11. Special and approximate methods intheory of plates

11. Special and approximate methods intheory of plates

CHAPTER 10

SPECIAL AND APPROXIMATE METHODS IN THEORY

OF PLATES

75 Singularities in Bending of Plates The state of stress in a plate

s said to have a singularity at a point! (xo0,yo) if any of the stress com- yonents at that point becomes infinitely large From expressions (101), 102), and (108) for moments and shearing forces we see that a singu- arity does not occur as long as the deflection w(z,y) and its derivatives 1p to the order four are continuous functions of z and y -

Singularities usually occur at points of application of concentrated forces and couples In certain cases a singularity due to reactive forces ean occur at a corner of a plate, irrespective of the distribution of the surface loading

In the following discussion, let us take the origin of the coordinates

at the point of the plate where the singularity occurs The expressions for the deflection given below yield (after appropriate differentiations) stresses which are large in comparison with the stresses resulting from loading applied elsewhere or from edge forces, provided x and y are small Single Force at an Interior Point of a Plate If the distance of the point under consideration from the boundary and from other concen- trated loads is sufficiently large, we have approximately a state of axial symmetry around the single load P Consequently, the radial shearing force at distance r from the load P is

Single Couple at an Interior Point of a Plate Let us apply a single

1 More exactly, at a point (20,Y0,2)

326 THEORY OF PLATES AND SHELLS

force —M,/Az at the origin and a single force +M,/Az at the point (—Az,0), assuming that Jf, isa known couple From the previous result [Eq (206)] the deflection due to the combined action of both forces is

_ Ms (2 + Ax)? + y? log [(z + Ax)? + y?}}

Mi 2+y (x? + ?)‡

As Ax approaches zero, we obtain the case of a couple Af; concentrated

at the origin (lig 168a) and the deflection is

In the case of the couple Af, shown in Fig 168b we have only to replace

6 by 0 + 7/2 in the previous formula to obtain the corresponding

SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 327 jue to a singularity of a higher order than that corresponding to a couple.! Substitution of expression (206), where rectangular coordinates may be ised temporarily, yields the deflection

mitted to the action of a transverse force or a 2

Single Load Acting in the Vicinity of a Burlt-in ⁄ | sles

Edge (Fig 170) The deflection of a semi-infinite cantilever plate carry-

ing a single load P at some point (£,7) is given by the expression

where r? = (a — £)? + (y — »)? We confine ourselves to the consider- ation of the clamping moment at the origin Due differentiation of expression (d) yields

at x = y = 0, provided £ and y do not vanish simultaneously It is seen that in general the clamping moment MM, depends only on the ratio n/£ 1'To make the nature of such a loading clear, let us assume a simply supported beam of a span L and arigidity EI with a rectangular moment diagram Az by M, sym- metrical to the center of beam and due to two couples Af applied at a distance Ar from each other Proceeding as before, 7.e., making Az — 0, however fixing the value of

H = M Az, we would arrive at a diagram of magnitude H concentrated at the middle

of beam Introducing a fictitious central load H/EI and using Mohr’s method, we would also obtain a triangular deflection diagram of the beam with a maximum ordi- nate HL/4EI A similar deflection diagram would result from a load applied at the center of a perfectly flexibie string

2See A Naddai, ‘‘llastische Platten,” p 203, Berlin, 1925

328 THEORY OF PLATES AND SHELLS

If, however, £ = » = 0 the moment M, vanishes, and thus the function M,(&,n) proves to be discontinuous at the origin

Of similar character is the action of a single load near any edge rigidly

or elastically clamped, no matter how the plate may be supported elsewhere This leads also to the characteristic shape of influence sur- faces plotted for moments on the boundary of plates clamped or continu- ous along that boundary (see Figs 171 and 173)

