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Theory of Plates
Part II: Plates in bending
Lecture Notes
Winter Semester 2012/2013
Prof. Dr.-Ing. Kai-Uwe Bletzinger
Lehrstuhl für Statik Technische Universität München
www.st.bv.tum.de
2
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
3
0 REFERENCES................................................................................................................................. 4
2 PLATES IN BENDING................................................................................................................... 6
2.1 INTRODUCTION........................................................................................................................... 8 2.1.1 IDEALIZATION OF A PLATE .................................................................................................... 8 2.1.2 ASSUMPTIONS OF THE KIRCHHOFF PLATE THEORY .............................................................. 9 2.1.3 THE PRINCIPAL LOAD CARRYING BEHAVIOR....................................................................... 10 2.2 THE DIFFERENTIAL EQUATION OF THE KIRCHHOFF THEORY .............................................. 11 2.2.1 REDUCTION TO THE MID-SURFACE...................................................................................... 11 2.2.2 THE DIFFERENTIAL EQUATION ............................................................................................ 13 2.2.3 PRINCIPAL MOMENTS – STRESS RESULTANTS IN ARBITRARY SECTIONS ............................. 20 2.2.4 THE DIFFERENTIAL EQUATION OF THE PLATE WITHOUT TWIST RESISTANCE...................... 20 2.2.5 VARYING TEMPERATURE LOADING AT TOP AND BOTTOM SURFACE................................... 21 2.2.6 CIRCULAR PLATES............................................................................................................... 25 2.3 BOUNDARY CONDITIONS ......................................................................................................... 28 2.3.1 THE EFFECTIVE SHEAR FORCE ............................................................................................. 28 2.3.2 VARIANTS OF BOUNDARY CONDITIONS (TABLE 2.9.) ......................................................... 32 2.4 SOLUTION METHODS................................................................................................................ 33 2.4.1 OVERVIEW........................................................................................................................... 33 2.4.2 ANALYTICAL SOLUTIONS .................................................................................................... 33 2.4.3 NUMERICAL METHODS ........................................................................................................ 47 2.5 STRUCTURAL BEHAVIOR ......................................................................................................... 48 2.5.1 POISSON RATIO ν ................................................................................................................. 48 2.5.2 ASPECT RATIO ..................................................................................................................... 50 2.5.3 CLAMPED EDGES ................................................................................................................. 51 2.5.4 FADE AWAY OF CLAMPED EDGE MOMENTS......................................................................... 53 2.5.5 TWISTING STIFFNESS ........................................................................................................... 54 2.5.6 PLATE, SUPPORTED AT THREE EDGES.................................................................................. 57 2.5.7 VARYING STIFFNESS............................................................................................................ 58 2.5.8 CONCENTRATED LOADS ...................................................................................................... 58 2.6 CONTINUOUS PLATES............................................................................................................... 61 2.6.1 INTRODUCTION.................................................................................................................... 61 2.6.2 THE METHOD OF LOAD REDISTRIBUTION (BELASTUNGSUMORDNUNGSVERFAHREN) ........ 61 2.6.3 THE METHOD BY PIEPER/MARTENS [15]............................................................................. 64 2.6.4 POINT SUPPORTED PLATES .................................................................................................. 66 2.7 INFLUENCE SURFACES ............................................................................................................. 69 2.7.1 INTRODUCTION.................................................................................................................... 69 2.7.2 DEFLECTIONS ...................................................................................................................... 69 2.7.3 STRESS RESULTANTS........................................................................................................... 70 2.7.4 EVALUATION ....................................................................................................................... 72 2.8 THE REISSNER/MINDLIN THEORY.......................................................................................... 74 2.8.1 INTRODUCTION.................................................................................................................... 74 2.8.2 KINEMATIC ASSUMPTIONS .................................................................................................. 74 2.8.3 BOUNDARY CONDITIONS..................................................................................................... 76 2.8.4 THE KIRCHHOFF CONSTRAINT............................................................................................. 78
4
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 Heine-mann, 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. Betonkalen-der, 1987 1990, 1993 I. Teil → drillsteife Platten mit Gleichlast und linear veränderli-cher Last
5
[17] Pieper, K., Martens, P.: Durchlaufende vierseitig gestützte Platten im Hochbau. Beton- und Stahlbetonbau (1966) 6, S. 158-162, Beton- und Stahlbetonbau (1967) 6, S. 150-151.
[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. Sprin-ger-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.
2 Plates in bending
7
8
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 sub-jected to loads acting within the system’s plane.
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.
moderately thick thin very thin
yx
t,tll
101to
51
501to
51
501
<
with transverse shear deformation
without transverse shear deformation, mostly used for practical applications
geometrically non-linear, 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 the-ory (e.g. von Karman theory) becomes necessary which is able to consider the membrane ac-tion. 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 de-formation 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.
t t
9
2.1.2 Assumptions of the Kirchhoff plate theory The assumptions are:
1.) geometrically linear: - small strains - - small deflections (”small” is problem depend-ent)
- 2.) linear material: - linear elastic (Hooke), - - in the most simple case homogeneous and isot-rope
- 3.) thin plate The latter assumption leads to the two Bernoulli hypotheses:
a) straight lines perpendicular to the mid-surface (i.e. transverse normals) before de-formation remain straight after deformation.
b) transverse normals rotate such that they remain perpendicular to the mid-surface af-ter 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.
