Zoomorphology

Zoomorphology (1985) 105:114-I 24

@] Springer-Verlag1985

Domes, arches and urchins: The skeletal architecture of echinoids (Echinodermata) Malcolm Telford Department of Zoology, University of Toronto, Ontario, M5S tAI, Canada

Summary. A combination of simple membrane theory and statical analysis has been used to determine how stresses are carried in echinoid skeletons. Sutures oriented circumferentially are subject principally to compression. Those forming radial zig-zags are subject to compression near the apex and tension near the ambitus. Radial and circumferential sutures in Eucidaris are equally bound with collagen fibers but in Diadema, Tripneustes, Psammechinus, Arbacia and other regular echinoids, most radial sutures are more heavily bound, and thus stronger in tension. Psammechinus, Tripneustes and several other echinoids have radial sutures thickened by ribs which increase the area of interlocking trabeculae. Ribs also increase flexural stiffness and carry a greater proportion of the stress. Further, ribs effectively draw stress from weaker areas pierced by podial pores, and increase the total load which can be sustained. Allometry indicates that regular echinoids become relatively higher at the apex as size increases, thus reducing ambital stresses. Some spatangoids with very high domes (eg Agassizia) maintain isometry, but others (eg Meoma) become flatter with size. Both holectypoids (Echinoneus) and cassiduloids (Apatopygus) maintain a constant height to diameter relationship. Flattening, and consequently ambital tensile stress, is greatest in the clypeasteroids. In this group the formation of internal buttresses which preferentially carry stress, reaches maximum development. A notable exception, however, is the high domed Clypeaster rosaCCHS.

In this analysis it was assumed that local buckling or bending does not occur. The test of some echinoids (e.g. Diadematoida) have relatively wide sutures swathed in collagen, which allows local deformation. Others (e.g. Arbacia) have rigid sutures with reduced collagen. In Psammechinus and other members of the Order Echinoida, in addition to rib formation, inner and outer surface trabeculae are thickened so that the individual plates are stiffened. Some spatangoids (Meoma, Paleopneustes) have extensive sutural collagen, but the cassiduloid Apatopygus has collagen confined to junctions of sutures, and elsewhere the joints are strengthened and stiffened by fusion of trabeculae. Fusion of surface trabeculae is almost complete in the holectypoid, Echinoneus, and the sutures are obscured.

in part to the 'frankly evangelical' work of Wainwright et al. (1976), to whom I now pay tribute. In the preface to 'Mechanical Design in Organisms', they remark that an understanding of some basic engineering concepts (strength, rigidity, elasticity etc) allows an interpretation of skeletal materials, structural elements and, ultimately, 'the overall design strategy of complete skeletal systems'. Very few analyses of whole skeletons have yet appeared and most of the literature concerning echinoid skeletons has been directed toward the properties of material (calcite) and its arrangement in the familiar porous stereom. For example, Raup (1962), Currey (1975), Emlet (1982), Klein and Currey (1970), Nichols and Currey (1968) and O'Neill (1981), to mention but a few, have all investigated the strength characteristics or crystalline structure of echinoderm calcite. Smith (1980) made a detailed study of the micromorphology of echinoid stereom and Weber et al. (1969) experimentally investigated strength of machined pieces of stereom from a number of echinoderms. The mechanical design of Diadema spines has been analysed by Burkhardt et al. (1983), whilst Eylers (1976) has investigated the skeletal mechanics of Asterias. In these pioneering studies, the skeletons were reduced to their functional components in order to determine the nature of the stresses sustained by each. Treating whole skeletons, Seilacher (1979) speculated that echinoid tests might be derived from pneu structures, but he made no analysis of the forces involved. In general, echinoids are domed, shell structures which can be treated by a combination of simple membrane theory and statical analysis. It will be shown that echinoid skeletons are adapted in many ways to resist externally applied forces. The analytical approach to structures is necessarily concerned with forces, materials and geometry which are related to each other as follows: External forces

