29 casti presentation
TRANSCRIPT
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Castiglianos nd Theorem Impact
Lecture : Energy Methods (IV) Castiglianos
theorem and impact loading
Yubao Zhen
Dec ,
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact
Review: Method of diagram multiplication
method ofdiagram multiplication
graphical way of Mohr integration
EI
L
mMdx=
SmC
EIgraphical operations:
area S fromMheight mC from (linear) m at xCofM
lengths and areas of basic graphs
applications
bending, bending with elastic foundation (spring)tricks
component diagrams, signs, organization
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact
Review: Method of diagram multiplication
method ofdiagram multiplication
graphical way of Mohr integration
EI
L
mMdx=
SmC
EIgraphical operations:
area S fromMheight mC from (linear) m at xCofM
lengths and areas of basic graphs
applications
bending, bending with elastic foundation (spring)tricks
component diagrams, signs, organization
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact
Review: Method of diagram multiplication
method ofdiagram multiplication
graphical way of Mohr integration
EI
L
mMdx=
SmC
EIgraphical operations:
area S fromMheight mC from (linear) m at xCofM
lengths and areas of basic graphs
applications
bending, bending with elastic foundation (spring)tricks
component diagrams, signs, organization
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact
Review: Method of diagram multiplication
method ofdiagram multiplication
graphical way of Mohr integration
EI
L
mMdx=
SmC
EIgraphical operations:
area S fromMheight mC from (linear) m at xCofM
lengths and areas of basic graphs
applications
bending, bending with elastic foundation (spring)tricks
component diagrams, signs, organization
Energy Methods (IV) Castiglianos Theorem
l d h
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Castiglianos nd Theorem Impact
Outline
Castiglianos nd theorem
form, derivation, original and modified versions impact problem
assumptions, physics, formula, applications
Energy Methods (IV) Castiglianos Theorem
C ti li d Th I t
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Castiglianos nd Theorem Impact
Outline
Castiglianos nd theorem
form, derivation, original and modified versions impact problem
assumptions, physics, formula, applications
Energy Methods (IV) Castiglianos Theorem
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. Castiglianos nd Theorem
( [] )
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Castigliano s nd Theorem Impact Theorem Derivation proof Usages Examples
An observation
Elastic strain energy in a cantilevered beam:
A
P
B A
x
L
Ue = L
M
EIdx=
L
(Px)EI
dx=P
EI
L
=
PL
EI
check this out:dUe
dP
=PL
EI
= A (Appendix G, p. , entry )
Observation:
the derivative of the strain energy with respect to the load is equal to
the deflection corresponding to the load.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Castigliano s nd Theorem Impact Theorem Derivation proof Usages Examples
Castiglianos nd Theorem
Carlos Alberto Pio Castigliano ( ), an Italian engineer.
Castiglianos (nd) Theorem
The partial derivative of the strain energy of a structure with respect to any
load is equal to the displacement corresponding to that load.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Castigliano s nd Theorem Impact Theorem Derivation proof Usages Examples
Castiglianos nd Theorem
Carlos Alberto Pio Castigliano ( ), an Italian engineer.
Castiglianos (nd) Theorem
The partial derivative of the strain energy of a structure with respect to any
load is equal to the displacement corresponding to that load.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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g p p g p
Castiglianos nd Theorem
Carlos Alberto Pio Castigliano ( ), an Italian engineer.
Castiglianos (nd) Theorem
The partial derivative of the strain energy of a structure with respect to any
load is equal to the displacement corresponding to that load.
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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The original form
The original form in his dissertation:
...... the partial derivative of the strain energy, considered as a function of the
applied forces acting on a linearly elastic structure, with respect to one of these
forces, is equal to the displacement in the direction of the force of its point of
application.
about the st theorem:
complementary to the nd theorem, it gives the loads on a structure in
terms of the partial derivatives of the strain energy with respect to the
displacements.
The st theorem is less commonly used than the nd theorem.
(thus neglected)
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. Proof of the theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Proof of the Castiglianos nd Theorem
statement of the problem:
n forces:P, P, ..., Pnare applied on a body, determine thedisplacement at a point of loading, say, for example, at Pj.
Pj
j
P1
P2
Pn
Pn1
An implied fact:
Ueis a state function ()
In thermodynamics, a state function,
or state quantity, is a property of
system that depends only on the
current state of the system, NOT on
the way the system reaches the state.
Ue =f
(P, P, ..., Pn
)Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Proof of the Castiglianos nd Theorem
statement of the problem:
n forces:P, P, ..., Pnare applied on a body, determine thedisplacement at a point of loading, say, for example, at Pj.
Pj
j
P1
P2
Pn
Pn1
An implied fact:
Ueis a state function ()
In thermodynamics, a state function,
or state quantity, is a property of
system that depends only on the
current state of the system, NOT on
the way the system reaches the state.
Ue =f
(P, P, ..., Pn
)Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Proof of the Castiglianos nd Theorem
statement of the problem:
n forces:P, P, ..., Pnare applied on a body, determine thedisplacement at a point of loading, say, for example, at Pj.
Pj
j
P1
P2
Pn
Pn1
An implied fact:
Ueis a state function ()
In thermodynamics, a state function,
or state quantity, is a property of
system that depends only on the
current state of the system, NOT on
the way the system reaches the state.
Ue =f
(P, P, ..., Pn
)Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Proof: Castiglianos nd Theorem (cont.)
Apply a differential increment dPjto Pjwith P, ..., Pnalready loaded
PjdPj
equilibrium configurationwith
equilibrium configurationwith and
P1Pn
P2
Pn1
P1, P2, , Pn dPj
P1, P2, , Pn
initial
Ue =Ue + dUj =U
e +
Ue
Pj
dPj
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Proof: Castiglianos nd Theorem (cont.)