For the shearing, or reactive, force acting at x = y = 0 in Fig 170

we obtain in similar manner

where r? = £2 + 7?,

76 The Use of Influence Surfaces in the Design of Plates In Art 29

we considered an influence function K(z,y,£,n) giving the deflection at some point (x,y) when a unit load is applied at a point (é,7) of a simply supported rectangular plate Similar functions may be constructed for any other boundary conditions and for plates of any shape We may also represent the influence surface K(é,n) for the deflection at some fixed point (z,y) graphically by means of contour lines By applying the principle of superposition to a group of n single loads P; acting at points (&:,n:) we find the total deflection at (x,y) as

w= > P:K (x,y, &i,n:) (a)

In a similar manner, a load of intensity p(é,y) distributed over an area A

of the surface of the plate gives the deflection

Of special practical interest are the influence surfaces for stress resultants! given by

a combination of partial derivatives of w(z,y) with respect to z and y To take an

1 Such surfaces have been used first by H M Westergaard, Public Roads, vol 11,

1930 See also F M Baron, J Appl Mechanics, vol 8, p A-3, 1941

SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 329 example, let us consider the influence surfaces for the quantity

By result (c) of Art 75 this latter expression yields the ordinates of a deflection surface

in coordinates ¢, 7 containing at — = 2,7 = ya singularity due to a ‘‘couple of second order’? H = 1 which acts at that point in accordance with Fig 169

The procedure of the construction and the use of influence surfaces may be illus- trated by the following examples.}

Influence Surface for the Edge Momeni of a Clamped Circular Plate? (Fig 171) By representing the deflection (197), page 293, in the form w = PK(z,0,é,6), we can con- sider K as the influence function for the deflection at some point (z,0), the momentary position of the unit load being (£,@) In calculating the edge couple M,atz = r/a = 1,

y = 0 we observe that all terms of the respective expressions (192), except for the following one, vanish along the clamped edge z = 1 The only remaining term yields

2 — ¢2\2

a öZ2 /z~: 4r ‡? — 2£ cos Ø +1 Eor brevity let us put ‡? — 2‡ cos 6 + 1 = 7»? and, furthermore, introduce the angle

¢ (Fig 171a) Then we have ¢? = 1 — 2n cos ¢ + 7? and

Influence Surface for the Bending Moment M, at the Center of a Simply Supported Square Plate.* It is convenient to use the influence surfaces for the quantities Mio = —D 0*w/dzx? and Myo = —D d*w/dy? with the purpose of obtaining the final result by means of Eqs (101)

The influence surface for Mz»9 may be constructed on the base of Fig 76 The influence of the single load P = 1 acting at point 0 is given by the first of the equations (151) and by Eq (152) This latter expression also contains the required singularity

of the type given by Eq (206), located at the point 0 The effect of other loads may

be calculated by means of the first of the equations (149), the series being rapidly convergent The influence surface is shown in Fig 172 with ordinates multiplied

330 THEORY OF PLATES AND SHELLS

SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 381 therefore it is simplest to calculate the effect of the load P; separately, by means of Eqs (163) and (165), in connection with Tables 26 and 27.! For this case we have

y = 0, v/u = k = 1, ¢ = 1.5708, y = 0, A = 2.669, and » = O, which yields N = 0 and a value of M calculated hereafter As for the effect of the load P2, it can be assumed as proportional to the ordinate 2.30 of the surface at the center of the loaded

~

area Introducing only the excesses of both single loads over the respective loads due

to g, we have to sum up the following contributions to the value of Afro:

1 Load P: from Eqs (163), (165), with € = ø/2, d = 0.1 V/2a,

, M_ P;—-0.0lga? 4 ÄMạẹ=—= 2 log —————= + 2.669 — 1.571

z9 2 Sir ( 06 O.la 4⁄2 T )

= 0.219(P; — 0.01ga?)