00
z
z
=σ=ε
As a consequence, the local effects of load application cannot be described by the theory. The assumptions 0z =ε and 0z =σ are in contradiction to materials with Poisson effect )0( ≠υ , however, this causes no further difficulties for analysis.
Bernoulli a + b: Kirchhoff theory xw
x ∂∂
−=ϕ
Bernoulli a: Reissner theory xzx xw
γ+∂∂
−=ϕ
Mindlin theory
10
Fig. 2.3: Comparison of Kirchhoff- and Reissner/Mindlin kinematics.
Neglecting the transverse shear strains xzγ and yzγ is inconsistent with the material law:
{ {b,aBernoulliofbecause0
G
0)stressshear(yzxz
yzxz
=
γ⋅=
≠
τ
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 resis-tant 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.
11
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:
real system
idealized system
Fig. 2.5: Idealization.
t
t
12
The integration of the stresses over the thickness of the plate gives the stress resultants which can be divided into force )qq( y,x and moment resultants ).m,m,m( xyyx
Fig. 2.6: Stress resultants.
stress resultants / unit length:
Fig.2.6: Stress resultants.
For example:
{ulusmodtionsec
mkNm
6t1
dzzm
1width
2Rx
2t
2t
xx ⎥⎦⎤
⎢⎣⎡⋅
⋅σ=σ=
↓
∫+
−
⎥⎦⎤
⎢⎣⎡⋅⋅⋅τ=τ= ∫
+
−mkNt1
32dzq M
xz
2t
2t
xzx
13
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:
( ) ( ) ( )
⎜⎜⎜⎜
⎝
⎛
+⋅++=+ ...dx
xfd!2
1
_______________
________________
dxdx
xdfxf)dxx(f2
02
000
higher order terms are neglected Fig. 2.7: Taylor-series expansion.
For example:
( ) dx
xmmdxxm x
xx ∂∂
+=+
Fig. 2.8: Taylor series expansion for the moment mx
14
The statical and geometrical equations are given for the infinitesimal element:
Fig. 2.9: Shear forces at the infinitesimal element
dydx|:0dydxpdxdyyq
qdxqdydxxqqydq:0V y
yyx
xx =+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
++−⎟⎠⎞
⎜⎝⎛
∂∂
++−=∑
0py
qx
q yx =+∂
∂+
∂∂
Fig. 2.10: Moments at the infinitesimal element.
0dxdyqdxdyy
mmdxmdydx
xm
mdym:0M xyx
yxyxx
xxy =−⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+−+⎟
⎠⎞
⎜⎝⎛
∂∂
++−=∑
0q
ym
xm
xyxx =−
∂
∂+
∂∂
∑ = :0M x 0qx
my
my
xyy =−∂
∂+
∂
∂
15
BEAM PLATE
∑ :V 0pyq
xq yx =+
∂
∂+
∂∂
∑ :yM 0qy
mx
mx
yxx =−∂
∂+
∂∂
Statical equations
+ Torsion MT
∑ :V 0pdxdQ
=+
∑ :M 0QdxdM
=−
∑ xM 0qx
my
my
xyy =−∂
∂+
∂
∂
5 unknowns 3 equations 2-times internally indetermined
Geometry
2
2
dxwd
dxd
−=ϕ
=κ
+ Twisting Γ due to torsion
Twisting:
change of angle ϕx in direction y
Curvature:
2
2x
x xw
x ∂∂
−=∂ϕ∂
=κ
2
2y
y yw
y ∂∂
−=∂
ϕ∂=κ
yxw
y
2x
xy ∂∂∂
−=∂ϕ∂
=κ
Table 2.2. Kirchhoff plate differential equation.
16
BEAM PLATE moment-curvature-relation
Material
moment-curvature-relation
A moment mx gives a curvature xmxκ in direction x and the curvature xm
yκ in direction y due to the Poisson effect (shortening at the upper surface and elongation at the bottom surface) from my:
Table 2.3. Kirchhoff plate differential equation.
yy my
mx κν−=κ
ymyκ
xκ ~ yx mm ν− xyy mm~ ν−κ
r1
=κ
M~κ
x
mx r
1x =κ xx m
xmy κν−=κ
from mx:
17
BEAM PLATE
Material
elastic bending stiffness: EI1
MEI1
=κ
or:
{stiffnessflexuralbeam
EIM κ=
moment twist relation:
elastic torsion stiffness:
TGI1
with: 3tbI
3
T =
{3
1width
h1E12
EI1
⋅⋅=
or: moment-twist-relation:
Due to equilibrium adjacent twisting moments across the corner of the infinitesimal element are of the same magnitude. It holds: xyyxxy m2mm =+
elastic torsion stiffness: ( )
( ){1width
3T 1K2
1t1
3E
12GI
1ν−
=⋅
ν+=
Table 2.4. Kirchhoff plate differential equation.
