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Chapter: Civil : Principles of Solid Mechanics : Rings, Holes, and Inverse Problems

Solution Tactics for Neutral Holes- Examples

For the inverse problem, a solution matching displacements at the interface using the integral equations, will necessarily be one of trial and error even for an assumed shape since the unknown geometric properties of the cross-section will, in general, be functions of position and must remain inside the integral.

Solution Tactics for Neutral Holes- Examples


For the inverse problem, a solution matching displacements at the interface using the integral equations, will necessarily be one of trial and error even for an assumed shape since the unknown geometric properties of the cross-section will, in general, be functions of position and must remain inside the integral. A better tactic is to enforce compatibility on a differential scale by

using Equations (7.34) with Equations (7.31) and (7.32). The displacements will then necessarily conform aside from initial conditions u0, v0, and _0. These along with any unknown integration constants in the equilibrium equations can then be found in the standard way once the geometry of the hole and liner is known.


The theory as presented essentially reduces to a “strength-of-materials” solution for a ring where the geometry of its deformation must match that of the sheet at the interface. It is, of course, approximate to the extent that all strength-of-material solutions are approximate. Thick-ring effects could eas-ily be incorporated if necessary by including the eccentricity of the neutral axis and adding the bending contribution at the interface.


However, as is usually done, the equations will be simplified further by neglecting the curvature contribution and reducing the liner to the “elastica” at the neutral axis. The significance of this “thin ring” assumption can be esti-mated by considering a circular liner of radius R in an isotropic field.


For this situation the ring simply expands uniformly and, since there is no   force

N/AE. Thus, the final term in the first two equations of Bresse will be negligible and will not be included hereafter.


Two special cases of obvious importance will be considered in the examples that follow. First, membrane reinforcement may be possible in certain fields by adjusting the hole shape to “eliminate” bending. This is the Mansfield solu-tion and, as shown previously, can occur only for that shape which is a level line of U given by Equation (7.32). This “short-circuits” the equations of Bresse in that if V =0, both μ and I are indeterminate. The area required can be found directly from Equations (7.34a) with (7.35a). However, an actual liner will have some bending stiffness inducing secondary moments (i.e., a gradient of normal stress through the thickness) violating the original membrane assumption. This dilemma, not considered by Mansfield, is a direct result of the strength-of-materials approach. Such secondary effects will be shown to be very small for thin liners.


Circular liners are a second important class of potential solutions. In this case the shape is prescribed and l, A l, and μ are to be determined to satisfy the neutral condition. The strains in the sheet in polar coordinates r, and θ become:



Similarly, since the radius is a constant, the differential equations of Bresse are easily solved for the required liner properties.



Isotropic Field


With equal normal stress in orthogonal directions, σm, the stress function is:

and the obvious shape is circular. Thus, by the equilibrium equations

which is not possible if M = 0. However, we know from Lamé’s exact solu-tion that there is, in fact, a constant moment in the ring since the hoop stress is not uniform through the thickness. Combining (c) and (d):

per unit thickness which we derived earlier.


There will be no requirement on the moment of inertia. Thus, a “neutral” liner will have a uniform cross-section and be “thin” as defined by d/r " 10 if Es/El(1-vs) <  10. This is nearly true for a steel liner in concrete or rock and true for a concrete or steel liner in soil.


Deviatoric Field


For the pure shear or deviatoric field with equal but opposite principal stresses, +- σd

Therefore, the membrane shape is a self-equilibriating system of parabolic “arch” and “suspension” segments which, while conceivable, is not likely in practice.


On the other hand, a circular, neutral liner may be feasible. Using cylindri-cal coordinates where

Through symmetry the moment at the springing line must be equal and opposite that at the crown. Thus, c must be zero. Therefore, the stress resultants in the liner are



These results are startling. It is possible to design a neutral circular liner com-pletely restoring the original field, which has constant A and I, and therefore, is easy to build. Checking shear compatibility from Equation (7.34c)


This is close to that required by Equation (7.42a) in order to match the hoop strain since

A 10 WF 17 would give the required moment of inertia, but over 100 times too much area, and slip joints to reduce the effective A would be necessary. Clearly, as one might expect for a circular shape in pure shear, bending dominates entirely and we might term this remarkable neutral liner with constant cross-section the flexural counterpart to the circular, axial-force liner for the isotro-pic field.




General Biaxial Field


A general biaxial field will be a combination of isotropic and deviatoric com-ponents. Assuming that the x axis is in the direction of the maximum princi-pal stress:

Thus, the membrane shape will be a combination of a circle and parabola. The hole boundary can only be closed if σm > σd, in which case an ellipse results with the principal axes in the ratio a/b = Rt(σ1 / σ2) . This important and practical case is given by Mansfield, but was actually found nearly a century before by Rankine using parallel projection.

