. Kollar and G. S. Springer, Mechanics of Composite Structures, Cambridge University Press (2003).

  7. J. M. Whitney, Structural Analysis of Laminated Anisotropic Plates, Technomic, Lancaster, Pennsylvania (1987).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Reshetnev Siberian State Aerospace University, Krasnoyarsk, Russia

    V. Nesterov

Authors
  1. V. Nesterov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Nesterov.

Additional information

Translated from Mekhanika Kompozitnykh Materialov, Vol. 47, No. 3, pp. 399–418, May-June, 2011.

Appendices

Appendices

  1. 1.

    Submatrices of the matrix S (35):

$$ \begin{array}{*{20}{c}} {{S_{\text{I}}} = \left[ {\begin{array}{*{20}{c}} 1 & x & y & {{x^2}} & {xy} & {{y^2}} & {{x^3}} \\ 0 & 1 & 0 & {2x} & y & 0 & {3{x^2}} \\ 0 & 0 & 1 & 0 & x & {2y} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]} & {{S_{\text{II}}} = \left[ {\begin{array}{*{20}{c}} {{x^2}y} & {x{y^2}} & {{y^3}} & {{x^3}y} & {x{y^3}} & 0 & 0 \\ {2xy} & {{y^2}} & 0 & {3{x^2}y} & {{y^3}} & 0 & 0 \\ {{x^2}} & {2xy} & {3{y^2}} & {{x^3}} & {3x{y^2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & x \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{S_{\text{III}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ y & {xy} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & x & y & {xy} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} & {{S_{\text{IV}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ x & y & {xy} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x & y & {0xy} \\ \end{array} .} \right]} \\ \end{array} $$
  1. 2.

    Submatrices of the matrix T (40):

\( \begin{array}{*{20}{c}} {{T_{{\text{I - I}}}} = \left[ {\begin{array}{*{20}{c}} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} & {{T_{{\text{I - II}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} & {{T_{{\text{I - III}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} \\ \end{array} \)

$$ \begin{array}{*{20}{c}} {{T_{{\text{I - IV}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \end{array} } \right],} & {{T_{{\text{II - I}}}} = \left[ {\begin{array}{*{20}{c}} 1 & a & 0 & {{a^2}} & 0 & 0 & 0 \\ 0 & 1 & 0 & {2a} & 0 & 0 & {3{a^2}} \\ 0 & 0 & 1 & 0 & a & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} & {{T_{{\text{II - II}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {{a^2}} & 0 & 0 & {{a^3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & a \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} \\ \end{array} $$

\( \begin{array}{*{20}{c}} {{T_{{\text{II - III}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} & {{T_{{\text{II - IV}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & a & 0 & 0 \\ \end{array} } \right],} & {{T_{{\text{III - III}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ b & {ab} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & a & b & {ab} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} \\ \end{array} \)

$$ \begin{array}{*{20}{c}} {{T_{{\text{III - I}}}} = \left[ {\begin{array}{*{20}{c}} 1 & a & b & {{a^2}} & {ab} & {{b^2}} & {{a^3}} \\ 0 & 1 & 0 & {2a} & b & 0 & {3{a^2}} \\ 0 & 0 & 1 & 0 & a & {2b} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} & {{T_{{\text{III - II}}}} = \left[ {\begin{array}{*{20}{c}} {{a^2}b} & {a{b^2}} & {{b^3}} & {{a^3}b} & {a{b^3}} & 0 & 0 \\ {2ab} & {{b^2}} & 0 & {3{a^2}b} & {{b^3}} & 0 & 0 \\ {{a^2}} & {2ab} & {3{b^2}} & {{a^3}} & {3a{b^2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & a \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{T_{{\text{III - IV}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a & b & {ab} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & a & b & {ab} \\ \end{array} } \right],} & {{T_{{\text{IV - I}}}} = \left[ {\begin{array}{*{20}{c}} 1 & 0 & b & 0 & 0 & {{b^2}} & 0 \\ 0 & 1 & 0 & 0 & b & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & {2b} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{T_{{\text{IV - III}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ b & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & b & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],} & {{T_{{\text{IV - IV}}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & b & 0 \\ \end{array} } \right].} \\ \end{array} $$
  1. 3.

