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An approximate theory of pseudo-arches
International Journal of Solids and Structures 48 (2011) 2960–2964
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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
An approximate theory of pseudo-arches M.J. Leitman a,⇑, P. Villaggio b,1 a b
Deppartment of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA Dipartimento di Ingegneria Strutturale, Universitá di Pisa, Pisa, Italy
a r t i c l e
i n f o
Article history: Received 28 December 2010 Received in revised form 13 May 2011 Available online 15 July 2011 Keywords: Stone walls Pseudo-arches Plane elasticity
a b s t r a c t A pseudo-arch is a triangular door built at the bottom of a heavy wall of stone or bricks. It represents a revolutionary static device, introduced at the beginning of the Neolithic Age, for creating an entrance through a wall without compromising the stability of the structure. We here formulate and solve approximately the problem of constructing a pseudo-arch in terms of plane elasticity. We show that the geometric proportions of some ancient pseudo-arches seem to reflect the predictions of the elastic approximate solution. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The oldest technical artifice, devised by Neolithic builders, for creating an opening at the bottom of a stone or brick wall is the so-called ‘‘pseudo-arch’’. The technique diffused in the Ægean Islands, Mycenæ, Italy and, until recently, the Alps. The most famous example of a pseudo-arch is the upper part of the Door of the Lions at Mycenæ (1300 BCE). A pseudo-arch is an isosceles triangular hole at the base of a wall of heavy rectangular blocks, as shown in Fig. 1. The blocks constituting the intrados of the arch are placed at decreasing distance from the base to the apex, where the two flanks of the intrados intersect. Each block behaves as a cantilevered beam partially supported by the blocks below and stabilized by the weight of the blocks above (Sparacio, 1999, 8, p. 66). Another, perhaps more technical, term for pseudo-arch is ‘‘corbelled arch’’ (Brown, 1993). The projection of each block was not the result of mathematical calculations but the outcome of empirical criteria. This technique improved over the centuries, sometimes at the expense of an unexpected collapse. This begs the question of analyzing the statical behavior of a pseudo-arch with a more precise method in order to check the soundness of the construction rules codified by tradition. Specifically, if the material is sufficiently strong and the loads not too severe, it is physically acceptable to regard the material as linearly elastic and study the stress diffusion in a heavy strip with a triangular notch at the bottom by means of the complex variable method of plane elasticity (see Fig. 2). ⇑ Corresponding author. Tel.: +1 216 368 2890; fax: +1 216 368 5163. E-mail addresses: [email protected], [email protected] (M.J. Leitman), [email protected] (P. Villaggio). 1 Tel.: +39 050 835711; fax: +39 050 554597. 0020-7683/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2011.06.013
The model shown in Fig. 2 ignores the toothed shape of the intrados and the detailed stress transmission between individual blocks. Nevertheless, the elastic solution will answer the following two questions. First, given the height of the wall, the specific weight of the material, and its elastic properties, find the stresses in the strip, in particular along the flanks of the wedge (intrados) and along the top of the wall (extrados). Second, given the limiting stresses in tension and compression of the material, how should the wedge angle and height of the door be chosen so that the structure enjoys a reasonable degree of safety? The surprising result is that the surviving pseudo-arches of antiquity have dimensions and proportions consistent with a tolerable stress level at critical points and also entail an economic use of material. 2. An elastic solution Modeling the pseudo-arch as a triangular notch in a heavy elastic strip, as shown in Fig. 2, is still a difficult problem in plane elasticity, mainly because the material has interior corners and the boundary conditions are mixed. To obtain a tractable explicit solution some further approximations are necessary. In this vein, we seek a semi-inverse solution that satisfies the boundary conditions along the flanks of the wedge. We will be able to satisfy the boundary condition at the top of the wall in the mean. Consider an infinite, elastic, concave wedge loaded by vertical, downward oriented gravitational body forces, the relevant part of which is shown in Fig. 2. We introduce a right-handed system of Cartesian (x, y)-axes with the origin at the vertex of the wedge and the positive x-axis oriented upward and the y-axis horizontal. The flanks of the wedge have angles ±x with the positive x-axis. We always suppose that p2 6 x 6 p. We will be concerned mainly with the rectangular region of width L and height H centered at
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As is customary in treating problems such as ours we split the stresses into two parts: a base state due only to the body forces acting in the entire (x, y) plane and another additional state induced by the presence of the triangular notch. see Savin (1961, 7, pp. 41– 42). A base state that renders the upper boundary at x = h1 traction free is easily found:
rx ¼ cðh1 xÞ; ry ¼ 0; sxy ¼ 0:
ð2:4Þ
In terms of polar coordinates (r, h) Eq. (2.4) become
Fig. 1. Disposition of blocks in a pseudo-arch.
rr ¼ rx cos2 h; rh ¼ rx sin2 h; srh ¼ rx sin h cos h:
ð2:5Þ
But the rays corresponding to angles ±x, the flanks of the wedge, must be traction free. Hence, we must superimpose an additional stress state, corresponding to zero body forces rx ; ry ; sxy or, in polar coordinates, rr ; rh ; srh such that
rh þ rh ¼ 0 and srh þ srh ¼ 0 for h ¼ x:
ð2:6Þ
The stresses for this additional state can be represented by the two analytic functions W⁄(z) and w⁄(z) through
rx þ ry ¼ W ðzÞ þ W ðzÞ; ry rx þ 2ısxy ¼ zW 0 ðzÞ þ w ðzÞ;
ð2:7Þ ð2:8Þ
or, in equivalent polar form (Milne-Thomson, 1960, 5, p. 31).
rr þ rh ¼ W ðzÞ þ W ðzÞ; rh rr þ 2ısrh ¼ e2ıh ðzW 0 ðzÞ þ w ðzÞÞ: Fig. 2. The pseudo-arch conceived as a heavy elastic strip with a triangular entail.
the origin as shown in the figure. The state of stress in this finite region is assumed to be a reasonable approximation of that in a portion of the wall with a triangular opening provided that the stress distribution predicted by the elastic solution along the base, the lower material boundary, may be regarded as a reasonable approximation of the effective boundary tractions. The elastic solution of a heavy plane wedge is classical. It seems that it was first found by Lévy (Lévy, 1898) and re-derived by Michell (1900) by a different method. It is recorded in the treatise by Timoshenko and Goodier (1970) and later proposed as an exercise in the theory of elasticity in the book by Szabó (1963) and in the collection by Rekach (1979, 6, p. 186). We here present a simpler approximate solution by means of the complex variable method. We adopt the formalism of Milne-Thomson (1960). Let rx, ry, and sxy denote the stress components in the wedge in the Cartesian system of Fig. 2. Let c denote the specific weight of the wall and assume the material is isotropic and linearly elastic with Lamé moduli k and l. Since the thickness of the wedge is small with respect to the other dimensions, we may assume a state of generalized plane stress; that is, the stresses are regarded as averages through the thickness. Following Milne-Thomson (1960), we introduce the complex variables z and its conjugate z:
z ¼ x þ ıy and z ¼ x ıy;
ð2:1Þ
and represent the stress components, in the absence of body forces, by means of the two analytic functions W(z) and w(z) such that
rx þ ry ¼ WðzÞ þ WðzÞ; ry rx þ 2ısxy ¼ zW 0 ðzÞ þ wðzÞ;
ð2:2Þ ð2:3Þ
where the bar, ðÞ, denotes conjugation and the prime, ()0 , denotes differentiation with respect to the variable in parenthesis. The stress state is completely determined once W(z) and w(z) are known.
