Topics in Ship Structural Design (Hull Buckling and Ultimate Strength)

Lecture 3 Plate Bending

Reference : Ship Structural Design Ch.09 NAOE Jang, Beom Seon

OPen INteractive Structural Lab 9.1 Small Deflection Theory "Long" Plates(Cylindrical Bending) . An equation relating the deflections to the loading can be developed as for the beam. Saddle shape . Cylindrical Bending:  A plate which is bent about one axis only(a>>b)  Transverse deformation does not occur, since such a deformation would required a saddle shape deformation.

 y  x  y =   x  y  x EE E εy -εx

 y  x  y  x  y    0 ε E E x -εy (1-2 )  = x x E E =  =E  x1- 2 x x Plate in cylindrical bending 2 OPen INteractive Structural Lab Reference Poisson Ratio

    trans  axial

Strain Δεy induced by εy     y x y E

1   (   ) x E x y z 1   (   ) y E y z x 1   (   ) z E z x y

OPen INteractive Structural Lab 9.1 Small Deflection Theory Formulas between a beam and a long plate . E΄ : effective modulus of elastictiy, always > E. E E  (1 2 ) → A plate is always “stiffer” than a row of beams . For a long prismatically loaded plate, the extra stiffness may be fully taken into account for by using E΄ in place of E.

. For simply supported plates: 55(1-)pbpb442   == max 38432E IEt 3 . For clamped plates: pb4 pb 4(1- 2 )  == max 384E I 32 Et3

4 OPen INteractive Structural Lab 9.1 Small Deflection Theory Formulas between a beam and a long plate . Moment curvature relation may be obtained from External bending moment = moment of stress force 23 tt/2/2 EzE t1 Mzdzdz  =()= tt/2/2 x 22 112(1-)rrxx E where     =/zr x (1 2 ) x xx

EDt13  3 M ==Et 2  D  2 12(1-) rrxx 12(1-) . D : the flexural rigidity of the plate the constant of proportionality between moment and curavature analogous with EI in beam theory EI M  

5 OPen INteractive Structural Lab 9.1 Small Deflection Theory Formulas between a beam and a long plate . The radius of curvature of the plate can be expressed in terms of the deflection w of the plate  2w 1  2w M    D  M  v  2 x2 x rx EI . Also for a unit strip in a long plate the max. stress is  maxmax=/McI where the section modulus I / c=(t/12)/(t/2)=t/632

. For a uniform pressure p, the maximum bending moment is proportional to pb2. The stress is expressed in 2 b  max =kp  t

. The coefficient k depends on the boundary conditions: k=3/4 for simply supported edges k=1/2 for clamped edges

6 OPen INteractive Structural Lab Reference Beam Theory stress – strain - curvature - Moment

1) strain-curvature 3) moment-curvature relation Geometric relation 2)strain-stress relation

M    x ydA A y     2 Ey 1   E   

1  3    Ey 1 M    4  EI 4) Stress-moment relation My    x I OPen INteractive Structural Lab Reference Beam theory 1) Strain-Curvature Relation Initial length of Line e-f = dx

Length after bending = L1 y L  (  y)d  dx  dx 1  dx d  compression  Natural axis No change in length

L1  dx y  x     y Tension dx 

OPen INteractive Structural Lab Reference Beam theory 3) moment-curvature relation

dM   x ydA

M    x ydA A

1 M  Ey2dA  E y2dA  EI  EI   Ey A A  1 M     EI : EI = bending rigidity

I  y 2dA : Moment of Inertia A OPen INteractive Structural Lab Reference Homework #1 Derive in detail Beam Theory : Relation between load, shear force, Moment

. Force equilibrium → Relation between shear force (V) and load (q) dV F  0 : V  qdx  (V  dV )  0   q  vert dx . Moment equilibrium → Relation between Moment (M) and shear force (V)

 dx  dM  M  0 :  M  qdx   (V  dV )dx  (M  dM )  0   V  2  dx . Combining two relations d 2M  q dx2 . Relation between deflection and distributed load

2 d  M d 4v q     dx2 EI dx4 EI

OPen INteractive Structural Lab 9.1 Small Deflection Theory Forces and moments in a plate element

. Force equilibrium → Relation between shear force (V) and load (q)

qx qy dV + +p =0  q q :shear force per unit length xy dx m : moment per unit length . Moment equilibrium mm m m dM xyy-+q=0 yx +x -q =0  V xyy yxx dx

. Combining two relations

2 22  m mm 2 x -2++p=0xyy d M 22  q xx  yy d 2 x . Relation between deflection and distributed load

 2w 2w   2w 2w  m  D  v  m  D  v  x  2 2  y  2 2  4 4 4 4  x y   y x   w  w  w p d v q 4  2 2 2  4     2w x x y y D dx4 EI m  D(1 v) xy xy OPen INteractive Structural Lab 9.1 Small Deflection Theory Derivation of the Plate Bending Equation  The previous theory is only applicable if : . Plane cross sections remain plane.

