Timoshenko beam theory

The Timoshenko beam theory was developed by Stephen Timoshenko early in the 20th century. The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams, or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter (in principle comparable to the height of the beam or shorter), and thus the distance between opposing shear forces decreases.Then, from the strain-displacement relations for small strains, the non-zero strains based on the Timoshenko assumptions areFrom equation (1), assuming appropriate smoothness, we have The Timoshenko beam theory was developed by Stephen Timoshenko early in the 20th century. The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams, or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter (in principle comparable to the height of the beam or shorter), and thus the distance between opposing shear forces decreases. If the shear modulus of the beam material approaches infinity—and thus the beam becomes rigid in shear—and if rotational inertia effects are neglected, Timoshenko beam theory converges towards ordinary beam theory. In static Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by where ( x , y , z ) {displaystyle (x,y,z)} are the coordinates of a point in the beam, u x , u y , u z {displaystyle u_{x},u_{y},u_{z}} are the components of the displacement vector in the three coordinate directions, φ {displaystyle varphi } is the angle of rotation of the normal to the mid-surface of the beam, and w {displaystyle w} is the displacement of the mid-surface in the z {displaystyle z} -direction. The governing equations are the following coupled system of ordinary differential equations: The Timoshenko beam theory for the static case is equivalent to the Euler-Bernoulli theory when the last term above is neglected, an approximation that is valid when