Thin Plate Bending and Buckling

Thin plate buckling

By

Dr. Jawad Khawar

Plate subjected to pure bending

(a) Direct stress on lamina of plate element; (b) radii of curvature of neutral plane.

Plate subjected to pure bending

D is known as the flexural rigidity of the plate

Plate subjected to bending and twisting

(a) Plate subjected to bending and twisting; (b) tangential and normal moments on an arbitrary plane.

Plate subjected to bending and twisting

Complementary shear stresses due to twisting moments Mxy.

Plate subjected to bending and twisting

Determination of shear strain γxy.

Plate Subjected to distributed transverse load

Plate element subjected to bending, twisting and transverse loads

Plate Subjected to distributed transverse load

Taking moment about x-axis

Taking moment about y-axis

Plate Subjected to distributed transverse load

or or

Boundary Conditions

• Simply supported edge at x=0

Boundary Conditions

• Built‐in edge (Fixed or Clamped) at x=0

• Free Edge at x=0

Equivalent vertical force system

Solution of Thin plate Bending Equation

Rectangular Thin plate simply supported at four edges 

Thin plate bending equation

BOUNDARY CONDITIONS

Assumed solution as a double trigonometric Fourier series

The applied transverse loading in terms of Fourier Series

Solution of Thin plate Bending Equation

Solution of Thin plate Bending Equation

Substituting w and q in terms of their Fourier series in the thin plate bending equation we have

Combined bending and in‐plane loading of a thin rectangular plate

Combined bending and in‐plane loading of a thin rectangular plate

Combined bending and in‐plane loading of a thin rectangular plate

Thin Plate BucklingSimply supported Loaded Edges and Freeunloaded edges

Thin Plate Buckling

Different loading and edge boundary conditions for thin plate buckling

Buckling coefficients for flat plates in compression

Different loading and edge boundary conditions for thin plate buckling

Buckling coefficients for flat plates in bending

Different loading and edge boundary conditions for thin plate buckling

Shear buckling coefficients for flat plates

Local instability

Secondary Instability or Local buckling

Primary instability or columnbuckling

Local instability

(a) Extruded angle; (b) formed channel; (c) extruded Z; (d) formed ‘top hat’.

Local instability

Local instability

Instability of stiffened panels

Instability of stiffened panelsInitial buckling of skin stringer panel Panel instability

Inter‐rivet buckling

Instability of stiffened panelsTorsional buckling of skin‐stringer panel

Flexural and torsional buckling of skin‐stringer panel

Instability of stiffened panelsInter‐rivet buckling or wrinkling of 

skin‐stringer panelWrinkling of skin‐stringer panel

Instability of stiffened panels

Simply supported on all four sides

Stiffeners may buckle as long plates simply supported on three sides with one edge free

or

Primary instability Secondary instability

Applied load

where

Failure stress in plates and stiffened panels

Average compressive stress Unloaded edge stress

Stowell,  Mayers and Budiansky failure criteria

σcy=σe

Failure stress in plates and stiffened panels

Gerard Method

Failure stress in plates and stiffened panels

• Diagonal tension webs are one of themost outstanding examples of methodsused in airframe stress analysis andstructural sizing.

• Standard structural practice has been toassume that the load carrying capacityof a shear web terminates when theweb buckles and stiffeners are usedmerely to raise the buckling stress ofthe web.

• Airframe design assumes that a thinweb with transverse stiffeners does notfail when it buckles. The web formsdiagonal folds and functions as a seriesof tension straps while the stiffeners actas compression posts. The web‐stiffenerthus acts as a truss and may be capableof carrying loads far greater than thoseproducing the buckling of the web.

Shear Panels (Tension Field Beams)

Typical Shear Panel Composition

Typical Spar construction of wing

Shear Panels

• Shear Resistant Webs

• Pure Diagonal Tension Webs

• Incomplete diagonal Tension Webs

Shear Resistant Web

• Shear resistant web are those that do not buckle under ultimated shear stress.

• Shear resistant web are not very common in aircraft structural design. However there are quite ubiquitous in other structures such as bridges and buildings

• In Shear resistant web the allowable web stresses can only be increased by– Increasing web thickness– Decreasing spacing between the stringers in shear resistant web

Shear Resistant Web

Pure or Complete Diagonal Tension Webs

Derivation of Stresses in Complete Tension Field webs

Determination of flange forcesBalancing Int. and ext. moments about the bottom flange

Resolving forces horizontally

Vertical Loading on stiffeners due to web stresses

Balancing vertical forces on the web

Compressive load on the vertical stiffeners

Experimentally determined empirical relation

Effective length of the stringers

Vertical loading on the flanges due to web stresses

AF is the cross-sectional area of the

flange

AS is the cross-sectional area of the

stiffeners

Derived using the principal of minimum strain energy

Alternate expression

Incomplete Complete Diagonal Tension Webs

Diagonal tension factor

Incomplete Complete Diagonal Tension Webs

Empirical factor

Modified flange stress is given by

Modified stiffener stress is given by

σ2

σ1

Incomplete Complete Diagonal Tension Webs

Effective length of stiffeners

Effect of Taper on Diagonal Tension Beam Calculations

Resolving forces vertically

Resolving forces horizontally

Taking moment about lower Flange

Solving above three equations simultaneously 

Effect of Taper on Diagonal Tension Beam Calculations

Compressive force acting on thevertical stiffeners/stringers isgiven by:

Shear force S acting on anysection of the beam is given by:

Buckling of curved plates

Compression Loading

Buckling of curved plates(Shear Loading)

Buckling of curved plates(Shear Loading)

Buckling of curved plates(Shear Loading)

Buckling of curved plates(Shear Loading)

Buckling of curved plates(Shear Loading)