[J. Appl. Mech. 12, A69A77, 1945]
Introduction
It is well known that the classical theory of bending of thin elastic plates normal to their original plane permits the satisfaction of fewer boundary conditions along the edges of a plate than can in reality be prescribed. For instance, along a free edge, one has the three conditions of vanishing vertical force and of vanishing bending and twisting couple. Kirchhoff (1) has shown that the assumptions underlying the classical theory are responsible for a contraction of the three conditions mentioned into two conditions, which are vanishing bending couple, and vanishing of the sum of vertical force and edgewise rate of change of twisting couple. The meaning of this reduction in the number of boundary conditions has been explained by Thomson and Tait (2). The history of the problem has been discussed by Love (3) and recently again by Stoker (4).
Because of the simplifying assumptions made in the development of the classical theory, it may lead to consequences such as the following.
1. There occur concentrated reactions at the corners of simply supported plates of polygonal shape.
2. Treatment of the St. Venant torsion problem of a rod with narrow rectangular cross section by means of plate theory, while leading to a fairly accurate torque-twist relation when the width of the plate is larger than, say, 10 times the thickness of the plate, does furnish insufficient information regarding the distribution of stress over the cross section.
3. Results for the magnitude of the stress concentration at the edge of holes in transversely bent plates become uncertain when the diameter of the hole is so small as to be of the order of magnitude of the plate thickness (5,6).
In the present paper, a theory of bending of plates is developed which, to a considerable extent, is free of the limitations just described. In this theory, three boundary conditions can and must be satisfied along the edges of a plate. The theory is applied (a) to the torsion problem of the rod with rectangular cross-section where very good agreement is reached with the results of the exact theory; (b) to the stress-concentration problem of the plate with circular hole. Here considerable deviations from the results of classical plate theory are obtained as soon as the diameter of the hole is less than 3 times the thickness of the plate.
The manner in which the equations of the theory are obtained consists in an application of Castigliano's theorem of least work, combined with the Lagrangian multiplier method of the calculus of variations. The physical basis of the present
results forms the device of not discarding the energy of the transverse shear stresses, in contrast to what is done in the different derivations of classical plate theory.
The results here obtained for flat plates may be extended so as to apply to curved shells.
Derivation of Fundamental Equations
As in the standard theory of thin plates, it is assumed that the bending stresses are distributed linearly over the thickness of the plate
In Equations [1] Mx and My are the bending couples, Hxy the twisting couple, h the thickness of the plate (which in what follows is assumed to be uniform), x, y are co-ordinates in the middle plane of the plate, and z the thickness co-ordinate (Figure 1).
Fig. 1.
Orientation of stress resultants and stress couples, and stress variation over plate thickness.
From Equations [1], there are obtained, by means of the differential equations of equilibrium, expressions for the transverse shear stresses which satisfy the conditions that the faces of the plate are free from shear stress
The shear stress resultants Vz and Vy depend upon the stress couples:
Substituting Equations [2] in the remaining differential equation of equilibrium, there results for the transverse normal stress the expression
which satisfies the loading conditions
The shear-stress resultants and the intensity of the vertical loading p are related by the equation
Combination of Equation [3] and Equation [6] results in one equation for the three quantities Mx, My, Hxy. To obtain further equations, use has to be made of the stress-strain relations. In view of the simplifying assumptions made in the writing of expressions for the stresses, this cannot be done in an exact manner. It is done in a rational manner in what follows by means of Castigliano's theorem of least work. This theorem states that, among all statically correct states of stress, the state of stress which also satisfies the stress-strain relations and the displacement boundary conditions is characterized by the condition that the variation of the following expression vanishes:
In Equation [7] the double integral extends over the cylindrical portion of the surface of the elastic body under consideration, un and us are displacement components parallel to the plane of the plate in normal and tangential direction, and w is the displacement component normal to the plane of the plate.
Substituting for the stresses from Equations [1], [2], and [4], and carrying out the integration with respect to z where possible there follows, with
It is consistent with the assumption of linear bending stress distribution to assume that the displacements un and us vary linearly over the thickness of the plate and that w does not vary over the thickness of the plate
The line integral in Equation [8] then becomes
where for a plate with built-in edges
For a plate with free edges , and are unprescribed.
