Mechanics of Materials by Beer, Johnston & DeWolf –
Solutions Manual (2025/2026) | Step-by-Step Answers for
Engineering Mechanics Problems
hooke's law
σ=E*ε
normal stress σ=
F/A
normal strain ε=
∆l/l
young's modulus E =
σ/ε
shear strain γ=
tanθ=w/h
shear modulus G =
𝜏/γ= E/2 *(1+ν)
bulk modulus K =
E/3*(1-2ν)
poisson's ratio ν=
-εlat/εaxial
poisson's ratio of 0 means
no lateral contraction
poisson's ratio of 0.1 means
low amounts of lateral contraction (diamond and cork)
poisson's ratio of 0.5 means
,constant volume strain
negative poissons ratio means
material is auxetic: expands transversely when stretched and contracts when
compressed
does temperature change affect elastic properties
not really, unless there is a phase change
the steeper the slope in an F vs x graph for a material, what does that mean
about the bond energy
dF/dx is the elastic modulus so it tells you about the stiffness
the larger the F at an x in an F vs x graph for a material, what does that mean
about the bond energy
that it is a higher energy bond (covalent vs metallic, covalent is higher)
resilience
area under curve up to yield point (end of elastic region) - the capacity of a
material to absorb energy when deformed elastically and release it upon
unloading
yield strength
the stress where a material begins to permanently deform (end of elastic
region)
strength
the maximum stress a material can withstand before failing, breaking, or
undergoing permanent deformation
toughness
a material's ability to absorb energy and deform plastically before fracturing.
the total area under the stress-strain curve
elastic properties depend mainly on (2)
, 1. interatomic bonding
2. crystal structure
isotropic cubic material strain (normal and shear)
εzz = (1/E) [σzz - ν(σyy + σxx)]
γx = γxy = τxy/G -- only the same if cubic
ν = poisson's ratio = -εyy/εzz
E = elastic modulus
stiffness matrix
cij = elastic stiffness
material's resistance to deformation
compliance matrix
sij = elastic compliance
how easily a material deforms under load
what happens to the constants in the compliance matrix for a cubic solid
s11=s22=s33 because of similar normal stress, s12=s21=s13=s31=s23=s32
(similar response from shear stress), s44=s55=s66 (similar response from shear
stress) and all other terms are zero
for a non-isotropic cubic solid, what are the compliance terms
s11, s12, s44
for a non-isotropic cubic solid, what are the stiffness terms
c11 =(s11+s22)/{(s11-s12)(s11+2s12)}
c12 = (-s12)/{(s11-s12)(s11+2s12)}
c44=1/(s44)
normal strain for a non-isotropic cubic solid
εzz= s12σxx + s12σyy + s11σzz
(the s11 term lines with the same item)
shear strain for a non-isotropic cubic solid
Solutions Manual (2025/2026) | Step-by-Step Answers for
Engineering Mechanics Problems
hooke's law
σ=E*ε
normal stress σ=
F/A
normal strain ε=
∆l/l
young's modulus E =
σ/ε
shear strain γ=
tanθ=w/h
shear modulus G =
𝜏/γ= E/2 *(1+ν)
bulk modulus K =
E/3*(1-2ν)
poisson's ratio ν=
-εlat/εaxial
poisson's ratio of 0 means
no lateral contraction
poisson's ratio of 0.1 means
low amounts of lateral contraction (diamond and cork)
poisson's ratio of 0.5 means
,constant volume strain
negative poissons ratio means
material is auxetic: expands transversely when stretched and contracts when
compressed
does temperature change affect elastic properties
not really, unless there is a phase change
the steeper the slope in an F vs x graph for a material, what does that mean
about the bond energy
dF/dx is the elastic modulus so it tells you about the stiffness
the larger the F at an x in an F vs x graph for a material, what does that mean
about the bond energy
that it is a higher energy bond (covalent vs metallic, covalent is higher)
resilience
area under curve up to yield point (end of elastic region) - the capacity of a
material to absorb energy when deformed elastically and release it upon
unloading
yield strength
the stress where a material begins to permanently deform (end of elastic
region)
strength
the maximum stress a material can withstand before failing, breaking, or
undergoing permanent deformation
toughness
a material's ability to absorb energy and deform plastically before fracturing.
the total area under the stress-strain curve
elastic properties depend mainly on (2)
, 1. interatomic bonding
2. crystal structure
isotropic cubic material strain (normal and shear)
εzz = (1/E) [σzz - ν(σyy + σxx)]
γx = γxy = τxy/G -- only the same if cubic
ν = poisson's ratio = -εyy/εzz
E = elastic modulus
stiffness matrix
cij = elastic stiffness
material's resistance to deformation
compliance matrix
sij = elastic compliance
how easily a material deforms under load
what happens to the constants in the compliance matrix for a cubic solid
s11=s22=s33 because of similar normal stress, s12=s21=s13=s31=s23=s32
(similar response from shear stress), s44=s55=s66 (similar response from shear
stress) and all other terms are zero
for a non-isotropic cubic solid, what are the compliance terms
s11, s12, s44
for a non-isotropic cubic solid, what are the stiffness terms
c11 =(s11+s22)/{(s11-s12)(s11+2s12)}
c12 = (-s12)/{(s11-s12)(s11+2s12)}
c44=1/(s44)
normal strain for a non-isotropic cubic solid
εzz= s12σxx + s12σyy + s11σzz
(the s11 term lines with the same item)
shear strain for a non-isotropic cubic solid
