Who started of with basic plate theories?
Gustav Kirchhoff
What is characteristic for a plate?
Thin walled h<<l,b
Loading only perpendicular to the plate plane
What are underlying assumptions of the Kirchhoff plate theory?
What other known theory is advanced on?
Plane state of stress (w.r.t. the thickness direction, as with discs)
No thickness change in deformed state (contradicts plane state of stress but is accepted)
Orthotropic linear-elastic behaviour
Geometric linearity
Plane cross sections in deformed state (no warping)
Normal hypothesis (all cross section normal to middle plane —> No shear)
The Kirchhoff plate theory (shear rigid plate) is a generalization of Euler-Bernoulli beam theory to plate structures
Which displacements are to be found at an arbitrary point in a plate?
u, v, w
Give the displacements at an arbitrary point in a plate
Which strains equate to zero in a Kirchhoff plate?
Why do those strains equate to zero?
As a direct result of the assumptions of the normal hypothesis, plane cross sections and no occuring thickness change,
epsilon-zz
gamma-xz
gamma-yz
equate to zero.
Give the strains that don´t equate to zero in a Kirchhoff plate.
How is the curvature introduced?
Give vector notations.
Give the stress strain relation for a plate in vector/ matrix notation.
Give the name for the stiffness matrix and why it is called like that.
The stiffness matrix is called “reduced stiffness” - reduced because the stiffness is reduced from the full 3D model due to the assumption of a plane state of stress.
How can the edge moment flows be calculated from stresses or deformations?
How can transverse force flows be determined in a Kirchhoff plate and what values are assumed by definition?
Transverse force flows Qx, Qy always equate to zero because gamma-xz and gamma-yz don´t appear by definition
Give the constitutive law for a Kirchhoff plate and introduce the Plate stiffnesses
Give the sum of the transverse shear forces for a Kirchhoff plate.
Give the Moment sums with respect to the x and the y axis in a Kirchhoff plate.
Give the Condensed plate equilibrium and the way to derrive it from the sum of transverse shear forces and the Moment sums.
Give the plate equation and the way to derrive it from the condensed plate equilibrium and the constitutive law.
How is the plate equation used in plate analysis?
If the differential equation can be solved for the given boundary conditions, internal forces can determined. Through internal forces, calculation of the deflection w_0(x,y) is possible.
What is described by a Navier Plate?
Rectangular plate
Simply supported at all edges (M=0,w=0)
Arbitrary load (p = p(x,y)
Give the boundary conditions for a Navier plate.
Simply supported —> Deflections and Moments and edges = 0:
Edges in y direction with x=0 & x=a for 0<y<b:
Edges in x direction with y=0 & y=b for 0<x<a
Remember: Mxx results from sigma-xx and accordingly Myy results from sigma-yy
How is the load p = p(x,y) formulated in a Navier Plate?
The load in a Navier Plate is formulated as a Fourier series:
What Ansatz is used for the deflection in a Navier Plate?
The Ansatz choosen for the deflection in a Navier Plate is a Fourier Series expansion
This Ansatz ensures, that the boundary conditions for deflection and moment = 0 at the edges are kept. (Due to Moment being Moments being described by second partial derrivatives, sin is kept)
Give the coefficient Wmn and how to now determine w0
w0: Insert Wmn in Ansatz
What is an issue of the Kirchhoff plate theory that other theories advance on?
The Kirchhoff plate theory works with the normal hypothesis, this automatically excludes shear.
Other plate theories allow for shear thereby being closer to the “real solution”
Which theories were presented that allow for shear deformation?
First order shear deformation theory (FSDT)
Third order shear deformation theory (TSDT)
What assumptions are underlying of FSDT?
The advancement from Kirchhoff to FSDT is aquivalent to the advancement from the Euler-Bernulli beam to the Timoshenko beam. —> The Normal hypothesis is dropped
Remaining assumptions:
Plane state of stress
Flat cross sections (no warping)
No thickness change
Linear elastic behaviour
What assumptions are underlying of TSDT?
The advancement from FSDT to TSDT is that flatness of the cross sections is not assumed anymore. —> Displacements u and v are developed as third order polynomials
Linear elasticity
Give a simple deformed state for FSDT, cut in the xz plane.
What independent degrees of freedom are to be found?
independent DOF:
w
psi-x
psi-y
Give a simple deformed state for TSDT, cut in the xz plane.
theta-x (used for development of u)
theat-y (used for development of v)
lambda-x (used for development of u)
lambda-y (used for development of v)
Give the displacement field for FSDT
Give the displacement field for TSDT
Give the constitutive law for FSDT
How does the constitutive law of TSDT stand out when compared to Kirchhoff ord FSDT?
With the additional Warping moment flows (P-Nm^3, R-Nm^2)
How are shear stresses distributed in FSDT?
Shear stresses in FSDT show a parabolic distribution.
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