An efficient finite element formulation for bending, free vibration and stability analysis of Timoshenko beams

Abstract

In this paper, a new three-node element is proposed for analysis of beams with shear deformation effect. In each node of this element, there exist translation and rotation degrees of freedom. The element’s formulation is based on the first-order shear deformation theory. For this aim, the displacement field of the element is approximated by a fifth-order polynomial. The shear strain is varied as a quadratic function within the element. It is worth noting that the quadratic function can be used for axial displacement field as well. By employing curvature and shear strain relations of Timoshenko beam theory, the exact and explicit shape functions of the displacement fields are obtained. By utilizing these shape functions, the stiffness matrix and the geometric stiffness matrix of the element are calculated. The mass matrices of the proposed element are derived from kinetic energy relation of the beam. Finally, several numerical tests are performed to assess the robustness of the developed element. The results of the numerical tests prove the absence of the shear locking and demonstrate high accuracy and efficiency of the proposed element for bending, free vibration and stability analysis of Timoshenko beams.

Appendix

The nonzero entries of the stiffness matrix $\left[ {K^{e} } \right]$ is:

$$\begin{aligned} k_{11} & = 4\text{EI}\left( {401587200\lambda^{3} + 19440000\lambda^{2} + 292140\lambda + 1273} \right) \\ k_{12} & = 2\text{EI}l\left( { - 1857945600\lambda^{4} + 76723200\lambda^{3} + 6814080\lambda^{2} + 119340\lambda + 569} \right) \\ k_{13} & = - 3584\text{EI}\left( {60\lambda + 1} \right)^{3} \\ k_{14} & = 1920\text{EI}l\left( {48\lambda + 1} \right)^{2} \left( {1680\lambda^{2} + 180\lambda + 1} \right) \\ k_{15} & = - 4\text{EI}\left( {208051200\lambda^{3} + 9763200\lambda^{2} + 130860\lambda + 377} \right) \\ k_{16} & = 2\text{EI}l\left( { - 1857945600\lambda^{4} - 20044800\lambda^{3} + 1975680\lambda^{2} + 38700\lambda + 121} \right) \\ k_{22} & = 4\text{EI}l^{2} \left( {3715891200\lambda^{5} + 6220800\lambda^{4} + 15949440\lambda^{3} + 967980\lambda^{2} + 16605\lambda + 83} \right) \\ k_{23} & = - 896\text{EI}l\left( {60\lambda + 1} \right)^{3} \\ k_{24} & = 320\text{EI}l^{2} \left( {48\lambda + 1} \right)^{2} \left( { - 40320\lambda^{3} - 2640\lambda^{2} + 156\lambda + 1} \right) \\ k_{26} & = 2\text{EI}l^{2} \left( {7431782400\lambda^{5} - 277862400\lambda^{4} - 14065920\lambda^{3} + 116520\lambda^{2} + 5490\lambda + 19} \right) \\ k_{33} & = 7168\text{EI}\left( {60\lambda + 1} \right)^{3} \\ k_{44} & = 1280\text{EI}l^{2} \left( {48\lambda + 1} \right)^{2} \left( {20160\lambda^{3} + 3840\lambda^{2} + 192\lambda + 1} \right). \\ \end{aligned}$$

(45)

The nonzero entries of the translation mass matrix $\left[ {M_{1}^{e} } \right]$ is:

$$\begin{aligned} m_{11} & = l\left( {2189721600\lambda^{4} + 179055360\lambda^{3} + 5828688\lambda^{2} + 88554\lambda + 523} \right) \\ m_{12} & = \frac{3}{2}l^{2} \left( { - 729907200\lambda^{5} + 36495360\lambda^{4} + 2871360\lambda^{3} + 93060\lambda^{2} + 2061\lambda + 19} \right) \\ m_{13} & = 44l\left( {60\lambda + 1} \right)^{2} \left( {12096\lambda^{2} + 540\lambda + 5} \right) \\ m_{14} & = - 40l^{2} \left( {48\lambda + 1} \right)^{2} \left( { - 23760\lambda^{3} - 792\lambda^{2} + 177\lambda + 1} \right) \\ m_{15} & = \frac{1}{2}l\left( {1368576000\lambda^{4} + 141419520\lambda^{3} + 4440816\lambda^{2} + 49668\lambda + 131} \right) \\ m_{16} & = - \frac{1}{4}l^{2} \left( {4379443200\lambda^{5} - 218972160\lambda^{4} - 120960\lambda^{3} + 526680\lambda^{2} + 9546\lambda + 29} \right) \\ m_{17} & = l^{3} \left( {35035545600\lambda^{6} + 291962880\lambda^{5} - 11335680\lambda^{4} + 193536\lambda^{3} + 6774\lambda^{2} + 129\lambda + 2} \right) \\ m_{18} & = 22l^{2} \left( {60\lambda + 1} \right)^{2} \left( {72\lambda + 1} \right) \\ m_{19} & = - 3l^{3} \left( {48\lambda + 1} \right)^{2} \left( {10137600\lambda^{4} + 929280\lambda^{3} - 9440\lambda^{2} + 76\lambda + 1} \right) \\ m_{110} & = \frac{3}{4}l^{3} \left( {46714060800\lambda^{6} + 389283840\lambda^{5} - 15114240\lambda^{4} + 258048\lambda^{3} - 4168\lambda^{2} - 268\lambda - 1} \right) \\ m_{111} & = 1408l\left( {60\lambda + 1} \right)^{2} \left( {3024\lambda^{2} + 108\lambda + 1} \right) \\ m_{112} & = 32l^{3} \left( {48\lambda + 1} \right)^{2} \left( {1900800\lambda^{4} + 411840\lambda^{3} + 24960\lambda^{2} + 288\lambda + 1} \right). \\ \end{aligned}$$

