Skip to main content
A first course on solid mechanics
Antonio Bilotta
Contents
Search Book
close
Search Results:
No results.
Dark Mode
Prev
Up
Next
\( \newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \renewcommand{\vec}[1]{\boldsymbol{#1}} \newcommand{\tens}[1]{\boldsymbol{#1}} \newcommand{\mat}[1]{\left[\boldsymbol{#1}\right]} \newcommand{\matWp}[2]{\left[\boldsymbol{#1}_{#2}\right]} \newcommand{\tensQ}[1]{\pmb{\mathbb{#1}}} \newcommand{\tensQc}[1]{\mathbb{#1}} \newcommand{\transp}[1]{{#1}^{\textrm{T}}} \newcommand{\veczero}{\vec{0}} \newcommand{\body}{\Omega} \newcommand{\calvec}[1]{\boldsymbol{\mathcal{#1}}} \newcommand{\commadiff}[2]{{#1,}_{#2}} \newcommand{\regulardiff}[2]{ \frac{\partial#1}{\partial#2} } \newcommand{\func}[2]{#1\left(#2\right)} \newcommand{\transp}[1]{{#1}^{\text{T}}} \newcommand{\transpm}[1]{{#1}^{-\text{T}}} \newcommand{\trace}[1]{\text{tr}#1} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Dedication
Acknowledgements
Preface
1
bodies’ kinematics
1.1
configurations
1.2
transformations
1.3
deformation gradient
1.3.1
the notion of tensor
1.4
displacement field
1.5
determinant of the deformation gradient
1.5.1
transformation formula of volume elements
1.5.2
area element transformation
1.6
polar decomposition
1.7
stretch tensor
\(\tens{U}\)
and principal directions (transformation of line elements)
1.7.1
1.7.2
1.7.3
1.7.4
spectral decomposition of the stretch tensor
1.7.5
eigenvalues and eigenvectors
1.8
strain measures
1.8.1
Cauchy-Green strain tensor
1.8.2
Green-Lagrange strain tensor
1.8.3
strain tensors and displacement gradient
1.9
the infinitesimal strain tensor
1.9.1
infinitesimal motions
1.10
exercises
1.10.1
1.10.2
1.10.3
1.10.4
1.10.5
1.10.6
1.10.7
1.10.8
1.10.9
1.10.10
1.10.11
1.11
answers to selected exercises
1.11.1
1.11.2
1.12
references and suggested readings
2
forces, equilibrium and stress
2.1
the continuity hypothesis
2.1.1
geometric description of the bodies
2.1.2
mass
2.1.3
forces
2.1.3.1
bulk force
2.1.3.2
traction
2.2
equilibrium equations
2.3
Cauchy stress tensor tensor
2.3.1
Cauchy’s postulate
2.3.2
Cauchy stress Theorem
2.3.2.1
equilibrium of Cauchy’s tetrahedron
2.3.2.2
principle of action and reaction
2.3.2.3
the tensor
2.3.2.4
differential form of equilibrium equations
2.3.2.5
symmetry of the tensor
2.4
analisys of states of stress
2.4.1
convention and meaning of the components of stress tensor
2.4.2
principal stresses and principal directions
2.4.3
basic states of stress
2.4.3.1
spheric or hydrostatic state
2.4.3.2
state of “pure” traction or compression
2.4.3.3
state of “pure” shear
2.5
principle of virtual work
2.5.1
from the differential formulation to the integral formulation
2.5.2
mechanical interpretation
2.6
exercises
2.6.1
2.6.2
2.6.3
2.6.4
2.6.5
2.6.6
2.6.7
2.6.8
2.6.9
2.7
answers to some of the proposed exercises
2.7.1
2.7.2
2.8
references and suggested readings
3
elasticity
3.1
motivation
3.1.1
static analysis
3.1.1.1
stress field assumption and solution
3.1.2
kinematic analysis
3.1.3
Principle of virtual work
3.1.4
conclusions
3.2
uniaxial states
3.2.1
uniaxial tensile or compressive test
3.2.2
uniaxial shear test
3.3
multi-axial states
3.3.1
the model as a generalization of Hooke’s law
3.3.1.1
properties of the constitutive tensor
3.3.2
experimental observation
3.3.3
the isotropic case
3.3.3.1
normal components
3.3.3.2