Γ= TT GIM
( ) ( )xy3yyx3x mmtE
12mmtE
12ν−=κν−=κ
( )( )xyy
yxx
Km
Km
κν+κ=
κν+κ=
( )2
3
112tEK
ν−=
( ) xyxy 1K2m2 κν−=
plate bending stiffness:
( )ν+=
12EG
bt
1t
18
BEAM PLATE
differential equations: elimination of curvatures
2
2
dxwdEIM −=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
ν+∂∂
−= 2
2
2
2
x yw
xwKm
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
ν+∂∂
−= 2
2
2
2
y xw
ywKm
( ) ( )yx
w1K2mm2
yxxy ∂∂∂
ν−−=+
2nd derivatives 4
4
2
2
dxwdEI
dxMd
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂ν+
∂∂
−=∂
∂22
4
4
4
2x
2
yxw
xwK
xm
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂ν+
∂∂
−=∂
∂22
4
4
4
2y
2
xyw
ywK
ym
( ) ( ) 22
4yxxy
2
yxw1K2
yxmm
∂∂∂
ν−−=∂∂
+∂
elimination of shear forces from equilibrium conditions
pdx
Md2
2
−= ( )
pym
yxmm
xm
2y
2yxxy
2
2x
2
−=∂
∂+
∂∂
+∂+
∂∂
elimination of moments EI
pdx
wd4
4
= Kp
yw
yxw2
xw
4
4
22
4
4
4
=∂∂
+∂∂
∂+
∂∂ or
Kpw =ΔΔ
Laplace operator ( ) ( ) ( )2
2
2
2
yx ∂∂
+∂
∂=Δ
Table 2.5. Kirchhoff plate differential equation.
back substitution
19
cons
titut
ive
equa
tions
2
2
x xw
∂∂
−=κ
2
2
y yw
∂∂
−=κ
yxw2
yxxy ∂∂∂
−=κ=κ
0py
qx
q yx =+∂
∂+
∂∂
0qy
mx
mx
yxx =−∂
∂+
∂∂
0qx
my
my
xyy =−∂
∂+
∂
∂
( )yxx Km κν+κ=
( )xyy Km κν+κ=
( ) xy
yxxy
1K2
mm
κν−=
=+
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
ν+∂∂
−= 2
2
2
2
x yw
xwKm
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
ν+∂∂
−= 2
2
2
2
y xw
ywKm
( )yx
w1Km2
yx ∂∂∂
ν−−=
( )p
ym
yxmm
xm
2y
2yxxy
2
2x
2
−=∂
∂+
∂∂
+∂+
∂∂
statical equa-tions
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+
∂∂
−= 2
3
3
3
x yxw
xwKq
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂∂
+∂∂
−=yx
wywKq
2
3
3
3
y
geometrical equations
material equations
pde:
of 4
th o
rder
in
hom
o gen
eous
, lin
ear
solution → w
stress resultants
yx q,q xyyx m,m,m
Kpwor
Kp
yw
yxw2
xw
4
4
22
4
4
4
=ΔΔ
=∂∂
+∂∂
∂+
∂∂
Laplace-operator: 2
2
2
2
yx ∂∂
+∂∂
=Δ
bipotential-operator: 4
4
22
4
4
4
yyx2
x ∂∂
+∂∂
∂+
∂∂
=ΔΔ
Table 2.6. Kirchhoff PDE construction scheme.
20
2.2.3 Principal moments – stress resultants in arbitrary sections
out of plane ⊗ into plane
Fig. 2.11: Stress resultants in arbitrary section
The internal forces are determined from the equilibrium conditions:
dxcosmdysinmdxsinmdycosmdtm yxxyyxn ϕ+ϕ+ϕ+ϕ=
( )
ϕ+ϕ−=
ϕ+ϕ=
ϕ+ϕ−==
ϕ−ϕ+ϕ=
ϕ+ϕ+ϕ=
cosqsinqq
sinqcosqq
2cosm2sinmm21mm
2sinmcosmsinmm
2sinmsinmcosmm
yxt
yxn
xyxytnnt
xy2
y2
xt
xy2
y2
xn
Principal moments from the condition mnt = 0:
( ) ( ) 2xy
2yxyxII,I m4mm
21mm
21m +−±+= where: mI ≥ mII
The angle ϕ which is related to the first principal moment mI is
( ) ( ) 2xy
2yxyx
xy
m4mmmm
m2tan
+−+−=ϕ
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 mo-ments can be developed. This is very close to the real behavior of ribbed plates („Rippen-platte)“ or coffered ceilings („Kassettendecken“) made from reinforced concrete.