ð2:9Þ ð2:10Þ
We want to determine complex stresses W⁄(z) and w⁄(z) so as to satisfy the two boundary conditions of Eq. (2.6). We assume the simplest possible representations, namely
W ðzÞ :¼ Az þ B and w ðzÞ :¼ Cz þ D;
ð2:11Þ
where A, B, C, D are complex constants. From Eqs. (2.9), (2.10) we have
rr þ rh ¼ Az þ Az þ B þ B; rh rr þ 2ısrh ¼ e2ıh ðAz þ Cz þ DÞ:
ð2:12Þ ð2:13Þ
From Eqs. (2.12), (2.13) the complex radial traction is
2 rh þ ısrh ¼ Az þ Az þ B þ B þ e2ıh ðAz þ Cz þ DÞ:
ð2:14Þ
For an arbitrary base state rx°, ry°, sxy° the induced complex traction is
2ðr h þ ıs rh Þ ¼ 2fr x sin hðsin h ı cos hÞþr y cos hðcos h þ ı sin hÞ s xy ðsin 2h ı cos 2hÞ : ð2:15Þ For our base state, given in Eq. (2.4), this reduces to
2ðr h þ ıs rh Þ ¼ 2cðh1 r cos hÞ sin hðsin h ı cos hÞ:
ð2:16Þ
Using Eqs. (2.14), (2.16), together with the boundary condition Eq. (2.6), we must have
2ðr h þ ıs rh Þ þ 2ðrh þ ısrh Þ h¼x n ¼ 2cðh1 r cos hÞ sin hðsin h ı cos hÞ þ Az þ Az þ B þ B þ e2ıh ðAz þ Cz þ DÞgjh¼x ¼ 0:
ð2:17Þ
Since Eq. (2.17) must hold for all values of r it induces four complex equations for the four complex constants A, B, C, D. Upon solving them we find that the four constants are real, which is not unexpected in view of the symmetry of the problem. These real values are
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g sinð2xÞ þ sinð2xgÞ ¼ 0:
1 cðsec2 x 2Þ; 4 1 B ¼ ch1 ; 2 1 C ¼ cðsec2 x þ 2Þ; 4 D ¼ ch1 :
A¼
ð2:18Þ
2ðrh þ ısrh Þ ¼ 2ðrr ısrh Þ ¼
2
c
2
r sec2 xðcos 2h cos 2xÞðcos h þ ı sin hÞÞ;
ð2:19Þ
r cos h sec2 xð2 þ cos 2x cos 2hÞ
c
ı r sin h sec2 xðcos 2x cos 2hÞ: 2
ð2:20Þ
The corresponding Cartesian state is surprisingly simple, namely
c c c rx ¼ x; ry ¼ x tan2 x; sxy ¼ y: 2
2
The roots g of Eq. (3.8) for each x, 2 6 x 6 p, are the eigenvalues for the elastic states. If (g, x) is an acceptable pair satisfying Eq. (3.8) we must have
c cosððg 1ÞxÞ ¼ : d cosððg þ 1ÞxÞ
We can now use Eqs. (2.18) together with Eqs. (2.4), (2.5), (2.7), (2.8) to determine the stress state. The complex tractions in polar form are
c
ð3:8Þ p
2
ð2:21Þ
It depends on the wedge angle x and the specific weight of the stone c, but not on the value of h1. Clearly the boundary at x = h1, the extrados, is not traction free. While sxy ¼ 2c y may be acceptably small near the x-axis, the normal stress, rx ¼ 2c h1 , is not. 3. Boundary conditions on the extrados Since the elastic state determined in Section 2 does not render the extrados, x = h1, traction free when h1 > 0 we seek another elastic state corresponding to zero body forces which also provides zero tractions on the flanks of the wedge at h = ±x. Ideally, we can superimpose such a state on the one determined in Section 2 so that the extrados, x = h1 > 0, is traction free. Unfortunately a state which will do the latter is unavailable. However, there is a class of stress states for the infinite wedge corresponding to zero body forces which also provide zero tractions on the flanks of the wedge at h = ±x. We can then superimpose one of them on the state in Section 2 so as to produce zero mean stress on the extrados. The state we choose will turn out to have an acceptable singularity at the origin. The family of states, rightly called ‘‘eigensolutions,’’ is widely discussed in the literature (see e.g. Timoshenko and Goodier (1970, p. 141)). Omitting the details, which may be found in the work just cited, it is found that, in polar coordinates, the extra stresses we seek, rer ; reh ; serh , have the form
rer ¼ rg1 ðf 00 ðhÞ þ ðg þ 1Þf ðhÞÞ; reh ¼ rg1 gðg þ 1Þf ðhÞ; serh ¼ rg1 gf 0 ðhÞÞ;
ð3:9Þ
Thus c, d are determined by single constant, say k:
c ¼ k cosððg 1ÞxÞ;
ð3:10Þ
d ¼ k cosððg þ 1ÞxÞ:
For each x; p2 6 x 6 p, we need to determine the acceptable roots of Eq. (3.8). To render the corresponding displacements continuous, we must have g > 0. We should also like the extra stresses, rer ; reh ; serh , to vanish at infinity, so we also require that g < 1. Such states must necessarily have a singularity at the origin of order rg1. At x ¼ p2 there is no wedge at all and the positive roots of Eq. (3.8) are g = 1, 2, 3, . . . ; none are acceptable. At x = p the wedge reduces to a crack and the positive roots of Eq. (3.8) are g ¼ 12 ; 1; 32 ; . . .; so ðg; xÞ ¼ 12 ; p is acceptable. In this case the stress singularity is of the order p1ffir, which is consistent with crack theory. Finally, for x, p2 < x < p, Eq. (3.8) has precisely one positive root g < 1; in fact, 12 < g < 1. In general there will be finitely many other positive roots g P 1, the number of which depends on the value on x. However, for each x; p2 < x 6 p, there is a unique eigenvalue g 2 ½12 ; 1Þ. Denote this relationship by x#g ¼ g^ ðxÞ. Fig. 3 is a graph ^ . Since all eigensolutions will oscillate with h, it is of the function g hopeless to match the condition rx ¼ 2c h1 for x = h1 > 0. For each x, p2 < x 6 p, Eq. (3.4), together with Eq. (3.10), induces a function.