tangential type components
3.3.3.3
summary and Voigt notation
3.4
the elastic problem
3.4.1
motivation (reprised): the prismatic solid simply stretched
3.4.2
the prismatic solid simply bent
3.4.2.1
static analysis
3.4.2.2
from stress to displacement solution
3.4.3
summary
3.5
one-dimensional models: stretched beam
3.5.1
premise
3.5.2
the reference given by the beam axis
3.5.3
internal work
3.5.4
elastic constitutive law
3.5.5
external work
3.5.6
virtual work principle and equilibrium equations
3.5.7
summary of the model
3.6
one-dimensional models: bent beam
3.6.1
the reference given by the beam axis
3.6.2
internal work
3.6.3
elastic constitutive law
3.6.4
external work
3.6.5
virtual work principle and equilibrium equations
3.6.6
summary of the model
3.7
applications of the stretched beam model
3.7.1
beam simply stretched
3.7.2
hyperstatic beam with discontinuity
3.7.3
hyperstatic beam subjected to an homogeneous termal increment
3.7.4
hyperstatic beam subjected to a sinusoidal distributed load
3.8
applications of the bent beam model
3.8.1
cantilever beam subjected to a shear force at the free end
3.8.2
cantilever beam subjected to a constant distributed load
3.8.3
supported beam subjected to a constant distributed load
3.9
exercises
3.9.1
3.9.2
3.9.3
3.9.4
3.9.5
3.9.6
3.9.7
3.9.8
3.9.9
3.9.10
3.10
answers to selected exercises
3.10.1
3.11
references and suggested readings
4
kinematics
4.1
basic kinematic description
4.1.1
linearization
4.2
constraints
4.2.1
simple external constraints
4.2.1.1
roller support
4.2.1.2
rotation lock
4.2.2
double external constraints
4.2.2.1
pinned support
4.2.2.2
roller support with rotation lock
4.2.3
triple external constraints
4.2.3.1
fixed support
4.2.4
internal constraints
4.2.4.1
internal hinge
4.3
kinematic analysis
4.3.1
kinematic matrix
4.3.2
kinematic classification
4.4
kinematic analysis of simple rigid body systems
4.4.1
simply supported beam
4.4.2
L shaped beam with 3 roller supports
4.4.3
system with three hinges
4.4.4
mechanism example
4.5
exercises
4.5.1
4.5.2
4.6
references and suggested readings
5
statics
5.1
equilibrium equations
5.1.1
static equilibrium conditions for a point system
5.1.2
scalar form of equilibrium equations
5.1.3
introduction of rigid body kinematics
5.1.4
moment of a force
5.2
2D case
5.3
constraints
5.3.1
absence of constraint
5.3.2
simple constraints
5.3.2.1
roller support
5.3.2.2
rotation lock
5.3.3
double constraints
5.3.3.1
hinge support
5.3.3.2
roller support with rotation lock
5.3.4
triple constraints
5.3.4.1
fixed support
5.3.5
summary
5.3.6
internal constraints
5.4
free body diagrams
5.4.1
5.5
static analysis
5.5.1
static matrix
5.5.2
static classification
5.6
application of static analysis to systems of bodies
5.6.1
cantilever beam
5.6.2
supported beam
5.6.3
simple frame
5.6.4
two bodies system
5.7
generalized section forces
5.7.1
generalized section forces sign convention
5.7.2
distributed loads
5.8
2D beam equilibrium differential equations
5.8.1
formulation of the problem
5.8.2
equilibrium conditions
5.8.3
general integrals
5.8.4
general integrals for constant distributed loads
5.9
generalized section forces calculation and diagrams
5.9.1
5.9.2
5.9.3
5.9.4
5.10
exercises
5.10.1
Colophon
Colophon
Website
My Website
©2020–2026 Antonio Bilotta
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit
CreativeCommons.org
🔗