21
Fig. 2.12: Infinitesimal plate element without twisting moments
0mxw
ywKm
yw
xwKm
xy
2
2
2
2
y
2
2
2
2
x
=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
ν+∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
ν+∂∂
−=
Together with the statical equations it follows:
Kp
yw
xw
4
4
4
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 tempera-ture 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 C/1T temperature coefficient
e) ( )y,xTT uu = temperature at bottom surface
f) ( )y,xTT oo = temperature at top surface
and the definition of curvature due to temperature expansion:
( )t
TT ouTT
−α=κ
we get the differential equation for the deflection w:
( ) ( ) T2T
2
2T
2
1yx
1w κΔν+−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂
κ∂+
∂κ∂
ν+−=ΔΔ
22
equilibrium
0yq
xq yx =
∂
∂+
∂∂
0qy
mx
mx
yxx =−∂
∂+
∂∂
0qx
my
my
xyy =−∂
∂+
∂
∂
2
2
x xw
∂∂
−=κ
2
2
y yw
∂∂
−=κ
yxw2
yxxy ∂∂∂
−=κ=κ
( )( )Tyxx 1Km κν+−κν+κ=
( )( )Txyy 1Km κν+−κν+κ=
( ) xyxy 1Km κν−+=
geometry
material
( )0
ym
yxmm
xm
2y
2yxxy
2
2x
2
=∂
∂+
∂∂
+∂+
∂∂
( )
( ) ⎥⎦
⎤⎢⎣
⎡κν++
∂∂
+∂∂
∂∂
−=
⎥⎦
⎤⎢⎣
⎡κν++
∂∂
+∂∂
∂∂
−=
T2
2
2
2
y
T2
2
2
2
x
1yw
xw
yKq
1yw
xw
xKq
( )
( )
( )yx
w1Km
1xw
ywKm
1yw
xwKm
2
xy
T2
2
2
2
y
T2
2
2
2
x
∂∂∂
ν−−=
⎥⎦
⎤⎢⎣
⎡κν++
∂∂
ν+∂∂
−=
⎥⎦
⎤⎢⎣
⎡κν++
∂∂
ν+∂∂
−=
( ) ( )
( )
( ) T
2T
2
2T
2
2T
2
2T
2
4
4
22
4
4
4
1w.resp
yx1w
.resp
0y
1x
1yw
yxw2
xw
κΔν+−=ΔΔ
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
κ∂+
∂κ∂
ν+−=ΔΔ
=∂
κν+∂+
∂κν+∂
+
+∂∂
+∂∂
∂+
∂∂
stress resultants yx q,q xyyx m,m,m
solution w
( )
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
===
κ=κ=κ=
tstanresulvanishing0mmm
curvatureetemperaturcurvaturetotal:control
xyyx
Tyx
Table 2.7. Differential equation for temperature loading ( )t
TT ouTT
−⋅α=κ
24
The special case of equal temperature distribution at any point of the plate leads to:
0w0yx 2
T2
2T
2
T =ΔΔ→=∂
κ∂+
∂κ∂
=κΔ
Considering a plate which is fully clamped at all edges we choose w(x, y) = 0 as a possible solution.
The differential equation 0w =ΔΔ and the boundary conditions are satisfied:
x)0y(
y)0x(
x)0y(
y)0x(
x0allfor0y
w
y0allfor0x
w
x0allfor0w
y0allfor0w
l
l
l
l
≤≤=∂
∂
≤≤=∂
∂
≤≤=
≤≤=
=
=
=
=
Although the plate does not deflect there exist stresses and resultants:
Tyx )1(Kmm κν+−= 0mxy =
0qq yx ==
The analogous result is known for the clamped beam subjected to a linear temperature distri-bution through the thickness:
tTT
ElM ouTjk
−α−=
jkki MM +=
Fig. 2.13: Clamped beam subjected to varying temperature distribution through thickness.
t
25
2.2.6 Circular plates For convenience polar coordinates are chosen in the following
Fig. 2.14: Infinitesimal element of a circular plate.
26
Differential equation: K
),r(pw ),r(ϕ
=ΔΔ ϕ
using the operator: 2
2
22
2
r1
rr1
r ϕ∂∂
+∂∂
+∂∂
=Δ
stress resultants:
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+ϕ∂
∂ν+
∂∂
−=rw
r1w
r1
rwKm 2
2
22
2
r
⎥⎦
⎤⎢⎣
⎡∂∂
+ϕ∂
∂+
∂∂
ν−=ϕ rw
r1w
r1
rwKm 2
2
22
2
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ϕ∂
∂∂∂
ν−−== ϕϕw
r1
r1Kmm rr
( )wr
Kqr Δ∂∂
−=
( )ϕ∂
Δ∂−=ϕ
wr1Kq
In the case of rotational symmetry the derivatives with respect to ϕ vanish:
( )drd
r1
drd
2
2
+=Δ
( )
Kp
drdw
r1
drwd
r1
drwd
r2
drwd r
32
2
23
3
4
4
=+−+
The resultants remain to be:
⎥⎦
⎤⎢⎣
⎡ν+−=
drdw
r1
drwdKm 2
2
r
⎥⎦
⎤⎢⎣
⎡+ν−=ϕ dr
dwr1
drwdKm 2
2
0mm rr == ϕϕ
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−+−=⎥
⎦
⎤⎢⎣
⎡+−=Δ−=
drdw
r1
drwd
r1
drwdK
drdw
r1
drwd
drdKw
drdKq 22
2
3
3
2
2
r
0q =ϕ
27
28
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 integra-tion 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
of deflection, e.g.: yw
y ∂∂
−=ϕ . That means, that if the displacements w(y) at the edge x = lx
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
reduction of resultants
ww
y
xx
→⎭⎬⎫ϕ
ϕϕ
xx
xy
xx
qforcesheareffective
qm
mm
→⎭⎬⎫
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 be-havior at the inner region.
