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

15/14M.Chrzanowski: Strength of Materials

SM2-09: Elastic energy

stop