T

Forces: Equilibrium equations

Internal forces

stress

t Material: Constitutive equations

+ Internal deformations - strain

A. Introduction Recent years have witnessed increasing interest in the mechanical analysis of living organisms. This is no doubt due

T Geometry: Capability equations External displacements

115 In the following treatment of echinoids, the terms 'internal' and 'external' will be used in this engineering sense as relative to structural components. Thus, a force oilginating from muscular activity within the coelom, for example, would be considered to be external to a test plate in just the same way as a force originating outside the organism, such as wave action or pressure from a sediment load. Dornes and arehes

An arch is a two dimensional structure which serves to transmit stress, from self-weight or applied load, to the abutments. A dome may be visualized, at least in part, as a three dimensional structure resulting from rotation of an arch or from a seiles of intersecting arches (Lin and Stotesbury 1981). This is a simplification, and it should be borne in mind that a dome structure differs from a seiles of arches in some important respects. Most notably, the material between the ribs contributes to the overall stiffness and load carrying capacity of the structure. None the less, it is useful to consider some features of arches before attempting the analysis of dornes. In a simple arch (Fig. 1 A) stress is carried in compression to the abutments. The resultant forces, R a and RB, placed on the abutments, can be resolved into vertical and horizontal components which are reacted by exactly equal but opposite forces, Va, Vn and H a, H» respectively. The two vertical reactions, Va and V~, are together equal to the combined weight of the arch and applied load. Their relative magnitudes depend upon the exact position of the load. If a point load is applied away from the center,

rain1 = vBr2

(1)

where L1 is the horizontal distance from the load to abutment A and L 2 is the distance to abutment B. Thus the mangitude of the vertical force increases with proximity of the load. Under self weight or loading at the apex only, Va and Vn will be equal if the arch is symmetrical. The horizontal components are called 'thrusts' and are due to the tendency of the ends of the arch to move apart. In a bridge, for example, the thrusts are reacted in compression by forces supplied by the abutments. If the ends of the arch are joined together by a tie rod, the thrusts can be reacted in tension. In a symmetilcal arch the thrusts are equal. Their actual magnitude depends on the vertical forces and on the relationship between height and span of the arch. To take a simple case, if the arch is loaded by weight, W, at the crown, then, ignoring self weight: Va = V» = W/2

(2)

and the thrust, H a = VA/tan 0

(3)

where than 0 is given by F/L (Fig. 1 A) (Lin and Stotesbury 1981). More generally, F is the vertical height of the loaded point, L~ and L2 the horizontal distances from this point to the abutments, A and B respectively, and VA/tan 0 = VB/tan (90-0).

(4)

Thrust is therefore inversely proportional to arch rise at the crown. This means that flat arches, in which height at the crown is small relative to the span, generate greater thrusts than relatively high arches. No structure is perfectly rigid. The arch under consideration, loaded at the crown

