Theory of Plates
PartDII:DPlatesDinDbendingD
LectureDNotesD
WinterDSemesterD2013/2014
Prof@DDr@MIng@DKaiMUweDBletzingerD
Lehrstuhl fürDStatikD
TechnischeDUniversitätDMünchenD
kub@bv@tum@de
www@st@bgu@tum@de
Many parts and figures of the present manuscript are taken from the German
lecture notes on “Platten” by Prof. E. Ramm [10], University of Stuttgart.
Lehrstuhl für Statik
Technische Universität München
80290 München
October 2008
2
0 REFERENCES................................................................................................................................. 4
2 PLATES IN BENDING................................................................................................................... 6
2.1
2.1.1
2.1.2
2.1.3
2.2
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
2.2.6
2.3
2.3.1
2.3.2
2.4
2.4.1
2.4.2
2.4.3
2.5
2.5.1
2.5.2
2.5.3
2.5.4
2.5.5
2.5.6
2.5.7
2.5.8
2.6
2.6.1
2.6.2
2.6.3
2.6.4
2.7
2.7.1
2.7.2
2.7.3
2.7.4
2.8
2.8.1
2.8.2
2.8.3
2.8.4
INTRODUCTION........................................................................................................................... 8
IDEALIZATION OF A PLATE .................................................................................................... 8
ASSUMPTIONS OF THE KIRCHHOFF PLATE THEORY .............................................................. 9
THE PRINCIPAL LOAD CARRYING BEHAVIOR ....................................................................... 10
THE DIFFERENTIAL EQUATION OF THE KIRCHHOFF THEORY .............................................. 11
REDUCTION TO THE MID-SURFACE ...................................................................................... 11
THE DIFFERENTIAL EQUATION ............................................................................................ 13
PRINCIPAL MOMENTS – STRESS RESULTANTS IN ARBITRARY SECTIONS ............................. 20
THE DIFFERENTIAL EQUATION OF THE PLATE WITHOUT TWIST RESISTANCE ...................... 20
VARYING TEMPERATURE LOADING AT TOP AND BOTTOM SURFACE ................................... 21
CIRCULAR PLATES ............................................................................................................... 25
BOUNDARY CONDITIONS ......................................................................................................... 28
THE EFFECTIVE SHEAR FORCE ............................................................................................. 28
VARIANTS OF BOUNDARY CONDITIONS (TABLE 2.9.) ......................................................... 32
SOLUTION METHODS ................................................................................................................ 33
OVERVIEW ........................................................................................................................... 33
ANALYTICAL SOLUTIONS .................................................................................................... 33
NUMERICAL METHODS ........................................................................................................ 47
STRUCTURAL BEHAVIOR ......................................................................................................... 48
POISSON RATIO ν ................................................................................................................. 48
ASPECT RATIO ..................................................................................................................... 50
CLAMPED EDGES ................................................................................................................. 51
FADE AWAY OF CLAMPED EDGE MOMENTS ......................................................................... 53
TWISTING STIFFNESS ........................................................................................................... 54
PLATE, SUPPORTED AT THREE EDGES .................................................................................. 57
VARYING STIFFNESS ............................................................................................................ 58
CONCENTRATED LOADS ...................................................................................................... 58
CONTINUOUS PLATES ............................................................................................................... 61
INTRODUCTION .................................................................................................................... 61
THE METHOD OF LOAD REDISTRIBUTION (BELASTUNGSUMORDNUNGSVERFAHREN) ........ 61
THE METHOD BY PIEPER/MARTENS [15]............................................................................. 64
POINT SUPPORTED PLATES .................................................................................................. 66
INFLUENCE SURFACES ............................................................................................................. 69
INTRODUCTION .................................................................................................................... 69
DEFLECTIONS ...................................................................................................................... 69
STRESS RESULTANTS ........................................................................................................... 70
EVALUATION ....................................................................................................................... 72
THE REISSNER/MINDLIN THEORY .......................................................................................... 74
INTRODUCTION .................................................................................................................... 74
KINEMATIC ASSUMPTIONS .................................................................................................. 74
BOUNDARY CONDITIONS ..................................................................................................... 76
THE KIRCHHOFF CONSTRAINT............................................................................................. 78
3
0 References
Books and papers in English Language:
[1]
Gould, Philipp L.: Analysis of Shells and Plates. Springer Verlag New York, 1988.
[2]
Pilkey, W.D., Wunderlich, W.: Mechanics of Structure: Variational and Computational
Methods. CRC Press, 1994.
[3]
Reddy, J. N.:Theory and Analysis of Elastic Plates. Taylor and Francis, London, 1999.
[4]
Szilard, R.: Theories and Applications of Plate Analysis. Wiley, 2004.
[5]
Timoshenko, S.P., Woinoswski-Krieger, S.: Theory of Plates and Shells. McGraw-Hill,
1987. (2. Aufl.)
[6]
Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Vol. 1: Basis, Vol. 2:
Solid Mechanics, Vol.3 Fluid Dynamics. 6. Auflage. Elsevier Butterworth and Heinemann, 2005.
[7]
Bischoff, M., Wall, W.A., Bletzinger, K.-U., Ramm, E., „Models and Finite Elements
for Thin-Walled Structures“, in (E.Stein, R.deBorst, T.J.R. Hughes, eds.) Encyclopedia
of Computational Mechanics, Vol. 2, Wiley, 59-137, 2004).
Plate theory (in German):
[8]
Girkmann, K.: Flächentragwerke. 6. Auflage. Springer-Verlag, Wien, 1963
[9]
Marguerre, K., Woernle, H.T.: Elastische Platten, BI Wissenschaftsverlag, Mannheim,
1975.
[10] Ramm, E.: Flächentragwerke: Platten. Vorlesungsmanuskript, Institut für Baustatik,
Universität Stuttgart, 1995.
[11] Hake, E. und Meskouris, K.: Statik der Flächentragwerke, Springer, 2001.
Finite Element Method (in German):
[12] Werkle, H.: Finite Elemente in der Baustatik. Vieweg Verlag, Wiesbaden, 1995.
[13] Bathe, K.-J.: Finite-Element-Methoden. Springer-Verlag, 1986.
[14] Link, M.: Finite Elemente in der Statik und Dynamik. Teubner-Verlag, Stuttgart, 1984.
[15] Ramm, E.: Finite Elemente für Tragwerksberechnungen. Vorlesungsmanuskript, Institut
für Baustatik, Universität Stuttgart, 1999.
Tables (in German):
[16] Czerny, F.: Tafeln für vierseitig und dreiseitig gelagerte Rechteckplatten. Betonkalender, 1987 1990, 1993 I. Teil → drillsteife Platten mit Gleichlast und linear veränderlicher Last
4
[17] Pieper, K., Martens, P.: Durchlaufende vierseitig gestützte Platten im Hochbau. Betonund Stahlbetonbau (1966) 6, S. 158-162, Beton- und Stahlbetonbau (1967) 6, S. 150151.
[18] Pucher, A.: Einflußfelder elastischer Platten. 2. Auflage. Springer-Verlag, Wien, 1958.
[19] Schneider, K.-J.: Bautabellen. 7. Auflage. Werner-Verlag, Düsseldorf, 1986.
[20] Schleeh, W.: Bauteile mit zweiachsigem Spannungszustand (Scheiben), Betonkalender
1978 (T2), Ernst & Sohn, Berlin.
[21] Stiglat, K., Wippel, H.: Platten. 2. Auflage. Ernst & Sohn, Berlin 1973
Concrete Design (in German):
[22] Leonhardt, F.: Vorlesungen über Massivbau, Teil 2: Sonderfälle der Bemessung. Springer-Verlag, Berlin, 1975.
[23] Leonhardt, F.: Vorlesungen über Massivbau, 3. Teil: Grundlagen zum Bewehren im
Stahlbetonbau. Springer-Verlag, 1974.
[24] Schlaich, J. und Schäfer, K.: Konstruieren im Stahlbetonbau. Betonkalender, 1993.
5
2 Plates in bending
7
2.1
Introduction
2.1.1 Idealization of a plate
The plate in bending is a plane structure. It is loaded laterally to its surface. The panel is subjected to loads acting within the system’s plane.
t
t
plate in bending
panel
Fig. 2.1: Plate structures.
Depending on the thickness-to-length relation several theories of plates have been developed:
Table 2.1. Plate theories.
t
t
,
lx ly
moderately thick
1
1
to
5
10
with transverse
shear deformation
thin
very thin
1
1
to
5
50
without transverse shear
deformation, mostly used
for practical applications
1
50
geometrically nonlinear, with membrane
deformation
<
theory
Reissner,
Mindlin
Kirchhoff
von Karman
related beam
theory
Timoshenko
Euler
Bernoulli
theory of second order
The displacements of very thin plates are usually so large that a geometrically non-linear theory (e.g. von Karman theory) becomes necessary which is able to consider the membrane action. On the other hand, the shear deformation of moderately thick plates has to be considered
(e.g. Reissner Mindlin theory). Most of the practical applications deal with thin plates. Within
the valid range of linear behavior a pure bending theory will be good enough and shear deformation can be neglected (Kirchhoff theory).
