13. Bending of plates under the combined action of lateral loads and forces in the middel plane of the plate

13. Bending of plates under the combined action of lateral loads and forces in the middel plane of the plate

BENDING OF PLATES UNDER THE COMBINED ACTION

OF LATERAL LOADS AND FORCES IN THE MIDDLE

PLANE OF THE PLATE

90 Differential Equation of the Deflection Surface In our previous discussion it has always been assumed that the plate is bent by lateral loads only If in addition to lateral loads there are forces acting in the middle plane of the plate, these latter forces may have a considerable effect on the bending of the plate and must be considered in deriving the corresponding differential equation of the deflection surface Proceed- ing as in the case of lateral loading (see Art 21, page 79), we consider the equilibrium of a small element cut from the plate by two pairs of planes parallel to the zz and yz coordinate planes (Fig 191) In addi- tion to the forces discussed in Art 21 we now have forces acting in the

middle plane of the plate We denote the

| magnitude of these forces per unit length by

mS N., N,, and Nz, = Ny,z, aS shown in the figure

Projecting these forces on the x and y axes and assuming that there are no body forces or

| | Net lì on tangential forces acting in those directions at

| I ° , the faces of the plate, we obtain the following

| Ny equations of equilibrium:

= pene ở ONz , 9Ne _ g

—+>> y ò

` Ny + Fay These equations are entirely independent of

(b) the three equations of equilibrium considered Fic 191 in Art 21 and can be treated separately, as

will be shown in Art 92

In considering the projection of the forces shown in Fig 191 on the

z axis, we must take into account the bending of the plate and the resulting small angles between the forces Nz and N, that act on the opposite sides of the element As a result of this bending the projection

378

of the normal forces N, on the z axis gives

N dz) & + —, dw , ð? 5x5 is) dy

After simplification, if the small quantities of higher than the second order are neglected, this projection becomes

—N: dụ 2+ (

n, 2 ® ae dy oN oe

In the same way the projection of the normal forces N, on the z axis gives

nN, 2 ae oat dy + ae ae dy (b)

Regarding the projection of the shearing forces N,, on the z axis, we observe that the slope of the deflection surface in the y direction on the two opposite sides of the element is dw/dy and dw/dy + (d?w/dx dy) dz Hence the projection of the shearing forces on the z axis is equal to

Ow, 4, ON xy OW

Nev ae ay YF ox ay OY

An analogous expression can be obtained for the projection of the shear- ing forces N, = Nz, on the z axis The final expression for the projec- tion of all the shearing forces on the z axis then can be written as

oN Ow ¡ vđy + ane ow

“3s iy Adding expressions (a), (b), and (c) to the load g dx dy acting on the ele- ment and using Eqs (216), we obtain, instead of Eq (100) (page 81), the following equation of equilibrium:

dx? mm ^ ôU? ~ - (4 + NSS +N Ge + 2N san )

Substituting expressions (101) and (102) for A/,, AL,, and ăz„, we obtain

aw dtw dw

oat T ” Ga? ay? * ays

| q+ N.S WIN Woy, Jw (217)

This equation should be used instead of Eq (103) in determining the deflection of a plate if in addition to lateral loads there are forces in the middle plane of the plate

If there are body forces! acting in the middle plane of the plate or tangential forces distributed over the surfaces of the plate, the differential equations of equilibrium

of the element shown in Fig 191 become

ON ONey

+ ——

ONey ON, =0

Ox oy —

Here X and Y denote the two components of the body forces or of the tangential forces per unit area of the middle plane of the plate

Using Eqs (218), instead of Eqs (216), we obtain the following differential equa- tion? for the deflection surface:

“— + N,— +2N.,—— —-X— - + vaya T ” ax day ox ay

(219)

Equation (217) or Eq (219) together with the conditions at the boundary (see Art 22,

page 83) defines the deflection of a plate loaded

° *T—_x laterally and submitted to the action of forces in the f— ime 3 middle plane of the plate

va -* P 91 Rectangular Plate with Simply Supported

„ _Ÿ Edges under the Combined Action of Uniform f€ ~-~~@ -==== >| Lateral Load and Uniform Tension Assume

y that the plate is under uniform tension in the tra, 192 z direction, as shown in Fig 192 The uniform lateral load g can be represented by the trigonometric series (see page 109)

Mar nny

Equation (217) thus becomes

O4w 04w 04w N, 0?w

ant | 7 3gt ay? * ay! D oz?