^f ðhÞ ¼ 2kfsinðxÞ sinðxg ^ ðxÞÞ cosðhÞ cosðhg ^ ðxÞÞ cosðxÞ ^ ðxÞÞ sinðhÞ sinðhg ^ ðxÞÞg cosðxg
ð3:11Þ
which, together with Eqs. (3.1), (3.2), (3.3) provides our desired extra elastic state. We record the result, although we will use another ^ method to complete our computations. For simplicity, we write g ^ ðxÞ for g
rer ¼ 2krg^1 g^ fsin h sin g^ hððg^ 1Þ cos x cos g^ x þ 2 sin x sin g^ xÞ ^ hððg ^ 1Þ sin x sin g ^ x þ 2 cos x cos g ^ xÞg; cos h cos g ð3:12Þ
reh ¼ 2krg^1 g^ ðg^ þ 1Þfcos h cos g^ h sin x sin g^ x ^ h cos x cos g ^ xg; sin h sin g
ð3:13Þ
ð3:1Þ ð3:2Þ ð3:3Þ
1.0
where g is a parameter to be determined and f(h) is of the form
f ðhÞ ¼ c cosððg þ 1ÞhÞ þ d cosððg 1ÞhÞ;
ð3:4Þ
for constants c, d. The condition that the flanks of the wedge at h = ±x be traction free reduces to
f ðxÞ ¼ f 0 ðxÞ ¼ f ðxÞ ¼ f 0 ðxÞ ¼ 0:
ð3:5Þ
In view of the obvious symmetry, we obtain the following two homogeneous equations for the constants c, d.
cosððg þ 1ÞxÞc þ cosððg 1ÞxÞd ¼ 0;
ð3:6Þ
ðg þ 1Þ sinððg þ 1ÞxÞc þ ðg 1Þ sinððg 1ÞxÞd ¼ 0:
ð3:7Þ
For Eqs. (3.6), (3.7) to have a non-zero solution the determinant of the coefficients must vanish. Thus,
0.9
0.8
0.7
0.6
100
120
140
160
180
^ ðxÞ, with x measured in degrees. Fig. 3. Graph of the function x#g ¼ g
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serh ¼ 2krg^1 g^ fsin h cos g^ hðg^ cos x cos g^ x þ sin x sin g^ xÞ ^ hðg ^ sin x sin g ^ x þ cos x cos g ^ xÞg: þ cos h sin g
c ð3:14Þ
1 Observe that the strength of the singularity increases from r10 to r1=2 as the wedge angle opens from p2 (no notch) to p (crack). Solutions of this type have been considered extensively by Williams (1952). Note that, using the eigenvalue condition Eq. (3.8), serh in Eq. (3.14) is zero on the flanks (h = ±x), as expected. Let us now compute the complex stress resultant, X + ıY, on that portion of the extrados corresponding to x ¼ h1 ; jyj 6 2L. Using Eq. (2.21), we have
X þ ıY ¼
c
Z
2
L 2
2L
ðh1 þ ıyÞdy ¼
c 2
h1 L:
c 2
^ þ 1ÞxÞðsin ðg ^ a1 Þ g ^ sin ððg ^ 2Þa1 ÞÞg ¼ 0 þ cos ððg
ð3:26Þ
So, in terms of r1 and a1, the value of k is
k¼
^ 1Þ cos ððg ^ 1Þx sin ðg ^ a1 Þ ðg g^
ch1 L 1 2
r1
1 ^ a1 Þ g ^ sinððg ^ 2Þa1 ÞÞ ^ þ 1ÞxÞðsinðg þ cos ððg
ð3:27Þ
Note that both X and Xe are of order h1 as h1 ? 0 but the constant k is not. 4. Three significant examples
h1 L:
ð3:16Þ
We now must compute the stress resultant Xe + ıYe corresponding to the extra stress state just established. Rather than compute the stresses rex ; rey ; sexy directly from rer ; reh ; serh given by Eqs. (3.12), (3.13), (3.14) and then computing the integral, we resort, once again, to the method of complex variables. As in Section 2, the extra stress state corresponds to two analytic stress functions We(z) and we(z). In this case it is easy to verify that
^ zg^ 1 ; W e ðzÞ ¼ 2dg
ð3:17Þ
^ ðg ^ 1Þzg^ þ1 ; we ðzÞ ¼ cg
ð3:18Þ
where c, d are given through the single constant k in Eq. (3.10) and g^ ¼ g^ ðxÞ. We can then use one of Milne–Thomson’s classic formulas (Milne-Thomson, 1960, 2.40) which asserts that the complex stress resultant on the segment x ¼ h1 ; jyj 6 2L is given by
X e þ ıY e ¼
g^ ^ þ 1Þ cos ððg ^ 1ÞxÞ sin ðg ^ a1 Þ h1 L þ kr1 fðg
ð3:15Þ
As expected, symmetry requires that this resultant be real. That is, Ye = 0 and we only have the normal force
X¼
2
z¼h1 þı2L ı 0 e W ðzÞ þ zW e ðzÞ þ 0 we ðzÞ ; 2 z¼h1 ı2L
ð3:19Þ
In order to elucidate our results numerically we examine three illustrious historical pseudo-arches. Specifically, we will compute the stresses at their key points: the vertices and at the lower edges of the flanks (the abutments). The values of these stresses should provide a useful indication of the stability of these monuments. Consider first the Door of the Lions in Mycenæ, mentioned in Section 1 and shown in Fig. 4. In this case there is no superstructure, so h1 = 0. We take x 56p ð¼ 150 Þ and the length of the flank qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 to be r 2 ¼ h2 þ 2l 3m. This triangular door is equilateral. The massive lions do not load the pseudo-arch since they are independently supported by a monolithic stone beam with a span of about 3m (Sparacio, 1999, 8, p. 60). Assume a specific weight of
c ¼ 2500 mkg3 for the stone. Using these values and Eqs. (2.19), (2.20) we get
kg at the vertex; r 2 ¼ 0; m2 pffiffiffi kg c rr ¼ r2 sec x ¼ 2500 3 2 at the abutments r2 ¼ 3m; m 2 5p h ¼ x ¼ : 6
rh ¼ 0
ð4:1Þ
0
where the prime in this case, (), denotes the indefinite integral with respect to the indicated variable. Immediately, we have 0 0
W e ðzÞ ¼ 2dzg^ ; e
ð3:20Þ g^
^ þ 1Þz : w ðzÞ ¼ cðg
ð3:21Þ
To evaluate the complex force resultant in Eq. (3.19), it is convenient to use polar form:
L ¼ r 1 eıa1 ; h1 ı 2
Adjusting the units, the compressive stress rr at the abutment is kg about 0:433 cm 2 , so it is very lightly loaded. A second eminent example is the upper part of the ‘‘Tomb of Clytemnestra,’’ also in Mycenæ(Brown, 1993, 1, p. 18). There is again no superstructure, so h1 = 0. Using the neat picture in p ð¼ 165 Þ and the length of the Brown’s book we estimate x 11 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 flank to be r2 ¼ h2 þ 2l =2 m. This door has a vertex angle about
ð3:22Þ
where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 L L 2 : r 1 ¼ h1 þ and a1 ¼ tan1 2 2h1
ð3:23Þ
Then, together with Eqs. (3.10), (3.22), (3.23), the complex force resultant Eq. (3.19) becomes g^ ^ þ 1Þ cos ððg ^ 1ÞxÞ sin ðg ^ a1 Þ X e ¼ kr1 fðg ^ a1 Þ g ^ sin ððg ^ 2Þa1 ÞÞg ^ þ 1ÞxÞðsin ðg þ cos ððg
ð3:24Þ
e
Again, by virtue of the symmetry, Y = 0, so we have only the normal force resultant Xe. The section of the extrados, x ¼ h1 ; jyj 6 2L will be traction free in the mean provided the constant k is chosen so that
X þ X e ¼ 0: Thus k must satisfy
ð3:25Þ Fig. 4. The Lions Lion_Gate.html).
