This document reviews thin beam theory for beams with piezoelectric materials. It shows that the moment equilibrium equation for such beams includes additional terms due to the direct and inverse piezoelectric effects. Specifically, the moment is equal to the product of the beam's equivalent bending stiffness and the second derivative of the deflection, plus terms proportional to the applied voltage that depend on the beam's geometry and material properties. In the absence of external loads, the piezoelectric effect generates an internal moment and corresponding distributed force within the beam.

1. Beams with Piezoelectric
Material

2. Review of Thin-Beam
Theory
• The Euler-Benoulli beam theory assumes that
the strain varies linearly through the thickness
of the beam and inversely proportional to the
radius of curvature.
2
d v
  y 2
dx
d 2v
  E   Ey 2
dx

3. Equilibrium
• The externally applied moment has to be in
equilibrium with the internally generated
moment. h/2 h/2
d 2v
M    bydy   Ey 2 bydy
h / 2 h/2
dx
• For homogeneous materials:
2 h/2
d v d 2v
M  E 2  y bdy  EI 2
2
dx h / 2 dx

4. Equilibrium
• Rearranging the terms:
M d 2v
 2
EI dx
My
 
I

5. With piezoelectric
materials
• Introducing change in the material property:
h/2
M
   ydy
b h / 2
t s / 2
 V
ts / 2
   Ea   a  d 31  ydy   Es s ydy

h / 2  ta 
 t s / 2
h/2
 V
  Ea   a  d 31  ydy

ts / 2  ta 


6. With piezoelectric
materials
• Expanding the integral
2 ts / 2 ts / 2
M d v V
 Ea 2 / 2y dy  Ea d31 ta  ydy
2
b dx h h / 2
2 ts / 2 2 h/2 h/2
d v d v V
 Es 2  y dy  Ea 2  y dy  Ea d 31
2 2
 ydy
dx t s / 2 dx t s / 2 ta ts / 2

7. With piezoelectric
materials
• Rearranging
 ts / 2 2
M d v2 ts / 2 h/2 
 2 Ea  y dy  Es  y dy  Ea  y dy 
2 2
b dx   h / 2
 t s / 2 ts / 2


t s / 2 h/2
V V
 Ea d 31
ta / 2ydy  Ea d31 ta
h
 ydy
ts / 2

8. With piezoelectric
materials
• Integrating
M

1 d 2v
b 24 dx 2
   
Ea h 3  t s  2 E s t s  Ea h 3  t s
3 3 3

 Ea d 31
V 2
8t a
 
t s  h  Ea d 31
2 V 2
8ta

h  ts
2

M 1 d 2v

b 12 dx 2
  
Ea h  t s  Est s 
3 3 3 Ea d31V 2
4ta

h  ts
2
 

9. Remember:
• For homogeneous structures: Eh 3 d 2 v M
2

12 dx b
• Thus, in the absence of the voltage:
EI Equivalent b
 3

Ea h  t s  E s t s
3 3
12
• OR: d 2v Eabd31V 2
M  EIEquivalent 2 
dx 4ta
h  ts
2
 

10. In the absence of load
 
2
d v Eabd31V
 h  ts
2 2
2
dx 4ta EIEquivalent
• Thus, the structure will feel a moment:
d 2v
M s  Es I s 2  
dx
Es I s Eabd31V 2
4ta EIEquivalent

h  ts
2


11. Piezoelectric forces
• The above is equivalent of having a force
applied by the piezoelectric material that is
equal to:
Fa 
Ms
ts
Es I s Eabd31V 2

2
4t s ta EIEquivalent
h  ts 

12. Homework #3
• Solve another 3 problems from chapter 2 of
the textbooks!