The analysis of plates can be reduced to a theory of a surface. The structural behavior can be
described by a bi-axial state of stresses and strains.
8
2.1.2 Assumptions of the Kirchhoff plate theory
The assumptions are:
1.) geometrically linear:
- small strains
- small deflections (”small” is problem depend-
-
2.) linear material:
- linear elastic (Hooke),
- in the most simple case homogeneous and isot-
-
3.) thin plate
ent)
rope
The latter assumption leads to the two Bernoulli hypotheses:
a) straight lines perpendicular to the mid-surface (i.e. transverse normals) before deformation remain straight after deformation.
b) transverse normals rotate such that they remain perpendicular to the mid-surface after deformation
Bernoulli a)
Bernoulli b)
Fig. 2.2: Bernoulli hypotheses.
and to a further presumption:
c) the transverse normals do not experience elongation (i.e., they are in-extensible),
i.e.
εz = 0
σz = 0
As a consequence, the local effects of load application cannot be described by the theory. The
assumptions ε z = 0 and σ z = 0 are in contradiction to materials with Poisson effect (υ ≠ 0) ,
however, this causes no further difficulties for analysis.
Bernoulli a + b: Kirchhoff theory
ϕx = −
∂w
∂x
Bernoulli a:
ϕx = −
∂w
+ γ xz
∂x
Reissner theory
Mindlin theory
9
Fig. 2.3: Comparison of Kirchhoff- and Reissner/Mindlin kinematics.
Neglecting the transverse shear strains γ xz and γ yz is inconsistent with the material law:
( shear stress )
τ xz = G ⋅ γ xz
yz
yz
{
{
≠0
=0
because of Bernoulli
a, b
Shear stresses exist because of the equilibrium of forces and stresses. Therefore, the related
material equations cannot be applied within the Kirchhoff theory. This is in contrast to the
Reissner/Mindlin-theory.
2.1.3 The principal load carrying behavior
The load carrying behavior of a plate can be compared with that of a beam grid which is resistant to bending and torsion. The related deformations are curvatures κx, κy and the twist κxy.
Fig. 2.4: Load carrying behavior of plates in bending.
10
2.2
The differential equation of the Kirchhoff theory
2.2.1 Reduction to the mid-surface
The plate will be described by its mid-surface:
t
t
real system
idealized system
Fig. 2.5: Idealization.
11
The integration of the stresses over the thickness of the plate gives the stress resultants which
can be divided into force (q x , q y ) and moment resultants (m x , m y , m xy ).
Fig. 2.6: Stress resultants.
stress resultants / unit length:
Fig.2.6: Stress resultants.
For example:
width 1
↓
+
t
2
1 ⋅ t 2 ⎡ kNm ⎤
m x = ∫ σ x z dz = σ ⋅
6 ⎢⎣ m ⎥⎦
t
−
2
{
sec tion mod ulus
+
R
x
t
2
2
⎡ kN ⎤
q x = ∫ τ xz dz = τ M
⋅ 1⋅ t ⎢ ⎥
xz ⋅
3
⎣m⎦
t
−
2
12
2.2.2 The differential equation
We will derive the differential equation analogously to the beam theory. A direct formulation
from continuum mechanics can be found in the literature [1]-[8].
The change of resultants from one edge to the other of an infinitesimal element is expressed
by a Taylor series expansion:
in general:
_______________
⎛
⎜
df (x 0 ) ⎜ 1 d 2 f ( x 0 )
f ( x 0 + dx ) = f (x 0 ) +
dx + ⋅
+ ...
⎜ 2!
dx
dx 2
⎜________________
⎝
higher order terms are neglected
Fig. 2.7: Taylor-series expansion.
For example:
m x (x + dx ) = m x +
Fig. 2.8: Taylor series expansion for the moment mx
13
∂m x
dx
∂x
The statical and geometrical equations are given for the infinitesimal element:
Fig. 2.9: Shear forces at the infinitesimal element
∑V = 0:
∂q
⎞
⎛
∂q
⎛
⎞
− q x dy + ⎜ q x + x dx ⎟ dy − q y dx + ⎜⎜ q y + y dy ⎟⎟ dx + p dx dy = 0 |: dx dy
∂x
∂y
⎝
⎠
⎠
⎝
∂q x ∂q y
+
+p=0
∂x
∂y
Fig. 2.10: Moments at the infinitesimal element.
∑M
y
= 0:
∂m yx
⎞
⎛
∂m x
⎛
⎞
dx ⎟ dy + m yx dx − ⎜⎜ m yx +
− m x dy + ⎜ m x +
dy ⎟⎟ dx − q x dy dx = 0
∂x
∂y
⎝
⎠
⎠
⎝
∑M
x
= 0:
∂m x
+
∂x
∂m y
+
∂y
14
∂m yx
∂y
∂m xy
∂x
− qx = 0
− qy = 0
BEAM
Statical
equations
+ Torsion MT
∑V :
∑M :
dQ
+p=0
dx
dM
−Q=0
dx
PLATE
∑V :
∂q x ∂q y
+p=0
+
∂y
∂x
∑M
y:
∂m x ∂m yx
+
− qx = 0
∂x
∂y
∑M
x
∂m y
∂y
+
∂m xy
∂x
5 unknowns
3 equations
2-times internally indetermined
− qy = 0
Curvature:
∂ϕ x
∂2w
=− 2
∂x
∂x
∂ϕ y
∂2w
=− 2
κy =
∂y
∂y
κx =
Geometry
dϕ
d2w
=− 2
dx
dx
+ Twisting Γ due to torsion
κ=
Twisting:
change of angle ϕx in direction y
Table 2.2. Kirchhoff plate differential equation.
15
κ xy =
∂ϕ x
∂2w
=−
∂y
∂x ∂y
BEAM
moment-curvature-relation
PLATE
moment-curvature-relation
from mx:
Material
κ mx x =
κ=
1
r
1
rx
κ my x = − ν κ mx x
A moment mx gives a curvature κ mx x in direction x and the curvature κ my x in
direction y due to the Poisson effect (shortening at the upper surface and
elongation at the bottom surface)
from my:
κ~M
κy y
κx ~ mx − ν my
κy ~ my − ν mx
m
Table 2.3. Kirchhoff plate differential equation.
16
m
κx y = − ν κy y
m
BEAM
elastic bending stiffness:
κ=
PLATE
1
EI
1
12
=
EI
E⋅ {
1 ⋅ h3
1
M
EI
κx =
width 1
m y = K (κ y + ν κ x )
M = EI
{κ
κy =
12
(m y − ν m x )
E t3
plate bending stiffness:
m x = K (κ x + ν κ y )
or:
or:
12
(m x − ν m y )
E t3
K=
E t3
12(1 − ν 2 )
beam flexural stiffness
moment twist relation:
Material
elastic torsion stiffness:
with:
IT =
b t3
3
M T = GI T Γ
E
G=
2 (1 + ν )
moment-twist-relation:
1
GI T
b
t
Due to equilibrium adjacent twisting moments across the corner of the
infinitesimal element are of the same magnitude. It holds:
m xy + m yx = 2 m xy
1
2 (1 + ν ) 3
1
=
=
3
elastic torsion stiffness: GI
E {
1⋅ t
2 K (1 − ν )
T
width 1
2 m xy = 2 K (1 − ν ) κ xy
Table 2.4. Kirchhoff plate differential equation.
17
1
t
BEAM
differential equations:
elimination of
curvatures
M = − EI
d2w
dx 2
PLATE
⎛ ∂2w
∂2w ⎞
m x = − K ⎜⎜ 2 + ν 2 ⎟⎟
∂y ⎠
⎝ ∂x
⎛ ∂ 2w
∂ 2w ⎞
m y = − K ⎜⎜ 2 + ν 2 ⎟⎟
∂x ⎠
⎝ ∂y
2
(m xy + m yx ) = − 2 K (1 − ν ) ∂ w
∂x ∂y
nd
2 derivatives
elimination of shear
forces from equilibrium
conditions
elimination of
moments
d 2M
d4w
=
−
EI
dx 2
dx 4
d 2M
= −p
dx 2
4
d w p
=
dx 4 EI
back substitution
⎛ ∂4w
∂ 2m x
∂4w ⎞
⎜
= − K ⎜ 4 + ν 2 2 ⎟⎟
∂x 2
∂x ∂y ⎠
⎝ ∂x
∂ 2m y
⎛ ∂4w
∂4w ⎞
⎜
⎟
=
−
+
ν
K
⎜ ∂y 4
∂y 2 ∂x 2 ⎟⎠
∂y 2
⎝
∂ 2 (m xy +m yx )
∂ 4w
= − 2 K (1− ν ) 2 2
∂x ∂y
∂x ∂y
2
2
∂ 2 m x ∂ (m xy + m yx ) ∂ m y
= −p
+
+
∂y 2
∂x ∂y
∂x 2
∂4w p
∂4w
∂4w
=
+2 2 2 +
∂y 4 K
∂x ∂y
∂x 4
Laplace operator Δ ( ) =
Table 2.5. Kirchhoff plate differential equation.