The variation of 3 according to Equation [8] is to be made equal to zero in such a way that the equilibrium Equation [6] remains satisfied. According to the rules of the calculus of variations, this is accomplished by multiplying Equation [6] by a Lagrange multiplier O and by combining Equations [8] and [6] in the following manner
Carrying out the variations
Integrating by parts in the last integral we have further
Substituting this in Equation [11], it is seen that, when GVn is arbitrary, then
As the same result can be obtained for any interior curve, it may be concluded that the Lagrange multiplier O is to be identified with the plate-deflection function w.
The variations of Vx and Vy, according to Equation [3], depend upon the variations of Mx, My, and Hxy. Integrating in Equation [11] further by parts there
follows
Equation [13] is the fundamental relationship of the present theory. From it follow the differential equations of the theory which hold in addition to the equilibrium equations and also the ''natural" boundary conditions of the problem, which under the assumptions made are either stress- or displacement-boundary conditions.
The stress-boundary conditions, which make the variations in the line integral vanish, are
The displacement-boundary conditions are
There are to be prescribed either Equation [14a] or Equation [15a], either Equation [14b] or Equation [15b], and either Equation [14c] or Equation [15c]. The significance of Equations [14] is evident. The same is true for Equation [15c]. Equations [15a] and [15b] indicate that, due to the effect of shear deformation, normal and tangential line elements in the middle surface do not remain perpendicular to the linear element which was before deformation perpendicular to the middle surface.
The double integral in Equation [12] is equivalent to three differential equations. They are, if the first two are solved for Mx and My
In addition to the three equations just given for the six unknowns Mx, My, Hxy, Vx, Vy, w, there hold the three equilibrium Equations [3] and [6]. In its present form, this system of equations is not readily solved. It will next be shown that it can be transformed in such a manner that the way to its solution is clear.
The first of the equations in its final form is the equilibrium Equation [6]
By means of Equation [I], Equations [16] are changed into
Equations [II] to [IV] will be used to determine Mx, My, Hxy when Vx, Vy, and w are known. From Equations [II] to [IV], there is next derived a system of two equations for Vx, Vy, and w. According to Equations [3], Vx and Vy are combinations of derivatives of Mx, My, and Hxy. In view of this, there follows from Equations [I] to [IV], by differentiation and combination:
In the foregoing equations
Equations [I], [V], and [VI] may be solved simultaneously for Vx, Vy, and w. Once this is done the stress couples Mx, My, and Hxy are obtained from Equations [II]
to [IV] without further integrations. The equations of the standard theory of thin plates are obtained by disregarding on the left of Equations [II] to [VI] all but the first terms. The possibility of satisfying three instead of two boundary conditions in the present theory derives from the presence of the 'V terms in Equations [V] and [VI].
Introduction of Stress Function
Considering now Equations [I] to [VI] when p = 0, it is seen that Equation [I] can be satisfied by means of a stress function F
Substituting Equations [17] in Equations [V] and [VI] these may be written in the form
Equations [18] are Cauchy-Riemann equations for the functions D'w and F (h2/10) 'F. Consequently, with two conjugate harmonic functions I and \, there is obtained
From
there follows
where \1 is the general solution of
Thus, the stress function F is a combination of a harmonic function and of a function defined by Equation [22]. And if the harmonic contribution to F is taken as the imaginary part of a complex function g(x+ iy) then D 'w is the corresponding real part of the same function g.
From
it follows further that w itself is a biharmonic function, the same as in the classical theory of plates without surface loading. Once the solution of the homogeneous system of equations is found, it is only necessary to obtain a particular integral to take care of the load function p.
The fact that in this formulation of the problem the only differential operators which occur are the invariant operators G and G 10/h2 indicates that explicit solutions of the theory may also be found in terms of plane polar and elliptical co-ordinates.
Before doing this, there will first be discussed, as an example of the scope of the theory, a relatively simple example of a plate problem with rectangular boundary.
Torsion of a Rectangular Plate
A rectangular plate of length 2l and width a is considered. The two sides y = ± a/2 are free of stress while the two sections x = ±l are assumed to rotate without distortion and to be free of normal stress. The condition of distortionless rotation means that line elements perpendicular to each other before deformation remain so after deformation so that along the rotated end sections . With this stipulation, the boundary conditions, Equations [14] and [15], become
To satisfy Equation [24a] let
As it is expected that the stresses will come to be independent of x and odd in y, the solution of Equation [V] is taken in the form
and as Vy vanishes all along the edges, the solution of Equation [VI] is taken as
From Equations [II] to [IV] follows then
As yet unsatisfied is the boundary condition, Equation [25b]. Substituting Equation [30] in Equation [25b]
The only nonvanishing stress couples and resultants are then
From Equations [32] and [33], there are obtained the values of the shear stresses by means of Equations [1] and [2], substituting the value of D and the
relation E = 2(1 + v)G
According to customary thin plate theory, corresponding expressions for the stresses would be
except at the edges y = ± a/2 where the stresses Wxz may be assumed to become infinite in such a way as to be equivalent to concentrated forces.