1 ‘The effect of the central load may also be calculated by means of influence lines

similar to those used in the next example or by means of Table 20

332 THEORY OF PLATES AND SHELLS

Owing to the square shape of the plate and the symmetry of the boundary conditions

we are in a position to use the same influence surface to evaluate Myo The location

of the load P2 corresponding to the location previously assumed for the surface Mzo

is given by dashed lines, and the contribution of the load P: now becomes equal to

M v0 = 0.035(P2 — 0.01ga*), while the contributions of P; and ø remain the same as before This yields

Myo = 0.219P: + 0.035P2 + 0.0344ga?

Now assuming, for example, vy = 0.2 we have the final result

M, = Mio + 0.2Myo = 0.263P; + 0.099P; -+ 0.0407qa?

Influence Surface for the Moment M, at the Center of Support between Two Interior Square Panels of a Plate Continuous in the Direction x and Simply Supported aty = +b/2 This case is encountered in the design of bridge slabs supported by many floor beams and two main girders Provided the deflection and the torsional rigidity of all sup- porting beams are negligible, we obtain the influence surface shown! in Fig 173

In the case of a highway bridge each wheel load is distributed uniformly over some rectangular area u by v For loads moving along the center line y = 0 of the slab a set of five influence lines (valid for v/b = 0.05 to 0.40) are plotted in the figure and their largest ordinates are given, which allows us to determine without difficulty the govern- ing position of the loading Both the surface and the lines are plotted with ordinates multiplied by 8r

EXAMPLE OF EVALUATION Let us assume a = b = 24 ft 0 in.; furthermore, for the rear tire P, = 16,000 lb, u = 18 in., v = 30 in., and for the front tire Py = 4,000 lb, u=18in.,v =15in The influence of the pavement and the slab thickness on the distribution of the single loads may be included in the values u and v assumed above For the rear tire we have v/b =~ 0.10 and for the front tire v/b ~ 0.05 Assuming the position of the rear tires to be given successively by the abscissas = 0.20a, 0.25a, 0.30a, 0.35a, and 0.40a, the respective position of the front tires is also fixed by the wheel base of 14 ft = 0.583a The evaluation of the influence surface for each par- ticular location of the loading gives a succession of values of the moment plotted in Fig 173 versus the respective values of — by a dashed line The curve proves to have

a maximum at about = 0.30a The procedure of evaluation may be shown for this

The influence lines marked 0.10 and 0.05, respectively, yield the contribution of both central loads (at y = 0) equal to

— (16,000 - 3.24 + 4,000 - 3.82) = —65,100 lb and the influence surface gives the contribution of the remaining six loads as

—16,000(1.66 + 2.25 + 0.44) — 4,000(1.59 + 2.25 + 0.41) = —86,600 Ib

1 For methods of its construction see references given in Art 52

334 THEORY OF PLATES AND SHELLS

Finally, taking into account the prescribed multiplier of 1/8r = 0.0398, we have the result

(Mz)min = —0.0398(65,100 + 86,600) = —6,040 lb-ft per ft

Maximum Shearing Force Due to a Load Uniformly Distributed over the Area of a [ectangle A load of this type, placed side by side with the built-in edge of an infinite cantilever plate, is shown in Fig 170 by dashed lines This problem is encountered also in the design of bridge slabs By using the result (210) and the principle of superposition we obtain the following shearing force at x = y = 0:

SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES O00

in which ÀA = p2u For some specific boundary conditions, solutions of Eq (6) exist only for a definite set of values Ai, A2, , Ax, - - - Of the parameter A, the so-called characteristic numbers (or eigenvalues) of the problem The respective solu-

tions form a set of characteristic functions wi(z,y), we(z,y), » Welty), These functions are mutually orthogonal; 2.e.,

[ J 10;(%,)0x(z,) dx dy = 0 (c)

A

fori ~ k, the integral being extended over the surface of the plate As the functions wz(xz,y) are defined except for a constant factor, we can ‘“normalize’’ them by choosing this factor such as to satisfy the condition