18
or
ΔΔw=
∂2 ( ) ∂2 ( )
+
∂y 2
∂x 2
p
K
∂q x ∂q y
+p=0
+
∂y
∂x
∂m x ∂m yx
− qx = 0
+
∂y
∂x
∂m y
∂y
+
∂m xy
∂x
⎛ ∂3w
∂ 3w ⎞
⎟
q x = − K ⎜⎜ 3 +
∂x ∂y 2 ⎟⎠
⎝ ∂x
− qy = 0
⎛ ∂3w
∂3w ⎞
⎟
+
q y = − K ⎜⎜
3
∂ x 2 ∂ y ⎟⎠
⎝ ∂y
∂2w
κx = −
∂x 2
κy = −
∂ 2w
∂y 2
κ xy = κ yx = −
∂2w
∂x ∂y
m x = K (κ x + ν κ y )
material
equations
m y = K (κ y + ν κ x )
constitutive equations
geometrical
equations
pde: of 4th order
inhomogeneous, linear
statical equations
2
2
∂ 2 m x ∂ (m xy + m yx ) ∂ m y
+
+
= −p
∂x ∂y
∂y 2
∂x 2
⎛ ∂2w
∂2w ⎞
m x = − K ⎜⎜ 2 + ν 2 ⎟⎟
∂y ⎠
⎝ ∂x
solution
→ w
⎛ ∂2w
∂2w ⎞
m y = − K ⎜⎜ 2 + ν 2 ⎟⎟
∂x ⎠
⎝ ∂y
m yx
stress resultants
∂2w
= − K (1− ν )
∂x ∂y
m xy + m yx =
= 2 K (1− ν ) κ xy
∂4w
∂4w
∂4w p
+
+
=
2
K
∂x 4
∂x 2 ∂y 2
∂y 4
p
or ΔΔw =
K
Laplace-operator:
∂2
∂2
Δ= 2 + 2
∂y
∂x
bipotential-operator: ΔΔ =
Table 2.6. Kirchhoff PDE construction scheme.
19
qx , qy
∂4
∂4
∂4
2
+
+
∂x 2 ∂y 2 ∂y 4
∂x 4
m x , m y , m xy
2.2.3 Principal moments – stress resultants in arbitrary sections
⊗ into plane
out of plane
Fig. 2.11: Stress resultants in arbitrary section
The internal forces are determined from the equilibrium conditions:
m n dt = m x cos ϕ dy + m y sin ϕ dx + m xy sin ϕ dy + m yx cos ϕ dx
m n = m x cos 2 ϕ + m y sin 2 ϕ + m xy sin 2ϕ
m t = m x sin 2 ϕ + m y cos 2 ϕ − m xy sin 2ϕ
1
(m y − m x ) sin 2ϕ + m xy cos 2ϕ
2
q n = q x cos ϕ + q y sin ϕ
m nt = m tn =
q t = − q x sin ϕ + q y cos ϕ
Principal moments from the condition mnt = 0:
m I, II =
1
(m x + m y ) ± 1
2
2
(m
− m y ) + 4 m 2xy
2
x
where: mI ≥ mII
The angle ϕ which is related to the first principal moment mI is
tan ϕ =
(m
x
− my ) +
2 m xy
(m
− m y ) + 4 m 2xy
2
x
2.2.4 The differential equation of the plate without twist resistance
The theoretical model of a plate without twist resistance and, therefore, without twisting moments can be developed. This is very close to the real behavior of ribbed plates („Rippenplatte)“ or coffered ceilings („Kassettendecken“) made from reinforced concrete.
20
Fig. 2.12: Infinitesimal plate element without twisting moments
⎛ ∂2w
∂2w ⎞
m x = − K ⎜⎜ 2 + ν 2 ⎟⎟
∂y ⎠
⎝ ∂x
⎛ ∂2w
∂2w ⎞
m y = − K ⎜⎜ 2 + ν 2 ⎟⎟
∂x ⎠
⎝ ∂y
m xy = 0
Together with the statical equations it follows:
∂4w ∂4w p
=
+
K
∂y 4
∂x 4
2.2.5 Varying temperature loading at top and bottom surface
It is assumed that the temperature distribution is linear through the plate thickness. Since we
are interested in the bending behavior of plates we assume further that the change of temperature at the mid surface vanishes, because this results in pure membrane stresses which are not
of interest at this print of discussion. With the abbreviations:
d)
α T [1 / C°]
temperature coefficient
e)
Tu = Tu (x , y )
temperature at bottom surface
f)
To = To (x, y )
temperature at top surface
and the definition of curvature due to temperature expansion:
κT =
α T (Tu − To )
t
we get the differential equation for the deflection w:
⎛ ∂ 2 κT ∂ 2 κT
ΔΔw = − (1 + ν ) ⎜⎜
+
2
∂y 2
⎝ ∂x
21
⎞
⎟⎟ = − (1 + ν ) Δκ T
⎠
equilibrium
geometry
∂q x ∂q y
=0
+
∂y
∂x
∂m x ∂m yx
− qx = 0
+
∂y
∂x
∂m y ∂m xy
− qy = 0
+
∂x
∂y
∂2w
κx = − 2
∂x
∂2w
κy = − 2
∂y
∂2w
κ xy = κ yx = −
∂x ∂y
m x = K (κ x + ν κ y − (1 + ν ) κ T )
material
m y = K (κ y + ν κ x − (1 + ν ) κ T )
2
2
∂ 2 m x ∂ (m xy + m yx ) ∂ m y
+
+
=0
∂y 2
∂x 2
∂x ∂y
∂4w
∂4w
∂4w
2
+
+
+
∂y 4
∂x 2 ∂y 2
∂x 4
⎡∂ 2 w ∂ 2 w
⎤
(
)
1
+
+
ν
κ
⎢ 2 +
⎥
T
∂y 2
⎣ ∂x
⎦
2
2
⎤
∂ ⎡∂ w ∂ w
(
)
+
+
ν
κ
qy = −K
1
⎢ 2 +
⎥
T
∂y ⎣ ∂x
∂y 2
⎦
qx = −K
∂
∂x
⎡ ∂ 2w
⎤
∂ 2w
+ (1 + ν ) κ T ⎥
mx = − K ⎢ 2 + ν
2
∂y
⎣ ∂x
⎦
⎡ ∂ 2w
⎤
∂2w
+ (1 + ν ) κ T ⎥
my = − K ⎢ 2 + ν
2
∂x
⎣ ∂y
⎦
2
∂ w
m xy = − K (1 − ν )
∂x ∂y
⎛ ∂ 2 κT ∂ 2 κT
+
ΔΔw = − (1 + ν ) ⎜⎜
2
∂y 2
⎝ ∂x
resp.
ΔΔw = − (1 + ν ) Δ κ T
solution w
stress resultants
qx , qy
m x , m y , m xy
m xy = + K (1 − ν ) κ xy
Table 2.7. Differential equation for temperature loading
∂ 2 (1 + ν ) κ T ∂ 2 (1 + ν ) κ T
=0
+
+
∂y 2
∂x 2
resp.
⎛ control : total curvature = temperature curvature (κ x = κ y = κ T )⎞
⎜
⎟
⎜
⎟
(
)
m
=
m
=
m
=
0
vanishing
resul
tan
ts
x
y
xy
⎝
⎠
α ⋅ (Tu − To )
κT = T
t
22
⎞
⎟⎟
⎠
The special case of equal temperature distribution at any point of the plate leads to:
Δ κT =
∂ 2κT ∂ 2 κT
=0
+
∂y 2
∂x 2
→
ΔΔw = 0
Considering a plate which is fully clamped at all edges we choose w(x, y) = 0 as a possible
solution.
The differential equation ΔΔw = 0 and the boundary conditions are satisfied:
w ( x =0) = 0
for all
0 ≤ y ≤ ly
w ( y=0) = 0
for all
0 ≤ x ≤ lx
=0
for all
0 ≤ y ≤ ly
=0
for all
0 ≤ x ≤ lx
∂w ( x = 0 )
∂x
∂w ( y = 0 )
∂y
Although the plate does not deflect there exist stresses and resultants:
m x = m y − K (1 + ν) κ T
m xy = 0
qx = qy = 0
The analogous result is known for the clamped beam subjected to a linear temperature distribution through the thickness:
M jk = − El α T
M ki = + M jk
Tu − To
t
t
Fig. 2.13: Clamped beam subjected to varying temperature distribution through thickness.
24
2.2.6 Circular plates
For convenience polar coordinates are chosen in the following
Fig. 2.14: Infinitesimal element of a circular plate.