Gate
at
Mycenæ.
(www.greatbuildings.com/buildings/
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M.J. Leitman, P. Villaggio / International Journal of Solids and Structures 48 (2011) 2960–2964
half that of the Door of the Lions. Again assume a specific weight of c ¼ 2500 mkg3 for the stone. At the same critical points
kg at the vertex; r2 ¼ 0; m2 c kg rr ¼ r2 sec x ¼ 2590 2 at the abutments r2 ¼ 2m; m 2 11p h ¼ x ¼ : 12
rh ¼ 0
ð4:2Þ
Again, adjusting the units, the compressive stress rr at the abutkg ment is about 0:259 cm 2 , so it is even more lightly loaded than the Door of the Lions. Our third archeological example of a pseudo-arch is the door of the ‘‘Terme di Baia’’ mentioned and statically commented upon by Sparacio (1999, p. 67). In this case there is a superstructure, with h1 2 m. The other dimensions are about the same as the precedp ð¼ 165 Þ and r = 2 m. Now we make ing example, namely x 11 2 12 use of the approximation of Section 3, in which we take p. L = l = 2r2sinx 1.04 m. Note also that r1 = 2 m and a1 ¼ 12 ^ corresponding to We need to compute the numerical value of g x ¼ 1112p by using Eq. (3.8). This value is about g^ ¼ 0:5 (The numer^ ¼ 0:501453. See also Fig. 3).2 At this point ical calculation yields g we can use Eq. (3.26) to evaluate the dimensionless constant k. For our data, this yields k = 7220. We next use Eq. (3.12) to evaluate the extra stress rer at the critical point (r1, x). The result is an extra stress rer ¼ 597 mkg2 . This value must be added to the value rr ¼ 2590 mkg2 from the previous example. (Recall that the elastic solution obtained in Section 2 did not depend upon the value of h1.) Adjusting for units, the total kg compressive stress at this critical point is now 0:319 cm 2 , slightly larger than the same door without the superstructure. It is natural to compare these numerical values of to stress components with those predicted by the elementary theory of beams, provided that its application to a blunt body of variable cross-section is admissible. For the sake of simplicity, we consider only the first two cases, in which h1 = 0. Here, the efficient part of the system is constituted by two opposite independent cantilevers, namely AOB and A0 OB0 of Fig. 2, clamped along the vertical edges AB and A0 B0 and subjected to their own weight. Using the notation of Fig. 2, the weight of each part is W ¼ 14 ch2 lb, where b is the thickness. The load acts vertically at a distance of 6l from the edge AB and generates a moment M ¼ 6l W on the clamped cross-section of area h2b. Thus, according to the theory of beams, the stress com-
2 Numerical computations and some figures were done using the MathimaticaÓsoftware.
kg ponent rx at B is rðr2 ; xÞ ¼ 0:830 cm 2 in the first case where kg r2 = 3 m,x = 150° and rx ðr2 ; xÞ ¼ 0:760 cm 2 in the second case where r2 = 2 m, x = 165°. Of course, the situation is the same on the other side. This result merits a short comment. The order of magnitude of the stress components is the same notwithstanding the differences between the two models but the signs of the two are qualitatively contrasting. The elementary theory of beams is more pessimistic as it predicts tensile stress at the points A and A0 .
5. Conclusion Pseudo-arches constitute a typical structural element, largely diffused in continental Greece and Southern Italy, inserted in the walls of fortified cities to permit ingress and egress. They were built by piling stone blocks, which in some cases weigh more than ten tons. It may seem that in the absence of a suitable connection between the blocks, the equilibrium of the structure would be precarious and the pressure on the underlying ground excessive. But an elastic analysis of the stress state around the intrados of the pseudo-arch shows that, except at the vertex where the tension may be singular and result in a localized crack, the compressive stresses at other points are far below the limit of rupture for the stone. This, perhaps, explains their permanence across the centuries. References Brown, D.J., 1993. Bridges. Mitchell Beazley, London. Lévy, M., 1898. Sur la Légimité de la Règle Dite du Trapèze dans l’Ètude de la Résistance des Barrages en Maçonnerie. Compt. Rend. 126, 1235. Michell, J.H., 1900. Elementary distributions of plane stress. Proc. London Math. Soc. 32, 35. Milne-Thomson, L.M., 1960. Plane Elastic Systems. Springer, Berlin nGöttingen nHeidelberg. Rekach, V.G., 1979. Manual of the Theory of Elasticity. Mir Publishers, Moscow. Savin, G.N., 1961. Stress Concentration Around Holes. Pergamon Press, Oxford nLondon nNew York. Sparacio, R. 1999. The Science and the Times of Building (in Italian). Torino: UTET. Szabó, I., 1963. Höhere Technische Mechanik. Springer, Berlin nGöttingen nHeidelberg. Timoshenko, S., Goodier, J.N., 1970. Theory of Elasticity, Third ed. McGraw-Hill, New York. Williams, M.L., 1952. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. Appl. Mech. 19, 526–528.