and prevented from spreading by abutments or a tie rod, will tend to bend (Figure 1 B). As the crown is pushed downwards ( + ve bending) the sides of the arch must compensate by bowing outwards ( - v e bending). In a semicircular arch, for example, maximum bowing occurs midway from the crown (Thadani 1964). The forces acting in a dome are similar. When loaded at the apex, as in the case of the arch, all of the stresses cariled radially are compressive. A dome is said to be a non-developable surface: when loaded apically the margins rend to spread resulting in tensile circumferential or hoop stresses (Fig. 1 C). Towards the center of the dome, corresponding approximately with the region of positive bending in the arch system, the surface is squeezed in compression. In shell structures stresses in the outer and inner surfaces are not equal. For this reason, engineers work with stress resultants (stress integrated through the thickness of the shell) which behave mathematically in the same way as stresses (Heyman 1977). Radially directed, compressive stress resultants (Fig. 1 D), at the margin of a hemispherical dome are double those at the center. Circumferentially directed stress resultants are compressive at the center, decrease to zero at about 40 ° latitude on the dome, after which they become tensile and at the margin are equal in magnitude to the radial stress resultants. Engineers strengthen dornes in a number of ways to meet these forces. Radially disposed ribs which locally increase flexural stiffness of the structure, carry a greater proportion of the stress. This effectively decreases both compressive and tensile stresses between the ribs. The center may be left open to avoid congestion of intersecting ribs (Lin and Stotesbury 1981), and a compression ring is then used to surround the aperture. Conversely, at the margin, a substantial tension ring may be installed to react circumferential tensile forces. If the dome can be represented by a rotated arch, the floor can be represented by a rotated tie rod. The floor can thus contribute to the function of the tension ring or even replace it. Additional rings or hoops may be installed to react circumferential forces and these, of course, would be most effectively placed in the lower part of the dome (between 0 ° and 40 ° latitude in a hemisphere). These additional hoops might also be necessary to stiften the dome between the ribs, thus reducing or eliminating bending. Just as the magnitude of thrusts from an arch depend on its shape, so too, do the hoop stresses in a dome. The shallower the dome, the greater the tensile stress towards the margin. It is hypothesized that echinoid skeletons, depending on their individual geometries, will show similar distributions and relative magnitudes of forces as engineered dornes. The features described above, radial ribs, hoops, compression rings, tension rings etc, all have their counterparts in the diverse skeletal structures of the Echinoidea. B. Materials and methods

Echinoid specimens: a wide variety of echinoids froln personal field collections and museums was selected from various locations, as follows: Cidaroida: Eucidaris tribuloides (Lamarck), Trinidad; Stylocidaris affinis Phillipi, Barbados. Diadematoida: Diadema antillarum Phillipi and Astropyga magnifica A.H. Clark, both from Barbados. Phymosomatoida: Arbacia punctulata (Lamarck), Trinidad.

116

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Fig. 1. A Diagram of forces transmitted by arch loaded at apex by force W. The resultant forces R« and R b resolve into vertical and horizontal components reacted by Vù, Vb and Hù, H b, respectively. B An arch tends to bend positively beneath the load and this is compensated by negative bending or bowing along the sides. C A dome is a non-developable surface. Loaded by selfweight or with an applied force, it tends to tear apart around the edges. D Radial stress resultants are entirely compressive. At the edge of the dome they are double the magnitude of stress resultants at the crown. Circumferential stress resultants are compressive towards the apex, tensile towards the ambitus, where they are equal in magnitude to radial stress resultants (after Heyrnan, 1977). E Length (x) and height (y) of thirteen species of regular echinoids, plotted on log coordinates. Differences in slope were not statistically significant but those in intercept values were (P<0.01). F Mean length (x) and height (y) of six species of irregular echinoids. Differences in slope were found to be statistically significant ( P < 0.01)