A

The form chosen for the right-hand side of (d) is appropriate in the case of a rectangular plate with the sides a and b, but whatever the contour of the plate may be, the dimen- sion of a length must be secured for w; The set of numbers \; and the corresponding set of normalized functions w:z(z,y) being established, it can be shown! that the expansion

As an example, let us take the rectangular plate with simply supported edges (Fig 59) Eigenfunctions which satisfy Iq (b) along with the boundary conditions

w = Aw = 0 and the condition (ở) are

we = 2 Vab sin ~~ sin (f)

m and n being two arbitrary integers The respective cigenvalue, from Iq (b), is

1 See, for instance, R Courant and D Hilbert, ‘‘ Methods of Mathematical Physics,’’ vol 1, p 370, New York, 1953

336 THEORY OF PLATES AND SHELLS

for which the modes of vibration, expressible in terms of Bessel functions, are well known

78 The Use of Infinite Integrals and Transforms Another method of treating the problems of bending of plates is the use of various transforms.! A few such transforms

Fourier Integrals In the case of infinite or semi-infinite strips with arbitrary condi- tions on the two parallel edges the method of M Lévy, described on page 113, can be used, but in doing so the Fourier series necessarily must be replaced by the respective infinite integrals In addition to the example considered in Art 50, the problem of an infinite cantilever plate (Fig 174) carrying a single load P may be solved in this way.?

Let w, be the deflection of the portion AB and wz the deflection of the portion BC

of the plate of width AC =a Then we have to satisfy the boundary conditions

fy) = - I, cos ay da I f(n) cos an dy (c)

Since the intensity of the loading is given by f(n) = P/v for —v/2 <q < v/2 and by

1 For their theory and application see I N Sneddon, ‘Fourier Transforms,’’ New York, 1951

2 The solution and numerical results hereafter given are due to T J Jaramillo,

J Appl Mechanics, vol 17, p 67, 1950 Making use of the Fourier transform,

H Jung treated several problems of this kind; see Math Nachr., vol 6, p 343, 1952

SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 337

zero elsewhere, we have

av sIn — COS ay

Ôn the other hand, the function f(y) is equal to the difference of the shearing forces Q:

at both sides of the section z = & Thus, by Eqs (108), we have

p— (aw: — Aw:) = fly) Ox (e)

onz = & In accordance with Eq (d) we represent the deflections w; and we by the

integrals

= I, Xi(z,a) cos ay da ~=1,2 (f)

in which the function

Xi(z,a) = (A; + Bix) cosh ax + (C; + Diz) sinh az

is of the same form as the function Y» on page 114

It remains now to substitute expressions (f) into Eqs (a), (b), and (e) in order to

determine the coefficients Ai, Bi, , De, independent of y but depending on a

Mellin Transform The application of this transform is suitable in the case of a wedge-shaped plate with any homogeneous conditions along the edges 6 = 0 and

338 | THEORY OF PLATES AND SHELLS

= a (Fig 176) To take an example let us consider the edge 6 = 0 as clamped and the edge @ = a, except for a single load P at r = ro, as free.}

We use polar coordinates (see Art 62) and begin by taking the general solution of the differential equation AAw = 0 in the form

where s is a parameter and

0(6,s) = A(s) cos 86 + B(s) sin s6 + C(s) cos (s + 2)@ + D(s) sin (s + 2)6 (h) The deflection and the slope along the clamped edge vanish if

[W(s)le-0 = 0 =| ee =0 ()

chon? AF le ip ON, S « \ The bending moment M, on the free edge

Poe \ \ vanishes on the condition that

Y _—

[ À [penne n-ne >| "Free ” art r or

eage

Fria 176 IG + r?_ a0? | 5 =0 @ J

Now, a function f(r) can be represented by means of Mellin’s formula as follows:

