Buckling and post-buckling of long pressurized elastic thin-walled ...
Thin Plate Bending and Buckling
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Transcript of 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)