25
Differential equation:
Δ Δ w ( r ,ϕ ) =
using the operator:
Δ=
p(r, ϕ)
K
∂2 1 ∂
1 ∂2
+
+
∂r 2 r ∂r r 2 ∂ϕ2
stress resultants:
⎡ ∂2w
⎛ 1 ∂ 2 w 1 ∂w ⎞⎤
⎟⎥
m r = − K ⎢ 2 + ν ⎜⎜ 2
+
2
r ∂r ⎟⎠⎦
⎝ r ∂ϕ
⎣ ∂r
⎡ ∂ 2 w 1 ∂ 2 w 1 ∂w ⎤
+
m ϕ = − K ⎢ν 2 + 2
⎥
r ∂ϕ 2 r ∂r ⎦
⎣ ∂r
m ϕr = m rϕ = − K (1 − ν )
∂
∂r
qr = − K
∂
(Δw )
∂r
qϕ = − K
1 ∂ (Δw )
r ∂ϕ
⎛ 1 ∂w ⎞
⎜⎜
⎟⎟
⎝ r ∂ϕ ⎠
In the case of rotational symmetry the derivatives with respect to ϕ vanish:
Δ( ) =
d2 1 d
+
dr 2 r dr
d 4 w 2 d 3 w 1 d 2 w 1 dw p (r )
+
− 2
+ 3
=
r dr 3
K
dr 4
r dr 2
r dr
The resultants remain to be:
⎡d2w
1 dw ⎤
mr = − K ⎢ 2 + ν
⎥
r dr ⎦
⎣ dr
⎡ d 2 w 1 dw ⎤
m ϕ = − K ⎢ν 2 +
⎥
r dr ⎦
⎣ dr
m ϕr = m rϕ = 0
q r = −K
2
3
2
d
(Δ w ) = −K d ⎡⎢ d w2 + 1 dw ⎤⎥ = − K ⎛⎜⎜ d w3 + 1 d w2 − 12 dw ⎞⎟⎟
dr
dr ⎣ dr
r dr ⎦
r dr
r dr ⎠
⎝ dr
qϕ = 0
26
27
2.3
Boundary Conditions
2.3.1 The effective shear force
2.3.1.1 The reason
As an effect of the Bernoulli hypotheses (plane cross section and normal condition ) the plate
differential equation is of fourth order. Consequently, in the solution there exist four integration constants for each direction, i.e. two at each edge. However at the edges there are three
independent resultants or displacements defined, respectively. I.e., they cannot be independent
from each other.
Fig. 2.15: Stress resultants and displacements at an edge.
From the Bernoulli hypotheses it follows that the rotations are equal to the negative gradient
∂w
of deflection, e.g.: ϕ y = −
. That means, that if the displacements w(y) at the edge x = lx
∂y
are given, the rotations ϕy are also known.
Therefore, and also with respect to energetic reasons, only two independent resultants can be
defined. Consequently, shear force and twisting moment are combined to the „effective shear
force“ q .
reduction of
displacements
ϕx
ϕx
ϕy ⎫
→ w
w ⎬⎭
reduction of
resultants
mx
mx
m xy ⎫
effective shear force
→
⎬
qx
qx ⎭
Table 2.8. Reduction of edge degrees of freedom, Kirchhoff plate theory
2.3.1.2 Illustrative argumentation
The effective shear force can be directly developed from an energy principle. However, we
choose an illustrative approach. The twisting moment is now replaced by a couple of forces.
Due to the principle of St.-Venant this modification at the edge does not affect the plate behavior at the inner region.
28
a)
Portion of the twisting moment
resultant force at element interface
b) Portion of the shear force
Addition of a) and b)
Sum:
distributed on dy → effective shear force
qx = q x +
Fig. 2.16: Effective shear force definition.
29
∂m xy
∂y
Fig. 2.17: Effective shear forces.
qx = qx +
∂m xy
⎛ ∂3w
∂3w
∂3w ⎞
⎟
(
)
= − K ⎜⎜ 3 +
−
−
ν
1
K
∂y
∂x ∂y 2
∂x ∂y 2 ⎟⎠
⎝ ∂x
⎡∂3w
∂3w ⎤
q x = − K ⎢ 3 + (2 − ν )
⎥
∂x ∂y 2 ⎦
⎣ ∂x
⎡∂3w
∂3w ⎤
q y = − K ⎢ 3 + (2 − ν ) 2
⎥
∂x ∂y ⎦
⎣ ∂y
Remarks:
-
support forces ( =ˆ effective shear forces) and shear forces at edges are not identical if
twisting moments exist
At free edges the effective shear forces vanish ( q = 0); however, in general the two components do not vanish. I.e., at free edges shear forces and twisting moments may exist.
The approximation takes effect only in a small layer near the boundary (boundary layer).
2.3.1.3 Concentrated corner force
A well anchored plate subjected to lateral load will develop tension support forces near the
corners. On the other hand, a plate without anchors will lift off the supports. The Kirchhoff
plate theory accumulates all tension forces near the corner to one concentrated tension force at
the corner.
Fig. 2.18: Corner force and support forces distribution.
30
2.3.2 Variants of boundary conditions (Table 2.9.)
boundary
conditions
Type
x
w=0
y
clamped
∂w
=0
∂x
x
w=0
y
simple supported
mx = 0
consequences
∂w
=0
∂y
∂2w
= 0 → m xy = 0 → sup port forces = q x
∂x ∂y
∂w
=0
∂y
∂2w
∂2w
=
=0
0
;
∂x 2
∂y 2
∂w
∂2w
≠ 0;
≠ 0 → m xy ≠ 0 → sup port forces = q x
∂x
∂x ∂y
x
qx = 0
y
free edge
mx = 0
x
w = f qx
y
both components qx and
∂m xy
∂y
cancel each other
often coupled elastic foundation, e.g. plate with edge
beam
mx = 0
f
elastic
x
fM
w=0
∂w
= f M mx
∂x
y
32
if edge beam is eccentric than membrane and bending
actions in the plate have to be considered
2.4
Solution methods
2.4.1 Overview
The problem is to solve the differential equation while the boundary conditions have to be
considered. There exist only very few special cases where both demands can be satisfied exactly. Mostly, the problem can be solved only approximately using numerical techniques.
exact
solutions
approximate
solutions
analytical
numerical
closed solutions
series expansions
Fig. 2.19: Solution methods.
finite differences
finite elements
For practical applications the methods are prepared for rules-of-thumb, diagrams, tables and
computer programs.
2.4.2 Analytical solutions
2.4.2.1 Circular plate
Considering the special case of rotational symmetrical loading, we have (compare paragraph
2.2.6):
p(r )
d 2 w 1 dw
where Δ w = 2 +
PDE: Δ Δ w =
K
r dr
dr
4
3
2
d w 2 d w 1 d w 1 dw p(r )
+
− 2
+ 3
=
r dr 3
K
r dr 2
r dr
dr 4
2
1
1
p(r )
d( )
′
or:
=( )
w IV + w ′′′ − 2 w ′′ + 3 w ′ =
where
r
r
r
K
dr
Fig. 2.20: Circular plate with rotational symmetrical loading.
33
The solution of the differential equation is:
w = w p + C1 + C 2 ρ 2 + C3 ρ 2 ln ρ + C 4 ln ρ
w ′ = w ′p + C 2
w ′′ = w ′p′ + C 2
2
1
ρ
ρ + C3 (2 ln ρ + 1) + C 4
a
a
aρ
2
1
(2 ln ρ + 3) − C 4 21 2
+ C3
2
a
a2
a ρ
w ′′′ = w ′p′′ + C 3
where: ρ =
2
a ρ
3
+ C4
2
a ρ3
3
r
and particular solution wp.
a
The integration constants Ci allow to satisfy the 2 x 2 boundary conditions at both edges of a
circular plate. The stress resultants are:
1 ⎞
⎛
m r = − K ⎜ w ′′ + ν w ′ ⎟
r ⎠
⎝
1 ⎞
⎛
m ϕ = − K ⎜ ν w ′′ + w ′ ⎟
r ⎠
⎝
1
1
⎛
⎞
q r = − K ⎜ w ′′′ + w ′′ − 2 w ′ ⎟
r
r
⎝
⎠
m ϕr = m rϕ = q ϕ = 0
Fig. 2.21: Stress resultants of a plate with rotational symmetrical loading
a) Circular plate with constant loading
Fig. 2.22: Clamped circular plate with constant loading.
boundary conditions:
particular solution:
plate center
clamped edge
ρ = 0:
ρ = 1: w = 0
w' = 0
( w finite)
w' = 0
p r4
wp =
64 K
qr = −
34
pa
2
From the condition w´ = 0 at the plate center (ρ = 0) the integration constant C4 is determined
to be C4 = 0. Otherwise w´ would be singular. From qr = - p · a/2 at the edge we get C3 = 0.
