elastic energy
DESCRIPTION
ELASTIC ENERGY. P H. R H. =P/A. P. Linear elastic material. =l/l. l. External work. =. Internal work. =. For a prismatic bar:. S. V. Law of conservation of energy (first law of thermodynamics):. Increment of:. Heat. External work. Adiabatic processes. Static processes. - PowerPoint PPT PresentationTRANSCRIPT
1/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
ELASTIC ENERGY
2/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
P
l
PH Linear elastic material
External work
=P/A
=l/l
RH
lP 2
1 2
1
Internal work
VVV
p dVE
dVE
dVW22
1
2
1 2
A
P
A
N
=
=
lPL 2
1
lAV
AE
lPlA
EA
PdV
EA
NW
V
p 222
2
2
2
2
2
EA
lPl
For a prismatic bar:
3/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
V S
iqq
)( iPP
p
V
ji
V
ij WdVTTdVu ,...
Law of conservation of energy (first law of thermodynamics):
dQ pdWdL kdWAdiabatic processes
Static processes
dt
dW
dt
dL p
...... dSuqdVuPL i
S
ii
V
i
Power = work done in a given time
Heat External work
Rate of potential energy
Potential energy Kinematical energy
Increment of:
4/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
dVDG
dVAK
dtWWVV
pp22
4
1
6
1...
For Hooke materials: KAA 3 GDD 2
dVdVdVdtWWVV
f
V
vpp ...
22
2
3
2
1
6
1 A
KAAA
Kv
22
2
1
4
1 GDDDD
Gf
Specific volumetric energy
Specific distortion energy
KAA 3 GDD 2
dt
d
5/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
212
1kkijijfv E
ijijkkijijkkijij EE
11
2122
1
Specific energy is a potential energy
A general form of specific energy for beams:
dxS
FW
l
p 0
2
2
1 F – cross-sectional force
S – beam stiffness
κ – shape coefficient
dxEA
N
AE
lPW
l
p 0
22
2
1
2
6/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
Components of elastic energy formula
Specific caseCross-sectional force
F
Beam stiffness
S
Shape coefficient
Tension N EA 1
Bending M EJ 1
Shear Q GA
Torsion Ms GJs
s
7/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
Maxwell-Mohr formula
8/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
Definitions of generalised force and generalized displacement:
Generalized force is any external loading in the form of point force, point moment, distributed loading etc.
Generalized displacement corresponding to a given generalized force is any displacement for which work of this force can be performed
The dimension of generalized displacement has to follow the rules of dimensional analysis taking into account that the dimension of work is [Nm].
dt
dW
dt
dL p dt
Corresponding elastic energy
External work: function of loading and displacement
pWL
9/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
Generalized displacement
Displacement dimension
Generalized force dimension
[1]
P
[Nm]
q
M
[N] u[m]
[N/m] [m2]
du/dx
udx
But also:
Corresponding generalized displacement is the sum of displacements u1+u2
P2P1
u1 u2
M2M1
Corresponding generalized displacement is the sum of rotation angles of neighbouring cross-sections
Generalized force
10/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
For linear elasticity the principle of superposition obeys:
ij
n
jji Pu
ij
n
jji uP or
where αij βij are influence coefficients for which Betti principle holds: αij = αji and i βij =βji
The work of external forces (generalized) Pi performed on displacements (generalized) ui is:
ij
n
ij
n
jiij
n
ij
n
jii
n
ii uuPPuPL
1 11 11 2
1
2
1
2
1
After expansion of the first term we have:
nni
n
ii uPuPuPuPuPL ...
2
1
2
1332211
1
P
l
lP 2
1
11/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
taking into account that: inniiij
n
jji PPPPu ...2211
nni
n
ii uPuPuPuPL ...
2
1
2
12211
1
nni
n
ii uPuPuPuP
PuP
PP
L...
2
1
2
1332211
1111
nnPPPu 11221111 ... nnPPPu 22222112 ...
11
22
1
111 0...0
2
1
P
uP
P
uP
P
uPu n
n
which after expansion reads:
nnnnnn PPPu ...2211…,
12121111 ...2
1nnPPPu
1u
1221
ii 11 1u
1P
12/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
Therefore, for any displacement we have:
ii
uP
L
pWL and since
u
P
PPW
P
ip
0
,
To find an arbitrary generalized displacement of any point of the structure one has to apply corresponding generalized force in this point, and
calculate internal energy associated with all loadings (real and generalized),
take derivative of this energy with respect to generalized force,
and finally set its true value equal to 0:
ii
p uP
W
dxS
FW
li
p 0
2
2
1
Where Fi is cross-sectional force for each case of internal forces reduction (normal force, shear force, bending moment, torsion moment)
u
13/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
Making use of superposition principle we have:
ipipp PFPPFWPFPFWW 1
where: FPF 1
Ppp FPFWW
PFPF
dxS
FW
l
p 0
2
2
1 With general formula for potential energy:
4
0
2
12
1
i
l
ii
Piin
jp dx
S
FPFW
or
we have:
where index i has been added for different reduction cases
xdenotes here
function of x
14/14M.Chrzanowski: Strength of Materials
SM2-09: Elastic energy
udx
S
FFdx
S
FFPF
P
W
i
l
ii
Piin
jPi
l
ii
iPiin
jP
p
4
010
4
010
2
2
1
4
01 i
l
iPii
n
j
dxS
FFu
This is Maxwell-Mohr formula for any generalized displacement .
Summation has to be taken over all structural members j and over
all internal cross-sectional forces i=4
4
0
2
12
1
i
l
ii
Piin
jp dx
S
FPFW
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SM2-09: Elastic energy
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