. The deflections of the plate are small(wmax not exceeding 3/4 t) . The maximum stress nowhere exceeds the plate yield stress(i.e., the material remains elastic)  A panel of plating will have curvature in two directions      = x  y  = y  x x EE y EE

v v2 v  y  x y E E

 x（1-2） =  +  E xy Differential element of plating

OPen INteractive Structural Lab 9.1 Small Deflection Theory Derivation of the Plate Bending Equation . Assumptions stated previously give rise to the strain-curvature relations

 2w   2w   x 2   (z)    (z)  （1-=+） xy x  2  y  2   x   y  E

E  2w 2w    (z)  v  x 2  2 2  1 v  x y 

2 2 t / 2 t / 2 E   w  w  2 mx   x zdz     v z dz t / 2 t / 2 2  2 2  1 v  x y 

3   2w  2w  Et m  D  v  D  x  2 2  2  x y  12(1-)

  2w  2w  m  D  v  y  2 2   y x  OPen INteractive Structural Lab Reference – Mechanics of Material Beam Theory : Deflection curve . Relation between deflection and distributed load From Geometric relation ds  dx d  ds 1 d     dx 1 d 2    1 d dv 2    tan     dx  ds dx

d 1 M  tan    dx  EI

d 2 M   dx2 EI d 4v q 2 4   d M dx EI  q dx2 OPen INteractive Structural Lab 9.1 Small Deflection Theory Forces and moments in a plate element . The lateral load pdxdy is carried by the distributed shear forces q acting on the four edges of the element. . In the general case of bending, twisting moments will also be generated on all the four faces

Without the lateral load pdxdy With the lateral load pdxdy

Forces and moments in a plate element

OPen INteractive Structural Lab 9.1 Small Deflection Theory Forces and moments in a plate element . Sine convention in beam theory – Determined how deform the material – Shear force : rotate an element clockwise (+) – Bending moment : shorten the upper skin (+)

Sine convention in beam theory Sine convention in plate bending OPen INteractive Structural Lab 9.1 Small Deflection Theory Forces and moments in a plate element . From equilibrium of vertical forces :

qx (qx  dx)dy  qxdy q q x x ++=0y p q xy  (q  y dy)dx  q dx  pdxdy  0 y y y . Taking moments parallel to the x-axis, at Point A

q :shear force per unit length m (m  xy dx)dy  m dy  m dx m : moment per unit length xy x xy y m  (m  y dy)dx  q dxdy  0 y y y Point A mm xyy-+q =0 xyy . Taking moments parallel to the y-axis m m yx +x -q =0 yxx OPen INteractive Structural Lab 9.1 Small Deflection Theory Forces and moments in a plate element . Because of the principle of complementary shear stress, it follows

that myx=-mxy, so that

mxy mx mmxyy myx mx +-q=0 -+-q=0 x -+q=0 y yxx yx xy

. Substituting for qx and qy

q q x ++=0y p xy

 m m  m m ( xy  x )  ( y  xy )  p  0 x y x y y x

2m 22mm x -2xy + y +p=0 x22  x  y  y OPen INteractive Structural Lab 9.1 Small Deflection Theory Forces and moments in a plate element . If a point A in the plate a distance z from the neutral surface is displaced a distance v in the y direction, . The change in slop of the line AB will be v v  dx  v v x  dx x

. The change in slop of the line AD u x . The shear strain is Shear strain and plate deflection  u  =+ xy Shear strain is induced by . The shear stress deflection. That is, the vertical deflection generates shear strain.  u  =G + xy OPen INteractive Structural Lab 9.1 Small Deflection Theory Forces and moments in a plate element . from the geometry of the slope of w in each direction (x and y): w w u  z v  z x y

