Unit 6- Plate Bending Theory.pdf

PLATE BENDING THEORY
STRESS STRAIN HYPOTHESIS
• SUNIL SHRESTHA ME07818
• UMESH LEKHAK ME07819
• NABIN SHRESTHA ME07821
• ABHISHEK CHHUNJU ME07822

Plate Bending Theory
• Plate bending theory is a branch of mechanics that
studies the behavior of thin, flat plates under external
loads.
• The theory is based on the assumption that the plate is
very thin compared to its other dimensions and that it
undergoes small deformations.

Stress-Strain Hypothesis
• The stress-strain hypothesis is one of the key concepts
in plate bending theory.
• It relates the stress in the plate to the strain it
undergoes.
• This hypothesis assumes that the stress in the plate is
proportional to its strain, and that the proportionality
factor, known as the modulus of elasticity, is constant.

• More specifically, the stress-strain hypothesis in plate
bending theory assumes that the normal stress in the
plate is proportional to its curvature.
• And the shear stress is proportional to the rate of
change of curvature.

• This can be expressed mathematically using the
following equations:
σx = E(y'' + νxy')
τxy = E(1 - ν)x'y'
• where σx is the normal stress in the x-direction, τxy is
the shear stress in the xy-plane, E is the modulus of
elasticity of the plate, ν is Poisson's ratio, and y is the
displacement of the plate in the y-direction.

• The equations above describe the stress-strain
relationship in the plate, which allows us to derive
equations that relate the external loads applied to the
plate to the resulting deformations and stresses.
• These equations can be derived using the principle of
virtual work, which states that the work done by
external forces on a system is equal to the work done
by the internal forces within the system.

• It is used in the analysis and design of a wide range of
engineering structures, such as bridges, ships, and
aircraft.
• By understanding the behavior of thin plates under
external loads, engineers can design more efficient and
reliable structures that can withstand the stresses and
strains that they will be subjected to in service.

PURBANCHAL UNIVERSITY
KHWOPA ENGINEERING COLLEGE
SOLID MECHANICS
PRINCIPLE OF VIRTUAL WORK
REENA SUWAL (ME07814)
SACHIN POKHAREL (ME07815)
SAMI DANGOL (ME07816)
SUNDAR BARTAULA (ME07817)
Assoc. Prof. Dr. Manjip Shakya
Department of Earthquake Engineering
8th March 2023

Introduction
Virtual displacement and virtual Work
• Virtual work is the work done by a real force acting through a virtual displacement or a
virtual force acting through a real displacement.
• A virtual displacement is any displacement consistent with the constraints of the
structure, i.e., that satisfy the boundary conditions at the supports.
• A virtual force is any system of forces in equilibrium.
Principle of virtual Work
The principle of virtual work states that in equilibrium the virtual work of the forces
applied to a system is zero. Newton's laws state that at equilibrium the applied forces are
equal and opposite to the reaction, or constraint forces. This means the virtual work of the
constraint forces must be zero as well.

F1 = 1
F1
F3
F2 M1
M2
M3
F2 = 2
F3 = 3
Virtual linear displacement
Virtual Work

= 2
F1
F3
F2 M1
M2
M3
M2
M3
Virtual angular displacement
M1
= 3
= 1
Virtual Work

F1
F2
F3
M1
M2
M3 Total Virtual Work

F1
F2
F3
M1
M2
M3
R
Equilibrium
du = 0 If R = 0
= 0

Mathematical expression
• Consider an elastic system subjected to a number of forces (including
moments) F1, F2, . . . , etc. Let 𝛿1, 𝛿2, . . ., etc. be the corresponding
displacements.
• These are the work absorbing components (linear and angular
displacements) in the corresponding directions of the force as shown
in figure.

• Let one of the displacements 𝛿1 be increased by a small quantity
∆𝛿1. During this additional displacement, all other displacements
where forces are acting are held fixed, which means that additional
forces may be necessary to maintain such a condition.
• Further, the small displacement ∆𝛿 1 that is imposed must be
consistent with the constraints acting. For example, if point ‘I’is
constrained in such a manner that it can move only in a particular
direction, then ∆𝛿1 must be consistent with such a constraint.
• A hypothetical displacement of such a kind is called a virtual
displacement. In applying this virtual displacement, the forces F1, F2,
. . ., etc. (except F1) do no work at all because their points of
application do not move (at least in the work-absorbing direction).
• The only force doing work is F1 by an amount F1 ∆𝛿1. plus a fraction
of∆F1 ∆𝛿1. , caused by the change in F1. This additional work is
stored in strain energy ∆𝑈.