29
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
ym
qq xyxx ∂
∂+=
Fig. 2.16: Effective shear force definition.
30
Fig. 2.17: Effective shear forces.
( ) 2
3
2
3
3
3xy
xx yxwK1
yxw
xwK
ym
qq∂∂
∂ν−−⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
∂+
∂∂
−=∂
∂+=
( ) ⎥⎦
⎤⎢⎣
⎡∂∂
∂ν−+
∂∂
−= 2
3
3
3
x yxw2
xwKq
( ) ⎥⎦
⎤⎢⎣
⎡∂∂
∂ν−+
∂∂
−=yx
w2ywKq 2
3
3
3
y
Remarks:
- support forces ( =̂ 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 com-ponents 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.
32
2.3.2 Variants of boundary conditions (Table 2.9.)
Type boundary conditions consequences
clamped
x
y
0
xw
0w
=∂∂
=
xxy
2
qforcesportsup0m0yx
w
0yw
=→=→=∂∂
∂
=∂∂
simple supported
x
y
0m
0w
x =
=
xxy
2
2
2
2
2
qforcesportsup0m0yx
w;0xw
0yw;0
xw0
yw
=→≠→≠∂∂
∂≠
∂∂
=∂∂
=∂∂
=∂∂
free edge
x
y
0m0q
x
x
==
both components qx and y
mxy
∂
∂cancel each other
f
x
y
0mqfw
x
x
==
elastic fM
x
y
xM mf
xw
0w
=∂∂
=
often coupled elastic foundation, e.g. plate with edge beam
if edge beam is eccentric than membrane and bending actions in the plate have to be considered
33
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 ex-actly. Mostly, the problem can be solved only approximately using numerical techniques.
exactsolutions
analytical
approximatesolutions
numerical
closed solutionsseries expansions
finite differencesfinite elements
Fig. 2.19: Solution methods.
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):
PDE: ( )Krpw =ΔΔ where
drdw
r1
drwdw 2
2
+=Δ
( )Krp
drdw
r1
drwd
r1
drwd
r2
drwd
32
2
23
3
4
4
=+−+
or: ( ) ( ) ( )′==′+′′−′′′+dr
dwhereKrpw
r1w
r1w
r2w 32
IV
Fig. 2.20: Circular plate with rotational symmetrical loading.
34
The solution of the differential equation is:
ρ+ρρ+ρ++= lnClnCCCww 42
32
21p
( )ρ
++ρρ
+ρ+′=′a1C1ln2
aC
a2Cww 432p
( ) 2242
322p a1C3ln2
a1C
a2Cww
ρ−+ρ++′′=′′
pww ′′′=′′′ρ
+ 33 a2C 334 a
2Cρ
+
where: ar
=ρ and particular solution wp.
The integration constants Ci allow to satisfy the 2 x 2 boundary conditions at both edges of a circular plate. The stress resultants are:
⎟⎠⎞
⎜⎝⎛ ′ν+′′−= w
r1wKmr
⎟⎠⎞
⎜⎝⎛ ′+′′ν−=ϕ w
r1wKm
⎟⎠⎞
⎜⎝⎛ ′−′′+′′′−= w
r1w
r1wKq 2r
0qmm rr === ϕϕϕ 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: plate center clamped edge
)finitew(
0'w:0 ==ρ
particular solution: K64
rpw4
p = 2apq
0'w0w:1
r −=
===ρ
35
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:
K64
ap2
CC4
21 =−=
solution: ( )222 raK64
pw −=
( )rarK16
pw 23 −=′
( )22 ar3K16
pw −=′′
rpK83w =′′′
where: K64
apw4
center =
stress resultants: ( ) ( )[ ]213
16apm 2
2
r −ρ−ν+=
( ) ( )[ ]ν−ρ−ν+=ϕ 213116apm 2
2
ρ−=2apq r
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 be-cause of the 2-dimensional load carrying behavior of the plate.
Fig. 2.23: Constant loaded circular plate (Poisson ratio ν = 0).
36
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
a2
Pq r π−=
particular solution: wp = 0
ρ+ρρ+ρ+= lnClnCCCw 42
32
21
Using w´ = 0 at the plate center, we get:
C4 = 0
and a2
Pq r π−= at the edge ρ = 1 yields:
K8
aPC2
3 π=
Finally, the remaining conditions give:
( )( )ν+π
ν+=−=
1K163aPCC
2
21
Back substituting the constants gives the following expressions for the deflection:
( ) ⎥⎦
⎤⎢⎣
⎡ρρ+ρ−
ν+ν+
π= ln21
13
K16aPw 22
2
and stress resultants:
( ) ρν+π
−= ln14Pmr
37
( ) ( )[ ]ρν+−ν+π
−=ϕ ln114Pm
ρπ−=
a2Pq r
Evaluating the expressions, we recognize at the plate center a finite deflection, however, sin-gular 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 wer-den..