The neutral shape and liner for no stress concentration is compared to the unlined harmonic shape (stress concentration of 1.5) in Figure 7.13a for a 2 to 1 biaxial field. If vs is 1/4, then the area of the membrane liner at the “spring-ing line” (i.e., y  = 0) would be about 2.5 times that required at the “crown” (i.e., x = 0). If E_/Es=1,000 and R is 10 ft, then the A_, at the springing line, should be 0.34 in.2/in. (219 mm2/mm). This is roughly that of the average corrugated plate section now used in practice for large-span elliptic culverts and arches in soil (Figure 7.13b). Therefore, if the axial stiffness were gradu-ally reduced by a factor of 2.5 from the springing line to the crown in such structures by slip joints or by using lighter plates, the moments could be “eliminated” and the design improved to the neutral case.*



For a shape other than the membrane ellipse, moments will be introduced and the liner must also have the correct flexural stiffness to achieve neutrality. Equations based on the general theory and conditions for their existence can be derived and studied. However, from a practical design standpoint, it is doubtful if such intermediate cases are as useful as the four special cases:


1.  Harmonic hole ellipse (7.25)


2.  Rigid harmonic inclusion ellipse (7.26)

3. Membrane neutral ellipse (a/b Rt(σ1/ σ2) )

4. Flexurally neutral circle (7.42)


If a hole is to be placed in a loaded body (e.g., a tunnel), the harmonic shape may be preferable to minimize stress concentration in the sheet and then rein-forcement provided to take active loads. For a sheet with a reinforced hole that is then loaded (e.g., buried pipe or most structural applications in plates and shells), the designer would normally choose the membrane configuration or, for ease in construction, a circle. Only if forced to use a rigid liner, would the designer choose the harmonic inclusion shape.


Gradient Fields with an Isotropic Component


In many engineering applications, there is an isotropic component in the stress field in addition to the gradient induced by bending or geostatic conditions. This is the general case in pressure vessels, shell structures or masonry walls, tunnels, pipe, or in other buried structures. For this case the stress function becomes



Although Mansfield does not consider this case, a simple closed membrane shape is possible if the stress function can be put in the form:

which, when equal to zero, gives the neutral hole shape. To do this let: x = fxo and y = yo, where xo, and yo refer to points on the circle of radius R and f is the “mapping function” which transforms this circle to the membrane shape. The stress function [Equation (7.44)] can then be rewritten:

limiting the possible gradient field for a membrane shape to G " 6.


A more stringent limitation is imposed by the area required of the liner for compatibility. The normal force in the liner is by Equation (7.31a)

giving a much more severe restriction on the size of a neutral membrane hole that can be put in any given gradient field.


The neutral “deloid” shape giving no stress concentration with a mem-brane liner, is compared in Figure 7.14a to the harmonic deloid for an unlined hole for the extreme case, G = 2. They are both of the same generic type, but the neutral shape is much closer to a circle. The non dimensionalized area required of the membrane liner for G=1.0, v=1/3 is shown in Figure 7.14b. Comparing the result to the membrane liner for a circular hole in an isotropic field where ElA/EsR = 1.5, this neutral deloid design is, for a gradient field, just as feasible. Even for a concrete liner in stiff soil where El/Es might be 100, the depth of a prismatic liner for a 20 ft diameter opening would be only 1.2 in. at the bottom, 1.8 in. at mid-height, and 6 in. at the top.


To achieve a circular neutral liner, rotation compatibility is critical in that


with a, b, and c all zero by symmetry arguments. The rotation of the sheet at the interface by Equation (7.36) is


Therefore, designing a liner for a circular neutral hole in a gradient field is not practical since the moment of inertia required is slightly negative near the neutral axis (unless vs = 0). However, this might be feasible in concrete or steel where the sheet near the neutral axis could be made thinner.




It has been possible to extend the Mansfield theory for membrane neutral lin-ers to the general case of flexural reinforcement. The resulting expressions for equilibrium and compatibility in terms of the stress function U for the sheet are, in fact, not limited to the neutral condition, but are valid for any thin rein-forcement and thus, are fundamental to the general interaction problem.


In the classical interaction problem, the total, U = Uo + U*, is unknown since the perturbation in the field U* due to the hole in unspecified. By inverting

the problem to a design mode where U* is specified (to be zero for the neutral condition), the stresses, strains, and displacements in the field are entirely known, as are the stress resultants in the liner. Thus, the tangential strain and the rotation of the liner can be found from the equations of Bresse and matched at the interface to the same known quantities in the field to give expressions for the required liner area and bending stiffness.


Closed-form solutions are given here for both the circular and the mem-brane shape for a variety of free fields. A review of these results reveals two interesting limitations to achieving, at least in a simple way, the neutral con-ditions with a thin continuous liner: Most important is the “Poisson ratio effect.” For a positive area of the liner: ( t)s must be exactly in phase with N, i.e., have the same sign at every point on the interface. This might seem obvi-ous, but apparently was not recognized by Mansfield in his work on mem-brane reinforcement. Similarly, if there is bending, the rotation change (dϕ/ds)s must be exactly in phase with M to have a positive moment of inertia. Thus, only in special cases such as the purely deviatoric field or biaxial bending can a neutral liner be found for a “simple” closed hole in a region of a field where either U or its gradient change sign or even approach zero.


Within this constraint, neutral liner designs are found for two basic situa-tions important in practice: (1) Circular holes in deviatoric and biaxial fields (flexural liner); and (2) membrane “deloid shape” and liner for the gradient field with an isotropic component. The second case is particularly useful in that it allows a design within a bending field and closely resembles the geo-static field condition for shallow pipes and tunnels. Moreover, since the “deloid” neutral shape is of the same generic type as the harmonic hole for minimum stress concentration with no liner, its use may serve a double pur-pose both in construction to reduce stresses around the hole and then with reinforcement to best withstand gradient service loads.

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