This equation and the boundary conditions at the simply supported edges

1 An example of a body force acting in the middle plane of the plate is the gravity force in the case of a vertical position of a plate

2 This differential equation has been derived by Saint Venant (see final note 73) in his translation of Clebsch, ‘‘Théorie de ]’élasticité des corps solides,”’ p 704, 1883.

will be satisfied if we take the deflection w in the form of the series

Mrz ney

Substituting this series in Eq (b), we find the following values for the coefficients Amn:

6n -

in which m and n are odd numbers 1, 3, 5, , and dm, = O1f m or n

or both are even numbers Hence the deflection surface of the plate is

5 3 >] Sin —— sin —- (e)

m n 2 Nm | a b

“ (2 + is) T a’ Da?

Comparing this result with solution (131) (page 110), we conclude from the presence of the term N,m?/x?Da? in the brackets of the denominator that the deflection of the plate is somewhat diminished by the action of the tensile forces N, This is as would be expected

By using M Lévy’s method (see Art 30) a solution in simple series may be obtained which is equivalent to expression (e) but more con- venient for numerical calculation The maximum values of deflection and bending moments obtained in this way! for v = 0.3 can be represented

in the form

qb"

a Eh3 (TM) max — 8q? (M,) max —= 8g)? (f)

The constants a, 8, and 8: depend upon the ratio a/b and a parameter

_ N,b?

Y ™ 42D

and are plotted in Figs 193, 194, and 195

If, instead of tension, we have compression, the force N, becomes

1H D Conway, J Appl Mechanics, vol 16, p 301, 1949, where graphs in the case

of compression are also given; the case N, = N, has been discussed by R F Morse and H D Conway, J Appl Mechanics, vol 18, p 209, 1951, and the case of a plate clamped all around by C C Chang and H D Conway, J Appl Mechanics, vol 19,

p 179, 1952 For combined bending and compression, see also J Lockwood Taylor, The Shipbuilder and Marine Engine Builder, no 494, p 15, 1950.

negative, and the deflections (e) become larger than those of the plate bent by lateral load only It may be seen also in this case that at cer- tain values of the compressive force Nz the denominator of one of the terms in series (e) may vanish This indicates that at such values of N, the plate may buckle laterally without any lateral loading

92 Application of the Energy Method The energy method, which was previously used in discussing bending of plates by lateral loading (see Art 80, page 342), can be applied also to the cases in which the

0.14

_———

_

O12

—

™ `

0.04

0.02

g

—

b

Fia 193

lateral load is combined with forces acting in the middle plane of the plate To establish the expression for the strain energy corresponding

to the latter forces let us assume that these forces are applied first to the unbent plate In this way we obtain a two-dimensional problem which can be treated by the methods of the theory of elasticity.1 Assuming that this problem is solved and that the forces N., N,, and N., are known

at each point of the plate, the components of strain of the middle plane

of the plate are obtained from the known formulas representing Hooke’s

1 See, for example, S Timoshenko and J N Goodier, ‘‘Theory of Elasticity,’’ 2d ed., p 11, 1951.

law, 012.,

hE (N, — vN,) Cụ — hi (Ny — „N,)

Yzu — hG

es =

The strain energy, due to stretching of the middle plane of the plate, 1s

V¥i= AS J (Nez + Nyey + N xyz) du dy

|

= | [N2 + N2 — 2vN,N, + 2(1 + r)N?, | dà dụ (220) where the integration is extended over the entire plate

Let us now apply the lateral load This load will bend the plate and produce additional strain of the middle plane In our previous discus- sion of bending of plates, this latter strain was always neglected Here,

0.14

0.10

ky

EZ

Z0

Fic 194

0.06

F—— _

_——

——

TT”

0

b

tg 195 however, we have to take it into consideration, since this small strain in combination with the finite forces Nz, N,, Nz, may add to the expression for strain energy some terms of the same order as the strain energy of bending The x, y, and z components of the small displacement that a point in the middle plane of the plate experiences during bending will be

ak 4x15

0

—9; = 9W dx

K Ox

U Fh~

\

z ut gu dx

Fig 196

, and w, respectively Considering a linear element AB of that plane in the z direction, it may be seen from Fig 196 that the elongation of the element

due to the displacement u is equal to

(du/dx) dx The elongation of the same element due to the displacement w is 4(dw/dx)? dx, as may be seen from the com-

denoted by u, 2,

parison of the length of the element A,B, in Fig 196 with the length of its projection on the z axis

tion of an element taken in the middle plane of the plate is

€

Thus the total unit elongation in the z direc-

Ou 1 fow\?