    Structure of submatrices and components of the matrix P (43):

$$ \begin{array}{*{20}{c}} {{P_{\text{I}}} = \left[ {\begin{array}{*{20}{c}} {{p_{11}}} & {{p_{12}}} & {{p_{13}}} & 0 & 0 & 0 & 0 \\ {{p_{21}}} & {{p_{22}}} & {{p_{23}}} & 0 & 0 & 0 & 0 \\ {{p_{31}}} & {{p_{32}}} & {{p_{33}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{p_{44}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{p_{55}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{p_{66}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {{p_{77}}} \\ \end{array} } \right],} & {{P_{\text{II}}} = \left[ {\begin{array}{*{20}{c}} {{p_{18}}} & {{p_{19}}} & {{p_{1,10}}} & 0 & 0 & 0 & 0 \\ {{p_{28}}} & {{p_{29}}} & {{p_{2,10}}} & 0 & 0 & 0 & 0 \\ {{p_{38}}} & {{p_{39}}} & {{p_{3,10}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{p_{4,11}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{p_{5,12}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{p_{6,13}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {{p_{7,14}}} \\ \end{array} } \right],} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{P_{\text{III}}} = \left[ {\begin{array}{*{20}{c}} {{p_{1,15}}} & {{p_{1,16}}} & {{p_{1,17}}} & 0 & 0 & 0 & 0 \\ {{p_{2,15}}} & {{p_{2,16}}} & {{p_{2,17}}} & 0 & 0 & 0 & 0 \\ {{p_{3,15}}} & {{p_{3,16}}} & {{p_{3,17}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{p_{4,18}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{p_{5,19}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{p_{6,20}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {{p_{7,21}}} \\ \end{array} } \right],} & {{P_{\text{I}}} = \left[ {\begin{array}{*{20}{c}} {{p_{1,22}}} & {{p_{1,23}}} & {{p_{1,24}}} & 0 & 0 & 0 & 0 \\ {{p_{2,22}}} & {{p_{2,23}}} & {{p_{2,24}}} & 0 & 0 & 0 & 0 \\ {{p_{3,22}}} & {{p_{3,23}}} & {{p_{3,24}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{p_{4,25}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{p_{5,26}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{p_{6,27}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {{p_{7,28}}} \\ \end{array} } \right],} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{p_{11}} = 1 - \frac{{xy}}{{ab}} + \frac{{3{x^2}y}}{{{a^2}b}} + \frac{{3x{y^2}}}{{a{b^2}}} + \frac{{2{x^3}y}}{{{a^3}b}} - \frac{{2x{y^3}}}{{{b^3}a}} - \frac{{3{x^2}}}{{{a^2}}} - \frac{{3{y^2}}}{{{b^2}}} + \frac{{2{x^3}}}{{{a^3}}} + \frac{{2{y^3}}}{{{b^3}}},} \\ {\begin{array}{*{20}{c}} {{p_{12}} = \frac{{2{x^2}y}}{{ab}} - \frac{{{x^3}y}}{{{a^2}b}} - \frac{{xy}}{b} + \frac{{{x^3}}}{{{a^2}}} - \frac{{2{x^2}}}{a} + x,} & {{p_{13}} = \frac{{2x{y^2}}}{{ab}} - \frac{{x{y^3}}}{{a{b^2}}} - \frac{{2{y^2}}}{b} + \frac{{xy}}{a} - \frac{{{y^3}}}{{b2}} + y,} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{18}} = \frac{{xy}}{{ab}} + \frac{{3{x^2}y}}{{{a^2}b}} - \frac{{3x{y^2}}}{{a{b^2}}} + \frac{{2{x^3}y}}{{{a^3}b}} + \frac{{2x{y^3}}}{{{b^3}a}} + \frac{{3{x^2}}}{{{a^2}}} - \frac{{2{x^3}}}{{{a^3}}},} & {{p_{19}} = \frac{{{x^2}y}}{{ab}} - \frac{{{x^3}y}}{{{a^2}b}} + \frac{{{x^3}}}{{{a^2}}} - \frac{{{x^2}}}{a}{p_{1,10}} = \frac{{2x{y^2}}}{{ab}} + \frac{{x{y^3}}}{{a{b^2}}} + \frac{{xy}}{a},} \\ \end{array} } \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{p_{1,15}} = - \frac{{xy}}{{ab}} + \frac{{3{x^2}y}}{{{a^2}b}} + \frac{{3x{y^2}}}{{a{b^2}}} - \frac{{2{x^3}y}}{{{a^3}b}} - \frac{{2x{y^3}}}{{{b^3}a}},} & {{p_{1,16}} = \frac{{ - {x^2}y}}{{ab}} + \frac{{{x^3}y}}{{{a^2}b}},\quad {p_{1,17}} = - \frac{{x{y^2}}}{{ab}} + \frac{{x{y^3}}}{{a{b^2}}},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{1,22}} = \frac{{xy}}{{ab}} - \frac{{3{x^2}y}}{{{a^2}b}} - \frac{{3x{y^2}}}{{a{b^2}}} + \frac{{2{x^3}y}}{{{a^3}b}} + \frac{{2x{y^3}}}{{{b^3}a}} + \frac{{3{y^2}}}{{{b^2}}} - \frac{{2{y^3}}}{{{b^3}}},} & {{p_{1,23}} = - \frac{{2{x^2}y}}{{ab}} + \frac{{{x^3}y}}{{{a^2}b}} + \frac{{xy}}{b}\;{p_{1,24}} = \frac{{x{y^2}}}{{ab}} - \frac{{x{y^3}}}{{a{b^2}}} - \frac{{{y^2}}}{b} + \frac{{{y^3}}}{{{b^2}}},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{21}} = - \frac{{6{x^2}y}}{{{a^3}b}} + \frac{{6xy}}{{{a^2}b}} - \frac{y}{{ab}} + \frac{{3{y^2}}}{{a{b^2}}} - \frac{{2{y^3}}}{{{b^3}a}} - \frac{{6x}}{{{a^2}}} + \frac{{6{x^2}}}{{{a^3}}},} & {{p_{22}} = 1 + \frac{{4xy}}{{{a^2}b}} - \frac{{3{x^2}y}}{{{a^2}b}} + \frac{{3{x^2}}}{{{a^2}}} - \frac{{4x}}{a} - \frac{y}{b},} & {{p_{23}} = } \\ \end{array} \frac{{2{y^2}}}{{ab}} - \frac{{{y^3}}}{{a{b^2}}} - \frac{y}{a},} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{p_{28}} = \frac{{6{x^2}y}}{{{a^3}b}} - \frac{{6xy}}{{{a^2}b}} + \frac{y}{{ab}} - \frac{{3{y^2}}}{{a{b^2}}} + \frac{{2{y^3}}}{{a{b^2}}} + \frac{{6x}}{{{a^2}}} - \frac{{6{x^2}}}{{{a^3}}},} & {{p_{29}} = \frac{{2xy}}{{ab}} - \frac{{3{x^2}y}}{{{a^2}b}} + \frac{{3{x^2}}}{{{a^2}}} - \frac{{2x}}{a},} & {{p_{2,10}} = - \frac{{2{y^2}}}{{ab}} + \frac{{{y^3}}}{{a{b^2}}} + \frac{y}{a},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{2,15}} = - \frac{{6{x^2}y}}{{{a^3}b}} + \frac{{6xy}}{{{a^2}b}} - \frac{y}{{ab}} + \frac{{3{y^2}}}{{a{b^2}}} - \frac{{2{y^3}}}{{{b^3}a}},} & {{p_{2,16}} = - \frac{{2xy}}{{ab}} + \frac{{3{x^2}y}}{{{a^2}b}},} & {{p_{2,17}} = - \frac{{{y^2}}}{{ab}} + \frac{{{y^3}}}{{a{b^2}}}} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{2,22}} = - \frac{{6{x^2}y}}{{{a^3}b}} - \frac{{6xy}}{{{a^2}b}} + \frac{y}{{ab}} - \frac{{3{y^2}}}{{a{b^2}}} + \frac{{2{y^3}}}{{{b^3}a}},} & {{p_{2,23}} = - \frac{{4xy}}{{ab}} + \frac{{3{x^2}y}}{{{a^2}b}} + \frac{y}{b}\quad {p_{2,24}} = \frac{{{y^2}}}{{ab}} - \frac{{{y^3}}}{{a{b^2},}}} \\ \end{array} } \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{p_{31}} = - \frac{{6x{y^2}}}{{{b^3}a}} - \frac{x}{{ab}} + \frac{{3{x^2}}}{{{a^2}b}} - \frac{{2{x^3}}}{{{a^3}b}} - \frac{{6y}}{{b2}} + \frac{{6{y^2}}}{{{b^3}}} + \frac{{6xy}}{{a{b^2}}},} & {{p_{32}} = \frac{{2{x^2}}}{{ab}} - \frac{{{x^3}}}{{{a^2}b}} - \frac{x}{b},} & {{p_{33}} = 1 + \frac{{4xy}}{{ab}} - \frac{{3x{y^2}}}{{a{b^2}}} + \frac{{3{y^2}}}{{{b^2}}} - \frac{x}{a} - \frac{{4y}}{b},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{38}} = \frac{{6x{y^2}}}{{{b^3}a}} + \frac{x}{{ab}} - \frac{{3{x^2}}}{{{a^2}b}} + \frac{{2{x^3}}}{{{a^3}b}} - \frac{{6xy}}{{a{b^2}}},} & {{p_{39}} = \frac{{{x^2}}}{{ab}} - \frac{{{x^3}}}{{{a^2}b}},} & {{p_{3,10}} = - \frac{{4xy}}{{ab}} + \frac{{3x{y^2}}}{{a{b^2}}} + \frac{x}{a}} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{3,15}} = - \frac{{6x{y^2}}}{{{b^3}a}} - \frac{x}{{ab}} + \frac{{3{x^2}}}{{{a^2}b}} - \frac{{2{x^3}}}{{{a^3}b}} + \frac{{6xy}}{{a{b^2}}},} & {{p_{3,16}} = - \frac{{{x^2}}}{{ab}} + \frac{{{x^3}}}{{{a^2}b}},} & {{p_{3,17}} = - \frac{{2xy}}{{ab}} + \frac{{3x{y^2}}}{{a{b^2}}},} \\ \end{array} } \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{p_{3,22}} = \frac{{6x{y^2}}}{{{b^3}a}} + \frac{x}{{ab}} - \frac{{3{x^2}}}{{{a^2}b}} + \frac{{6y}}{{{b^2}}} - \frac{{6{y^2}}}{{{b^3}}} - \frac{{6xy}}{{a{b^2}}},} & {{p_{3,23}} = - \frac{{2{x^2}}}{{ab}}} \\ \end{array} + \frac{{{x^3}}}{{{a^2}b}} + \frac{x}{b}\;{p_{3,24}} = \frac{{2xy}}{{ab}} - \frac{{3x{y^2}}}{{a{b^2}}} + \frac{{3{y^2}}}{{{b^2}}} - \frac{{2y}}{b},} \\ {\begin{array}{*{20}{c}} {{p_{44}} = 1 + \frac{{xy}}{{ab}} - \frac{x}{a} - \frac{y}{b},} & {{p_{4,11}} = - \frac{{xy}}{{ab}} + \frac{x}{a},} & {{p_{4,18}} = \frac{{xy}}{{ab}},} & {{p_{4,25}} = - \frac{{xy}}{{ab}} + \frac{y}{b},} & {{p_{55}} = 1 + \frac{{xy}}{{ab}} - \frac{x}{a} - \frac{y}{b},} & {} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{5,12}} = - \frac{{xy}}{{ab}} + \frac{x}{a},} & {{p_{5,19}} = \frac{{xy}}{{ab}},} & {{p_{5,26}} = - \frac{{xy}}{{ab}} + \frac{y}{b},} & {{p_{66}} = 1 + \frac{{xy}}{{ab}} - \frac{x}{a} - \frac{y}{b},} & {{p_{6,13}} = - \frac{{xy}}{{ab}} + \frac{x}{a},} & {{p_{6,20}} = \frac{{xy}}{{ab}},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{p_{6,27}} = - \frac{{xy}}{{ab}} + \frac{y}{b},} & {{p_{77}} = 1 + \frac{{xy}}{{ab}} - \frac{x}{a} - \frac{y}{b},} & {{p_{7,14}} = - \frac{{xy}}{{ab}} + \frac{x}{a},} & {{p_{7,21}} = \frac{{xy}}{{ab}},} & {{p_{7,28}} = - \frac{{xy}}{{ab}} + \frac{y}{b}} \\ \end{array} } \\ \end{array} $$
  1. 4.

