Download Part II: Applications of plate and shell theories

Lecture 5
Lecture 5 (Additional materials):
1. Yield-line theories
2. Size effects in plate structures
3. Plate vibration
4. Fracture mechanics of plates
S. Mahmoud Mousavi- Rak-54.3110 Plate and shell structures
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Lecture 5
Introduction to Yield-line theories
Introduction to Yield-line theories
Theory of plasticity: The deformation of a material undergoing non-reversible changes of shape in
response to applied forces.
Application the theory of plasticity
I.
Beam constructions
II. Two-dimensional continua
III. Three-dimensional continua
In the application of the theory of plasticity three different solution techniques can be distinguished:
1. The incremental (stepwise) elastic-plastic calculation
2. Application of the lower-bound theorem, which is based on the equilibrium equations (equilibrium
system)
3. Application of the upper-bound theorem, which is based on a mechanism (yield-line theory)
S. Mahmoud Mousavi- Rak-54.3110 Plate and shell structures
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Lecture 5
Introduction to Yield-line theories
Mechanism method for beams and frames
Yield line theory for Plates
The upper-bound theorem
The upper-bound theorem is quite well developed, especially for the application of reinforced concrete slabs.
The calculation procedure is known as the yield-line theory.
A yield line in a plate is similar to a plastic hinge in a frame.
In the yield line theory, the plate is assumed to yield along the so called yield lines so that the plate becomes a
mechanism with one degree of freedom.
The internal virtual work done by the bending moments at the yield lines as well as the virtual work done by
the external load, the sum of which must vanish for a plate in equilibrium, is determined as the mechanism
moves. This leads to an estimate of the plastic collapse load (or in limit state design: an estimate of the
plastic moment). The difficulty is to ascertain that the yield condition of the plate is not violated at any point.
As is apparent from the above, the yield line theory is analogical to the mechanism method of beams and
frames.
Ref:
Rudolph Szilard, 2004, Theories and Applications of Plate Analysis Classical, Numerical and Engineering Methods, John Wiley & Sons, Inc.
Ref: Vrouwenvelder, Witteveen, 2003, The plastic behaviour and the calculation of plates subjected to bending, Technical University Delft.
S. Mahmoud Mousavi- Rak-54.3110 Plate and shell structures
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Lecture 5
II: Size effects in Structures
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Lecture 5
Size effects in Structures
Classical continuum mechanics cannot capture size effects.
In order to capture size effects, the generalized continuum mechanics is introduced.
Different types of generalization of classical continuum mechanics:
Higher-grade
Classical continuum
mechanics
Gradient Elasticity
Nonlocal Elasticity
Higher-order
Micropolar Elasticity
(Cosserat Theory)
Gradient Micropolar Elasticity
Nonlocal Micropolar Elasticty
S. Mahmoud Mousavi- Rak-54.3110 Plate and shell structures
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Lecture 5
Size effects in Structures
Classical continuum mechanics:
1
1
strain energy density : W   ij  ij   x x   y y   z z   xy xy   xz xz   yz yz 
2
2
1
1
U    :  dV    x x   y y   z z   xy xy   xz xz   yz yz  dxdydz
2V
2 V
First strain gradient elasticity:
1
1
W   ij  ij  l 2 ij ,k  ij ,k
2
2
Second strain gradient elasticity:
l, c1, c2: Characteristic length scales
1
1
1
W   ij  ij  c12 ij ,k  ij ,k  c2 4 ij ,kl  ij ,kl
2
2
2
Mousavi S.M., Paavola J., Reddy J.N., 2015, Variational approach to dynamic analysis of third-order shear deformable plates within gradient elasticity. Meccanica 50:1537-1550.
Tahaei Yaghoubi S., Mousavi S.M., Paavola J., 2015, Strain and velocity gradient theory for higher-order shear deformable beams. Archive of Applied Mechanics, 85:877–892.
S. Mahmoud Mousavi- Rak-54.3110 Plate and shell structures
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Lecture 5
III: Plate vibration
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Lecture 5
Vibration of structures
Vibration of structures:
I.
Forced vibration: Forced vibration of a dynamic system is produced by external forces during the vibration.
II. Free vibration: Free vibration of a dynamic system is produced by introducing an initial velocity or
displacement into the system.