Using the conditions w = w´ = 0 at the edge the constants C1 and C2 can be determined:
C2 p a 4
=
2
64 K
2
p
w=
a2 − r2
64 K
p
w′ =
r3 − a 2 r
16 K
p
w ′′ =
3 r2 − a2
16 K
3
w ′′′ =
pr
8K
C1 = −
solution:
stress resultants:
(
)
where: w center =
(
)
(
)
[
) ]
p a2
(3 + ν ) 1 − ρ 2 − 2
16
p a2
(1 + 3 ν ) (1 − ρ2 ) − 2 ν
mϕ =
16
pa
qr = −
ρ
2
mr =
p a4
64 K
(
[
]
Setting Poisson’s ratio ν = 0 we get the following distributions for deflection w, moments mr
and mϕ and shear force qr. For comparison the results for the simply supported plate are also
given. In contrast to a beam the deflections and moments are reduced by the factor 3/8 because of the 2-dimensional load carrying behavior of the plate.
Fig. 2.23: Constant loaded circular plate (Poisson ratio ν = 0).
35
b) Circular plate with concentrated load of plate center
Fig. 2.24: Simply supported circular plate.
bondary conditions:
edge
plate center
ρ = 1:
w=0
ρ = 0: w´ = 0
mr = 0
qr = −
particular solution:
P
2πa
wp = 0
w = C1 + C 2 ρ 2 + C3 ρ 2 ln ρ + C 4 ln ρ
Using w´ = 0 at the plate center, we get:
C4 = 0
and q r = −
P
2πa
at the edge ρ = 1 yields:
C3 =
P a2
8πK
Finally, the remaining conditions give:
C1 = − C 2 =
P a 2 (3 + ν )
16 π K (1 + ν )
Back substituting the constants gives the following expressions for the deflection:
P a2
w=
16 π K
⎡3 + ν
⎤
(
1 − ρ 2 ) + 2 ρ 2 ln ρ⎥
⎢
⎣1 + ν
⎦
and stress resultants:
mr = −
P
(1 + ν ) ln ρ
4π
36
mϕ = −
qr = −
P
[(1 + ν ) − (1 + ν ) ln ρ]
4π
P
2πaρ
Evaluating the expressions, we recognize at the plate center a finite deflection, however, singular values of the stress resultants.
Fig. 2.25: Concentrated load at center of a circular plate (Poisson ratio ν = 0).
More solutions for circular and annular plates are given in e.g. [3], Fehler! Verweisquelle
konnte nicht gefunden werden. and Fehler! Verweisquelle konnte nicht gefunden werden..
2.4.2.2 Rectangular plates
By means of series expansion the partial differential equation ΔΔw(x,y) = p(x,y)/K can be decoupled into ordinary differential equations. It depends on whether one or both directions are
expanded that it is called single or double series expansion. As an example we shall deal with
the single series expansion in the following.
The single series expansion uses an ansatz which satifies the Navier conditions (w = 0 and
Δw = 0) of two opposite edges. Therefore, it can only be applied for plates which are simply
supported at two opposite edges. The support conditions of the remaining edges can be chosen arbitrarily.
a) Plate strip
Load distribution and deflection are constant in y-direction. The differential equation further
reduces to:
37
d 4 w (x ) p(x )
=
K
dx 4
This is equivalent to the differential equation of a beam except for the Poisson ratio ν which
contributes to the plate stiffness K.
Load p and deflection w are expanded by a Fourier series. The expansion can be reduced to
the odd components (p(x) = - p(-x)) with the period L = 2 a:
Fig. 2.26: Plate strip.
It follows for the load expansion
2nπx
= ∑ p n sin α n x
L
n
p(x ) = ∑ p n sin
n
where pn are the Fourier coefficients and
αn =
2nπ nπ
=
L
a
Multiplying by sin αm x and integrating yields:
a
a
∫ sin (a x ) p(x ) dx = ∫ sin (a x ) ∑ p
m
0
sin (α n x ) dx
144444244444
3
m
n
n
0
for m ≠ n
=0
= pn
a
2
for m = n
The Fourier coefficient for the n-th series component is:
38
a
2
p n = ∫ p(x ) sin (α n x ) dx
a 0
Analogously, the deflection w(x) is represented by:
w (x ) = ∑ w n sin (α n x )
n
which implies the Navier conditions (w = 0 and w´´ = 0) at both edges
Inserting load and deflections into the differential equation:
∑α
4
n
w n sin (α n x ) =
n
1
K
∑p
n
sin (α n x )
Comparison of coefficients yields:
wn =
pn
α 4n K
and the general solution:
w (x ) =
a4
K π4
pn
∑n
n
4
nπ
a
n = 1, 2, 3, ...
sin (α n x )
αn =
Back substitution gives the stress resultants:
mx = − K
d2w
dx 2
my = ν mx
qx = − K
d3w
dx 3
The deflection w(x) converges very fast because of n4 in the denominator. The stress resultants converges not as good because of the factors 1/n2 and 1/n for moments and shear force,
respectively.
The special case of constant area load yields:
a
2
4p
p n = p ∫ sin (α n x ) dx =
a 0
nπ
w (x ) =
4pa2
K π5
1
∑n
5
sin (α n x )
39
n = 1, 3, 5, ...
n = 1, 3, 5, ...
The load is approximated as:
real load
approximation by one series component n = 1
p=
4p
π
sin x
π
a
approximation by two series components n = 1, 3
p=
4p⎛
3π ⎞
π
1
x⎟
⎜ sin x + sin
π ⎝
a
3
a ⎠
Fig. 2.27: Approximation of loading.
The even series components n = 2, 4, 6, ... are neglected since they would cause antisymmetric distributions.
exact
deflection w
moment m
0.013021
1 series component
(N = 1)
p a4
K
0.125 p a 2
2 series components
(n = 1, n = 3)
p a4
K
p a4
K
p a4
− 0.000054
K
p a4
= 0.013017
K
0.129001 p a 2
0.129006 p a 2
0.013071
0.013071
− 0.004778 p a 2
= 0.124228 p a 2
Table 2.10. Comparison exact solution — series expansion at position x = ½ a
For constant distributed load even one series component gives sufficient accuracy as the comparison with the exact solution shows. More varying load distributions require more series
components, respectively.
40
b) Rectangular plate with two opposite Navier supported edges
The homogenous differential equation:
∂4w
∂4w
∂4w
=0
2
+
+
∂y 4
∂x 2 ∂y 2
∂x 4
is solved by the ansatz in product form:
w h = ∑ Fn (y ) sin (α n x ) ;
αn =
nπ
a
which implicitly satisfies the Navier conditions. The ansatz represents the function wn by a
sin-expansion in x-direction and by a function F(y) which is dependent on y only.
arbitrary
boundary conditions
Fig. 2.28: Rectangular plate and boundary conditions.
By the product form of the ansatz the partial differential equation decouples into a system of
4th order differential equations:
2
⎞
⎛ d 4 Fn (y )
2 d Fn ( y )
⎜⎜
2
−
α
+ α 4n Fn (y )⎟⎟ sin α n = 0
n
4
2
dy
dy
⎝14
44444424444444
3⎠ 123
=0
≠0
A possible solution is given by making use of the e λy function:
Fn (y ) =
1
(A n cosh α n y + α n y Bn sinh α n y + C n sinh α n y +
a 2n
+ α n y D n cosh α n y ) ;
αn =
nπ
a
And, therefore:
wh = ∑
n
1
(A n cosh α n y + α n y Bn sinh α n y +
α 2n
C n sinh α n y + α n y D n cosh α n y ) sin α n x
14444
444
4244444444
3
Fn ⎛⎜⎝ y ⎞⎟⎠
41
The 4 integration constants allow to satisfy 4 boundary conditions in y-direction, i.e. 2 at each
edge
•
particular solution:
As particular solution the plate strip is chosen which is simply supported at the edges x = 0
and x = a, respectively
•
total solution:
Superposition of homogenous and particular solutions yields:
α4
w=
K π4
pn
∑n
n
4
sin α n x + ∑
n
1
(A n cosh α n y Bn sinh α n y +
α 2n
+ C n sinh α n y + α n y D n cosh α n y ) sin α n x
With constant distributed load p = constant:
w=
4p
Ka
1
∑a
n
5
n
sin α n x + ∑
n
1
(A n cosh α n y + α n y Bn sinh α n y +
a 2n
+ C n sinh α n y + α n y D n cosh α n y ) sin α n x
The following types of plates are covered by this solutions
Fig. 2.29: Rectangular plates with two opposite Navier-supported edges.
It is assumed that p = p(x) or p = constant. For arbitrary distributed load p = p(x,y) only the
particular solution has to be replaced.
42
c) Some examples
The following examples, which had been determined by series expansions, are taken from
tables [16], [17].
Among others there exist tables for various types of support conditions under constant load,
triangular and trapezoidal load which approximates the distribution of soil pressure. Mostly
the Poisson ratio ρ = 0 is used for the generation of table values.