Apply a differential increment dPjto Pjalone first, then the P , ..., Pn
Pj
dPj
dj
j
P1
P2
Pn
Pn1
initial
final
intermediate
Ue =Ue +
dPjdj + dPjj
(higher order termdPjd
j neglected)Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Two approximations:
Ue =Ue + Ue
PjdPj
Ue =Ue + dPjj
state function requires a match of the two forms
dPjj =Ue
PjdPj
j =Ue
Pj
Castiglianos nd Theorem
Energy Methods (IV) Castiglianos Theorem
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. Applications
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
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Application to beams general remarks
the generalized displacement at where Pjis applied:
=U
eP (note: is along P)
At the location where the displacement is requested:
Ueis quadratic about loads displacements are linear about
loads A generalized force Pis required at the point of interest
if physical loading is present, then change it to a variable Pif not, apply an fictitious force P
Calculate the internal energy as a function of the load P Apply =
Ue
P , then set
P=Pj(for physical loading)P= (for fictitious loading)
Energy Methods (IV) Castiglianos Theorem
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Application to beams general remarks
the generalized displacement at where Pjis applied:
=Ue
P (note: is along P)
At the location where the displacement is requested:
Ueis quadratic about loads displacements are linear about
loads A generalized force Pis required at the point of interest
if physical loading is present, then change it to a variable Pif not, apply an fictitious force P
Calculate the internal energy as a function of the load P Apply =
Ue
P , then set
P=Pj(for physical loading)P= (for fictitious loading)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
A li i b l k
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Application to beams general remarks
the generalized displacement at where Pjis applied:
=Ue
P (note: is along P)
At the location where the displacement is requested:
Ueis quadratic about loads displacements are linear about
loads
A generalized force Pis required at the point of interest
if physical loading is present, then change it to a variable Pif not, apply an fictitious force P
Calculate the internal energy as a function of the load P Apply =
Ue
P , then set
P=Pj(for physical loading)P= (for fictitious loading)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
A li i b l k
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Application to beams general remarks
the generalized displacement at where Pjis applied:
=Ue
P (note: is along P)
At the location where the displacement is requested:
Ueis quadratic about loads displacements are linear about
loads
A generalized force Pis required at the point of interest
if physical loading is present, then change it to a variable Pif not, apply an fictitious force P
Calculate the internal energy as a function of the load P Apply =
Ue
P , then set
P=Pj(for physical loading)P= (for fictitious loading)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
A li ti t b l k
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Application to beams general remarks
the generalized displacement at where Pjis applied:
=Ue
P (note: is along P)
At the location where the displacement is requested:
Ueis quadratic about loads displacements are linear about
loads
A generalized force Pis required at the point of interest
if physical loading is present, then change it to a variable Pif not, apply an fictitious force P
Calculate the internal energy as a function of the load P Apply =
Ue
P , then set
P=Pj(for physical loading)P= (for fictitious loading)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
A li ti t b l k
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Application to beams general remarks
the generalized displacement at where Pjis applied:
=Ue
P (note: is along P)
At the location where the displacement is requested:
Ueis quadratic about loads displacements are linear about
loads
A generalized force Pis required at the point of interest
if physical loading is present, then change it to a variable Pif not, apply an fictitious force P
Calculate the internal energy as a function of the load P Apply =
Ue
P , then set
P=Pj(for physical loading)P= (for fictitious loading)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Application to beams general remarks
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Application to beams general remarks
the generalized displacement at where Pjis applied:
=Ue
P (note: is along P)
At the location where the displacement is requested:
Ueis quadratic about loads displacements are linear about
loads
A generalized force Pis required at the point of interest
if physical loading is present, then change it to a variable Pif not, apply an fictitious force P
Calculate the internal energy as a function of the load P Apply =
Ue
P , then set
P=Pj(for physical loading)P= (for fictitious loading)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Application to beams general remarks
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Application to beams general remarks
the generalized displacement at where Pjis applied:
=
Ue
P (note: is along P)
At the location where the displacement is requested:
Ueis quadratic about loads displacements are linear about
loads
A generalized force Pis required at the point of interest
if physical loading is present, then change it to a variable Pif not, apply an fictitious force P
Calculate the internal energy as a function of the load P Apply =
Ue
P , then set
P=Pj(for physical loading)P= (for fictitious loading)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Application to beams the modified version
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Application to beams the modified version
Practically, two operations will GREATLY simply the process:
taking differentiation before integrationUe =
L
N
EA +
M
EI +
fsV
GA +
T
GIpdx
Ue
P =
P
L
N
EA +
M
EI +
fsV
GA +
T
GIpdx
Ue
P =
L
NN
P
EA +
M
M
P
EI +
fsV
V
P
GA +
T
T
P
GIp dxnote: functions N(P, x),M(P, x), V(P, x) and T(P, x)
value substitution ofPbefore integration
P=
Pj(for physical) and P=
(for fictitious) loading
L
NN
P
EA +M
M
P
EI +fs V
V
P
GA +T
T
P
GIp
P=Pj ,or
dx
idea: differentiation/substitution before integration
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Application to beams the modified version
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Application to beams the modified version
Practically, two operations will GREATLY simply the process:
taking differentiation before integrationUe =
L
N
EA +
M
EI +
fsV
GA +
T
GIpdx
Ue
P =
P
L
N
EA +
M
EI +
fsV
GA +
T
GIpdx
Ue
P =
L
NN
P
EA +
M
M
P
EI +
fsV
V
P
GA +
T
T
P
GIp dxnote: functions N(P, x),M(P, x), V(P, x) and T(P, x)
value substitution ofPbefore integration
P=
Pj(for physical) and P=
(for fictitious) loading
L
NN
P
EA +M
M
P
EI +fs V
V
P
GA +T
T
P
GIp
P=Pj ,or
dx
idea: differentiation/substitution before integration
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. Examples
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : a cantilevered beam
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Example : a cantilevered beam
Given: A cantilevered beam withPandM;
Determine: A,Aand C.
C
P
BAM0
x
L/2 L/2
C
BA C
Q
PM0
illustration: A, A(ok), C?