. Making these substitutions gives  w u=-z u(), z(), (), x x  2w   2Gz xy (+) convention

τ . The twisting moment 부호 정의 필요 2 3 2 t / 2  w 2 Gt  w E mxy    2G z dz  t / 2 G  xy 6 xy 2(1 v) τ

Et 3  2w Et 3 (1 v)  2w  2w m    D(1 v) xy 12(1 v) xy 12(1 v)(1 v) xy xy

-τ

+τ OPen INteractive Structural Lab 9.1 Small Deflection Theory Forces and moments in a plate element

. Substitute mx, my, mxy  2w 2w   2w 2w  m  D  v  x  2 2  my  D  v   x y   y 2 x2    2 22  m mmxy y  2w x -2 + +p=0 m  D(1 v) x22  x  y  y xy xy . Finally  2    2w  2w   2   2w   2    2w  2w   D  v   2 D(1 v)   D  v    p 2   2 2    2   2 2  x   x y  xy  xy  y   y x  . Equation of equilibrium for the plate  4w  4w  4w p  2   4w  p / D x4 x2y 2 y 4 D

. Assumption • small deflection, max deflection < ¾ t • no stretching of the middle plane ( no membrane effect)

OPen INteractive Structural Lab 9.1 Small Deflection Theory Boundary Conditions . The types of restraint around the boundary of a plate can be idealized as follows: 1. Simply supported: edges free to rotate and to move in the plane

2. Pinned: edges free to rotate but not free to move in the plane

3. Clamped but free to slide: edges not free to rotate but free to move in the plane of the plate;

4. Rigidly clamped: edges not free to rotate or move in the plane

. In most plated frame structures (particularly in vehicles and free- standing structure) : little restraint against edge pull-in. → BC. 1 and 3 are generally applicable

OPen INteractive Structural Lab 9.1 Small Deflection Theory Simply Supprted Plates . The general expression for the load  mynx p( x , yA )=sinsin mn mn11 ab

. The coefficient Amn can be obtained by Fourier analysis for any particular load condition. For a uniform pressure p0. 16 p A = 0 mn  2mn Homework #2 Prove it. . For the biharmonic equation, a sinusoidal load distributi on produces a sinusoidal deflection mn== m yn x  4w  4w  4w p =sinsinBmn 4  2 2 2  4  mn11 ab x x y y D

  4m4 2 4m2n2  4n4  16 p B      0 mn  4 2 2 4  6  a a b b   Dmn

16 p0 Bmn = 222 6 mn  Dmn22+ ab

OPen INteractive Structural Lab 9.1 Small Deflection Theory Simply Supprted Plates . Due to the symmetry of the problem, M and n only take odd values. Homework #3 Plot the cur mn 16 p my mx ve using any kinds of soft w  0 sin sin  2 2 ware.  m n  a b m1 n1  6 Dmn    2 2   a b 

. Effect of boundary condition Clamped < Simply supported about 4~5 times.

. Effect of aspect ratio when a=b, the deflection is minimum

Maximum deflection of rectangular plates under uniform pressure OPen INteractive Structural Lab 9.1 Small Deflection Theory Simply Supported Plates . To determine the bending stress in the plate, calculate the bending stress.

 2w mn 16 p  2n2 my nx mn 16 p n my nx   0 sin sin   0 sin sin 2  2 2 2 2  2 2 2 x m1 n1  m n  b a b m1 n1  m n  a b  6 Dmn    4 Dmb2     2 2   2 2   a b   a b   2w mn 16 p m my nx   0 sin sin 2  2 2 2 y m1 n1  m n  a b  4 Dna2     2 2   a b  22mn== 16 p nmm22 y n x =+sinsin 0  mD=-+   2 22 x 22 22 baab xymn114 mn   mn22+ ab

. The bending moment will always be greater across the shorter span. Homework #4 Prove this. . The symbol b is used for shorter dimension, then a/b> 1

. mx is the larger of the two moments and has its maximum value in the center of the plate.