1'The problem was discussed by S Woinowsky-Krieger, Ingr.-Arch., vol 20, p 391,

1952 Some corrections are due to W T Koiter, Ingr.-Arch., vol 21, p 381, 1953 For a plate with two clamped edges see Y 8 Uflyand, Doklady Akad Nauk S.S.S.R., vol 84, p 463, 1952 See also W T Koiter and J B Alblas, Proc Koninkl N ed Akad Weterschap., ser B, vol 57, no 2, p 259, 1954 |

SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 339

We finally equate expressions (J) and (0) and thus obtain, in addition to Eqs (z) and

(j), a fourth condition to determine the quantities A(s), B(s), C(s), and D(s) Sub-

stitution of these coefficients in the expressions (h) and (m) and introduction of a new

variable wu = —(s + 1)2, where2 = v —1, yields the following expression for the deflection of the plate:

G cos ( log 2 + Hsin ( log —

The variation of the deflections along the free edge and the distribution of the moments Af, along the edge @ = O in the particular case of a = 7/4 anda = z/2 1s shown in Fig 177

Hankel Transform Let a circular plate with a radius a be bent to a surface of revolution by a symmetrically distributed load g(r) We multiply the differential equation AAw = q/D of such a plate by rJo(Ar) dr and integrate by parts between

340 THEORY OF PLATES AND SHELLS

r =QOandr = » Provided w = 0 for r > a, the result is

4 I " w(r)rJ (ar) dr = g(a) (q) where

gd) = (Cy + C2)To(da) + (Cs + 8G) Ji (Aa) +5 i q(ø)pJo(Aø) dp — (r)

Jo and J; are Bessel functions of the order zero and one, and C; are constants Appli-

cation of the Hankel inversion theorem to Eq (q) gives -

The constants C; now are obtainable from the conditions on the boundary r = a of the plate and from the condition that the function g(A)/A‘ must be bounded The expression (r) must be slightly modified in the case of an annular plate.! Examples

of the application of solutions of the type (s) to the problem of elastically supported plates are given in Art 61

Sine Transform In the case of rectangular plates we have used solutions of the form

w(xz,y) = ZY(y,a) sin ax and in the case of sectorial plates those of the form

tp(r,0) = >(r,8) sin 86

The finite sine transforms of the function w, taken with respect to x and 6, respec- tively, and introduced together with transformed derivatives of w and the trans- formed differential equation of the plate, then prove useful in calculating the constants

of the functions Y and ?# from the given boundary conditions of the plate.”

79 Complex Variable Method By taking 2 = x + iy and Z = x — ty for inde- pendent variables the differential equation (104) of the bent plate becomes

where ¢ and x are functions which are analytic in the region under consideration

Usually the derivative y = dx/dz is introduced along with x

1 For the foundation of the method and an extensive list of transforms needed in its application see H Jung, Z angew Math Mech., vol 32, p 46, 1952

2 The application of the method is due to L 1 Deverall and Ơ J Thorne, J Appl Mechanics, vol 18, pp 152, 359, 1951

3 denotes the real part of the solution This form of the solution of the bipoten- tial equation is due to E Goursat, Bull Soc Math France, vol 26, p 236, 1898

SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 341

In the case of a single load P acting at zo = to + iyo the solution wo may be chosen

in the form

P

s 167D

Wo (2 — 20)(2 — Zo) log [(z — 20)(2 — 2o)] (c)

which is substantially equivalent to expression (206) For a uniform load

_ g2???