Fig. 2.30: Choice of boundary conditions and loads as found in tables ( 1.0 ≤ ly/lx ≤ 2.0 )
43
Fig. 2.31: Example from Czerny, Betonkalender 1987
44
rectangular plates
under constant
area load
simple supported
rectangular plate
under triangular
area load
simple supported
rectangular plate
simple supported
at three edges
Fig. 2.32: Stress resultant distributions from Czerny: Plattentafeln, Betonkalender, 1987
45
Fig. 2.33: Example from Stiglat/Wippel: Platten
46
2.4.3 Numerical methods
2.4.3.1 Introduction
Numerical methods discretize either the differential equation (local or strong formulation) or
an energy expression (global or weak formulation). The continuous problem is transferred
into a system with a finite number of degrees of freedom. Among many other techniques we
restrict ourselves to mention the finite difference method, the finite element method, and the
boundary element method. The finite difference method is very effective for plates of regular
geometry, however, had been replaced by the finite element method almost entirely. The finite element method is a little bit more expensive, but much more flexible for general problems. The boundary element method requires the solution of the differential equation and discretizes only the edge of the plate. Therefore, the method can be superior to the finite element
method. On the other hand, however, it looses its advantages if many information about the
interior of the plate region is to be determined. In the following we will concentrate on the
finite element method.
2.4.3.2 The finite element method (FEM) – a short overview
A characteristic feature of the method is that the structure is divided into many elements of
finite size. The originally continuously distributed stresses and displacements are represented
by discrete values which are defined at the nodes of the elements (discretization). By these
nodes adjacent elements are coupled and can be composed to the complete structure. Proper
shape functions on element level insure that common edges of adjacent elements do not gape
during deformation. The accuracy of a finite element calculation depends on how good the
mechanical behavior of the whole structure can be approximated by the local shape functions
on element level.
real system
finite element model
deflections: wFE < w
Fig. 2.34: Discretization by finite elements, using linear displacement interpolation
A coarse discretization usually gives too stiff results, i.e. too small displacements.
The results can be improved either by a finer discretization or by higher order interpolation
functions (linear → quadratic → cubic → ...) on element level.
47
Fig. 2.35: Mesh refinement and higher order elements
The FEM is suited for computer application only since by the discretization procedure very
large systems of equations are generated with a very large number of unknowns.
In practice, the FEM appeared to be successful. During the last years increasingly powerful
and comfortable programs have been developed which assist usage by graphical interfaces for
pre- and post-processing as well as by links to CAD- and design packages.
The application of the FEM affords a careful choice of suitable elements and discretization.
Reentrant corners, point supports, concentrated loads, and the modeling of the boundary conditions required special considerations. Convergence studies on the accuracy and error analyses can give valuable information. For reading see [6], [12]-[15] and FEM-manuscript.
2.5
Structural behavior
The structural behavior of plates is very much different from that of plane frames. Some of
the most important parameters will, therefore, be discussed in the sequel.
2.5.1 Poisson ratio ν
The basic plate theory presumes homogeneous and isotropic cross sections. For reinforced
concrete this approximately applies only for the uncracked state (“Zustand I”). However, the
structure is designed for the cracked state (“Zustand II”). In the cracked tension zone, where
all forces are carried by the steel, exists no lateral contraction effect any more. Bittner Fehler!
Verweisquelle konnte nicht gefunden werden. has shown that tension reinforcement in RC
plates should be designed for ν = 0. However, it should be noted that the compression forces
will be underestimated by this simplification. DIN 1045 allows to assume ν = 0.2 for the
evaluation of the stresses. For simplification ν = 0.0 can also be taken.
The influence of the Poisson effect is obvious for the infinitely long, simply supported plate
strip under constant load. In addition to the moment, which already is known from the beam
theory, an additional moment in the lateral direction occurs which depends on the Poisson
effect. By this effect the displacements of the plate strip are less than those of the equivalent
beam.
48
beam
moments
displacement
M=
w=
pl
8
quadratic plate (ν = 0)
p l2
mx =
8
my = ν mx
2
5 p l4
384 EI
w=
5 p l4
5 p l4
=ˆ
(
1− ν2 )
384 K
384 EI
Fig. 2.36: Comparison beam – plate strip
Using the formulas which are given in Fehler! Verweisquelle konnte nicht gefunden werden. the resultants for different Poisson ratios can be calculated.
49
2.5.2 Aspect ratio
The load carrying behavior of plates is mainly determined by the geometry of the plate. The
aspect ratio ly/lx is the most important parameter of rectangular plates. The load is carried
identically in both directions for quadratic plates (mx = my). With increasing ratio ly/lx the
load is carried more and more across the shorter span lx. From the ratio ly/lx ≥ 2 on the oneaxial behavior of plate strips (mx, my = ν⋅mx) can be assumed.
Fig. 2.37: Influence of the aspect ratio ly/lx on the field moments mx, my of the simple supported rectangular plate, from Czerny-tables, ν = 0.
50
2.5.3 Clamped edges
Compared to the beam, clamped edges have much less effect on the field moments.
beam
quadratic plate (υ = 0)
simple
supported
clamped
ratio of
field moments
p l2
8
p l2
24
p l2
27.2 ≈ 2
p l2
56.8
= 3
Fig. 2.38: Influence of clamped suports of quadratic plates and beams
The reason are the twisting moments myx = mxy. For the clamped plate they not only vanish at
the axes of symmetry but also at the edges. Therefore, the favorable influence of the center
term of the differential equation disappears. Or, in other words, the simple supported plate is
still clamped along the diagonals by the twisting action.
51
simple
supported
clamped
Fig. 2.39: Twisting moments of the quadratic plate.
The next example shows the influence of the aspect ratio of a rectangular plate which is
clamped at one long edge.
Plate
Beam
_____
all edges
simple supported
________
one long
edge
clamped
field moments
support moments
Fig. 2.40: Influence of aspect ratio on the effect of clamped supports.
52
The influence of the relatively large clamped edge moment mxem reduces the field moments
only with increasing aspect ratio ly/lx when the structural behavior becomes one-axial anyhow.
Next, for a plate of given aspect ratio ly/lx the effect of additional clamped supports on field
and support moments is demonstrated.
Fig. 2.41: Effect of different combinations of clamped and simple supported edges on fieldand support moments.
It is obvious that the support moments mxr and myr are comparatively large, the field moments, however, do not reduce by the same factors.
2.5.4 Fade away of clamped edge moments
Bending moments fade away very fast from the clamped edge to the interior of the plate.
Their effect on the field moments and on the edge moments at the opposite edge is restricted.
On the other hand the improvement of plate stiffness by additional adjacent fields is limited.
53
Fig. 2.42: Moment transmission for beam and plate
2.5.5 Twisting stiffness
The twisting stiffness of a plate is its ability to react on principle moments which are not parallel to the edges. The amount of the twisting moments is a measure of how much the directions of the principle moments deviate from the edge parallel directions x and y.
The so called coffered ceilings (“Kassettendecken”) are exceptions which cannot resist twisting moments because of their design, or prefabricated plates with additional on site concrete
layer (“Fertigplatten mit statisch mitwirkender Ortbetonschicht), which – with reference to
DIN 1045 – must be analyzed disregarding the twisting stiffness.
2.5.5.1 Twisting resistant plate
The simple supported, not anchored plate is mainly supported at the inner edge regions. The
corners lift off. The lift off can be omitted by extra load or anchors at the corners. As reaction
negative principal moments m1 develop near the corners with tension at the plate top surface
and in lateral direction positive principal moments m2 with tension at the bottom. The directions of the principal moments change at the corner by 45° which is caused by twisting moments mxy of the same absolute value as m1 and m2. The corner anchor force is A = 2⋅mxy.
extra
load at corner
at the corner –m1 = m2 = |mxy|
without anchorage
A = 2 m xy
corner
force
with anchorage
Fig. 2.43: Twisting resistant square plate.
54
twisting moments
principal moments
along diagonal
principal
directions
Fig. 2.43 (cont.): Twisting resistant square plate.
If the corners are not anchored, or the reinforcement at the corner is insufficient, or there are
relative large openings near the corner the twisting resistance is reduced. Consequently the
field moments increase. Regarding Heft 240, DafStb, the field moments which have been
evaluated considering full twisting resistance have to be enlarged by the factors mentioned
there.
δ1
δ2
Mx
My
for field moments evaluated without Poisson
effect (ν = 0.0)
for field moments evaluated with (ν = 0.2)
produces stresses in xdirection
produces stresses in ydirection
Fig. 2.44: Field moment enlargement factors for the simply supported plate with reduced
twisting resistance w.r.t. Heft 240, DafStb Fehler! Verweisquelle konnte nicht gefunden werden.
55
2.5.5.2 Plate without twisting resistance
Regarding DIN 1045 no twisting resistance is allowed to be considered for pre-fabricated
plates or for two axial ribbed plates (compare section 2.4). Consequently, the field moments
increase considerably as can be seen in comparison to the twisting resistant plate for several
aspect ratios
Fig. 2.45: Comparison of field moments of not twist resistant and twist resistant rectangular
plates (Czerny tables [16]; Stiglat/Wippel: [21])
It can be seen that mx increases even beyond the beam value 18 (p l 2x ) . This is possible because the group of beams (2) in one direction is elastically supported by the beams (1) in the
other direction. It, therefore effects as additional load.
56
Fig. 2.46: Increased field moments larger than beam values.