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
An approximate theory of pseudo-arches M.J. Leitman a,⇑, P. Villaggio b,1 a b
Deppartment of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA Dipartimento di Ingegneria Strutturale, Universitá di Pisa, Pisa, Italy
a r t i c l e
i n f o
Article history: Received 28 December 2010 Received in revised form 13 May 2011 Available online 15 July 2011 Keywords: Stone walls Pseudo-arches Plane elasticity
a b s t r a c t A pseudo-arch is a triangular door built at the bottom of a heavy wall of stone or bricks. It represents a revolutionary static device, introduced at the beginning of the Neolithic Age, for creating an entrance through a wall without compromising the stability of the structure. We here formulate and solve approximately the problem of constructing a pseudo-arch in terms of plane elasticity. We show that the geometric proportions of some ancient pseudo-arches seem to reflect the predictions of the elastic approximate solution. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The oldest technical artifice, devised by Neolithic builders, for creating an opening at the bottom of a stone or brick wall is the so-called ‘‘pseudo-arch’’. The technique diffused in the Ægean Islands, Mycenæ, Italy and, until recently, the Alps. The most famous example of a pseudo-arch is the upper part of the Door of the Lions at Mycenæ (1300 BCE). A pseudo-arch is an isosceles triangular hole at the base of a wall of heavy rectangular blocks, as shown in Fig. 1. The blocks constituting the intrados of the arch are placed at decreasing distance from the base to the apex, where the two flanks of the intrados intersect. Each block behaves as a cantilevered beam partially supported by the blocks below and stabilized by the weight of the blocks above (Sparacio, 1999, 8, p. 66). Another, perhaps more technical, term for pseudo-arch is ‘‘corbelled arch’’ (Brown, 1993). The projection of each block was not the result of mathematical calculations but the outcome of empirical criteria. This technique improved over the centuries, sometimes at the expense of an unexpected collapse. This begs the question of analyzing the statical behavior of a pseudo-arch with a more precise method in order to check the soundness of the construction rules codified by tradition. Specifically, if the material is sufficiently strong and the loads not too severe, it is physically acceptable to regard the material as linearly elastic and study the stress diffusion in a heavy strip with a triangular notch at the bottom by means of the complex variable method of plane elasticity (see Fig. 2). ⇑ Corresponding author. Tel.: +1 216 368 2890; fax: +1 216 368 5163. E-mail addresses: [email protected], [email protected] (M.J. Leitman), [email protected] (P. Villaggio). 1 Tel.: +39 050 835711; fax: +39 050 554597. 0020-7683/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2011.06.013
The model shown in Fig. 2 ignores the toothed shape of the intrados and the detailed stress transmission between individual blocks. Nevertheless, the elastic solution will answer the following two questions. First, given the height of the wall, the specific weight of the material, and its elastic properties, find the stresses in the strip, in particular along the flanks of the wedge (intrados) and along the top of the wall (extrados). Second, given the limiting stresses in tension and compression of the material, how should the wedge angle and height of the door be chosen so that the structure enjoys a reasonable degree of safety? The surprising result is that the surviving pseudo-arches of antiquity have dimensions and proportions consistent with a tolerable stress level at critical points and also entail an economic use of material. 2. An elastic solution Modeling the pseudo-arch as a triangular notch in a heavy elastic strip, as shown in Fig. 2, is still a difficult problem in plane elasticity, mainly because the material has interior corners and the boundary conditions are mixed. To obtain a tractable explicit solution some further approximations are necessary. In this vein, we seek a semi-inverse solution that satisfies the boundary conditions along the flanks of the wedge. We will be able to satisfy the boundary condition at the top of the wall in the mean. Consider an infinite, elastic, concave wedge loaded by vertical, downward oriented gravitational body forces, the relevant part of which is shown in Fig. 2. We introduce a right-handed system of Cartesian (x, y)-axes with the origin at the vertex of the wedge and the positive x-axis oriented upward and the y-axis horizontal. The flanks of the wedge have angles ±x with the positive x-axis. We always suppose that p2 6 x 6 p. We will be concerned mainly with the rectangular region of width L and height H centered at
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As is customary in treating problems such as ours we split the stresses into two parts: a base state due only to the body forces acting in the entire (x, y) plane and another additional state induced by the presence of the triangular notch. see Savin (1961, 7, pp. 41– 42). A base state that renders the upper boundary at x = h1 traction free is easily found:
rx ¼ cðh1 xÞ; ry ¼ 0; sxy ¼ 0:
ð2:4Þ
In terms of polar coordinates (r, h) Eq. (2.4) become
Fig. 1. Disposition of blocks in a pseudo-arch.
rr ¼ rx cos2 h; rh ¼ rx sin2 h; srh ¼ rx sin h cos h:
ð2:5Þ
But the rays corresponding to angles ±x, the flanks of the wedge, must be traction free. Hence, we must superimpose an additional stress state, corresponding to zero body forces rx ; ry ; sxy or, in polar coordinates, rr ; rh ; srh such that
rh þ rh ¼ 0 and srh þ srh ¼ 0 for h ¼ x:
ð2:6Þ
The stresses for this additional state can be represented by the two analytic functions W⁄(z) and w⁄(z) through
rx þ ry ¼ W ðzÞ þ W ðzÞ; ry rx þ 2ısxy ¼ zW 0 ðzÞ þ w ðzÞ;
ð2:7Þ ð2:8Þ
or, in equivalent polar form (Milne-Thomson, 1960, 5, p. 31).
rr þ rh ¼ W ðzÞ þ W ðzÞ; rh rr þ 2ısrh ¼ e2ıh ðzW 0 ðzÞ þ w ðzÞÞ: Fig. 2. The pseudo-arch conceived as a heavy elastic strip with a triangular entail.
the origin as shown in the figure. The state of stress in this finite region is assumed to be a reasonable approximation of that in a portion of the wall with a triangular opening provided that the stress distribution predicted by the elastic solution along the base, the lower material boundary, may be regarded as a reasonable approximation of the effective boundary tractions. The elastic solution of a heavy plane wedge is classical. It seems that it was first found by Lévy (Lévy, 1898) and re-derived by Michell (1900) by a different method. It is recorded in the treatise by Timoshenko and Goodier (1970) and later proposed as an exercise in the theory of elasticity in the book by Szabó (1963) and in the collection by Rekach (1979, 6, p. 186). We here present a simpler approximate solution by means of the complex variable method. We adopt the formalism of Milne-Thomson (1960). Let rx, ry, and sxy denote the stress components in the wedge in the Cartesian system of Fig. 2. Let c denote the specific weight of the wall and assume the material is isotropic and linearly elastic with Lamé moduli k and l. Since the thickness of the wedge is small with respect to the other dimensions, we may assume a state of generalized plane stress; that is, the stresses are regarded as averages through the thickness. Following Milne-Thomson (1960), we introduce the complex variables z and its conjugate z:
z ¼ x þ ıy and z ¼ x ıy;
ð2:1Þ
and represent the stress components, in the absence of body forces, by means of the two analytic functions W(z) and w(z) such that
rx þ ry ¼ WðzÞ þ WðzÞ; ry rx þ 2ısxy ¼ zW 0 ðzÞ þ wðzÞ;
ð2:2Þ ð2:3Þ
where the bar, ðÞ, denotes conjugation and the prime, ()0 , denotes differentiation with respect to the variable in parenthesis. The stress state is completely determined once W(z) and w(z) are known.
