Chapter 4
Investigation of Timoshenko Beams in
the Elastic Range
4.1 The Basics of a Timoshenko Beam
A thick or Timoshenko beam is defined as a long prismatic body whose axial dimension is much larger than its transverse dimensions [13, 16]. This structural member
is only loaded perpendicular to its longitudinal body axis by forces (single forces F Z
or distributed loads q Z ) or moments (single moments M Y or distributed moments
m Y ). Perpendicular means that the line of application of a force or the direction of a
moment vector forms a right angle with the X -axis, see Fig. 4.1. As a result of this
loading, the deformation occurs only perpendicular to its main axis. The formulation
is a shear-flexible theory which means that the shear forces contribute to the bending
deformation.
Derivations are restricted many times to the following simplifications:
• only applying to straight beams,
• no elongation along the X -axis,
• no torsion around the X -axis,
• deformations in a single plane (here: X -Z ), i.e. symmetrical bending,
• infinitesimally small deformations and strains,
• simple cross sections, and
• the material is linear-elastic, i.e., Youngs’s modulus E and shear modulus G.
The three basic equations of continuum mechanics, i.e the kinematics relationship,
the constitutive law and the equilibrium equation, as well as their combination to the
describing partial differential equations are summarized in Table 4.1. It should be
noted here that the deflection u Z and the rotation φ Y are now independent variables
and both represented in the coupled differential equations.
Different formulations of the coupled differential equations are collected in
Table 4.2 where different types of loadings, geometry and bedding are differentiated. The last case in Table 4.2 refers again to the elastic or Winkler foundation of
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2_4
89
Investigation of Timoshenko Beams in
the Elastic Range
4.1 The Basics of a Timoshenko Beam
A thick or Timoshenko beam is defined as a long prismatic body whose axial dimension is much larger than its transverse dimensions [13, 16]. This structural member
is only loaded perpendicular to its longitudinal body axis by forces (single forces F Z
or distributed loads q Z ) or moments (single moments M Y or distributed moments
m Y ). Perpendicular means that the line of application of a force or the direction of a
moment vector forms a right angle with the X -axis, see Fig. 4.1. As a result of this
loading, the deformation occurs only perpendicular to its main axis. The formulation
is a shear-flexible theory which means that the shear forces contribute to the bending
deformation.
Derivations are restricted many times to the following simplifications:
• only applying to straight beams,
• no elongation along the X -axis,
• no torsion around the X -axis,
• deformations in a single plane (here: X -Z ), i.e. symmetrical bending,
• infinitesimally small deformations and strains,
• simple cross sections, and
• the material is linear-elastic, i.e., Youngs’s modulus E and shear modulus G.
The three basic equations of continuum mechanics, i.e the kinematics relationship,
the constitutive law and the equilibrium equation, as well as their combination to the
describing partial differential equations are summarized in Table 4.1. It should be
noted here that the deflection u Z and the rotation φ Y are now independent variables
and both represented in the coupled differential equations.
Different formulations of the coupled differential equations are collected in
Table 4.2 where different types of loadings, geometry and bedding are differentiated. The last case in Table 4.2 refers again to the elastic or Winkler foundation of
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2_4
89