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : A, A the modified version
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Example :A, A the modified version
(for bending only): = L
M
EIM
P
dx
modified Castiglianos theorem
C
P
BAM0
x
L/2 L/2
C BA C
Q
P
M0
M= PxM, M
P
= x,M
M=
A =
EI
L
(PxM)(x)dx= PL
EI +
ML
EI
A =
EI
L
(PxM
)(
)dx=
PL
EI +
ML
EI
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : C the modified version
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Example :C the modified version
(for bending only): = L
M
EIM
Pdx
C
P
BAM0
x
L/2 L/2
C BA C
Q
P
M0
M=
PxM , xL
PxM Q(x L
), L
xL
M
Q =
,
(x L
),
C= L
MEI M
Qdx=
EI L
L(PxM Q(x L))[(x L)]dx
set Q =,C=
EI
L
L
(PxM
)[
(x L
)]dx=
PL
EI +
ML
EI
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : A and A the original version
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Example :AandA the original version
A
P
BAM0
A xL
method of superpositionAppendix G, p. ,
entry : concentrated P
v= PL
EI,=
PL
EI
entry : concentratedM
v= ML
EI ,=
ML
EI
Solution:
M= PxM
Ue = L
M
EIdx=
EI
L
(PxM)dx
Ue =PL
EI
+
PML
EI
+
ML
EI
A =Ue
P =
PL
EI +
ML
EI
A =Ue
M=
PL
EI +
ML
EI
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example :AandA the original version
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p A A g
A
P
BAM0
A xL
method of superpositionAppendix G, p. ,
entry : concentrated P
v= PL
EI,=
PL
EI
entry : concentratedM
v= ML
EI ,=
ML
EI
Solution:
M= PxM
Ue = L
M
EIdx=
EI
L
(PxM)dx
Ue =PL
EI
+
PML
EI
+
ML
EI
A =Ue
P =
PL
EI +
ML
EI
A =Ue
M=
PL
EI +
ML
EI
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example :AandA the original version
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p A A g
A
P
BAM0
A xL
method of superpositionAppendix G, p. ,
entry : concentrated P
v= PL
EI,=
PL
EI
entry : concentratedM
v= ML
EI ,=
ML
EI
Solution:
M= PxM
Ue = L
M
EIdx=
EI
L
(PxM)dx
Ue =PL
EI
+
PML
EI
+
ML
EI
A =Ue
P =
PL
EI +
ML
EI
A =Ue
M=
PL
EI +
ML
EI
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example :C the original version
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p C g
C
P
BAM0
x
L/2 L/2
C
BA C
Q
PM0
Solution:
M= PxM , xLPxM Q(x L), L xL
UACe =
L
Mdx
EI =
PL
EI +
PML
EI +
ML
EI
UCBe =
L
L
Mdx
EI =
PL
EI +
PML
EI +
PQL
EI +
ML
EI +
MQL
EI +
QL
EI
Ue=
U
AC
e +
U
CB
e =
PL
EI +
PML
EI +
PQL
EI +
ML
EI +
MQL
EI +
QL
EI
C=Ue
Q =
PL
EI +
ML
EI +
QL
EI
set Q =, we haveC=PL
EI +
ML
EI
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example :C the original version
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p g
C
P
BAM0
x
L/2 L/2
C
BA C
Q
PM0
Solution:
M= PxM , xLPxM Q(x L), L xL
UACe =
L
Mdx
EI =
PL
EI +
PML
EI +
ML
EI
UCBe =
L
L
Mdx
EI =
PL
EI +
PML
EI +
PQL
EI +
ML
EI +
MQL
EI +
QL
EI
Ue=
U
AC
e +
U
CB
e =
PL
EI +
PML
EI +
PQL
EI +
ML
EI +
MQL
EI +
QL
EI
C=Ue
Q =
PL
EI +
ML
EI +
QL
EI
set Q =, we haveC=PL
EI +
ML
EI
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example :C the original version
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C
P
BAM0
x
L/2 L/2
C
BA C
Q
PM0
Solution:
M= PxM , xLPxM Q(x L), L xL
UACe =
L
Mdx
EI =
PL
EI +
PML
EI +
ML
EI
UCBe =
L
L
Mdx
EI =
PL
EI +
PML
EI +
PQL
EI +
ML
EI +
MQL
EI +
QL
EI
Ue=
U
AC
e +
U
CB
e =
PL
EI +
PML
EI +
PQL
EI +
ML
EI +
MQL
EI +
QL
EI
C=Ue
Q =
PL
EI +
ML
EI +
QL
EI
set Q =, we haveC=PL
EI +
ML
EI
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Remarks about example
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modified version does give simplified calculations
real load, fictitious load
.real load:
given as value: change to a variable
given in symbol: stick with it
.fictitious load:
only if there is no real load at the point of interest
question:
for a point with real load, can we add a fictitious load to it?
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Remarks about example
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modified version does give simplified calculations
real load, fictitious load
.real load:
given as value: change to a variable
given in symbol: stick with it
.fictitious load:
only if there is no real load at the point of interest
question:
for a point with real load, can we add a fictitious load to it?
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Remarks about example
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modified version does give simplified calculations
real load, fictitious load
.real load:
given as value: change to a variable
given in symbol: stick with it
.fictitious load:
only if there is no real load at the point of interest
question:
for a point with real load, can we add a fictitious load to it?
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : Example -, p. , an overhanging beam
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Given: overhanging beam, distributed q, concentrated P.
Solve: CandC
AB
C
L L/2
Pq
A B C
C
C
Analysis:
available methods:
deflection curve
moment-area methodsuperposition
energy methods:
. unit-load method;
. Castiglianos theorem
common:
all require the bending moment
diagram
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : Example -, p. , an overhanging beam
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Given: overhanging beam, distributed q, concentrated P.
Solve: CandC
AB
C
L L/2
Pq
A B C
C
C
Analysis:
available methods:
deflection curve
moment-area methodsuperposition
energy methods:
. unit-load method;
. Castiglianos theorem
common:
all require the bending moment
diagram
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : Example -, p. , an overhanging beam
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Given: overhanging beam, distributed q, concentrated P.
Solve: CandC
AB
C
L L/2
Pq
A B C
C
C
Analysis:
available methods:
deflection curve
moment-area methodsuperposition
energy methods:
. unit-load method;
. Castiglianos theorem
common:
all require the bending moment
diagram
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : Example -, p. , an overhanging beam
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Given: overhanging beam, distributed q, concentrated P.
Solve: CandC
AB
C
L L/2
Pq
A B C
C
C
Analysis:
available methods:
deflection curve
moment-area methodsuperposition
energy methods:
. unit-load method;
. Castiglianos theorem
common:
all require the bending moment
diagram
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : Example -, p. , an overhanging beam
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Given: overhanging beam, distributed q, concentrated P.
Solve: CandC
AB
C
L L/2
Pq
A B C
C
C
Analysis:
available methods:
deflection curve
moment-area methodsuperposition
energy methods:
. unit-load method;
. Castiglianos theorem
common:
all require the bending moment
diagram
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example : Example -, p. , an overhanging beam
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Given: overhanging beam, distributed q, concentrated P.