2.4.2.2 Rectangular plates By means of series expansion the partial differential equation ΔΔw(x,y) = p(x,y)/K can be de-coupled 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 cho-sen arbitrarily.
a) Plate strip
Load distribution and deflection are constant in y-direction. The differential equation further reduces to:
38
( ) ( )Kxp
dxxwd
4
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
( ) ∑∑ α=π
=n
nnn
n xsinpL
xn2sinpxp
where pn are the Fourier coefficients and
an
Ln2
nπ
=π
=α
Multiplying by sin αm x and integrating yields:
( ) ( ) ( ) ( )44444 344444 21
dxxsinpxasindxxpxasinn
nnm
a
0
a
0m ∑∫∫ α=
= 0 for m ≠ n
= 2apn for m = n
The Fourier coefficient for the n-th series component is:
39
( ) ( ) dxxsinxpa2p n
a
0n α= ∫
Analogously, the deflection w(x) is represented by:
( ) ( )∑ α=n
nn xsinwxw
which implies the Navier conditions (w = 0 and w´´ = 0) at both edges
Inserting load and deflections into the differential equation:
( ) ( )∑∑ α=αα xsinpK1xsinw nnnn
n
4n
Comparison of coefficients yields:
K
pw 4n
nn α
=
and the general solution:
( ) ( )∑ απ
=n
n4n
4
4
xsinnp
Kaxw a
nn
π=α
n = 1, 2, 3, ... Back substitution gives the stress resultants:
2
2
x dxwdKm −=
xy mm ν=
3
3
x dxwdKq −=
The deflection w(x) converges very fast because of n4 in the denominator. The stress resul-tants 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:
( )π
=α= ∫ np4dxxsinp
a2p
a
0nn n = 1, 3, 5, ...
( ) ( )∑ απ
= xsinn1
Kap4xw n55
2
n = 1, 3, 5, ...
40
The load is approximated as:
approximation by one series component n = 1
xa
sinp4p ππ
=
real load
approximation by two series components n = 1, 3
⎟⎠⎞
⎜⎝⎛ π
+π
π= x
a3sin
31x
asinp4p
Fig. 2.27: Approximation of loading.
The even series components n = 2, 4, 6, ... are neglected since they would cause anti-symmetric distributions.
exact 1 series component (N = 1)
2 series components (n = 1, n = 3)
deflection w Kap013021.0
4
Kap013071.0
4
Kap013017.0
Kap000054.0
Kap013071.0
4
4
4
=
−
moment m 2ap125.0 2ap129001.0
2
2
2
ap124228.0ap004778.0ap129006.0
=
−
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 com-parison with the exact solution shows. More varying load distributions require more series components, respectively.
41
b) Rectangular plate with two opposite Navier supported edges
The homogenous differential equation:
0yw
yxw2
xw
4
4
22
4
4
4
=∂∂
+∂∂
∂+
∂∂
is solved by the ansatz in product form:
( ) ( )a
n;xsinyFw nnnhπ
=αα= ∑
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.
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:
( ) ( ) ( ) 0
0
sin
0
yFdy
yFd2dy
yFdnn
4n2
n2
2n4
n4
=
≠
α
=
⎟⎟⎠
⎞⎜⎜⎝
⎛α+α−
3214444444 34444444 21
A possible solution is given by making use of the yeλ function:
( ) (
)a
n;ycoshDy
ysinhCysinhByycoshAa1yF
nnnn
nnnnnnn2n
n
π=ααα+
+α+αα+α=
And, therefore:
(
) xsin
yF
ycoshDyysinhC
ysinhByycoshA1w
n
n
nnnnn
nnnnnn
2n
h
ααα+α
+αα+αα
=
⎟⎠⎞⎜
⎝⎛
∑
44444444 344444444 21
arbitrary boundary conditions
42
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:
(
) xsinycoshDyysinhC
ysinhBycoshA1xsinnp
Kw
nnnnnn
nnnnn
2n
nn
4n
4
4
ααα+α+
+ααα
+απ
α= ∑∑
With constant distributed load p = constant:
(
) xsinycoshDyysinhC
ysinhByycoshAa1xsin
a1
aKp4w
nnnnnn
nnnnnn
2n
nn
5n
ααα+α+
+αα+α+α= ∑∑
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.
43
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 )
44
Fig. 2.31: Example from Czerny, Betonkalender 1987
45
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
46
Fig. 2.33: Example from Stiglat/Wippel: Platten
47
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 fi-nite element method is a little bit more expensive, but much more flexible for general prob-lems. The boundary element method requires the solution of the differential equation and dis-cretizes 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.
48
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 con-ditions required special considerations. Convergence studies on the accuracy and error analy-ses 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.