This is the principle of virtual work

The work done by the forces must be equal to the strain energy
which is stored up in the system. This fact can be expressed in
another form known as the principle of virtual work, i.e.

• In the equations above,
V denotes the material domain,
S the surface completely enclosing V,
and  the variational symbol signifies a virtual quantity
upon applying the divergence theorem
➔

We know, WE = WI
Substituting the value of WE and WI
………(i)
………(ii) a
………(ii) b
Eqn (i) is a field eqn
Eqn (ii) specify boundary conditions.

Conclusion
This principle is applicable to any elastic body
• linear elastic materials
• Non-linear elastic materials

BOUNDARY CONDITION
GROUP -3
NIWESH TAMRAKAR(ME07810)
PRESHIT MAHATO(ME07811)
RAJU RIJAL(ME07812)
RAMESH PRAJAPATI(ME07813)

Boundary Condition
In physics and engineering, boundary conditions are the set of
conditions that must be satisfied at the edges or boundaries of a
physical system.
Boundary conditions can take many forms, depending on the type
of system being analyzed.
For example, in a heat transfer problem, boundary conditions
might specify the temperature or heat flux at the boundaries of a
system.
In a fluid dynamics problem, boundary conditions might specify
the velocity or pressure at the boundaries of a system.

Boundary Condition for Plate Theory
❑ These boundary conditions are important because they
define the support and loading conditions that the
plate is subjected to, and therefore they determine
the deformation and stress distribution in the plate.
❑ In plate bending theory, the boundary conditions
depend on the type of support at the edges of the
plate.

Theory of Thin Plate Bending
Equilibrium Equation along z-
Direction:

POISSON’S BOUNDARY CONDITIONS:
• Clamped Edge
• Simply Supported Edge
• Free edge

Clamped Edges:
❖ Edges of the Plate are fixed and do not move.

Free Edges:
 Bending Moment:

Free Edges:
 Moment Mxy and Shear Forces
 Q + Q ’
 ∎
The net force acting on the
face
Q’x=-Mxy=
𝜕Mxy
𝜕𝑦
+Mxy
Q’x=-
𝜕Mxy
𝜕𝑦
Total shear force
Vx=Qx+Q’x

Free Edges:
 Total Shear Force

Free Edges:
 The Forces, Vx and Vy => Reduced, or Kirchoff’s or Effective Shear
Forces
 In case of a free edge,

Prepared By:
Anjana Kasula (ME07801) Bidhan Thusa (ME07803)
Argen Karmacharya (ME07802) Bikesh Tamrakar (ME07804)
1

Plates :-
▪ body whose lateral dimensions are large compared to the separation
between the surfaces.
▪ are initially flat structural elements.
▪ are subjected to traverse loads (that are
normal to its mid-surface) supported by
bending and shear action.
▪ Thin plates – t < 20b where, b = smallest side
Thick plates – t > 20b
▪ Small deflection – w ≤ t/5
▪ Thin plate theory – Kirchhoff's Classical Plate Theory (KCPT)
Thick plate theory – Reissner - Mindlin Plate Theory (MPT)
2

Assumptions :-
▪ Thickness is smaller than other dimensions
▪ Normal stresses in transverse direction are small compared with other
stresses, so neglected
▪ Governing equation is based on undeformed geometry
▪ Material is linearly-elastic body, thus follows Hooke’s law
▪ Middle surface remains unstrained during bending , so taken as neutral
surface
▪ Normal to mid-surface before deformation remain normal to the same
surface after deformation
▪ Traverse shear strains are negligible, doesn’t apply shear across section
is zero.Traverse shear strain have negligible contribution to
deformations.
3