Natural frequency is frequency of free (unforced) vibrations.
Fundamental frequency is the smallest value of the natural frequency.
Resonant frequency: Frequencies at which the response amplitude is a relative maximum.
Mode Shape is the pattern of motion of the normal modes.
In forced vibration, when damping is small, the resonant frequency is approximately equal to the natural frequency
of the system.
 Boundary conditions determine the mode shapes and the natural frequencies
 Initial conditions determine the contribution of each mode to the total response (or, in other words, the
contribution of each mode to the total response depends on how the system has been started into motion).
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Lecture 5
Vibration of structures
Formulation:
 Dynamic equilibrium methods
 Energy methods: (Governing equations, Boundary Conditions and Initial Conditions)
t2
Hamilton’s principle:
 L  dt  0

t1
Lagrangian function: L  K  U  W
Strain energy:
U
1
1
 x x   y y   z z   xy xy   xz xz   yz yz  dxdydz

:

dV

2 
2 

Kinetic energy: K  1   u  u dV
2
t t
External work: W   f  udV   t  udS

N
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Lecture 5
Vibration of Plates
Considering bending or membrane analysis:
I. Flexural Vibration of Plates
II. Transverse Vibration of Membranes
Flexural Vibration of Thin Plates
 2 w( x, y, t )
4
Motion equation of plates:  D w( x, y, t )  pz ( x, y, t )  m
& B.C. & I .C.( Initial Condition)
2
m : mass of the plate per unit area.
Initial conditions include: w( x, y, 0) &
t
w
( x, y, 0)
t
Inertia force
Rotatory inertia, shear forces and damping are neglected
Free Flexural Vibration of Thin Plates:
Free vibration of a dynamic system is produced by introducing an initial velocity or displacement into the
system.
2
Free vibration: pz=0  D 4 w( x, y, t )  m  w( x2, y, t ) & B.C. & I .C.( Initial Condition)
t
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Lecture 5
Free Flexural Vibration of Plates
 Free Flexural Vibration of Rectangular Plates
 D 4 w( x, y, t )  m
 2 w( x, y, t )
t 2
Separation of variables: w( x, y, t )  W ( x, y) (t )  X ( x)Y ( y) (t )
 Free Flexural Vibration of Circular Plates
 2 w( x, y, t )
 D w(r ,  , t )  m
t 2
4
Separation of variables:
w(r ,  , t )  R(r )( ) (t )
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Lecture 5
Free Flexural Vibration of Rectangular Plates
Solution: Free Flexural Vibration of Rectangular Plates
Using separation of variables, the solution is assumed: w( x, y, t )  W ( x, y ) (t )  X ( x)Y ( y ) (t )
Assuming harmonic time dependence (according to I.C.) :  (t )  sin t  or cos t 
 X Y  2 X Y   XY  
m 2
XY  0
D
W ( x, y ) : Mode Shape
 : Natural Frequency
Solution methods:
• Equilibrium method:
Navier’s solution for simply supported rectangular plates
Levy’s solution for rectangular plates
•
Energy methods
Ritz method
S. Mahmoud Mousavi- Rak-54.3110 Plate and shell structures
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Lecture 5
Free Flexural Vibration of Rectangular Plates
Example1- Simply supported rectangular plates: Navier’s solution
Governing equation:
X Y  2 X Y   XY  
m 2
XY  0
D
Mode Shapes


W ( x, y )  X ( x)Y ( y )   Wmn sin
m 1 n 1
m x
n y
sin
;
a
b
Wmn  Amplitude of Vibration
4
2
2
4
2
2
2

 m 
 m   n   n  m
n  D
2 m


2



0

Natural
Frequency
:




    
mn


 
 

D
 a 
 a   b   b 
 a   b   m
Ref:
Rudolph Szilard, 2004, Theories and Applications of Plate Analysis Classical, Numerical and Engineering Methods, John Wiley & Sons, Inc.
J.N. Reddy, 2006, Theory and analysis of elastic plates and shells, CRC Press.
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Lecture 5
Fracture mechanics of plates
Fracture mechanics of plates: Kirchhoff Theory.
The simplest approach to the out-of-plane fracture problems is
to assume small deflection, Kirchhoff plate theory.
Consider an infinite plate:
Near crack tip stress and displacement fields for a crack in an infinite plate
where h is the plate thickness and E is Young’s modulus. The stress intensity factors k1 and k2 for symmetric loading (bending)
and anti-symmetric loading (twisting) are defined by
Ref: Alan T. Zehnder, Mark J. Viz, 2005, Fracture Mechanics of Thin Plates and Shells Under Combined Membrane, Bending, and Twisting Loads,
Applied Mechanics Reviews, 58(1), 37-48.
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