ð2:9Þ ð2:10Þ
We want to determine complex stresses W⁄(z) and w⁄(z) so as to satisfy the two boundary conditions of Eq. (2.6). We assume the simplest possible representations, namely
W ðzÞ :¼ Az þ B and w ðzÞ :¼ Cz þ D;
ð2:11Þ
where A, B, C, D are complex constants. From Eqs. (2.9), (2.10) we have
rr þ rh ¼ Az þ Az þ B þ B; rh rr þ 2ısrh ¼ e2ıh ðAz þ Cz þ DÞ:
ð2:12Þ ð2:13Þ
From Eqs. (2.12), (2.13) the complex radial traction is
2 rh þ ısrh ¼ Az þ Az þ B þ B þ e2ıh ðAz þ Cz þ DÞ:
ð2:14Þ
For an arbitrary base state rx°, ry°, sxy° the induced complex traction is
2ðr h þ ıs rh Þ ¼ 2fr x sin hðsin h ı cos hÞþr y cos hðcos h þ ı sin hÞ s xy ðsin 2h ı cos 2hÞ : ð2:15Þ For our base state, given in Eq. (2.4), this reduces to
2ðr h þ ıs rh Þ ¼ 2cðh1 r cos hÞ sin hðsin h ı cos hÞ:
ð2:16Þ
Using Eqs. (2.14), (2.16), together with the boundary condition Eq. (2.6), we must have
2ðr h þ ıs rh Þ þ 2ðrh þ ısrh Þ h¼x n ¼ 2cðh1 r cos hÞ sin hðsin h ı cos hÞ þ Az þ Az þ B þ B þ e2ıh ðAz þ Cz þ DÞgjh¼x ¼ 0:
ð2:17Þ
Since Eq. (2.17) must hold for all values of r it induces four complex equations for the four complex constants A, B, C, D. Upon solving them we find that the four constants are real, which is not unexpected in view of the symmetry of the problem. These real values are
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g sinð2xÞ þ sinð2xgÞ ¼ 0:
1 cðsec2 x 2Þ; 4 1 B ¼ ch1 ; 2 1 C ¼ cðsec2 x þ 2Þ; 4 D ¼ ch1 :
A¼
ð2:18Þ
2ðrh þ ısrh Þ ¼ 2ðrr ısrh Þ ¼
2
c
2
r sec2 xðcos 2h cos 2xÞðcos h þ ı sin hÞÞ;
ð2:19Þ
r cos h sec2 xð2 þ cos 2x cos 2hÞ
c
ı r sin h sec2 xðcos 2x cos 2hÞ: 2
ð2:20Þ
The corresponding Cartesian state is surprisingly simple, namely
c c c rx ¼ x; ry ¼ x tan2 x; sxy ¼ y: 2
2
The roots g of Eq. (3.8) for each x, 2 6 x 6 p, are the eigenvalues for the elastic states. If (g, x) is an acceptable pair satisfying Eq. (3.8) we must have
c cosððg 1ÞxÞ ¼ : d cosððg þ 1ÞxÞ
We can now use Eqs. (2.18) together with Eqs. (2.4), (2.5), (2.7), (2.8) to determine the stress state. The complex tractions in polar form are
c
ð3:8Þ p
2
ð2:21Þ
It depends on the wedge angle x and the specific weight of the stone c, but not on the value of h1. Clearly the boundary at x = h1, the extrados, is not traction free. While sxy ¼ 2c y may be acceptably small near the x-axis, the normal stress, rx ¼ 2c h1 , is not. 3. Boundary conditions on the extrados Since the elastic state determined in Section 2 does not render the extrados, x = h1, traction free when h1 > 0 we seek another elastic state corresponding to zero body forces which also provides zero tractions on the flanks of the wedge at h = ±x. Ideally, we can superimpose such a state on the one determined in Section 2 so that the extrados, x = h1 > 0, is traction free. Unfortunately a state which will do the latter is unavailable. However, there is a class of stress states for the infinite wedge corresponding to zero body forces which also provide zero tractions on the flanks of the wedge at h = ±x. We can then superimpose one of them on the state in Section 2 so as to produce zero mean stress on the extrados. The state we choose will turn out to have an acceptable singularity at the origin. The family of states, rightly called ‘‘eigensolutions,’’ is widely discussed in the literature (see e.g. Timoshenko and Goodier (1970, p. 141)). Omitting the details, which may be found in the work just cited, it is found that, in polar coordinates, the extra stresses we seek, rer ; reh ; serh , have the form
rer ¼ rg1 ðf 00 ðhÞ þ ðg þ 1Þf ðhÞÞ; reh ¼ rg1 gðg þ 1Þf ðhÞ; serh ¼ rg1 gf 0 ðhÞÞ;
ð3:9Þ
Thus c, d are determined by single constant, say k:
c ¼ k cosððg 1ÞxÞ;
ð3:10Þ
d ¼ k cosððg þ 1ÞxÞ:
For each x; p2 6 x 6 p, we need to determine the acceptable roots of Eq. (3.8). To render the corresponding displacements continuous, we must have g > 0. We should also like the extra stresses, rer ; reh ; serh , to vanish at infinity, so we also require that g < 1. Such states must necessarily have a singularity at the origin of order rg1. At x ¼ p2 there is no wedge at all and the positive roots of Eq. (3.8) are g = 1, 2, 3, . . . ; none are acceptable. At x = p the wedge reduces to a crack and the positive roots of Eq. (3.8) are g ¼ 12 ; 1; 32 ; . . .; so ðg; xÞ ¼ 12 ; p is acceptable. In this case the stress singularity is of the order p1ffir, which is consistent with crack theory. Finally, for x, p2 < x < p, Eq. (3.8) has precisely one positive root g < 1; in fact, 12 < g < 1. In general there will be finitely many other positive roots g P 1, the number of which depends on the value on x. However, for each x; p2 < x 6 p, there is a unique eigenvalue g 2 ½12 ; 1Þ. Denote this relationship by x#g ¼ g^ ðxÞ. Fig. 3 is a graph ^ . Since all eigensolutions will oscillate with h, it is of the function g hopeless to match the condition rx ¼ 2c h1 for x = h1 > 0. For each x, p2 < x 6 p, Eq. (3.4), together with Eq. (3.10), induces a function.