2.5.6 Plate, supported at three edges
The principal moment direction and the load carrying behavior extremely depend on the aspect ratio ly/lx.
x
y
at corners:
m1 positiv
m2 negativ
Fig. 2.47: Principal moment directions of plates simply supported at three edges.
The moment trajectories show:
ly ≤ lx:
ly ≥ lx
•
•
•
•
•
loads are transferred to the corners
m1 and m2 are almost as large as mxfrm (moment in x-direction at free edge)
if ly < 0.5 lx than m1, m2 > mxfrm
one axial load carrying behavior near the free edge
the plate is still heavily twisted at regions far from the free edge y ≤ ly
57
2.5.7 Varying stiffness
Although “stiffness attracts forces” the stresses decrease.
system and load
stress resultants
moment scale
maximum stresses
σy =
m1y:3
m1y:1
= 1.48
my ⋅ 6
h2
→
σ1y:3
σ1y:1
= 0.16
Fig. 2.48: Plate with varying stiffness compared to plate of constant thickness
2.5.8 Concentrated loads
Concentrated loads on plates induce the largest stresses at the loading point. The theory gives
even singular solutions. This is shown very obviously by means of the influence surface of a
bending moment (compare paragraph 7).
58
Fig. 2.49: Influence surface for mx at field center of a simply supported plate strip, [18]
If the points of loading and bending moment deviate only a little the volume of the influence
surface below the load, i.e. the moments value, decreases very rapidly. In the close vicinity of
the loading point the stress state is almost rotational symmetrical.
section in x-axis
section in
y-axis
Fig. 2.50: Moment distributions in a plate strip under concentrated loads.
a) Field moments
The maximum field moments always appear at the loading point. The absolute value of the
moments are mainly influenced by the ratio of loading area to span width. Because of the fading behavior of stress resultants (Fig. 2.50) the load position, aspect ratio, and boundary conditions are less important.
59
Therefore, the moment under the load decreases much more slower compared to the beam if
the load position is moved towards the support. It can also be seen from the mx-influence surface that for most cases several different concentrated loads do not effect each other. Consequently, dimensioning of plates is very much dominated by the concentrated load and its vicinity rather than by the whole structure.
b) Support moments
The load position dominates the amount of the support moments. Concentrated loads near
clamped edges result in large support moments. It is recommended to evaluate influence surfaces for heavy loads.
Fig. 2.51: Influence of load position on the support moment of a strip [21], Fehler! Verweisquelle konnte nicht gefunden werden. compared to the beam.
60
2.6
Continuous plates
2.6.1 Introduction
Continuous plates exist in multiple combinations, e.g. see fig 6.1.
Fig. 2.52: Examples of continuous plates
The maximum stress resultants of those plate systems can be approximated by methods which
are based on the solution of single plates and which are discussed in the sequel. We presume
that (1) the edge supports are vertically fixed, i.e. supporting beams are infinitely rigid, (2)
that the plate structure freely rotates on the supports, and (3) that the loading can be represented by field wise combinations of uniform loading patterns.
More precise results can be obtained by numerical methods (FEM, BEM, FDM) especially for
plate systems of general geometry.
2.6.2 The method of load redistribution (Belastungsumordnungsverfahren)
The assumptions are:
-
regular grid of plates, difference of spare width in one direction max 25 %
simply supported peripheral edges
constant thickness everywhere in the plate system
field wise constant distributed load
61
2.6.2.1 Field moments
To determine the minimum and maximum field moments the plate system is uniformly loaded
by the dead load g and in a checkerboard manner by the life load p.
i max. moment
k min. moment
g uniformly distributed
p checkerboard
Fig. 2.53: Load distribution for min/max field moment.
In the next step the load is re-distributed and is split into a symmetrical and an antisymmetrical portion.
q′ = g +
p
2
(uniformly distributed)
q ′′ = ±
p
2
(checkerboard)
Fig. 2.54: Load re-distribution.
The maximum field moments for both load parts q’ and q” are taken from the tables by
Czerny and are superposed. It is ignored that the maximum moments occur at different positions. The boundary conditions for the single one span plates are:
-
load case q’: all continuity edges clamped
load case q”: all edges simply supported
62
max./min.
field moment
p
p
⎛
⎜g +
max m = l 2 ⎜
2± 2
x
min F
m
mf "
⎜ f'
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
2.6.2.2 Support moments
The maximum support moment at the common edge of two adjacent plate fields is due to
loading of both adjacent fields by g and p. In the remaining fields p is distributed in a checkerboard pattern.
max. support
moment at
common edge
of fields
i , k
g (uniformly distributed
p (checkerboard modified)
Fig. 2.55: Load combination for maximum support moment.
q′ = g +
p
(uniformly distributed)
2
q ′′ = ±
p
2
(checkerboard)
Fig. 2.56: Load re-distribution for maximum support moment
63
Again, the support moments are taken from Czerny’s tables. The boundary conditions are:
-
load case q’: all continuity edges clamped
load case q”: the considered edge clamped, all others simply supported
The mean value of both adjacent fields is taken.
Max. support moment:
1
ms = −
2
⎡ ⎛
p
p ⎞
p
p ⎞⎤
⎛
⎟
⎜g +
⎟
⎢ 2 ⎜g + 2
2 ⎟ + l2 ⎜
2 + 2 ⎟⎥
⎜
+
l
⎢ x ,i
⎥
x ,k
m′s′,i ⎟
m′s′,k ⎟⎥
⎜ m′s ,k
⎢ ⎜⎜ m′s ,i
⎟
⎜
⎟
⎢⎣ ⎝
⎠
⎝
⎠⎥⎦
144
42444
3 144
42444
3
field i
field k
2.6.3 The method by Pieper/Martens [17]
The authors had studied many practical plate systems of different spans, aspect ratios, and
combinations of field wise constant loading (p ≤ 2/3 q) and had evaluated how large field and
support moments really are. From the results they derived rules for a simple and fast approximated calculation.
They found that in general field moments can be determined as the mean value of the field
moments for a simply supported plate and those of a plate which is clamped at the interior
continuity edges. The mean values are listed in tables for several types of plates. Those cases
where after two small fields the third is larger need extra considerations.
q ⋅ l 2x
mean values (listed in tables) m f =
f
Fig. 2.57: Field moments regarding Pieper/Martens
The support moments are determined as the mean value of the clamped edge support moments of both adjacent plate fields but not less than 75 % of the larger clamped edge moment
(ratio of spans smaller than 5 : 1). If the ratio of spans is larger than 5 : 1 the clamped edge
moment of the larger field has to be taken.
64
Fig. 2.58: Support moments regarding Pieper/Martens
Fig. 2.59: Method by Pieper/Martens [17] taken from Schneider Bautabellen [19]
65
2.6.4 Point supported plates
Roof slabs (“Flachdecken”) are often point wise supported by columns.
The method of re-distributed loads can be used to determine the min/max stress resultants and
support reactions. Analogously to paragraph 6.2 the dead load g is equally distributed over the
entire plate structure where the life load p is circularly arranged (Fig. 2.60 – 2.63).
i max V hatched fields loaded
k min V white fields loaded
Fig. 2.60: Load arrangement for min./max. support force V
66
i max ms
k min ms
Fig. 2.61: Load arrangement for min./max. support moment ms
i max mF,G
k min mF,G
Fig. 2.62: Load arrangement for min./max. field moment mF,G in column axis (“Gurt”)
67
i max mF,F
k min mF,F
Fig. 2.63: Load arrangement for min./max. field moment at field center
68
2.7
Influence surfaces
2.7.1 Introduction
Influence surfaces are analogous to influence lines of beam structures. The influence surface
represents the influence of a moving concentrated load P(x,y) = 1 at position x,y on displacement or stress resultants at control point A(u,v). It is used for detailing, especially for concentrated loads, moving loads, or extreme local area loads. The influence surface is a spatial
function z(x,y). Usually it is represented by contour lines projected on the ground-plan. Influence surfaces are numerically evaluated.
2.7.2 Deflections
Using Betti-Maxwell proposition the influence surface is identical to the deflection surface
due to a concentrated load P = 1 at control point A(u,v).
a) Beams
w(u;x)
Betti
Maxwell
location cause
w(x;u)
location cause
”w” ≡ w(x;u)
Fig. 2.64: Influence line for the deflection “w”
The influence line for the deflection “w” at control point A due to a moving load at P(x) is
identical to the deflection due to a Load 1 at control point A(u).
b) Plate:
w(u,v;x,y)
location
Betti
w(x,y;u,v)
Maxwell
cause
location
cause
”w” ≡ w(x,y;u,v)
Fig. 2.65: Influence surface for deflection “w”
69
The influence surface for the deflection “w” at control point A(u,v) due to a
moving load at P(x,y) is identical to the deflection surface due to a load 1 at
control point A(u,v).
2.7.3 Stress resultants
The kinematical method used to evaluate influence lines for beams cannot be applied for
plates. An alternative approach makes use of the deflection influence line or surface.
a) Beams
curvature at
position u
Fig. 2.66: Influence line for Mu at control point A due to a load at P.