In the present theory, according to Equations [34] and [35], the stress Wxy is substantially constant over the width of the plate, except near the edges y = ±a/2 where it decreases to zero within a distance of the order of magnitude of the plate thickness h. The stress Wxz has its largest values when y = ±a/2 and drops down nearly to zero values a distance away from the edges y = ±a/2 which is again of the order of magnitude of the thickness h.
The results of Equations [34] and [35] may be compared with the results of the St. Venant torsion theory, and the agreement is remarkably close even for plates so thick that the designation of "plate" is no longer appropriate.
Taking first the case of a square cross section, there follows
In the exact theory, the numerical factors would be equal to each other and have the value 1.35 (ref. 7). It is of some interest to note that the average of the two values, 1.33, is remarkably close to the exact value. If one did not know the exact value, it would have suggested itself to consider this average as the true approximation rather than either one of the values in Equation [36].
For a plate twice as wide as thick, there results for the maximum shear stress
which differs by less than 2 per cent from the exact value 1.86.
The limiting values of the stresses for very large values of a/h are Wxy(0, h/2) = 2GT(h/2) and Wxz(a/2, 0) = 1.58GT(h/2). An expression for the resultant torque is obtained from
As
it follows that, as in the exact theory, the stresses Wxy and Wxz contribute in equal measure to the value of T. With Hxy from Equation [32]
The values of k1, according to Equation [40] compare with the exact values of k1 and the values of k1 according to customary plate theory as follows:
a/h 1 2 f
k1 0.139 0.228 0.333
k1,ex 0.1406 0.229 0.333
k1,pl 0.333 0.333 0.333
For values of a/h which are larger than three, k1 of Equation [40] becomes (1/3)(1 .63h/a) which is a well known approximation formula.
Polar-Co-Ordinate Solutions of Equations of the Theory
Introduction of a stress function, according to Equations [17], and the subsequent integration of the system of equations in terms of the functions I, \, and \1, as defined by Equations [19] to [23], inclusive, indicates the way to obtain explicit solutions in polar co-ordinates r,T.
The shear-stress resultants are now expressed in terms of the stress function F as follows
As before
and
where I + i\ = g(x + iy) and now
For D'w may be written
The conjugate of this is
Suitable solutions of Equation [22] are
In Equation [44], In and Kn are the modified Bessel functions (8). The functions In become rapidly large for large values of their arguments, while the functions Kn become rapidly small for large values of their argument. For small values of the argument, In stays finite and Kn becomes infinite.
Equation [42] is integrated to
The starred constants depend upon the unstarred constants as follows
For each term in the trigonometric series, Equations [43], [44], and [45], there are six constants of integration, so that three boundary conditions can be satisfied along both edges of a circular-ring plate.
In order to evaluate the foregoing solutions, it is necessary to obtain expressions for the stress couples Mr, MT, and HrT, which correspond to Equations [II] to [IV]
for Mx, My, and Hxy. This may be done as follows: Observe that, in Equations [II] to [IV], the shear-stress resultants Vx and Vy occur in the same manner in which the displacement components u and v occur in the components of strain Hx,Hy,Jxy. This suggests that, in the equations for Mr, MT, and HrT, the quantities Vr and VT may occur in the same manner in which radial and circumferential displacement components occur in the expressions for Hr,HT, and JrT. The correctness of this statement may be proved by deriving the equations of the theory directly, introducing curvilinear co-ordinates before applying Castigliano's theorem. This calculation is omitted in the present paper.
Equations [II] to [IV] then have the following analogues in polar co-ordinates
Substituting the stress function F from Equations [41], there may be written, for the case of absent surface loads
The foregoing results permit a number of applications to problems involving circular boundaries. Two of these are made in what follows.