Solve: CandC
AB
C
L L/2
Pq
A B C
C
C
Analysis:
available methods:
deflection curve
moment-area methodsuperposition
energy methods:
. unit-load method;
. Castiglianos theorem
common:
all require the bending moment
diagram
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example :C
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A
B
C
L L/2
Pq
x1 x2RA
Equili.:M
(B
)= RA =
qL
P
Solution:
use different coordinates tosimplify the process
M =RAx
qx =
qx
(L x
)
Px
(for
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A
B
C
L L/2
Pq
x1 x2RA
Equili.:M
(B
)= RA =
qL
P
Solution:
use different coordinates tosimplify the process
M =RAx
qx =
qx
(L x
)
Px
(for
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A
B
C
L L/2
Pq
x1 x2RA
Equili.:M
(B
)= RA =
qL
P
Solution:
use different coordinates to
simplify the process
M =RAx
qx =
qx
(L x
)
Px
(for
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A
B
C
L L/2
Pq
x1 x2RA
Equili.:M
(B
)= RA =
qL
P
Solution:
use different coordinates to
simplify the process
M =RAx
qx =
qx
(L x
)
Px
(for
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Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Example :C
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AB C
L L/2
Pq
x1 x2RA
MC
Solution:
no real moment applied at C,
addMC(virtual,MC=)
Equili.:M(B) =
RA =qL
P
MC
L
M =RAx qx =
qx
(L x) Px
MCx
L(for
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AB C
L L/2
Pq
x1 x2RA
MC
Solution:
no real moment applied at C,
addMC(virtual,MC=)
Equili.:M(B) =
RA =qL
P
MC
L
M =RAx qx =
qx
(L x) Px
MCx
L(for
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AB C
L L/2
Pq
x1 x2RA
MC
Solution:
no real moment applied at C,
addMC(virtual,MC=)
Equili.:M(B) =
RA =qL
P
MC
L
M =RAx qx =
qx
(L x) Px
MCx
L(for
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AB C
L L/2
Pq
x1 x2RA
MC
Solution:
no real moment applied at C,
addMC(virtual,MC=)
Equili.:M(B) =
RA =qL
P
MC
L
M =RAx qx =
qx
(L x) Px
MCx
L(for
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AB C
L L/2
Pq
x1 x2RA
MC
Solution:
no real moment applied at C,
addMC(virtual,MC=)
Equili.:M(B) =
RA =qL
P
MC
L
M =RAx qx =
qx
(L x) Px
MCx
L(for
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AB C
L L/2
Pq
x1 x2RA
MC
Solution:
no real moment applied at C,
addMC(virtual,MC=)
Equili.:M(B) =
RA =qL
P
MC
L
M =RAx qx =
qx
(L x) Px
MCx
L(for
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A
B
C
L L/2
PqA B C
C
C
C=PL
EI
qL
EI
, C=PL
EI
qL
EIC> P>qLthen
P>qL
, Cmoves downward
P P>qLthen
P>qL, Crotates clockwiseP
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A
B
C
L L/2
PqA B C
C
C
C=PL
EI
qL
EI
, C=PL
EI
qL
EIC> P>qLthen
P>qL
, Cmoves downward
P P>qLthen
P>qL, Crotates clockwiseP
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A
B
C
L L/2
PqA B C
C
C
C=PL
EI
qL
EI
, C=PL
EI
qL
EIC> P>qLthen
P>qL
, Cmoves downward
P P>qLthen
P>qL, Crotates clockwiseP
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A
B
C
L L/2
PqA B C
C
C
C=PL
EI
qL
EI
, C=PL
EI
qL
EIC> P>qLthen
P>qL
, Cmoves downward
P P>qLthen
P>qL, Crotates clockwiseP
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A
B
C
L L/2
PqA B C
C
C
C=PL
EI
qL
EI
, C=PL
EI
qL
EIC> P>qLthen
P>qL
, Cmoves downward
P P>qLthenP>qL, Crotates clockwiseP
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A
B
C
L L/2
PqA B C
C
C
C=PL
EI
qL
EI
, C=PL
EI
qL
EIC> P>qLthen
P>qL
, Cmoves downward
P P>qLthenP>qL, Crotates clockwiseP
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procedure: if point of interest has no applied generalized force, apply one evaluate
Ue apply modified Castiglianos theorem
=Ue
P=
L
N
N
P
EA+M
M
P
EI+fs V
V
P
GA+ T
T
P
GIp
P=Pj ,or
dx
applicable areas: deflection and slope at a single point
advantages:
less cumbersome than the integration method
less tricky than the moment-area method
fewer formulas to memorize than the method of superpositionconnection to the unit-load method:
= nNAEdx+ mM
EI dx+
fsvV
GAdx+ tTGIp dx Mohr integration
s =S
P, where S =(N,M, V, T), s =(n, m, v, t)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Sub-summary on Castiglianos theorem
procedure:
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procedure: if point of interest has no applied generalized force, apply one evaluate
Ue apply modified Castiglianos theorem
=Ue
P=
L
N
N
P
EA+M
M
P
EI+fs V
V
P
GA+ T
T
P
GIp
P=Pj ,or
dx
applicable areas: deflection and slope at a single point
advantages:
less cumbersome than the integration method
less tricky than the moment-area method
fewer formulas to memorize than the method of superpositionconnection to the unit-load method:
= nNAEdx+ mM
EI dx+
fsvV
GAdx+ tTGIp dx Mohr integration
s =S
P, where S =(N,M, V, T), s =(n, m, v, t)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Sub-summary on Castiglianos theorem
procedure:
-
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procedure: if point of interest has no applied generalized force, apply one evaluate
Ue apply modified Castiglianos theorem
=Ue
P=
L
N
N
P
EA+M
M
P
EI+fs V
V
P
GA+ T
T
P
GIp
P=Pj ,or
dx
applicable areas: deflection and slope at a single point
advantages:
less cumbersome than the integration method
less tricky than the moment-area method
fewer formulas to memorize than the method of superpositionconnection to the unit-load method:
= nNAEdx+ mM
EI dx+
fsvV
GAdx+ tTGIp dx Mohr integration
s =S
P, where S =(N,M, V, T), s =(n, m, v, t)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Sub-summary on Castiglianos theorem
procedure:
-
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p if point of interest has no applied generalized force, apply one evaluate U
e apply modified Castiglianos theorem
=Ue
P=
L
N
N
P
EA+M
M
P
EI+fs V
V
P
GA+ T
T
P
GIp
P=Pj ,or
dx
applicable areas: deflection and slope at a single point
advantages:
less cumbersome than the integration method
less tricky than the moment-area method
fewer formulas to memorize than the method of superpositionconnection to the unit-load method:
= nNAEdx+ mM
EI dx+
fsvV
GAdx+ tTGIp dx Mohr integration
s =S
P, where S =(N,M, V, T), s =(n, m, v, t)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Sub-summary on Castiglianos theorem
procedure:
-
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p if point of interest has no applied generalized force, apply one evaluate U
e apply modified Castiglianos theorem
=Ue
P=
L
N
N
P
EA+M
M
P
EI+fs V
V
P
GA+ T
T
P
GIp
P=Pj ,or
dx
applicable areas: deflection and slope at a single point
advantages:
less cumbersome than the integration method
less tricky than the moment-area method
fewer formulas to memorize than the method of superposition
connection to the unit-load method:
= nNAEdx+ mM
EI dx+
fsvV
GAdx+ tTGIp dx Mohr integration
s =S
P, where S =(N,M, V, T), s =(n, m, v, t)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Sub-summary on Castiglianos theorem
procedure:
-
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p if point of interest has no applied generalized force, apply one evaluate U
e apply modified Castiglianos theorem
=Ue
P=
L
N
N
P
EA+M
M
P
EI+fs V
V
P
GA+ T
T
P
GIp
P=Pj ,or dx
applicable areas: deflection and slope at a single point
advantages:
less cumbersome than the integration method
less tricky than the moment-area method
fewer formulas to memorize than the method of superposition
connection to the unit-load method:
= nNAEdx+ mM
EI dx+
fsvV
GAdx+ tTGIp dx Mohr integration
s =S
P, where S =(N,M, V, T), s =(n, m, v, t)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Theorem Derivation proof Usages Examples
Sub-summary on Castiglianos theorem
procedure:
-
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if point of interest has no applied generalized force, apply one evaluate U
e apply modified Castiglianos theorem
=Ue
P=
L
N
N
P
EA+M
M
P
EI+fs V
V
P
GA+ T
T
P
GIp
P=Pj ,or dx
applicable areas: deflection and slope at a single point
advantages:
less cumbersome than the integration method
less tricky than the moment-area method
fewer formulas to memorize than the method of superposition
connection to the unit-load method:
= nNAEdx+ mM
EI dx+
fsvV
GAdx+ tTGIp dx Mohr integration
s =S
P, where S =(N,M, V, T), s =(n, m, v, t)
Energy Methods (IV) Castiglianos Theorem
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. Impact ()
