Chapter 2
Investigation of Rods in the Elastic Range
2.1 The Basics of a Rod
A rod is defined as a prismatic body whose axial dimension is much larger than its
transverse dimensions [1, 3–5, 7, 9, 10]. This structural member is only loaded in
the direction of the main body axes (X ), see Fig. 2.1. As a result of this loading, the
deformation occurs only along its main axis.
Derivations are restricted many times to the following simplifications:
• only applying to straight rods,
• displacements are (infinitesimally) small,
• strains are (infinitesimally) small, 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 (PDE) are summarized in Table 2.1.
Alternative formulations of the partial differential equation for a rod are collected
in Table 2.2. It should be noted here that some of the different cases given in Table 2.2
can be combined. The last case in Table 2.2 refers to the case of elastic embedding
of a rod where the embedding modulus k has the unit of force per unit area.
Under the assumption of constant material (E = const.) and geometric (A =
const.) properties, the first differential equation in Table 2.1 can be easily integrated
twice for constant distributed load ( p X = p 0 = const.) to obtain the general solution
of the problem [6]:
u X (X ) =
1
E A
−
1
2
p 0 X
2
+ c 1 X + c 2
,
(2.1)
where the two constants of integration c i (i = 1, 2) must be determined based on the
boundary conditions (see Table 2.3). The following equation for the internal normal
force N X was obtained based on one-time integration of the PDE and might be useful
to determine some of the constants of integration:
© 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_2
11
Investigation of Rods in the Elastic Range
2.1 The Basics of a Rod
A rod is defined as a prismatic body whose axial dimension is much larger than its
transverse dimensions [1, 3–5, 7, 9, 10]. This structural member is only loaded in
the direction of the main body axes (X ), see Fig. 2.1. As a result of this loading, the
deformation occurs only along its main axis.
Derivations are restricted many times to the following simplifications:
• only applying to straight rods,
• displacements are (infinitesimally) small,
• strains are (infinitesimally) small, 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 (PDE) are summarized in Table 2.1.
Alternative formulations of the partial differential equation for a rod are collected
in Table 2.2. It should be noted here that some of the different cases given in Table 2.2
can be combined. The last case in Table 2.2 refers to the case of elastic embedding
of a rod where the embedding modulus k has the unit of force per unit area.
Under the assumption of constant material (E = const.) and geometric (A =
const.) properties, the first differential equation in Table 2.1 can be easily integrated
twice for constant distributed load ( p X = p 0 = const.) to obtain the general solution
of the problem [6]:
u X (X ) =
1
E A
−
1
2
p 0 X
2
+ c 1 X + c 2
,
(2.1)
where the two constants of integration c i (i = 1, 2) must be determined based on the
boundary conditions (see Table 2.3). The following equation for the internal normal
force N X was obtained based on one-time integration of the PDE and might be useful
to determine some of the constants of integration:
© 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_2
11