49
beam
quadratic plate (ν = 0)
moments 8
pM2l
= xy
2
x
mm8
pm
ν=
=l
displacement EI
p384
5w4l
= ( )244
1EI
p384
5ˆ
Kp
3845w ν−==
ll
Fig. 2.36: Comparison beam – plate strip
Using the formulas which are given in Fehler! Verweisquelle konnte nicht gefunden wer-den. the resultants for different Poisson ratios can be calculated.
50
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 one-axial 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 sup-ported rectangular plate, from Czerny-tables, ν = 0.
51
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
3
24lp8lp
2
2
= 2
8.56lp
2.27lp
2
2
≈
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.
52
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 sup-ported
________ one long
edge clamped
field moments
support moments
Fig. 2.40: Influence of aspect ratio on the effect of clamped supports.
53
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 any-how.
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 field- and support moments.
It is obvious that the support moments mxr and myr are comparatively large, the field mo-ments, 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.
54
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 par-allel to the edges. The amount of the twisting moments is a measure of how much the direc-tions of the principle moments deviate from the edge parallel directions x and y.
The so called coffered ceilings (“Kassettendecken”) are exceptions which cannot resist twist-ing 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 direc-tions of the principal moments change at the corner by 45° which is caused by twisting mo-ments mxy of the same absolute value as m1 and m2. The corner anchor force is A = 2⋅mxy.
extra load at corner without anchorage at the corner –m1 = m2 = |mxy|
xym2A = corner force
with anchorage
Fig. 2.43: Twisting resistant square plate.
55
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 evalu-ated without Poisson effect (ν = 0.0) for field moments evalu-ated with (ν = 0.2) produces stresses in x-direction produces stresses in y-direction
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 ge-funden werden.
56
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 ( )2x8
1 pl . This is possible be-cause 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.
57
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 as-pect 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: • 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 ly ≥ lx • one axial load carrying behavior near the free edge • the plate is still heavily twisted at regions far from the free edge y ≤ ly
58
2.5.7 Varying stiffness Although “stiffness attracts forces” the stresses decrease.
system and load
stress resultants
moment scale maximum stresses
2y
y h6m ⋅
=σ
16.048.1mm
1:1y
3:1y
1:1y
3:1y =
σ
σ→=
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).
59
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.
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 fad-ing behavior of stress resultants (Fig. 2.50) the load position, aspect ratio, and boundary con-ditions are less important.
section in y-axis
section in x-axis
60
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 sur-face that for most cases several different concentrated loads do not effect each other. Conse-quently, dimensioning of plates is very much dominated by the concentrated load and its vi-cinity 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 sur-faces for heavy loads.
Fig. 2.51: Influence of load position on the support moment of a strip [21], Fehler! Verweis-
quelle konnte nicht gefunden werden. compared to the beam.
61
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 repre-sented 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
62
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.
k
max. moment
min. moment
i
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 anti-symmetrical portion.
2pgq +=′ (uniformly distributed)
2pq ±=′′ (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 posi-tions. The boundary conditions for the single one span plates are:
- load case q’: all continuity edges clamped - load case q”: all edges simply supported
63
max./min. field moment
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
±+
="m
2p
'm2pg
mminmax
ff
2xF l
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 check-erboard pattern.
max. supportmoment atcommon edgeof fields , ki
g (uniformly distributed p (checkerboard modified)
Fig. 2.55: Load combination for maximum support moment.
2pgq +=′ (uniformly distributed)
2pq ±=′′ (checkerboard)
Fig. 2.56: Load re-distribution for maximum support moment
64
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:
444 3444 21444 3444 21
ll
kfieldifield
m2p
m2pg
m2p
m2pg
21m
k,sk,s
2k,x
i,si,s
2i,xs
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
′′+
′
++
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
′′+
′
+−=
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 ap-proximated 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.
mean values (listed in tables) flqm
2x
f⋅
=
Fig. 2.57: Field moments regarding Pieper/Martens
The support moments are determined as the mean value of the clamped edge support mo-ments 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.
65
Fig. 2.58: Support moments regarding Pieper/Martens
Fig. 2.59: Method by Pieper/Martens [17] taken from Schneider Bautabellen [19]
66
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).
max V hatched fields loaded
min V white fields loadedk
i
Fig. 2.60: Load arrangement for min./max. support force V
67
max ms
min msk
i
Fig. 2.61: Load arrangement for min./max. support moment ms
max mF,G
min mF,Gk
i
Fig. 2.62: Load arrangement for min./max. field moment mF,G in column axis (“Gurt”)
68
max mF,F
min mF,Fk
i
Fig. 2.63: Load arrangement for min./max. field moment at field center
69
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 displace-ment or stress resultants at control point A(u,v). It is used for detailing, especially for concen-trated 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. Influ-ence 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) w(x;u)
location cause
”w” ≡ w(x;u)
BettiMaxwell
location cause
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) w(x,y;u,v)
location cause
”w” ≡ w(x,y;u,v)
BettiMaxwell
location cause
Fig. 2.65: Influence surface for deflection “w”
70
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 corre-sponding 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)
71
b) Plate It is assumed that the influence surface for the deflection “w” = w(x,y;u,v) is known. The in-fluence surfaces for curvature and twisting can be determined by second order derivatives with respect to the coordinates of the control point, e.g.:
2
2
u u"w"""
∂∂
−=κ
Then, the moment influence surface is:
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂ν+
∂∂
−=κν+κ= ν 2
2
2
2
uu v"w"
u"w"K""""K"m"
The influence surfaces of other resultants are determined analogously
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 con-
trol point.