^f ðhÞ ¼ 2kfsinðxÞ sinðxg ^ ðxÞÞ cosðhÞ cosðhg ^ ðxÞÞ cosðxÞ ^ ðxÞÞ sinðhÞ sinðhg ^ ðxÞÞg cosðxg
ð3:11Þ
which, together with Eqs. (3.1), (3.2), (3.3) provides our desired extra elastic state. We record the result, although we will use another ^ method to complete our computations. For simplicity, we write g ^ ðxÞ for g
rer ¼ 2krg^1 g^ fsin h sin g^ hððg^ 1Þ cos x cos g^ x þ 2 sin x sin g^ xÞ ^ hððg ^ 1Þ sin x sin g ^ x þ 2 cos x cos g ^ xÞg; cos h cos g ð3:12Þ
reh ¼ 2krg^1 g^ ðg^ þ 1Þfcos h cos g^ h sin x sin g^ x ^ h cos x cos g ^ xg; sin h sin g
ð3:13Þ
ð3:1Þ ð3:2Þ ð3:3Þ
1.0
where g is a parameter to be determined and f(h) is of the form
f ðhÞ ¼ c cosððg þ 1ÞhÞ þ d cosððg 1ÞhÞ;
ð3:4Þ
for constants c, d. The condition that the flanks of the wedge at h = ±x be traction free reduces to
f ðxÞ ¼ f 0 ðxÞ ¼ f ðxÞ ¼ f 0 ðxÞ ¼ 0:
ð3:5Þ
In view of the obvious symmetry, we obtain the following two homogeneous equations for the constants c, d.
cosððg þ 1ÞxÞc þ cosððg 1ÞxÞd ¼ 0;
ð3:6Þ
ðg þ 1Þ sinððg þ 1ÞxÞc þ ðg 1Þ sinððg 1ÞxÞd ¼ 0:
ð3:7Þ
For Eqs. (3.6), (3.7) to have a non-zero solution the determinant of the coefficients must vanish. Thus,
0.9
0.8
0.7
0.6
100
120
140
160
180
^ ðxÞ, with x measured in degrees. Fig. 3. Graph of the function x#g ¼ g
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serh ¼ 2krg^1 g^ fsin h cos g^ hðg^ cos x cos g^ x þ sin x sin g^ xÞ ^ hðg ^ sin x sin g ^ x þ cos x cos g ^ xÞg: þ cos h sin g
c ð3:14Þ
1 Observe that the strength of the singularity increases from r10 to r1=2 as the wedge angle opens from p2 (no notch) to p (crack). Solutions of this type have been considered extensively by Williams (1952). Note that, using the eigenvalue condition Eq. (3.8), serh in Eq. (3.14) is zero on the flanks (h = ±x), as expected. Let us now compute the complex stress resultant, X + ıY, on that portion of the extrados corresponding to x ¼ h1 ; jyj 6 2L. Using Eq. (2.21), we have
X þ ıY ¼
c
Z
2
L 2
2L
ðh1 þ ıyÞdy ¼
c 2
h1 L:
c 2
^ þ 1ÞxÞðsin ðg ^ a1 Þ g ^ sin ððg ^ 2Þa1 ÞÞg ¼ 0 þ cos ððg
ð3:26Þ
So, in terms of r1 and a1, the value of k is
k¼
^ 1Þ cos ððg ^ 1Þx sin ðg ^ a1 Þ ðg g^
ch1 L 1 2
r1
1 ^ a1 Þ g ^ sinððg ^ 2Þa1 ÞÞ ^ þ 1ÞxÞðsinðg þ cos ððg
ð3:27Þ
Note that both X and Xe are of order h1 as h1 ? 0 but the constant k is not. 4. Three significant examples
h1 L:
ð3:16Þ
We now must compute the stress resultant Xe + ıYe corresponding to the extra stress state just established. Rather than compute the stresses rex ; rey ; sexy directly from rer ; reh ; serh given by Eqs. (3.12), (3.13), (3.14) and then computing the integral, we resort, once again, to the method of complex variables. As in Section 2, the extra stress state corresponds to two analytic stress functions We(z) and we(z). In this case it is easy to verify that
^ zg^ 1 ; W e ðzÞ ¼ 2dg
ð3:17Þ
^ ðg ^ 1Þzg^ þ1 ; we ðzÞ ¼ cg
ð3:18Þ
where c, d are given through the single constant k in Eq. (3.10) and g^ ¼ g^ ðxÞ. We can then use one of Milne–Thomson’s classic formulas (Milne-Thomson, 1960, 2.40) which asserts that the complex stress resultant on the segment x ¼ h1 ; jyj 6 2L is given by
X e þ ıY e ¼
g^ ^ þ 1Þ cos ððg ^ 1ÞxÞ sin ðg ^ a1 Þ h1 L þ kr1 fðg
ð3:15Þ
As expected, symmetry requires that this resultant be real. That is, Ye = 0 and we only have the normal force
X¼
2
z¼h1 þı2L ı 0 e W ðzÞ þ zW e ðzÞ þ 0 we ðzÞ ; 2 z¼h1 ı2L
ð3:19Þ
In order to elucidate our results numerically we examine three illustrious historical pseudo-arches. Specifically, we will compute the stresses at their key points: the vertices and at the lower edges of the flanks (the abutments). The values of these stresses should provide a useful indication of the stability of these monuments. Consider first the Door of the Lions in Mycenæ, mentioned in Section 1 and shown in Fig. 4. In this case there is no superstructure, so h1 = 0. We take x 56p ð¼ 150 Þ and the length of the flank qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 to be r 2 ¼ h2 þ 2l 3m. This triangular door is equilateral. The massive lions do not load the pseudo-arch since they are independently supported by a monolithic stone beam with a span of about 3m (Sparacio, 1999, 8, p. 60). Assume a specific weight of
c ¼ 2500 mkg3 for the stone. Using these values and Eqs. (2.19), (2.20) we get
kg at the vertex; r 2 ¼ 0; m2 pffiffiffi kg c rr ¼ r2 sec x ¼ 2500 3 2 at the abutments r2 ¼ 3m; m 2 5p h ¼ x ¼ : 6
rh ¼ 0
ð4:1Þ
0
where the prime in this case, (), denotes the indefinite integral with respect to the indicated variable. Immediately, we have 0 0
W e ðzÞ ¼ 2dzg^ ; e
ð3:20Þ g^
^ þ 1Þz : w ðzÞ ¼ cðg
ð3:21Þ
To evaluate the complex force resultant in Eq. (3.19), it is convenient to use polar form:
L ¼ r 1 eıa1 ; h1 ı 2
Adjusting the units, the compressive stress rr at the abutment is kg about 0:433 cm 2 , so it is very lightly loaded. A second eminent example is the upper part of the ‘‘Tomb of Clytemnestra,’’ also in Mycenæ(Brown, 1993, 1, p. 18). There is again no superstructure, so h1 = 0. Using the neat picture in p ð¼ 165 Þ and the length of the Brown’s book we estimate x 11 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 flank to be r2 ¼ h2 þ 2l =2 m. This door has a vertex angle about
ð3:22Þ
where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 L L 2 : r 1 ¼ h1 þ and a1 ¼ tan1 2 2h1
ð3:23Þ
Then, together with Eqs. (3.10), (3.22), (3.23), the complex force resultant Eq. (3.19) becomes g^ ^ þ 1Þ cos ððg ^ 1ÞxÞ sin ðg ^ a1 Þ X e ¼ kr1 fðg ^ a1 Þ g ^ sin ððg ^ 2Þa1 ÞÞg ^ þ 1ÞxÞðsin ðg þ cos ððg
ð3:24Þ
e
Again, by virtue of the symmetry, Y = 0, so we have only the normal force resultant Xe. The section of the extrados, x ¼ h1 ; jyj 6 2L will be traction free in the mean provided the constant k is chosen so that
X þ X e ¼ 0: Thus k must satisfy
ð3:25Þ Fig. 4. The Lions Lion_Gate.html).