▪ Equilibrium of forces along Z-direction
𝜕𝑄𝑥
𝜕𝑥
. 𝑑𝑥𝑑𝑦 +
𝜕𝑄𝑦
𝜕𝑦
. 𝑑𝑥𝑑𝑦 + 𝑞. 𝑑𝑥𝑑𝑦 = 0 →
𝜕𝑄𝑥
𝜕𝑥
+
𝜕𝑄𝑦
𝜕𝑦
+ 𝑞 = 0 −−− −(𝑖)
▪ Mx = -D(
𝜕2𝑤
𝜕𝑥2 + υ
𝜕2𝑤
𝜕𝑦2 ), My = -D(
𝜕2𝑤
𝜕𝑦2 + υ
𝜕2𝑤
𝜕𝑥2 ), Mxy = -Myx = -D 1 − υ
𝜕2𝑤
𝜕𝑥𝜕𝑦
▪ D = flexural rigidity of the plate=
𝐸𝑡3
12(1−υ2)
, q = distributed loading on the plate
▪ Mx, My = Bending Moment, Mxy, Myx = Twisting Moment, Qx, Qy = Shear Forces
4

▪ Moment along x-axis,
Mxy +
𝜕𝑀𝑥𝑦
𝜕𝑥
𝑑𝑥 𝑑𝑦 − 𝑀𝑥𝑦. 𝑑𝑦 + 𝑀𝑦. 𝑑𝑥 − (𝑀𝑦 +
𝜕𝑀𝑦
𝜕𝑦
𝑑𝑦)𝑑𝑥 + Q𝑦. 𝑑𝑥𝑑𝑦 = 0
𝜕𝑀𝑥𝑦
𝜕𝑥
−
𝜕𝑀𝑦
𝜕𝑦
+ 𝑄𝑦 = 0 − − −(𝑖𝑖)
▪ Moment along y-axis,
Myx +
𝜕𝑀𝑦𝑥
𝜕𝑦
𝑑𝑦 𝑑𝑥 − 𝑀𝑦𝑥. 𝑑𝑦 + 𝑀𝑥. 𝑑𝑦 − 𝑀𝑥 +
𝜕𝑀𝑥
𝜕𝑥
𝑑𝑥 𝑑𝑦 − Q𝑥. 𝑑𝑥𝑑𝑦 = 0
𝜕𝑀𝑦𝑥
𝜕𝑦
+
𝜕𝑀𝑥
𝜕𝑥
− 𝑄𝑥 = 0 − − − (𝑖𝑖𝑖)
▪ After substituting value in eqn. (i) from (ii) and (iii), we get
𝜕4𝑤
𝜕𝑥4 + 2 ∗
𝜕4𝑤
𝜕𝑥2𝜕𝑦2 +
𝜕4𝑤
𝜕𝑦4 =
𝑞
𝐷
→
𝜕2
𝜕𝑥2 +
𝜕2
𝜕𝑦2
𝜕2𝑤
𝜕𝑥2 +
𝜕2𝑤
𝜕𝑦2 =
𝑞
𝐷
→ 𝛻2
𝛻2
𝑤 =
𝑞
𝐷
5

𝜵𝟒
𝒘 =
𝒒
𝑫
→ 𝐃 𝜵𝟒
𝒘(𝒙, 𝒚) = 𝒒(𝒙, 𝒚)
▪ 𝜵𝟒𝒘 is known as Laplace Operator or Biharmonic Operator.This expression is
the basic differential equation for plate bending theory.
▪ This plate equation solution requires specification of appropriate boundary
condition that involves deflection & its derivatives at the edges of the plate.
▪ Kirchhoff's plate equation is widely used model in plate bending theory & has
numerous applications in engineering & physics such as, design of aircraft wings
and analysis of seismic waves in the earth’s crust.
6

Three basic boundary conditions
1. Built in edge(Clamped)
𝑤 𝑥 = 𝑎 = 0,
𝜕𝑤
𝜕𝑥
𝑥 = 𝑎 = 0
2. Simply supported
𝑤 𝑥 = 𝑎 = 0,
𝜕2𝑤
𝜕𝑥2 + υ
𝜕2𝑤
𝜕𝑦2 𝑥 = 𝑎 = 0 →
𝜕2𝑤
𝜕𝑥2 = 0 and
𝜕2𝑤
𝜕𝑥2 = 0
3. Free edge
Mx(x=a)=0, Mxy(x=a)=0, Qx(x=a)=0
7

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

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.

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.

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.

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.

(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.

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.

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.

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 .

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.

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.

Unit 6- Plate Bending Theory.pdf

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