(46)

The nonzero entries of the rotary mass matrix $\left[ {M_{2}^{e} } \right]$ is:

$$\begin{aligned} m_{21} &= \frac{12}{l}\left( {403200\lambda^{2} + 14880\lambda + 139} \right) \hfill \\ m_{22} & = - \left( {55987200\lambda^{3} + 1969920\lambda^{2} + 15444\lambda - 39} \right) \hfill \\ m_{23} &= - \frac{1536}{l}\left( {60\lambda + 1} \right)^{2} \hfill \\ m_{24} &= - 240\left( {48\lambda + 1} \right)^{2} \left( {60\lambda - 1} \right) \hfill \\ m_{25} &= \frac{12}{l}\left( {57600\lambda^{2} + 480\lambda - 11} \right) \hfill \\ m_{26} &= - \left( { - 2073600\lambda^{3} + 207360\lambda^{2} + 5076\lambda + 9} \right) \hfill \\ m_{27} &= 4l\left( {219801600\lambda^{4} + 9711360\lambda^{3} + 153144\lambda^{2} + 1263\lambda + 7} \right) \hfill \\ m_{28} &= 48\left( {60\lambda + 1} \right)^{2} \left( {336\lambda - 1} \right) \hfill \\ m_{29} &= - 8l\left( {48\lambda + 1} \right)^{2} \left( { - 18000\lambda^{2} + 600\lambda + 1} \right) \hfill \\ m_{210} &= l\left( {8294400\lambda^{4} + 2557440\lambda^{3} + 9936\lambda^{2} - 924\lambda - 5} \right) \hfill \\ m_{211} &= \frac{3072}{l}\left( {60\lambda + 1} \right)^{2} \hfill \\ m_{212} &= 256l\left( {48\lambda + 1} \right)^{2} \left( {3600\lambda^{2} + 60\lambda + 1} \right). \hfill \\ \end{aligned}$$

(47)

The nonzero entries of the geometric stiffness matrix $\left[ {K_{g} } \right]$ is:

$$\begin{aligned} k_{g1} & = \frac{12}{l}\left( {2409523200\lambda^{4} + 137721600\lambda^{3} + 2953800\lambda^{2} + 30156\lambda + 139} \right) \\ k_{g2} & = - 3\left( {22295347200\lambda^{5} + 597196800\lambda^{4} + 2695680\lambda^{3} + 80640\lambda^{2} + 1152\lambda - 13} \right) \\ k_{g3} & = - \frac{1536}{l}\left( {60\lambda + 1} \right)^{2} \left( {2520\lambda^{2} + 84\lambda + 1} \right) \\ k_{g4} & = 240\left( {48\lambda + 1} \right)^{2} \left( {241920\lambda^{3} + 26640\lambda^{2} + 228\lambda + 1} \right) \\ k_{g5} & = - \frac{12}{l}\left( {1248307200\lambda^{4} + 60307200\lambda^{3} + 880200\lambda^{2} + 4044\lambda + 11} \right) \\ k_{g6} & = - 9\left( {7431782400\lambda^{5} + 199065600\lambda^{4} - 714240\lambda^{3} - 7680\lambda^{2} + 576\lambda + 1} \right) \\ k_{g7} & = 4l\left( {66886041600\lambda^{6} + 398131200\lambda^{5} - 17418240\lambda^{4} + 483840\lambda^{3} + 15174\lambda^{2} + 471\lambda + 7} \right) \\ k_{g8} & = 48\left( {60\lambda + 1} \right)^{2} \left( {84\lambda - 1} \right) \\ k_{g9} & = - 8l\left( {48\lambda + 1} \right)^{2} \left( {29030400\lambda^{4} + 2592000\lambda^{3} - 30960\lambda^{2} + 420\lambda + 1} \right) \\ k_{g10} & = l\left( {267544166400\lambda^{6} + 1592524800\lambda^{5} - 69672960\lambda^{4} + 1935360\lambda^{3} - 55944\lambda^{2} - 2076\lambda - 5} \right) \\ k_{g11} & = \frac{3072}{l}\left( {60\lambda + 1} \right)^{2} \left( {2520\lambda^{2} + 84\lambda + 1} \right) \\ k_{g12} & = 256l\left( {48\lambda + 1} \right)^{2} \left( {1814400\lambda^{4} + 388800\lambda^{3} + 23040\lambda^{2} + 240\lambda + 1} \right). \\ \end{aligned}$$

(48)