117 Temnopleuroida: Tripneustes edulis (Leske), Barbados; Lytechinus variegatus (Lamarck), Barbados; L. williamsi Chesher, Jamaica. Echinoida: Echinus gracilis A. Agassiz, N o r t h Carolina; Echinometra lucunter (Linnaeus), Florida; E. viridis A. Agassiz, Jamaica; Psammechinus miliaris (P. L.S. Muller), Scotland; Strongylocentrotus droebachiensis O.F. Muller, New Brunswick; Pseudoboletia occidentalis A.H. Clark, Barbados. Holectypoida: Echinoneus cyclostomus Leske, U.S. National Museum, Bermuda (El1799), Bahamas (E5594), Antigua (E4338); Florida Department of Natural Resources, Bahamas (E J-71-20), Eniwetok Atoll (EJ-68-107). Cassiduloida: Apatopygus recens Milne-Edwards, U.S. National Museum, New Zealand (E10131, El1084, E16325 and E16329). Clypeasteroida: Clypeaster rosaceus (Linnaeus), Florida. Spatangoida: Paleopneustes cristatus A. Agassiz and Agassizia excentricus A. Agassiz, Barbados; M e o m a ventricosa (Lamarck), Florida. Here, and throughout this paper, the classification followed by Smith (1984) has been adopted. Collection catalogue numbers given in parentheses. Fixatives: Specimens from field collections were injected with, and then fixed by immersion in 10% buffered formalin, which was changed after 24 h. The specimens of Stylocidaris, Diadema, Arbacia and Tripneustes had been air dried after fixation. Museum collections of Apatopygus were all dried with spines, pedicellaria and podia etc in situ, and were clearly fresh, live specimens when obtained. SEM preparation: Spines were removed, the external surfaces cleaned to ensure good adhesion to stubs and the specimens were broken open. Dried specimens were hydrated in distilled watet for at least 24 h. After removal of internal organs, the coelomic lining was carefully peeled away to expose the surface of the underlying plates. To ensure that it was fully removed, preparations were stained with fast green and examined by light microscopy. Selected pieces of test with radially and/or circumferentially oriented sutures, were washed again in distilled water, mounted and freeze-dried. Before sputter coating with gold, the pieces of test were ringed with conductive paint to avoid electron charging during subsequent examination. M o r p h o m e t r y : Height and diameter o f test (without spines) was measured to the nearest 0.05 m m by Vernier calipers. The irregular echinoids, and the genus Echinometra, have clear longitudinal and transverse axes and, in these, both diameters were taken and averaged. Regression analyses were by least squares o f log-transformed data. Slopes and intercepts were compared by analysis of covariance. All statistical treatments followed Sokal and Rohlf (1981). C. Results

M o r p h o m e t r y : Height and diameter were measured for a total of 132 regular echinoids belonging to 13 species. A sufficient size range was available to allow regression analysis of the data for five of the species separately. Although some differences in slopes and intercepts are apparent (Table 1), these echinoids are remarkably alike in gross form (Fig. 1 E). Height/diameter ratios were calculated from individual regression equations, or, in some cases, from individ-

Table 1. Regression analyses for log-transformed data of height (y) and diameter (x) from five species of regular echinoids Species

n

Stylocidaris affinis Tripneustes edulis Psammechinus miliaris Strongylocentrotus droebachiensis Echinometra lucunter

tO 11 36 30 15

Intercept (a)

Slope (b)

Coeff. det (r 2)

0.55 0.42 0.46 0.47 0.37

1.06 1.07 1.08 1.03 1.12

1.00 0.99 0.99 0.98 0.96

Table 2. Height/Diameter ratios of regular echinoids. Ratios were

determined at 100% and 10% of maximum observed diameter, from regression equations of Table 1. For other species, ratios were determined for smallest and largest individuals available Species

10%

100%

Stylocidaris affinis Psammechinus miliaris Tripneustes edulis Echinometra lucunter Strongylocentrotus droebachiensis

0.65 0.52 0.56 0.46 0.50

0.75 0.63 0.62 0.60 0.53

Smallest

Largest

0.58 0.55 0.57 0.50 0.54 0.45 0.46 ---

0.67 0.60 0.60 0.60 0.58 0.53 0.51 0.54 0.39

Eucidaris tribuloides Lytechinus variegatus Lytechinus williamsi Arbacia punctulata Echinometra viridis Pseudoboletica occidentalis Strongylocentrotus purpurata Diadema antillarum Astropyga magnifica

Table 3. Regression analyses of log-transformed data of height (y) and mean diameter (x) for six species or irregular echinoids Species

n

Intercept Slope (a) (b)

Coeff. det. (r z)

Echinoneus cyclostomus Apatopygus recens Clypeaster rosaceus Meoma ventricosa Paleopneustes cristatus Agassizia excentricus

31 42 12 15 7 23

0.54 0.45 0.15 1.14 0.41 0.83

0.99 0.97 0.98 0.98 0.97 0.95

1.0t 1.00 1.22 0.84 1.11 1.02

Table 4. Height/Mean Diameter ratios of irregular echinoids. Ratios were determined from regression equations of Table 3, for individuals at 10% and 100% of the maximum observed mean diameter Species

10%

100%

Agassizia excentricus Paleopneustes cristatus Echinoneus cyclostomus Meoma ventricosa Apatopygus recens Clypeaster rosaceus

0.83 0.54 0.55 0.74 0.45 0.28

0.87 0.69 0.56 0.51 0.45 0.44

118 Table 5. Thickness of ambulaeral plates at geometrically equivalent intervals passing radially around the test of regular echinoid species.