_ 64D

Wo

If the outer or the inner boundary of the plate is a circle we always can replace it

by a unit circle z = e9 or briefly z = o The boundary conditions on z = o must be

expressed in complex form also The functions ¢ and y may be taken in the form of a

power series, with additional terms, if necessary, depending on the value of stress resultants taken along the inner edge of the plate Multiplication of the boundary conditions by the factor [2ri(¢ — z)]"! do and integration along z = then yields the required functions ¢ and y.*

For boundaries other than a circle a mapping function z = œ(ÿ) = w(pe**) may be

used so as to map the given boundary line onto the unit circle ÿ = e'? =o The determination of the functions ¢:(¢) = ¢(z) and ¡(‡) = ý) from the boundary conditions on ¢ = o then is reduced to the problem already considered The Musch- elishvili method outlined above is especially efficient in cases concerning stress distribution around holes; the function w(¢) then has to map the infinite region of the plate into the interior of the unit circle

The complex variable method also allows us to express Green’s functions of a circular plate with various boundary conditions in closed form.’ In other cases, such as that

of a clamped square plate, we must rely on an approximate determination of the Green functions.®

When expressible by a double trigonometric series, the deformation of the plate can also be represented in a simpler form by making use of the doubly periodic properties

of the elliptic functions For the quantity Aw, satisfying the potential equation A(Aw) = 0, such a representation becomes particularly convenient because of the close connection between the Green function for the expression Aw and the mapping function of the region of the given plate into the unit circle Once Aw is determined

*For evaluation of integrals of the Cauchy type implied in this procedure see N 1 Muschelišhvili, “Some Basic Problems of the Mathematical Theory of Elasticity,’ Groningen, 1953

1 An extensive application of the method to the problem of stress concentration is due to G N Savin; see his ‘“‘Stress Concentration around Holes,” Moscow, 1951 See also Yi-Yuan Yu, J Appl Mechanics, vol 21, p 129, 1954, and Proc Ninth Intern Congr Appl Mech., vol 6, p 378, Brussels, 1957; also L I Deverall, J Appl Mechanics, vol 24, p 295, 1957 A somewhat different method, applicable as well

to certain problems of the thick-plate theory, was used by A C Stevenson, Phil Mag.,

2E Reissner, Math Ann., vol 111, p 777, 1935; A Lourye, Priklad Mat Mekhan., vol 4, p 93, 1940

3F Schultz-Grunow, Z angew Math Mech., vol 33, p 227, 1953

4 Courant and Hilbert, op cit., vol 1, p 377 Elliptic functions have been used in particular by A Nddai, Z angew Math Mech., vol 2, p 1, 1922 (flat slabs); by

F Tölke, Ingr.-Arch., vol 5, p 187, 1934 (rectangular plates); and also by B D Aggarwala, Z angew Math Mech., vol 34, p 226, 1954 (polygonal plates and, in particular, triangular plates)

342 THEORY OF PLATES AND SHELLS

the shearing forces of the plate are readily given by the derivatives of that function by

virtue of Eqs (108)

80 Application of the Strain Energy Method in Calculating Deflec- tions Let us consider again the problem of the simply supported rectangular plate From the discussion in Art 28 it is seen that the deflection of such a plate (Fig 59) can always be represented in the form

of a double trigonometric series:!

of virtual displacements may be used In the application of this principle

we need the expression for strain energy (see page 88):

1Í ?w+ z4 ?n and n ¥ n’, we conclude that in calculating the integral (c)

we have to consider only the squares of terms of the infinite series in the parentheses Using the formula

2 f° mre i nry 2 ae ¬ ay — ab

I I sin* —— sin _ dx dy = tr the calculation of the integral (c) gives

' The terms of this series are characteristic functions of the plate under consideration (see Art 77)

SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES 343 From the fact that

ƒ I sin: ““Z Ễ sin: “# _ re dx dy =[ I cos: ““” Ễ cog? ` đụ -2

it can be concluded that the second term under the integral sign in expression (b) is zero after integration Hence the total strain energy

in this case is given by expression (c) and is

From the principle of virtual displacements it follows that this work must

be equal to the change in potential energy (d) due to the variation Sam's’

Hence

m'nE nian sin b “ 88x OV

P 6Qm’n’ SiN Substituting expression (d) for V, we obtain

If a system is in a position of stable equilibrium, its total energy is a