Chapter 3
Investigation of Euler–Bernoulli Beams
in the Elastic Range
3.1 The Basics of an Euler–Bernoulli Beam
A thin or Euler–Bernoulli beam is defined as a long prismatic body whose axial
dimension is much larger than its transverse dimensions [2, 6, 8, 12, 13]. 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. 3.1. As a result of this loading, the deformation occurs only perpendicular to its
main axis.
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., constant Young’s modulus E.
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 equation are summarized in Table 3.1.
Alternative formulations of the of the fourth order partial differential equations
for a beam are collected in Table 3.2 where different types of loadings, geometry and
bedding are differentiated. The last case in Table 3.2 refers to the elastic foundation
of a beam which is also know in the literature as Winkler foundation [15]. The elastic
foundation or Winkler foundation modulus k has in the case of beams
1 the unit of
force per unit area.
Under the assumption of constant material (E = const.) and geometric (I Y =
const.) properties, the differential equation in Table 3.1 can be integrated four times
1 In the general case, the unit of the elastic foundation modulus is force per unit area per unit length,
i.e.
N
m 2 /m =
N
m 3 .
© 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_3
37
Investigation of Euler–Bernoulli Beams
in the Elastic Range
3.1 The Basics of an Euler–Bernoulli Beam
A thin or Euler–Bernoulli beam is defined as a long prismatic body whose axial
dimension is much larger than its transverse dimensions [2, 6, 8, 12, 13]. 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. 3.1. As a result of this loading, the deformation occurs only perpendicular to its
main axis.
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., constant Young’s modulus E.
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 equation are summarized in Table 3.1.
Alternative formulations of the of the fourth order partial differential equations
for a beam are collected in Table 3.2 where different types of loadings, geometry and
bedding are differentiated. The last case in Table 3.2 refers to the elastic foundation
of a beam which is also know in the literature as Winkler foundation [15]. The elastic
foundation or Winkler foundation modulus k has in the case of beams
1 the unit of
force per unit area.
Under the assumption of constant material (E = const.) and geometric (I Y =
const.) properties, the differential equation in Table 3.1 can be integrated four times
1 In the general case, the unit of the elastic foundation modulus is force per unit area per unit length,
i.e.
N
m 2 /m =
N
m 3 .
© 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_3
37