    Expressions for components of the main diagonal of the stiffness matrix K e of the finite element of the plate:

$$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{k_{11}} = \frac{2}{5} \cdot \frac{{14{D_{33}}{a^2}{b^2} + 5{D_{12}}{b^2}{a^2} + 10{D_{11}}{b^4} + 10{D_{22}}{a^4}}}{{{a^3}{b^3}}},} & {{k_{2,2}} = \frac{4}{{15}} \cdot \frac{{5{D_{11}}{b^2} + 2{D_{33}}{a^2}}}{{ab}},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{k_{3,3}} = \frac{4}{{15}} \cdot \frac{{2{D_{33}}{b^2} + 5{D_{22}}{a^2}}}{{ba}},} & {{k_{4,4}} = \frac{1}{9} \cdot \frac{{3{D_{11}}{b^2} + {K_x}{a^2}{b^2} + 3{D_{33}}{a^2}}}{{ab}},} & {{k_{5,5}} = \frac{1}{9} \cdot \frac{{3{D_{22}}{a^2} + {K_y}{a^2}{b^2} + 3{D_{33}}{b^2}}}{{ab}}} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{k_{6,6}} = \frac{1}{3} \cdot \frac{{{B_{33}}{a^2} + {B_{11}}{b^2}}}{{ab}},} & {{k_{7,7}} = \frac{1}{3} \cdot \frac{{{B_{33}}{b^2} + {B_{22}}{a^2}}}{{ab}},} & {{k_{8,8}} = } \\ \end{array} \frac{2}{5} \cdot \frac{{14{D_{33}}{a^2}{b^2} + 5{D_{12}}{b^2}{a^2} + 10{D_{11}}{b^4} + 10{D_{22}}{a^4}}}{{{a^3}{b^3}}},} \\ {\begin{array}{*{20}{c}} {{k_{9,9}} = \frac{4}{{15}} \cdot \frac{{5{D_{11}}{b^2} + 2{D_{33}}{a^2}}}{{ab}},} & {{k_{10,10}} = \frac{4}{{15}} \cdot \frac{{2{D_{33}}{b^2} + 5{D_{22}}{a^2}}}{{ba}},} & {{k_{11,11}} = \frac{1}{9} \cdot \frac{{3{D_{11}}{b^2} + {K_x}{a^2}{b^2} + 3{D_{33}}{a^2}}}{{ab}},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{k_{12,12}} = \frac{1}{9} \cdot \frac{{3{D_{22}}{a^2} + {K_y}{a^2}{b^2} + 3{D_{33}}{a^2}}}{{ab}},} & {{k_{13,13}} = \frac{1}{3} \cdot \frac{{{B_{33}}{a^2} + {B_{11}}{b^2}}}{{ab}},} & {{k_{14,14}} = \frac{1}{3} \cdot \frac{{{B_{33}}{b^2} + {B_{22}}{a^2}}}{{ab}}} \\ \end{array}, } \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{k_{15,15}} = \frac{2}{5} \cdot \frac{{14{D_{33}}{a^2}{b^2} + 