Buckling problems

1. Plate Bending
Theory
Buckling Problem
MEDHASMI KHATIWADA ME07805
NABIN SHRESTHA ME07807
NEHA KASHYAPATI ME07820
NIRAJ KAYASTHA ME07821

2. Plate
 A plate is a flat structural element for which the thickness is small compared with the surface
dimensions.
 The thickness is usually constant but may be variable and is measured normal to the middle surface
of the plate.

3. Basic theory of thin plates
Assumptions:
 One dimension (thickness) is much smaller than the other two dimensions
(width and length) of the plate i.e.; t << Lx, Ly
 Shear stress is small; shear strains are small i.e; σz = 0; εz = εxz = εyz = 0
 Thin plates must be thin enough to have small shear deformations but thick
enough to accommodate in-plane/membrane forces.

4. Assumptions of Plate Theory
 Let the plate mid-surface lie in the x y plane and the z – axis be along the thickness
direction, forming a right handed set, Fig. 6.1.4.
 The stress components acting on a typical element of the plate are shown in Fig. 6.1.5.

5. Assumptions of Plate Theory
The following assumptions are made:
(i) The mid-plane is a “neutral plane”
 The middle plane of the plate remains free of in-plane stress/strain. Bending of
the plate will cause material above and below this mid-plane to deform in-plane.
 The mid-plane plays the same role in plate theory as the neutral axis does in the
beam theory.
(ii) Line elements remain normal to the mid-plane
 Line elements lying perpendicular to the middle surface of the plate remain
perpendicular to the middle surface during deformation, Fig. 6.1.6; this is similar
the “plane sections remain plane” assumption of the beam theory.

6. Assumptions of Plate Theory
(iii) Vertical strain is ignored
 Line elements lying perpendicular to the mid-surface do not change length
during deformation, so that εzz = 0 throughout the plate. Again, this is similar
to an assumption of the beam theory.
 These three assumptions are the basis of the Classical Plate Theory or the
Kirchhoff Plate Theory.

7. Buckling:
 In structural engineering, buckling is the sudden change in shape (deformation) of a
structural component under load, such as the bowing of a column under compression or
the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing
load, when the load reaches a critical level, a member may suddenly change shape and
the structure and component is said to have buckled.
 Buckling may occur even though the stresses that develop in the structure are well
below those needed to cause failure in the material of which the structure is composed.
Further loading may cause significant and somewhat unpredictable deformations,
possibly leading to complete loss of the member's load-carrying capacity.
 However, if the deformations that occur after buckling do not cause the complete
collapse of that member, the member will continue to support the load that caused it to
buckle.
 If the buckled member is part of a larger assemblage of components such as a building,
any load applied to the buckled part of the structure beyond that which caused the
member to buckle will be redistributed within the structure. Some aircraft are designed
for thin skin panels to continue carrying load even in the buckled state.

8. Buckling:
 Most of steel or aluminum structures are made of tubes or welded plates.
Airplanes, ships and cars are assembled from metal plates pined by welling
riveting or spot welding.
 Plated structures may fail by yielding fracture or buckling.
 Buckling of thin plates occurs when a plate moves out of plane under
compressive load, causing it to bend in two directions.
 The Bucking behavior of thin plates is significantly different from buckling
behavior of a column.
 Buckling in a column terminates the members ability to resist axial force and as
a result , the critical load is the member’s failure load.

9. Buckling:
 The same cannot be said for the buckling of thin plates due to the membrane
action of the plate
 Plates under compression will continue to resist increasing axial force after
achieving the critical load , and will not fail until a load far greater than the
critical load is attained.
 That shows that a plate’s critical load is not the same as its failure load .

10. Plastic Buckling:
 When a material is loaded in compression it may buckle when a critical load is
applied.
 If loading is performed at constant strain-rate, this initial buckling will be
elastic and will be recoverable when the applied compressive stress is reduced.
 If loading is continued under these conditions, the buckled material may
deform enough to cause local plastic deformation to occur. This deformation is
permanent and cannot be recovered when the load is removed.

11. Example of Plastic Buckling :
 The photograph shows a thin wall carbon-steel tube that has been buckled in
compression. The tube has a square section, and the plastic deformation is
self-constraining. Initially, the material deformed elastically. Upon reaching
the buckling threshold, it bowed out and plastic deformation was initiated at
the region of maximum curvature.
 This "plastic hinge" can be folded at a lower applied stress than that needed
to initiate the buckle. When the material has closed on itself, a second hinge
is generated as the next tube section starts to buckle and plastically deform.
This process is repeated until the deformation is discontinued.

12. Example of Plastic Buckling :