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
(Quasi-)Static loading versus dynamic loading
Two types of loads:
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static () and quasi-static () loadings
dead load () or increases in a infinitely slow mannerdynamic loadings ():
vary with time. e.g., collision. impact loading () induce much much greater internal forces and stresses in a structure
than static loading
missile
(dynamic + explosive)
dead load
quasi-static load
impact zone
dynamic loading
impact loading
N
V
V
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
(Quasi-)Static loading versus dynamic loading
Two types of loads:
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static () and quasi-static () loadings
dead load () or increases in a infinitely slow mannerdynamic loadings ():
vary with time. e.g., collision. impact loading () induce much much greater internal forces and stresses in a structure
than static loading
missile
(dynamic + explosive)
dead load
quasi-static load
impact zone
dynamic loading
impact loading
N
V
V
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
(Quasi-)Static loading versus dynamic loading
Two types of loads:
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static () and quasi-static () loadings
dead load () or increases in a infinitely slow mannerdynamic loadings ():
vary with time. e.g., collision. impact loading () induce much much greater internal forces and stresses in a structure
than static loading
missile
(dynamic + explosive)
dead load
quasi-static load
impact zone
dynamic loading
impact loading
N
V
V
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Basic assumptions on the analysis of impact loadings
Assumptions to simplify the analysis:
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Assumptions to simplify the analysis:
the incoming object sticks with the structure after impact no energy lost during impact
the incoming moving object taken as rigid body, deformation only in
impacted structure
kinetic energy of previously stationary body neglected zeromass of inertia
materials behave linear-elastically
the consequence:
at an instant, system is in momentary rest (), mechanicalenergies (kinetic + potential) transform fully into elastic strain energy in
the bars/beams.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Basic assumptions on the analysis of impact loadings
Assumptions to simplify the analysis:
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Assumptions to simplify the analysis:
the incoming object sticks with the structure after impact no energy lost during impact
the incoming moving object taken as rigid body, deformation only in
impacted structure
kinetic energy of previously stationary body neglected zero
mass of inertia
materials behave linear-elastically
the consequence:
at an instant, system is in momentary rest (), mechanicalenergies (kinetic + potential) transform fully into elastic strain energy in
the bars/beams.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Basic assumptions on the analysis of impact loadings
Assumptions to simplify the analysis:
-
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Assumptions to simplify the analysis:
the incoming object sticks with the structure after impact no energy lost during impact
the incoming moving object taken as rigid body, deformation only in
impacted structure
kinetic energy of previously stationary body neglected zero
mass of inertia
materials behave linear-elastically
the consequence:
at an instant, system is in momentary rest (), mechanicalenergies (kinetic + potential) transform fully into elastic strain energy in
the bars/beams.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Basic assumptions on the analysis of impact loadings
Assumptions to simplify the analysis:
-
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Assumptions to simplify the analysis:
the incoming object sticks with the structure after impact no energy lost during impact
the incoming moving object taken as rigid body, deformation only in
impacted structure
kinetic energy of previously stationary body neglected zero
mass of inertia
materials behave linear-elastically
the consequence:
at an instant, system is in momentary rest (), mechanicalenergies (kinetic + potential) transform fully into elastic strain energy in
the bars/beams.
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Basic assumptions on the analysis of impact loadings
Assumptions to simplify the analysis:
-
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Assumptions to simplify the analysis:
the incoming object sticks with the structure after impact no energy lost during impact
the incoming moving object taken as rigid body, deformation only in
impacted structure
kinetic energy of previously stationary body neglected zero
mass of inertia
materials behave linear-elastically
the consequence:
at an instant, system is in momentary rest (), mechanicalenergies (kinetic + potential) transform fully into elastic strain energy in
the bars/beams.
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Vertical impact of a mass-spring system
Note: at system is at rest and F = k
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Note: atmax, system is at rest and Fmax =kmaxenergy conservation: Upot. =Ue
W (h + max) = (kmax)maxW (h + max) =
kmax
max W
k max
W
k
h =
max =W
k +W
k + W
kh
Most important generalization:
static displacement (dead load ofWon the spring)st =W
k
max = st + + hst Special case:h = max =st
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Vertical impact of a mass-spring system
Note: at system is at rest and F = k
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Note: atmax, system is at rest and Fmax =kmaxenergy conservation: Upot. =Ue
W (h + max) = (kmax)maxW (h + max) =
kmax
max W
k max
W
k
h =
max =W
k +W
k + W
kh
Most important generalization:
static displacement (dead load ofWon the spring)st =W
k
max = st + + hst Special case:h = max =st
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Vertical impact of a mass-spring system
Note: at max, system is at rest and Fmax = kmax
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Note: atmax, system is at rest and Fmax kmaxenergy conservation: Upot. =Ue
W (h + max) = (kmax)maxW (h + max) =
kmax
max W
k max
W
k
h =
max =W
k +W
k + W
kh
Most important generalization:
static displacement (dead load ofWon the spring)st =W
k
max = st + + hst Special case:h = max =st
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Horizontal impact of a mass-spring system
energy conservation: K U
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energy conservation:K=Ue
Wg v = + kmaxmax =
Wv
gk
vertical static disp. : st=
W
k
max =stv
g
remarks:
direct application ofenergy conservation
less commonly used
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Horizontal impact of a mass-spring system
energy conservation: K = U
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energy conservation:K=Ue
Wg v = + kmaxmax =
Wv
gk
vertical static disp. : st=
W
k
max =stv
g
remarks:
direct application ofenergy conservation
less commonly used
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Horizontal impact of a mass-spring system
energy conservation: K = U
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energy conservation:K=Ue
Wg v = + kmaxmax =
Wv
gk
vertical static disp. : st=
W
k
max =stv
g
remarks:
direct application ofenergy conservation
less commonly used
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Horizontal impact of a mass-spring system
energy conservation: K = Ue
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energy conservation:K Ue
Wg v = + kmaxmax =
Wv
gk
vertical static disp. :
st=
W
k
max =stv
g
remarks:
direct application ofenergy conservation
less commonly used
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
On the extension from mass-spring to deformable bodies
basic idea:W=Ue(always works but ad
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column
(equiv. spring stiff.k =AE
L )
simple beam, k =EI L
repeated process)
more convenient: the equivalent spring
() approach
. impact of the column:
st =PL
AE
=P
(AEL) =
P
k
(static)
. impact of the simple beam:
st =PL
EI =
P