Bending and Twisting of an Infinite Plate with Circular Hole
The solution of these problems by means of classical thin plate theory has been given by Goodier (5). They have been investigated by Bickley (9) as problems of the theory of moderately thick plates. Experimental results (10) have confirmed the results of thin plate theory for a hole diameter plate thickness ratio of about seven.
Taking first the case of plain bending, the boundary conditions in the present theory are
Instead of Equations [47] and [48], there may be written
The conditions at infinity suggest that the following expressions for w and F be taken
From Equation [52] there follows for the shear-stress resultants with, according to (8)
For the stress couples, there results after some calculations
Substituting Equations [54], [56], and [58] in the boundary conditions, Equations [49], and [50], and with the notation
the following expressions for the constants in Equations [51] to [58] are obtained
The stress-concentration factor of the problem is obtained from the value of the tangential edge stress couple
The value of this function is greatest for T = S/2
For large values of the following asymptotic expressions for K2 and K0 may be used (8)
Hence
which is in agreement with the result obtained by means of standard thin plate theory (5,6).
For small values of P the function K2 becomes infinite of a higher order than the function K0 and consequently
It is noteworthy that in the limit of vanishing hole diameter the value of the stress-concentration factor in bending becomes almost twice as large as in the limit of vanishing plate thickness, and moreover equal to the value of the stress-concentration factor in plane stress.
For intermediate values of a/h the value of kB has been calculated by means of tables for the functions K2 and K0 (8). Figure 2 contains a graph of kB versus 2a/h. It is seen that even for holes 3 times as wide as the plate is thick the value of kB, according to the present calculations, is still more than 10 per cent greater than the value obtained by the application of standard plate theory.
That taking into account the shear deformability of the plate leads to higher values of the stress-concentration factor than not taking into account this effect becomes physically evident when it is recognized that the assumed loading condition of the plate would lead to independent states of plane stress in every layer of the plate for an ideal orthotropic material offering no resistance to the transverse shear stresses Wrz,WTz, that is, for a material for which Grz = GTz = 0. In contrast to this the results of the customary theory may be thought of as exact results for a material for which Grz = GTz = f.
Relatively simple expressions are obtained for the shear-stress resultants Vr and VT, by substituting the constants B2 and D2 in Equations [54] and [55]
For large values of a/h, for which the first terms of the asymptotic expressions of K2 and K0 may be used, this reduces to
Letting r = a + nh and consequently , it follows from the asymptotic formulas for Vr and VT that, for very thin plates and for a distance of the order of magnitude of the plate thickness away from the edge of the hole, the shear-stress resultants have the values
These expressions coincide with the expressions which according to the standard theory of thin plates hold throughout the plate (5, 6).
Comparison of Equations [71] and [73a] shows that the resultant Vr increases from its true edge value zero very rapidly to the edge value of thin plate theory.
Also noteworthy is the behavior of the function representing VT. From Equation [72] it follows
This shows that, in the present theory, the edge value of VT is of opposite sign from the value according to Equation [73b]. Moreover, for thin plates, VT(a,T), according to the present theory, is of an entirely different order of magnitude than according to the usual plate theory. For given plate thickness, its value no longer decreases
with increasing hole diameter. Substituting
there results for the maximum transverse edge shear stress
Thus, with transverse-shear deformation taken into account, there are portions of the plate where the transverse shear stress is of the same order of magnitude as the primary bending stress V0, no matter how thin the plate may be.
From Equation [70] there follows for the variation of edge shear with diameter-thickness ratio
which compares with the constant value 4M0/(3 + Q)a, according to the standard theory. Figure 2 contains a graph of this function.
Plate Subject to Pure Twist
The results for this case may be obtained from the preceding results by superposition of a state of plain positive bending about the y-axis, as given, and a state of plain negative bending about the x-axis. Hence, for
Fig. 2.
Stress-concentration factors and edge shear-stress resultant versus ratio of hole diameter to plate thickness.
pure twist, every stress and displacement quantity gT is expressed as follows in terms of the corresponding quantity gB for plain bending about the y-axis:
From Equation [66] there follows
Values of kT as function of the ratio 2a/h are plotted in Fig. 2. Limiting values of kT are
which agrees with the result of standard thin plate theory, and
which is the same value which occurs when adjacent layers of the plate can slide freely with respect to each other. It is evident that, for the twisted plate, the effect of
which is the same value which occurs when adjacent layers of the plate can slide freely with respect to each other. It is evident that, for the twisted plate, the effect of