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Analysis of PlatedStructures
Flat Plate - very small thickness in comparison
to the planar dimensions
Lateral Loading - forces are perpendicular to the
plane of the plate
Resists the applied load by bending in twodirections and twisting moment - BendingBehaviour Dominates
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Stress Resultants in a Plate
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Stresses
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Plate Theory
Plate theory is to calculate the deformation andstresses in a plate subjected to loads.
Due to very small thicness in comparison to the
planar dimensions! reduces the full three-dimensionalsolid mechanics problem to a two-dimensional
problem.
"hin Plate "heory#
• thickness to width ratio < 0.1
•
maximum deection < one tenth of thickness
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!ssumptions
1. The line normal to the neutral axis "efore "endin#
remains strai#ht after "endin#.
$. The normal stress in thickness direction is
ne#lected.' ( ) *
%. The transverse shearin# strains gx& and gyz are
assumed to "e &ero.
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'imitations.
1. (irchho) plate element cannot rotateindependently of the position of the mid*surface.Pro"lems occur at "oundaries where theunde+ned transverse shear stresses are
necessary especially for thick plates
$. is only applica"le for analysis of plates withsmaller deformations as hi#her order terms ofstrain*displacement relationship cannot "ene#lected for lar#e deformations.
%. ,nly for small deformations the transversesti)ness can "e assumed to "e constant
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"hic Plate "heory
Reissner - indlin or indlin plate theory is applied
for analysis of thick plates where the sheardeformations are considered rotation and lateraldeections are decoupled.
/ross*sections need not "e perpendicular to the axialforces after deformation.
+ssumptions#
1. The deections of the plate are small.
$. ormal to the plate mid*surface "efore
deformation remains strai#ht "ut not necessarily
normal to it after deformation.
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PL+", ,lements
+pplications
Floor Panels hear alls
"riangular elements
Four or ,ight /oded 0uadrilateral ,lements
D1F ) 2 3Displacement and "wo Rotations4 , , at
each node
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ulation of Rectangular Plate Bending Element – THIC
5soparametric formulation
x
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ulation of Rectangular Plate Bending Element – THIC
!. oment -/urvature Relation
6ompactForm
+nalogous to tress 7 trainRelation
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23 is the strain displacement matrix
4di5 is the nodal displacement vector
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/3p
6
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2. Strain 7isplacement Relation
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atrix8orm
8our ode 9uadrilateral case
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/ompact 8orm
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:xplicit form
5n general for any i/ 3p 2i3 6
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The "endin# and shear terms are separated andwritten as
/ompact 8orm
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Element !ti"ne## $atrix
8auss 0uadrature integration rule is used tocompute the sti&ness matri9 :;.