= 5 ¬+- 5 &) (221)

Similarly the strain in the y direction is

,_ v , 1 (aw)?

| fy = » + 5 (0y) (222)

Considering now the shearing strain in the middle plane due to bend- ing, we conclude as before (see Fig 23) that the shearing strain due to the displacements u and v is du/dy + dv/dx To determine the shear- ing strain due to the displacement w we take two infinitely small linear elements OA and OB in the x and y directions, asshown in Fig 197 Because of displacements in the z direction these elements come to the positions O1A; and O,B; The difference between the angle 7/2 and the angle A,0,B, is the shearing strain corresponding to the displacement w

To determine this difference we con-

sider the right angle B;O+4;, in which 0 dx

B.O, is parallel to BO Rotating the 3 |

plane B.O,A, about the axis 0,A, by 0,

the angle dw/dy, we bring the plane y

B,0,Ai into coincidence with the 2, A Ow gs

plane B,O,A;* and the point B: to posi- % ox

tionC Thedisplacement B.C'is equal cá + ow

to (dw/dy) dy and is inclined to the ver- i wy Ol

tical B.B, by the angle dw/dz Hence Ox be ;

BiC is equal to (dw/dx)(dw/dy) dy, Fic 197

and the angle CO,B,, which repre-

sents the shearing strain corresponding to the displacement w, is (dw/dx)(dw/dy) Adding this shearing strain to the strain produced by the displacements wu and v, we obtain

> =

\ ——>ke-~—=—

a, ><

——> a =

Ow Ow

Formulas (221), (222), and (223) represent the components of the addi- tional strain in the middle plane of the plate due to small deflections Considering them as very small in comparison with the components «:, €,, and yz, used in the derivation of expression (220), we can assume that the forces N., Ny, Nz, remain unchanged during bending With this assumption the additional strain energy of the plate, due to the strain produced in the middle plane by bending, is

Vo = Sf(Nae, + Nye, + Nevyi,) dx dy

Substituting expressions (221), (222), and (223) for «¿, «,, and y;,, we

* The angles dw/dy and dw/dzx correspond to small deflections of the plate and are

regarded as small quantities.

finally obtain

-LIxŒ}+w(3) ceEB]zx m

It can be shown, by integration by parts, that the first integral on the right-hand side of expression (224) is equal to the work done during bend- - ing by the forces acting in the middle plane of the plate Taking, for example, a rectangular plate with the coordinate axes directed, as shown

in Fig 192, we obtain for the first term of the integral

Proceeding in the same manner with the other terms of the first integral

in expression (224), we finally find

Jo Lr [NS + esp tM (Gy + az) Jee a b

0

“LÍ, u (SE + vt) av ay — [ I 0 ( su) ae dy

The first integral on the right-hand side of this expression is evidently equal to the work done during bending by the forces applied at the edges

= Qand z = a of the plate Similarly, the second integral is equal to the work done by the forces applied at the edges y = Oandy = b The last two integrals, by virtue of Eqs (218), are equal to the work done during bending by the body forces acting in the middle plane These integrals each vanish in the absence of such corresponding forces

Adding expressions (220) and (224) to the energy of bending [see ka (117), page 88], we obtain the total strain energy of a bent plate under the combined action of lateral loads and forces acting in the middle plane

of the plate This strain energy is equal to the work T, done by the lateral load during bending of the plate plus the work T; done by the forces acting in the middle plane of the plate Observing that this latter work is equal to the strain energy V; plus the strain energy represented

by the first integral of expression (224), we conclude that the work pro-

Nu

uced by the lateral forces is

.pplying the principle of virtual displacement, we now give a variation 6w

o the deflection w and obtain, from Eq (225),

‘he left-hand side in this equation represents the work done during the irtual displacement by the lateral load, and the right-hand side is the orresponding change in the strain energy of the plate The application

f this equation will be illustrated by several examples in the next article

93 Simply Supported Rectangular Plates under the Combined Action

f Lateral Loads and of Forces in the Middle Plane of the Plate Let us egin with the case of a rectangular plate uniformly stretched in the direction (ig 192) and carrying a concentrated load P at a point with oordinates and 7 The general expression for the deflection that satis-

es the boundary conditions is

TW]

sin sỶ b (a)

_ Marx

m=1,2,3, n=1,2,3,

‘o obtain the coefficients ann in this series we use the general equation 226) Since N, = N., = 0 in our case, the first integral on the right- and side of Eq (225), after substitution of series (a) for w, is

‘he strain energy of bending representing the second integral in Kq 225) is [see Eq (d), page 343]