OPen INteractive Structural Lab 9.1 Small Deflection Theory Clamped Plates . The usual methods are the energy(or Ritz) method and the method of Levy . The energy method, while giving only approximate results . The Levy type solution is achieved by the superposition of three loading systems applied to a simply supported plate :

(1) uniformly distributed along the short edges : deflection w1

(2) moments distributed along the short edges : deflection w2

(3) moments distributed along the long edges : deflection w3

short edges : 12  3   +  +   0 y y a/2   y  y  y  a /2 long edges : 123  + + 0 xxb /2   x  x xb /2 ( where x=0, y=0 is at the center of the plate)

OPen INteractive Structural Lab 9.1 Small Deflection Theory Clamped Plates 2  b  Homework #5 Plot Max. S Stress  kp  tress curve in simply sup  t  ported plates.  k depends on : . plate edge conditions . Plate side ratio a/b . Position of point considered

The effect of aspect ratio is smaller for clamped plates, and beyond about a/b=2

such a plate behaves essentially as a clamped strip and the influence of aspect aspect ratio is negligible.

Stresses in rectangular plates under uniform lateral pressure

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Large-Deflection Plate Theory . Membrane stress arises when the deflection becomes large and/or when the edges are prevented from pulling in.

. Small-deflection theory fails to allow for membrane stresses.

. As the deflection increases, an increasing prortion of the load is carried by this membrane action.

. The lateral load is supported by both bending and membrane action.

. Amore comprehensive plate theory, usually referred to as "large- deflection" plate theory.

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Membrane Tension(Edges Restrained Against Pull-in)  What is membrane tension? . Membrane stress : Stresses that act tangentially to the curved surface of a shell . Pressure vessel : closed structures containing liquids or gases under pressure . Gage pressure (Internal – external pressure) is resisted by membrane tension

Spherical Pressure vessel Cylindrical Pressure vessel

OPen INteractive Structural Lab Reference Membrane stress  Spherical Pressure vessel

 Fhoriz  0 2 (2rmt)  p(r ) t r  r  m 2 pr2 pr    2rmt 2t

OPen INteractive Structural Lab Reference Membrane stress  Cylindrical Pressure vessel

σ1 : circumferential stress hoop stress

1(2bt)  2pbr  0 pr   1 t

σ2 : longitudinal stress 2  2 (2rt)  pr  0 pr   2 2t

1  2 2

OPen INteractive Structural Lab Reference Membrane stress  Ship & Offshore Structure

Hull Girder Global Stress Bending

Membrane stress on plating Double Bottom Bending

Membrane stress Bending stress Local Stress Local Bending of Longitudinal Bending stress on plating

# of elements should be more than thee (3) Plate between boundaries. Bending

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Large-Deflection Plate Theory by von Karman

. Nx, Ny : membrane tension per unit length

. Nxy : membrane shear force per unit length . Vertical component of the tension force at side (1)

 N xdy sin(slope)  N xdy tan(slope)  N xdy(w / x)

Vertical component w  N dy( ) x x

Vertical component  +Nx dydx  . Net force in the x-direction xxx 

 2w  N dy( ) x x2

. Net force in the y-direction  2w  N dx( ) y y 2

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Large-Deflection Plate Theory by von Karman . The vertical component of the shear force along sides (3) and (4) is   NdxdyNdxxyxy  xyxx  Vertical component w   w   N xydx   dy  x y  x   Vertical component

w N dx( ) xy x

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Large-Deflection Plate Theory by von Karman . Total vertical forces

  2w  2w  2w   N  2N  N dxdy  x 2 xy y 2   x xy y  . New terms are added

 4w  4w  4w 1   2w  2w  2w   4w  4w  4w p  2    p  N  2N  N   2   4 2 2 4  x 2 xy y 2  4 2 2 4 x x y y D  x xy y  x x y y D

1   2w  2w  2w  4w   p  N  2N  N   x 2 xy y 2  D  x xy y 

. Nx, Ny, Nxy are functions of x and y. . Except for a few simple cases, precise mathematical solutions are very difficult.

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Membrane Tension(Edges Restrained Against Pull-in) . The relative magnitude of membrane effects depends on : the degree of lateral deflection or "curvature" of the plate surface the degree to which the edges are restrained form pulling in.

. If the edges are restrained and large deflection (w>1.5 t) the contribution of membrane tension > bending, vice versa.

. In ship plating : little restraint against edge pull-in for large deflections, supporting stiffeners or beams fails earlier.

. Nevertheless, some situations in which large deflections can be permitted, → the use of membrane tension can give substantial weight savings.