EIL = Pk (static)calculation ofst(vertical):
max = st + + hst
(note:k is not required,stis enough)
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
On the extension from mass-spring to deformable bodies
basic idea:W=Ue(always works but ad )
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column
(equiv. spring stiff.k =AE
L )
simple beam, k =EI L
repeated process)
more convenient: the equivalent spring
() approach
. impact of the column:
st =PL
AE
=P
(AEL) =
P
k
(static)
. impact of the simple beam:
st =PL
EI =
P
EIL = Pk (static)calculation ofst(vertical):
max = st + + hst (note:k is not required,stis enough)
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
On the extension from mass-spring to deformable bodies
basic idea:W=Ue(always works but at d )
-
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column
(equiv. spring stiff.k =AE
L )
simple beam, k =EI L
repeated process)
more convenient: the equivalent spring
() approach
. impact of the column:
st =PL
AE
=P
(AEL) =
P
k
(static)
. impact of the simple beam:
st =PL
EI =
P
EIL = Pk (static)calculation ofst(vertical):
max = st + + hst (note:k is not required,stis enough)
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
On the extension from mass-spring to deformable bodies
basic idea:W=Ue(always works but arepeated process)
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column
(equiv. spring stiff.k =AE
L )
simple beam, k =EI L
repeated process)
more convenient: the equivalent spring
() approach
. impact of the column:
st =PL
AE =
P
(AEL) =P
k (static)
. impact of the simple beam:
st =PL
EI =
P
EI
L
=P
k (static)
calculation of
st(vertical):
max = st + + hst
(note:k is not required,stis enough)
Energy Methods (IV) Castiglianos Theorem Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
On the extension from mass-spring to deformable bodies
basic idea:W=Ue(always works but arepeated process)
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column
(equiv. spring stiff.k =AE
L )
simple beam, k =EI L
repeated process)
more convenient: the equivalent spring
() approach
. impact of the column:
st =PL
AE =
P
(AEL) =P
k (static)
. impact of the simple beam:
st =PL
EI =
P
EI
L
=P
k (static)
calculation of
st(vertical):
max = st + + hst
(note:k is not required,stis enough)
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
The impact factor ()
atmax(an instant static system), the force developed in the equivalent
spring
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p g
Pmax =kmax, equivalent static load ()the applied static load:P=kst
define: the impact factor n =Pmax
P (other notation:kd)
n=
Pmax
P =
max
st=
+ +
h
stphysical significance of the impact factor n:
the magnification of a statically applied load
Pmax =nP, max =nst
more important: on the induced stresses:
max =nst (all stress components)
are tightly related to the safety of the materials.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
The impact factor ()
atmax(an instant static system), the force developed in the equivalent
spring
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p g
Pmax =kmax, equivalent static load ()the applied static load:P=kst
define: the impact factor n =Pmax
P (other notation:kd)
n=
Pmax
P =
max
st=
+ +
h
stphysical significance of the impact factor n:
the magnification of a statically applied load
Pmax =nP, max =nst
more important: on the induced stresses:
max =nst (all stress components)
are tightly related to the safety of the materials.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Remarks
equivalent spring depends on the impact location
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equivalent spring depends on the impact location
critical step: calculation ofst
whole structure is involved, best method:energy method
especially the unit load method
if there is two-force-member (), contribution of its axial loadNshould be includedfor beam that resist bending, V, Ncan be ignored.
versions ofn =Pmax
P =
max
st=
max
stcan be used to
check safety
determine max. loadsize design
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Remarks
equivalent spring depends on the impact location
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equivalent spring depends on the impact location
critical step: calculation ofst
whole structure is involved, best method:energy method
especially the unit load method
if there is two-force-member (), contribution of its axial loadNshould be includedfor beam that resist bending, V, Ncan be ignored.
versions ofn =Pmax
P =
max
st=
max
stcan be used to
check safety
determine max. loadsize design
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Remarks
equivalent spring depends on the impact location
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equivalent spring depends on the impact location
critical step: calculation ofst
whole structure is involved, best method:energy method
especially the unit load method
if there is two-force-member (), contribution of its axial loadNshould be includedfor beam that resist bending, V, Ncan be ignored.
versions ofn =Pmax
P =
max
st=
max
stcan be used to
check safety
determine max. loadsize design
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Remarks
equivalent spring depends on the impact location
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equivalent spring depends on the impact location
critical step: calculation ofst
whole structure is involved, best method:energy method
especially the unit load method
if there is two-force-member (), contribution of its axial loadNshould be includedfor beam that resist bending, V, Ncan be ignored.
versions ofn =Pmax
P =
max
st=
max
stcan be used to
check safety
determine max. loadsize design
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Remarks
equivalent spring depends on the impact location
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q p g p p
critical step: calculation ofst
whole structure is involved, best method:energy method
especially the unit load method
if there is two-force-member (), contribution of its axial loadNshould be includedfor beam that resist bending, V, Ncan be ignored.
versions ofn =Pmax
P =
max
st=
max
stcan be used to
check safety
determine max. loadsize design
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Remarks
equivalent spring depends on the impact location
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q p g p p
critical step: calculation ofst
whole structure is involved, best method:energy method
especially the unit load method
if there is two-force-member (), contribution of its axial loadNshould be includedfor beam that resist bending, V, Ncan be ignored.
versions ofn =Pmax
P =
max
st=
max
stcan be used to
check safety
determine max. loadsize design
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Remarks
equivalent spring depends on the impact location
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critical step: calculation ofst
whole structure is involved, best method:energy method
especially the unit load method
if there is two-force-member (), contribution of its axial loadNshould be includedfor beam that resist bending, V, Ncan be ignored.
versions ofn =Pmax
P =
max
st=
max
stcan be used to
check safety
determine max. loadsize design
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: Aluminum pipe, a load ofW= kip;
Solve: of top of the pipe for
(a) static load, (b) impact load at h =
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(a) static load, (b) impact load at h
Solution:
(a) static load
st =WL
AE =
( .
)
(
) =
. in.
(b) impact load
impact factor:n = +
+
h
st
=
max
=nst=. in.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: Aluminum pipe, a load ofW= kip;
Solve: of top of the pipe for
(a) static load, (b) impact load at h =
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(a) static load, (b) impact load at h
Solution:
(a) static load
st =WL
AE =
( .
)
(
) =
. in.
(b) impact load
impact factor:n = +
+
h
st
=
max
=nst=. in.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: Aluminum pipe, a load ofW= kip;
Solve: of top of the pipe for
(a) static load, (b) impact load at h =
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( ) , ( ) p h
Solution:
(a) static load
st =WL
AE =
( .
)
(
) =
. in.