The influence line of a stress resultant due to a moving local P(x) is identical to the corresponding derivative of the deflection influence line with respect to the coordinate u of the
control point A.
Fig. 2.67: Influence line for “M” at A due to a moving load P(x)
70
b) Plate
It is assumed that the influence surface for the deflection “w” = w(x,y;u,v) is known. The influence surfaces for curvature and twisting can be determined by second order derivatives
with respect to the coordinates of the control point, e.g.:
"κu " = −
∂ 2 " w"
∂u 2
Then, the moment influence surface is:
⎛ ∂ 2 " w"
∂ 2 " w" ⎞
⎟
+
ν
" m u " = K (" κ u " + ν " κ ν ") = − K ⎜⎜
2
∂v 2 ⎟⎠
⎝ ∂u
The influence surfaces of other resultants are determined analogously
The influence surfaces of resultants are determined by the
related differentiation of the deflection influence surface
(i.e. the deflection surface under a concentrated unit load)
with respect to the control point coordinates.
The following figure shows the influence surface for moment mx at the center of a rectangular
simply supported plate. It is represented by contour lines and an isometric map.
Fig. 2.68: Bending moment influence surface. The surface approaches infinity over the control point.
71
Fig. 2.68 (cont.): Bending moment influence surface. The surface approaches infinity over the
control point. (plot is arbitrarily truncated)
2.7.4 Evaluation
A resultant at control point A(u,v) is the sum of all products of load value and influence coordinate η, e.g. for a bending moment m:
m(u , v ) = ∫∫ p F (x , y ) " m(x , y )" dx dy + ∫ p L (s ) " m(s )" ds + ∑ Pi " m i "
{
1
424
3
123
concentrated load
area load
line load
For constant area load pF the integration reduces to the evaluation of the influence surface
volume above the loaded area.
Fig. 2.69: Evaluation of an influence surface
72
Numerical methods can be used for integration, e.g. Simpson’s rule. Influence values which
tend towards infinity at the control point, can be exactly integrated and be used as correction
of the numerical results. In many tables correction values are given. See [18] for influence
surfaces and their application.
Often it is possible to approximate distributed loads by equivalent concentrated loads and to
use these for evaluation.
73
2.8
The Reissner/Mindlin Theory
2.8.1 Introduction
A relaxation of Kirchhoff assumptions was made by Reissner in 1945 and in a slightly different manner by Mindlin in 1951. These modified theories extend the field of application of the
theory to thick plates.
2.8.2 Kinematic assumptions
As in Kirchhoff’s theory cross sections are assumed to stay plane during deformation, however, they are allowed to change the angle relative to the mid plane. That means that the total
state of deformation can be described by the displacement w0 and the rotations ϕx and ϕy of
the mid plane. The local displacements at any point in the directions of x and z are taken as
(Fig. 1):
w(x,y,z) = w0(x,y) and u(x,y,z) = z ϕx(x,y)
v(x,y,z) = z ϕy(x,y)
x, u
P
ϕx
γxz
−
∂w
∂x
w(x) = w0(x)
z, w
u(x) = z ϕx(x)
Fig. 2.70: Plate deformation (section parallel x – z plane)
The strains are:
εx =
∂ϕ
∂u
=z⋅ x
∂x
∂x
εy =
∂ϕ y
∂v
=z⋅
∂y
∂y
εz =
∂w
=0
∂z
74
γ xz =
∂u ∂w
∂w
+
= ϕx +
∂z ∂x
∂x
γ yz =
∂v ∂w
∂w
+
= ϕy +
∂z ∂y
∂y
γ xy =
∂ϕ y ⎞
⎛ ∂ϕ
∂u ∂v
⎟
+
= z ⎜⎜ x +
∂ y ∂x
∂x ⎟⎠
⎝ ∂y
Together with the plane stress assumption (σz = 0):
E ⎡1 ν ⎤ ⎧ε x ⎫
⎧σ x ⎫
⎨σ ⎬ =
⎨ ⎬
2
⎩ y ⎭ 1 − ν ⎢⎣ν 1 ⎥⎦ ⎩ε y ⎭
and
τij = G γ ij
the stress resultants are obtained:
t
2
n x = ∫ σ x dz = 0 ; n y = 0
−
t
2
“membrane forces”
t
2
∫τ
n xy =
xy
dz = 0
t
−
2
qx =
t
2
∫τ
xz
t
−
2
qy =
t
2
∫τ
−
t
2
yz
∂w ⎞
⎛
dz = α G t ⎜ ϕx +
⎟ ;
∂x ⎠
⎝
shear correction factor α = 5/6
⎛
∂w ⎞
⎟
dz = α Gt ⎜⎜ ϕ y +
∂y ⎟⎠
⎝
t
2
∂ϕ y ⎞
⎛ ∂ϕ
⎟ = K (κ x + ν κ y )
m x = ∫ σ y z dz = K ⎜⎜ x + ν
∂x
∂y ⎟⎠
t
⎝
−
2
t
2
⎛ ∂ϕ y
∂ϕ ⎞
+ ν x ⎟⎟ = K (κ y + ν κ x )
m y = ∫ σ y z dz = K ⎜⎜
∂x ⎠
t
⎝ ∂y
−
2
t
2
m xy = ∫ τ xy z dz =
−
t
2
∂ϕ ⎞ 1 − ν ⎛ ∂ϕ x ∂ϕ y ⎞
⎛ ∂ϕ
1
⎟ = (1 − ν ) K κ xy
+
Gt 3 ⎜⎜ x + y ⎟⎟ =
K⎜
⎜
⎟
∂
∂
∂
ϕ
∂
12
y
x
2
x
⎝
⎠
⎝ y
⎠
75
Since we allow shear deformation the rotations ϕ x and ϕ y are independent from the derivatives of w. That means, that the stress resultants are expressed as functions of three independent kinematical variables: w, ϕ x , ϕ y .
The relations between ϕ x , ϕ y and curvatures and twisting are:
κx =
∂ϕ x
∂x
κy =
∂ϕ y
κ xy =
∂y
1 ⎛ ∂ϕ x ∂ϕ y ⎞
⎜
⎟
+
2 ⎜⎝ ∂y
∂x ⎟⎠
2.8.3 Boundary conditions
At an edge the number of statical and kinematical variables are balanced. E.g. at an edge
x = const.:
mx, mxy, qx
ϕx , ϕy , w
3 statical variables:
3 kinematical variables:
Note, even if w = 0 (no deflection) the rotation normal to the edge may exist. The introduction
of effective shear forces is unnecessary.
Examples: Combinations of boundary conditions at edge x = const.
boundary conditions comment
x
free
mx = 0
mxy = 0
qx = 0
y
x
clamped
no problem with
interpretation of
effective shear
ϕx = 0
y
ϕy = 0
as with Kirchhoff
w=0
x
simple
soft support
mx = 0
y
m xy = 0
SS
w=0
x
simple
hard support
mx = 0
y
ϕy = 0
HS
w=0
Shear forces are
equivalent to
Kirchhoff effective
shear
shear forces are
equivalent to
Kirchhoff shear
The additional twisting degree of freedom makes it possible to separate the effects of clamped
twist and vertical support. With the Kirchhoff theory they are coupled and result in effective
76
shear forces. Hard support is difficult to construct (clamped twisting). Simple soft support
gives either tension support forces near the corners or lift off.
The following figure illustrates the differences of soft- and hard support:
the problem
square plate, uniform area load, l/t = 10, simple supported
z
x
y
A
B
shear force qx
moments mxy
section A-A
B
A
hard support
2.01
section B-B
167.00
1.49
152.20
-6.60
section A-A
13.88
-9.55
section B-B
soft support
2.71
1.49
148.90
The consequences for the design are:
-
take twisting moment and shear forces from hard support
take support forces from soft support
77
2.8.4 The Kirchhoff constraint
The Kirchhoff theory assumes no shear deformation, i.e.
γ xz = ϕ x +
∂w
=0
∂x
↔
ϕx = −
∂w
∂x
γ yz = ϕ y +
∂w
=0
∂y
↔
ϕy = −
∂w
∂y
which means an explicit coupling of rotations and displacement. Considering the Reissner/Mindlin theory the shear resistance increases with decreasing plate thickness. At the limit
the shear deformation vanishes as the thickness approaches zero. De facto rotations and displacements are coupled again: the so called “Kirchhoff constraint” applies. As a consequence
numerical problems with the Reissner/Mindlin theory occur for thin plates because the partitions of shear and bending stiffness are unbalanced (“shear locking”). Those problems can be
overcome by several procedures (see FEM).
Consequently, the curvatures and the twist approach at the “Kirchhoff limit” t → 0 the values:
∂ϕ x
∂2w
=− 2
∂x
∂x
∂ϕ
∂2w
κy = y = − 2
∂y
∂y
κx =
∂ϕ ⎞
⎛ ∂ϕ
∂2w
κ xy = 12 ⎜⎜ x + y ⎟⎟ = −
∂x ⎠
∂x ∂y
⎝ ∂y
78