2. Plate bending refers to the deflection of
a plate perpendicular to the plane of the plate
under the action of external forces and moments.
The amount of deflection can be determined by
solving the differential equations of an
appropriate plate theory.
The stresses in the plate can be calculated from
these deflections.
Once the stresses are known, failure theories can
be used to determine whether a plate will fail under
a given load.
3. Small elastic displacement Large elastic deflection
Maximum deflections
are small compared to
the plate thickness
The effect of direct
tensile forces on
deflections are small
Maximum deflections
are large compared to
the plate thickness
The effect of direct
tensile forces become
relatively large for
deflections greater
then the plate
thickness.
4. For a circular plate with radius a and thickness h,
consider the polar coordinates with origin at the
center of the plate
Radius= a
Thickness=h
5. By considering the axisymmetrical case in which
the plate loaded and supported symmetrically with
respect to the z axis (then the dependency on θ
vanishes)
6. With P = Po = constant
• Where A1,A2,B1 and B2 are constants of integration.
These are determined by the boundary conditions
at r=a and the regularity conditions w, ωr, Mrr and
Vr must be finite at the center of the plate
7.
8. Where the subscripts (r, θ) on w denote partial
differentiation accordingly for the solid plate we
conclude that A2=B2=0 for axi symmetric
conditions
9. Support and
Loading
Principal Stress
Point of
maximum
stress
Maximum
Deflection
Edge simply
supported; load
uniform(r0=a)
center
Edge fixed; load
uniform(r0=a)
Edge
Edge simply
supported; load
at the center
Center
Fixed edge;
load at the
center
Center
10. Q .A plate made of mild steel (E=200GPa, u=0.29
and yield stress 315 MPa) has a thickness h=10mm
and covers a circular opening having a diameter of
200mm. The plate is fixed at the edges and is
subjected to a uniform pressure p
A) determine the magnitude of yield pressure Py and
deflection Wmax at the center of the plate when
this pressure is applied.
B) determine a working pressure based on a FOS=2.
11. A) The maximum stress in the plate is a radial
flexural stress at the outer edge of the plate
given by
12. The magnitude of Py by the maximum shear stress
theory of failure is obtained by setting stress
maximum equal to Y
13. The maximum deflection of the plate when
this pressure is applied is given by the
appropriate equation
14. B) let Pw be the working pressure its value is
based on Py and so
15. E= 200GPa
μ= 0.29
Thickness= 10mm
Element type = shell type- membrane 41
Quadrilateral meshing
Fully constrained at its boundaries
16.
17.
18. A clamped circular plate of diameter 100mm is
subjected to a uniform pressure of 420Mpa
The central deflections of the circular plate is found
to be 3.5mm
Theoretical result of transverse deflection= 3.5mm
Ansys result for analytical displacement along the
z-axis is given by = 13mm