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.
72
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 coor-dinate η, e.g. for a bending moment m:
( ) ( ) ( ) ( ) ( ){
loadedconcentratloadlineloadarea
"m"Pds"sm"spdydx"y,xm"y,xpv,um iiLF
32143421∑∫∫∫ ++=
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
73
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.
74
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 differ-ent 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, how-ever, 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)
Px, u
z, w
w(x) = w0(x)
u(x) = z ϕx(x)
ϕx
γxz xw
∂∂
−
Fig. 2.70: Plate deformation (section parallel x – z plane)
The strains are:
x
xx z
xu
∂ϕ∂
⋅=∂∂
=ε
yz
yv y
y ∂
ϕ∂⋅=
∂∂
=ε
0zw
z =∂∂
=ε
75
xw
xw
zu
xxz ∂∂
+ϕ=∂∂
+∂∂
=γ
yw
yw
zv
yyz ∂∂
+ϕ=∂∂
+∂∂
=γ
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
ϕ∂+
∂ϕ∂
=∂∂
+∂∂
=γxy
zxv
yu yx
xy
Together with the plane stress assumption (σz = 0):
⎭⎬⎫
⎩⎨⎧εε
⎥⎦⎤
⎢⎣⎡ν
νν−
=⎭⎬⎫
⎩⎨⎧σσ
y
x2
y
x1
11
E and ijij G γ=τ
the stress resultants are obtained:
∫−
==σ=2t
2t
yxx 0n;0dzn
∫−
=τ=2t
2t
xyxy 0dzn
∫−
⎟⎠⎞
⎜⎝⎛
∂∂
+ϕα=τ=2t
2t
xxzx xwtGdzq ; shear correction factor α = 5/6
∫−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+ϕα=τ=2t
2t
yyzy ywGtdzq
( )∫−
κν+κ=⎟⎟⎠
⎞⎜⎜⎝
⎛∂ϕ∂
ν+∂ϕ∂
=σ=2t
2t
yxyx
yx Kyx
Kdzzm
( )∫−
κν+κ=⎟⎟⎠
⎞⎜⎜⎝
⎛∂ϕ∂
ν+∂ϕ∂
=σ=2t
2t
xyxy
yy Kxy
Kdzzm
( ) xyy
y
x2t
2t
yx3xyxy K1
xK
21
xyGt
121dzzm κν−=⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂ϕ∂
+ϕ∂ϕ∂ν−
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂ϕ∂
+∂ϕ∂
=τ= ∫−
“membrane forces”
76
Since we allow shear deformation the rotations xϕ and yϕ are independent from the deriva-tives of w. That means, that the stress resultants are expressed as functions of three independ-ent kinematical variables: w, xϕ , yϕ .
The relations between xϕ , yϕ and curvatures and twisting are:
⎟⎟⎠
⎞⎜⎜⎝
⎛∂ϕ∂
+∂ϕ∂
=κ∂ϕ∂
=κ∂ϕ∂
=κxy2
1yx
yxxy
yy
xx
2.8.3 Boundary conditions At an edge the number of statical and kinematical variables are balanced. E.g. at an edge x = const.:
3 statical variables: mx, mxy, qx 3 kinematical variables: w,, yx ϕϕ 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
free
x
y
mx = 0 mxy = 0 qx = 0
no problem with interpretation of effective shear
clamped
x
y
0w
00
y
x
=
=ϕ=ϕ
as with Kirchhoff
simple soft support
x
ySS
0w
0m0m
xy
x
=
==
Shear forces are equivalent to Kirchhoff effective shear
simple hard support
x
y
HS
0w
00m
y
x
=
=ϕ=
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
77
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
x
yzA
A
B
B
moments mxy shear force qx
167.00
2.01
sect
ion
A-A
hard
supp
ort
152.20
1.49
sect
ion
B-B
13.88
2.71
-6.60 sect
ion
A-A
soft
supp
ort
148.90
1.49
-9.55 sect
ion
B-B
The consequences for the design are:
- take twisting moment and shear forces from hard support - take support forces from soft support
78
2.8.4 The Kirchhoff constraint The Kirchhoff theory assumes no shear deformation, i.e.
xw0
xw
xxxz ∂∂
−=ϕ↔=∂∂
+ϕ=γ
yw0
yw
yyyz ∂∂
−=ϕ↔=∂∂
+ϕ=γ
which means an explicit coupling of rotations and displacement. Considering the Reiss-ner/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 dis-placements 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 parti-tions 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:
yxw
xy
yw
y
xw
x
2yx
21
xy
2
2y
y
2
2x
x
∂∂∂
−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂
ϕ∂+
∂ϕ∂
=κ
∂∂
−=∂ϕ∂
=κ
∂∂
−=∂ϕ∂
=κ