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Membrane Tension(Edges Restrained Against Pull-in)  The case of a unit-width strip . Laterally loaded . Edges are prevented from approaching . the difference between the arc length of the deflected strip and the original straight length

2 2 b d b 1 d  =11 dx   dx 0 dx 0   2 dx

. Membrane force is unidirectional (Ny=Nxy=0) and constant over the length. . Extension due to the tension

2 Etd b  Tdx  2bdx0 

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Membrane Tension(Edges Restrained Against Pull-in)

. w0 initial deflection, w1 due to the load  x ()()sinx  01 b . The total change in length is 2 b 1 x ()cos 22dx 0 2 01bb2  2 ()2 4b 01

. The change in length due to the initial deflection

2 2  w0 4b

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Membrane Tension(Edges Restrained Against Pull-in) . The change in length due to the loading is the difference between

these two: 2    (2) 2 1011 4b . Strain energy due to bending 2 4 4 2 b b EI  d w  EI 2  2 x  EIw1    dx  w1  sin dx  0  2  0 4 3 2  dx  2 b b 4b . Strain energy due to tension: 11AE T 2 22p b 1 . The work done

b b 1 1  x  pw1b W  pwdx  pw1 sin dx  0 2 0 2  b   . The work done by the load is equal to the total strain energy, hence:

pw b AE 2  4 EI(w2 )  4 AE  4 EI(w2 ) 1  1  1  (2w w  w2 )2  1  2b 4b3 32 b3 0 1 1 4b3

OPen INteractive Structural Lab Reference – Mechanics of Material Strain Energy of Bending . When bending moment M varies along its length 1 d 2v d  dx  dx  dx  dx2 Md M d 2v M M M 2dx dU   dx  dx  or 2 2 dx2 2 EI 2EI 2 2 EI  d 2v  EI  d 2v  dU   dx    dx  2   2  2dx  dx  2  dx  . if integrated

2 M 2dx EI  d 2v  U  or U    dx    2  2EI 2  dx 

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Membrane Tension(Edges Restrained Against Pull-in) . The work done by the load is equal to the total strain energy, hence:

43AE 4242AEAE   4 E Ipb 1  0 10  0 3333 1 32884bbbb  . A=t, E’= E/(1-v) and I=t3/12 per unit width 24 322 232tpb 10  1014+ 40 25 31Et

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Effect of Initial Deformation . If there is no initial deflection: 232tpb24 3  113125Et

. For the initial stages of loading the deflection w1 will be small relative to the thickness, and hence the first term may be neglected: 2 44 1 48(1) pbpb 5  0.143 tEtEt  

. The result obtained from small-deflection theory, ignoring membrane action:

2 4 4 1 5(1) pb pb   0.142  tEt 32  Et

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Effect of Initial Deformation

Mean magnitude of max. Bending . Membrane action requires stress ignoring membrane effect some deflection, either initial or Mean magnitude of max. Bending stress with due to load membrane effect Surface stress on . if there is no initial deflection tension side

→ membrane action does not Surface stress on become significant until the compression side deflection due to load approaches the plate thickness. Membrane stress . Welding deformation & permanent zero initial deflection set give a beneficial influence on elastic strength of plates. Membrane stress

Membrane stress Bending stress 0.5t initial deflection

OPen INteractive Structural Lab 9.2 Combined Bending and Membrane Stresses-Elastic Range Effect of Initial Deformation . The beneficial effect of initial deformation.

Type of Initial Deformation Elastic Source of Increase in Strength Elastic Strength Flat plate 1.59 - Initial deflection (stress-free) equal to 2.58 Membrane action plating thickness Initially flat plate dished to a permanent set 4.50 Membrane action plus equal to plating thickness residual stresses

. Initial lateral deformation is beneficial only when the plate edges are at least partly restrained from pulling in, thus allowing the development of in-plane tension.

OPen INteractive Structural Lab Which one is in front? Di Vinci's Last Supper

• Famous for not only artistic value but also fine command of perspective Westerner try to see and Asian try to be

• The law of perspective is one of representative characteristics • Object has the meaning of observation and all things • Objective is not subjective. • I see = I understand • Seeing is believing, Westerner makes seeing very important. • From the observer’s perspective, farmost thing indicates that in front. • In the • Closer thing is drawn larger and the farther thing smaller. -> reverse perspective. • Western artists draw picture seeing an object in front of him. • Asian artists draw picture after seeing and feeling the object and returning.

• Universe is covered by Indra net. Each marble is mirroring others OPen INteractive Structural Lab