Gate
at
Mycenæ.
(www.greatbuildings.com/buildings/
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half that of the Door of the Lions. Again assume a specific weight of c ¼ 2500 mkg3 for the stone. At the same critical points
kg at the vertex; r2 ¼ 0; m2 c kg rr ¼ r2 sec x ¼ 2590 2 at the abutments r2 ¼ 2m; m 2 11p h ¼ x ¼ : 12
rh ¼ 0
ð4:2Þ
Again, adjusting the units, the compressive stress rr at the abutkg ment is about 0:259 cm 2 , so it is even more lightly loaded than the Door of the Lions. Our third archeological example of a pseudo-arch is the door of the ‘‘Terme di Baia’’ mentioned and statically commented upon by Sparacio (1999, p. 67). In this case there is a superstructure, with h1 2 m. The other dimensions are about the same as the precedp ð¼ 165 Þ and r = 2 m. Now we make ing example, namely x 11 2 12 use of the approximation of Section 3, in which we take p. L = l = 2r2sinx 1.04 m. Note also that r1 = 2 m and a1 ¼ 12 ^ corresponding to We need to compute the numerical value of g x ¼ 1112p by using Eq. (3.8). This value is about g^ ¼ 0:5 (The numer^ ¼ 0:501453. See also Fig. 3).2 At this point ical calculation yields g we can use Eq. (3.26) to evaluate the dimensionless constant k. For our data, this yields k = 7220. We next use Eq. (3.12) to evaluate the extra stress rer at the critical point (r1, x). The result is an extra stress rer ¼ 597 mkg2 . This value must be added to the value rr ¼ 2590 mkg2 from the previous example. (Recall that the elastic solution obtained in Section 2 did not depend upon the value of h1.) Adjusting for units, the total kg compressive stress at this critical point is now 0:319 cm 2 , slightly larger than the same door without the superstructure. It is natural to compare these numerical values of to stress components with those predicted by the elementary theory of beams, provided that its application to a blunt body of variable cross-section is admissible. For the sake of simplicity, we consider only the first two cases, in which h1 = 0. Here, the efficient part of the system is constituted by two opposite independent cantilevers, namely AOB and A0 OB0 of Fig. 2, clamped along the vertical edges AB and A0 B0 and subjected to their own weight. Using the notation of Fig. 2, the weight of each part is W ¼ 14 ch2 lb, where b is the thickness. The load acts vertically at a distance of 6l from the edge AB and generates a moment M ¼ 6l W on the clamped cross-section of area h2b. Thus, according to the theory of beams, the stress com-
2 Numerical computations and some figures were done using the MathimaticaÓsoftware.
kg ponent rx at B is rðr2 ; xÞ ¼ 0:830 cm 2 in the first case where kg r2 = 3 m,x = 150° and rx ðr2 ; xÞ ¼ 0:760 cm 2 in the second case where r2 = 2 m, x = 165°. Of course, the situation is the same on the other side. This result merits a short comment. The order of magnitude of the stress components is the same notwithstanding the differences between the two models but the signs of the two are qualitatively contrasting. The elementary theory of beams is more pessimistic as it predicts tensile stress at the points A and A0 .
5. Conclusion Pseudo-arches constitute a typical structural element, largely diffused in continental Greece and Southern Italy, inserted in the walls of fortified cities to permit ingress and egress. They were built by piling stone blocks, which in some cases weigh more than ten tons. It may seem that in the absence of a suitable connection between the blocks, the equilibrium of the structure would be precarious and the pressure on the underlying ground excessive. But an elastic analysis of the stress state around the intrados of the pseudo-arch shows that, except at the vertex where the tension may be singular and result in a localized crack, the compressive stresses at other points are far below the limit of rupture for the stone. This, perhaps, explains their permanence across the centuries. References Brown, D.J., 1993. Bridges. Mitchell Beazley, London. Lévy, M., 1898. Sur la Légimité de la Règle Dite du Trapèze dans l’Ètude de la Résistance des Barrages en Maçonnerie. Compt. Rend. 126, 1235. Michell, J.H., 1900. Elementary distributions of plane stress. Proc. London Math. Soc. 32, 35. Milne-Thomson, L.M., 1960. Plane Elastic Systems. Springer, Berlin nGöttingen nHeidelberg. Rekach, V.G., 1979. Manual of the Theory of Elasticity. Mir Publishers, Moscow. Savin, G.N., 1961. Stress Concentration Around Holes. Pergamon Press, Oxford nLondon nNew York. Sparacio, R. 1999. The Science and the Times of Building (in Italian). Torino: UTET. Szabó, I., 1963. Höhere Technische Mechanik. Springer, Berlin nGöttingen nHeidelberg. Timoshenko, S., Goodier, J.N., 1970. Theory of Elasticity, Third ed. McGraw-Hill, New York. Williams, M.L., 1952. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. Appl. Mech. 19, 526–528.