(b) impact load
impact factor:n = +
+
h
st
=
max =nst =. in.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: () A- steel beam, () a weight ofW=. kip dropped from
h =. in., () Est = ksi
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Determine:
max,
max
Solution:
st =WL
EI =. in.
impact factor
n = + + hst
=.somax =nst =. in.
equivalent static load Pmax =nW
Mmax =PmaxL
, max =
Mmaxc
I =
PmaxLc
I =
nWLc
I =. ksi
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: () A- steel beam, () a weight ofW=. kip dropped from
h =. in., () Est = ksi
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Determine:
max,
maxSolution:
st =WL
EI =. in.
impact factor
n = + + hst
=.somax =nst =. in.
equivalent static load Pmax =nW
Mmax =PmaxL
, max =
Mmaxc
I =
PmaxLc
I =
nWLc
I =. ksi
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: () A- steel beam, () a weight ofW=. kip dropped from
h =. in., () Est = ksi
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Determine:
max,
maxSolution:
st =WL
EI =. in.
impact factor
n = + + hst
=.somax =nst =. in.
equivalent static load Pmax =nW
Mmax =PmaxL
, max =
Mmaxc
I =
PmaxLc
I =
nWLc
I =. ksi
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: () A- steel beam, () a weight ofW=. kip dropped from
h =. in., () Est = ksi
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Determine:
max,
maxSolution:
st =WL
EI =. in.
impact factor
n = + + hst
=.somax =nst =. in.
equivalent static load Pmax =nW
Mmax =PmaxL
, max =
Mmaxc
I =
PmaxLc
I =
nWLc
I =. ksi
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: () A- steel beam, () a weight ofW=. kip dropped from
h =. in., () Est = ksi
Determine: ,
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Determine: max
,max
Solution:
st =WL
EI =. in.
impact factor
n = + + hst
=.somax =nst =. in.
equivalent static load Pmax =nW
Mmax =PmaxL
, max =
Mmaxc
I =
PmaxLc
I =
nWLc
I =. ksi
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example
Given: () A- steel beam, () a weight ofW=. kip dropped from
h =. in., () Est = ksi
Determine: ,
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max,
maxSolution:
st =WL
EI =. in.
impact factor
n = + + hst
=.somax =nst =. in.
equivalent static load Pmax =nW
Mmax =PmaxL
, max =
Mmaxc
I =
PmaxLc
I =
nWLc
I =. ksi
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example : (cont.)
Alternatively:
energy conservation: W=Ue
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W=W(h + max)Ue =
Pmaxmax
finding relation between Pmaxand max
max =PmaxL
EI Pmax =
EImax
L
substitute parameters:
.max .max . =
max =. in.
rest steps follow those already given.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example : (cont.)
Alternatively:
energy conservation: W=Ue
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W=W(h + max)Ue =
Pmaxmax
finding relation between Pmaxand max
max =PmaxL
EI
Pmax =EImax
L
substitute parameters:
.max .max . =
max =. in.
rest steps follow those already given.
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example : (cont.)
Alternatively:
energy conservation: W=Ue
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W=W(h + max)Ue =
Pmaxmax
finding relation between Pmaxand max
max =PmaxL
EI
Pmax =EImax
L
substitute parameters:
.max .max . =
max =. in.
rest steps follow those already given.
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example : (cont.)
energy conservation:K=Ue
=
A
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mv Pmax( A)max relation between Pmaxand (A)max(A)max = PmaxLACEI
Pmax =EI(A)max
LAC
(A)max =mvLAC
EI =. mm
deflection analysis
Pmax =EI(A)max
LAC=. kN
A = PmaxL
ACEI
=. rad
Bmax = Amax + ALAB =. mm
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example : (cont.)
energy conservation:K=Ue
=
PA
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mv Pmax( A)max relation between Pmaxand (A)max(A)max = PmaxLACEI
Pmax =EI(A)max
LAC
(A)max =mvLAC
EI =. mm
deflection analysis
Pmax =EI(A)max
LAC=. kN
A = PmaxL
ACEI
=. rad
Bmax = Amax + ALAB =. mm
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example : (cont.)
energy conservation:K=Uemv =
P
maxA max
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max( A)max relation between Pmaxand (A)max(A)max = PmaxLAC
EI Pmax =
EI(A)maxLAC
(A)max =mvLAC
EI =. mm
deflection analysis
Pmax =EI(A)max
LAC=. kN
A = PmaxL
AC
EI =. rad
Bmax = Amax + ALAB =. mm
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example : (cont.)
energy conservation:K=Ue
mv =
P
maxA max
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max( A)max relation between Pmaxand (A)max(A)max = PmaxLAC
EI Pmax =
EI(A)maxLAC
(A)max =mvLAC
EI =. mm
deflection analysis
Pmax =EI(A)max
LAC=. kN
A = PmaxL
AC
EI =. rad
Bmax = Amax + ALAB =. mm
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example : (cont.)
energy conservation:K=Ue
mv =
Pmax A max
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( ) relation between Pmaxand (A)max(A)max = PmaxLAC
EI Pmax =
EI(A)maxLAC
(A)max =mvLAC
EI =. mm
deflection analysis
Pmax =EI(A)max
LAC=. kN
A = PmaxL
ACEI
=. rad
Bmax = Amax + ALAB =. mm
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example part from final exam (B)
Given:
() beamAB, pin-supported; two-force-member BD;
() dAB = mm, dBD = mm, E = GPa;
() weight W N falls from h mm onto C
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() weight W= N falls from h = mm onto C
Determine: equivalent static load in rod BD
A Bh
D
W
C
0.8m
0.4m
0.
6m
analysis:
beam and two-force-member
impact load leads the static
load magnified by n
all relies on calculation ofst
unit load method
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example part from final exam (B)
Given:
() beamAB, pin-supported; two-force-member BD;
() dAB = mm, dBD = mm, E = GPa;
() weight W = N falls from h = mm onto C
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() weight W= N falls from h = mm onto C
Determine: equivalent static load in rod BD
A Bh
D
W
C
0.8m
0.4m
0.
6m
analysis:
beam and two-force-member
impact load leads the static
load magnified by n
all relies on calculation ofst
unit load method
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example part from final exam (B)
Given:
() beamAB, pin-supported; two-force-member BD;
() dAB = mm, dBD = mm, E = GPa;
() weight W = N falls from h = mm onto C
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() weight W= N falls from h = mm onto C
Determine: equivalent static load in rod BD
A Bh
D
W
C
0.8m
0.4m
0.
6m
analysis:
beam and two-force-member
impact load leads the static
load magnified by n
all relies on calculation ofst
unit load method
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example part from final exam (B)
Given:
() beamAB, pin-supported; two-force-member BD;
() dAB = mm, dBD = mm, E = GPa;
() weight W = N falls from h = mm onto C
-
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() weight W= N falls from h = mm onto C
Determine: equivalent static load in rod BD
A Bh
D
W
C
0.8m
0.4m
0.
6m
analysis:
beam and two-force-member
impact load leads the static
load magnified by n
all relies on calculation ofst
unit load method
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example part from final exam (B)
Given:
() beamAB, pin-supported; two-force-member BD;
() dAB = mm, dBD = mm, E = GPa;
() weight W = N falls from h = mm onto C
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() weight W N falls from h mm onto CDetermine: equivalent static load in rod BD
A Bh
D
W
C
0.8m
0.4m
0.