5{D_{12}}{b^2}{a^2} + 10{D_{11}}{b^4} + 10{D_{22}}{a^4}}}{{{a^3}{b^3}}},} & {{k_{16,16}} = \frac{4}{{15}} \cdot \frac{{5{D_{11}}{b^2} + 2{D_{33}}{a^2}}}{{ab}},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{k_{17,17}} = \frac{4}{{15}} \cdot \frac{{2{D_{33}}{b^2} + 5{D_{22}}{a^2}}}{{ba}},} & {{k_{18,18}} = \frac{1}{9} \cdot \frac{{3{D_{11}}{b^2} + {K_x}{a^2}{b^2} + 3{D_{33}}{a^2}}}{{ab}},} & {{k_{19,19}} = \frac{1}{9} \cdot \frac{{3{D_{22}}{a^2} + {K_y}{a^2}{b^2} + 3{D_{33}}{b^2}}}{{ab}}} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{k_{20,20}} = \frac{1}{3} \cdot \frac{{{B_{33}}{a^2} + {B_{11}}{b^2}}}{{ab}},} & {{k_{21,21}} = \frac{1}{3} \cdot \frac{{{B_{33}}{b^2} + {B_{22}}{a^2}}}{{ab}},} & {{k_{22,22}} = \frac{2}{5} \cdot \frac{{14{D_{33}}{a^2}{b^2} + 5{D_{12}}{b^2}{a^2} + 10{D_{11}}{b^4} + 10{D_{22}}{a^4}}}{{{a^3}{b^3}}}} \\ \end{array}, } \\ {\begin{array}{*{20}{c}} {{k_{23,23}} = \frac{4}{{15}} \cdot \frac{{5{D_{11}}{b^2} + 2{D_{33}}{a^2}}}{{ab}},} & {k24,24 = \frac{4}{{15}} \cdot \frac{{2{D_{33}}{b^2} + 5{D_{22}}{a^2}}}{{ba}},} & {{k_{25,25}} = \frac{1}{9} \cdot \frac{{3{D_{11}}{b^2} + {K_x}{a^2}{b^2} + 3{D_{33}}{a^2}}}{{ab}},} \\ \end{array} } \\ {\begin{array}{*{20}{c}} {{k_{26,26}} = \frac{1}{9} \cdot \frac{{3{D_{22}}{a^2} + {K_y}{a^2}{b^2} + 3{D_{33}}{b^2}}}{{ab}},} & {{k_{27,27}} = \frac{1}{3} \cdot \frac{{{B_{33}}{b^2} + {B_{11}}{a^2}}}{{ab}},} & {{k_{28,28}} = \frac{1}{3} \cdot \frac{{{B_{33}}{b^2} + {B_{22}}{a^2}}}{{ab}}} \\ \end{array} } \\ \end{array} $$

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nesterov, V. Stiffness matrix of the finite element of a plate compliant in transverse shear. Mech Compos Mater 47, 271–284 (2011). https://doi.org/10.1007/s11029-011-9207-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11029-011-9207-9

Keywords

Search

Navigation