6m
analysis:
beam and two-force-member
impact load leads the static
load magnified by n
all relies on calculation ofst
unit load method
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example part from final exam (B)
Given:
() beamAB, pin-supported; two-force-member BD;
() dAB = mm, dBD = mm, E = GPa;
() weight W = N falls from h = mm onto C
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() weight W N falls from h mm onto CDetermine: equivalent static load in rod BD
A B
h
D
W
C
0.8m
0.4m
0.
6m
analysis:
beam and two-force-member
impact load leads the static
load magnified by n
all relies on calculation ofst
unit load method
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution: calculation ofst
Solution:
for static load Wapplied at C, evaluate:
.M-diagram for beamAB; . axial load for rod BD repeat for unit load
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p
AB
D
WL
4
C
AB
D
L
4
C
5
6W
5
6
(a) (b)
Mohr integration by diagram multiplication.
Cst = L WL L + (W) () LBDEA
Cst =
WL
EI +
WLBD
EA
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution: calculation ofst
Solution:
for static load Wapplied at C, evaluate:
.M-diagram for beamAB; . axial load for rod BD repeat for unit load
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p
AB
D
WL
4
C
AB
D
L
4
C
5
6W
5
6
(a) (b)
Mohr integration by diagram multiplication.
Cst = L WL L + (W) () LBDEA
Cst =
WL
EI +
WLBD
EA
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution: calculation ofst
Solution:
for static load Wapplied at C, evaluate:
.M-diagram for beamAB; . axial load for rod BD repeat for unit load
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p
AB
D
WL
4
C
AB
D
L
4
C
5
6W
5
6
(a) (b)
Mohr integration by diagram multiplication.
Cst = L WL L + (W) () LBDEA
Cst =
WL
EI +
WLBD
EA
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution (cont.)
A h
W
C0.
4m
Solution (cont.):
(cont.)Cst = WL
EI + WLBD
EA
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AB
D
C
0.8m
.
0.
6m
( ) st EIEA
substitute values in:
st =. m + .
m =. m
the impact factor:
n = +
+ h
Cst
=.
the equivalent static load
NBD =
W n =.
N
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution (cont.)
AB
h
W
C0.4m
Solution (cont.):
(cont.)Cst = WL
EI + WLBD
EA
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AB
D
C
0.8m
.
0.
6m
st EI EAsubstitute values in:
st =. m + .
m =. m
the impact factor:
n = +
+ h
Cst
=.
the equivalent static load
NBD =
W n =.
N
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution (cont.)
AB
h
W
C0.4m
Solution (cont.):
(cont.)Cst = WL
EI + WLBD
EA
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B
D
0.8m
0.
6m
st EI EAsubstitute values in:
st =. m + .
m =. m
the impact factor:
n = +
+ h
Cst
=.
the equivalent static load
NBD =
W
n =.
N
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution (cont.)
AB
h
W
C0.4m
Solution (cont.):
(cont.)Cst = WL
EI + WLBD
EA
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B
D
0.8m
0.
6m
EI EAsubstitute values in:
st =. m + .
m =. m
the impact factor:
n = +
+ h
Cst
=.
the equivalent static load
NBD =
W
n =.
N
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution (cont.)
AB
h
W
C0.
4m
Solution (cont.):
(cont.)Cst = WL
EI + WLBD
EA
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B
D
0.8m
0.
6m
EI EAsubstitute values in:
st =. m + .
m =. m
the impact factor:
n = +
+ h
Cst
=.
the equivalent static load
NBD =
W
n =.
N
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example solution (cont.)
AB
h
W
C0.
4m
Solution (cont.):
(cont.)Cst = WL
EI + WLBD
EA
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B
D
0.8m
0.
6m
EI EAsubstitute values in:
st =. m + .
m =. m
the impact factor:
n = +
+ h
Cst
=.
the equivalent static load
NBD =
W
n =.
N
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example final exam (A)
W 1 m
rigid bar
C
z Given:
solid circular L-shaped barABCwith d = mm
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1 m
2 m
0.1 m
A
B
x
y
D
with d mm
fixed atA, Cfixed to a rigid bar CD
W= N falls with zero initial
velocity
E = GPa, G = GPa,[] = MPa neglect effect of transverse shear
Check strength ofABCwith the th strength theory
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example final exam (A)
W 1 m
rigid bar
C
z Given:
solid circular L-shaped barABCwith d= mm
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1 m
2 m
0.1 m
A
B
x
y
D
fixed atA, Cfixed to a rigid bar CD
W= N falls with zero initial
velocity
E = GPa, G = GPa,[] = MPa neglect effect of transverse shear
Check strength ofABCwith the th strength theory
Energy Methods (IV) Castiglianos Theorem
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Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example final exam (A)
W 1 m
rigid bar
C
z Given:
solid circular L-shaped barABCwith d= mm
-
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1 m
2 m
0.1 m
A
B
x
y
D fixed atA, Cfixed to a rigid bar CD
W= N falls with zero initial
velocity
E = GPa, G = GPa,[] = MPa neglect effect of transverse shear
Check strength ofABCwith the th strength theory
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example final exam (A)
W 1 m
rigid bar
C
z Given:
solid circular L-shaped barABCwith d= mm
-
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1 m
2 m
0.1 m
A
B
x
y
D fixed atA, Cfixed to a rigid bar CD
W= N falls with zero initial
velocity
E = GPa, G = GPa,[] = MPa neglect effect of transverse shear
Check strength ofABCwith the th strength theory
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example final exam (A)
W 1 m
rigid bar
0 1
C
z Given:
solid circular L-shaped barABCwith d= mm
-
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1 m
2 m
0.1 m
A
B
x
y
D fixed atA, Cfixed to a rigid bar CD
W= N falls with zero initial
velocity
E = GPa, G = GPa,[] = MPa neglect effect of transverse shear
Check strength ofABCwith the th strength theory
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example final exam (A)
W 1 m
rigid bar
0 1
C
z Given:
solid circular L-shaped barABCwith d= mm
-
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1 m
2 m
0.1 m
A
B
x
y
D fixed atA, Cfixed to a rigid bar CD
W= N falls with zero initial
velocity
E = GPa, G = GPa,[] = MPa neglect effect of transverse shear
Check strength ofABCwith the th strength theory
Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impact loading effect Basic systems Examples
Example (cont.)
rigid bar z
T
M
W
W
Solution:
impact + combined loadings +
strength theories
Dst
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W
1 m
2 m
1 m
g d ba
0.1 m
A
C
x
y
D B
W
W
Dst =W
EI +
W
EI +
W GIp
= ( +
)WEI + WGIpwhere I=
d
Ip =
d
D
st=
. m
the impact factor
n = +
+ hDst =.Energy Methods (IV) Castiglianos Theorem
Castiglianos nd Theorem Impact Impac