Measurements at apex in gin, all others expressed as percentage increase from apical thickness. R/t=thickness ratio. Ocular, genital plates and perignathic girdle not included Species

R/t

Apex

2

3

4

5

Peristome

Stylocidaris affinis Echinometra lucunter Strongyloeentrotus droebachiensis Echinusgracilis Tripneustes edulis Diademaantillarum

22 24 29 34 36 38

850 950 950 700 700 400

+24 +42 +16 +29 +29 +25

+ 41 + 58 + 32 + 57 + 134 + 63

+ 53 + 37 + 47 + 93 + 100 + 50

+ 65 + 37 + 37 +100 + 79 + 13

+ 100 + 26 + 58 + 57 + 79 +150

ual specimens (Table 2). Length, width and height were measured for 256 irregular echinoids belonging to 8 different species. Regression analyses and plots of height and length are shown in Table 3 and Figure 1 F, height/diameter ratios are given in Table 4. Plate thickness and rib formation: Test thickness increases from near the apex to the ambitus and, in some instances, to the peristome (Table 5). In m a n y echinoids the test plates are thickened bordering radially oriented sutures, sometimes forming distinct radial ribs. Interradial sutures (between interambulacral plate rows) are often thickened in this way (Table 6), but the extent of thickening is greater in adradial (interambulacral/ambulacral) and perradial (ambulacral/ambulacral) sutures (terminology of Phelan 1977). The ambulacral plates are themselves usually thicker than the interambulacrals, especially around the podial pores. Bordering circumferential sutures the plates are not thickened to any appreciable extent. The peristomial margin may be thick, but even more, the perignathic girdle ( H y m a n 1955) forms a reinforcing pallisade just inside the peristomial edge. Towards the apex, the genital (and sometimes the ocular) plates form a thickened ring (not included in Table 5). In S. droebachiensis the periproctal ring, although small, is 30-35% thicker than the next radially adjacent plates. In those echinoids with large periprocts, such as the Cidaroida and Diadematoida, the surrounding ring is especially p r o m i n e n t and thick. In the Diadematoida, the ambulacra are bowed outwards, forming blunt, rounded corrugations of the dome surface. Ambulacral and interambulacral thicknesses of the test a r o u n d the ambitus for four irregular echinoids are shown in Table 7. These urchins are somewhat asymmetrical and i n t e r a m b u l a c r u m 5 (Lovén's system) is wider than the others. Two measurements of thickness were taken here, orte at the posterior margin, the other midway between that and a m b u l a c r u m 5. Scanning electron microscopy: Most sutures between plates are more or less heavily b o u n d with fibers. M a n y histological and uttrastructural studies (eg Moss and Meehart 1967; Travis 1970; Harold 1985) have shown that these

Table 6. Ambital test thicknesses for regular echinoids with ribs.

The center of the interambulacral plate was taken as reference thickness in gin; other thicknesses are expressed as percent changes from the reference dimension. I = interambulacrum; A = ambulacrum; sutures hyphenated, I - I =interradial, I - A =adradial, A - A = perradial Species

I

I-I

!-A

A

A-A

Echinus gracilis Strongylocentrotus droebachiensis Echinometralucunter Tripneustes edulis

900 950

+ 22 +16

+ 39 +47

+ 11 + 5

+ 61 +58

1,150 750

-- 4 + 13

+ 9 + 27

+ 4 + 7

+26 + 47

fibers are of collagen. Radial sutures are generally b o u n d by more fibers than circumferential ones. Prominent, wide sutures are tied by greater a m o u n t s of collagen than closefitting, inconspicuous sutures, and, in places where fusion of trabeculae has occurred, coUagen can be totally absent. Cidaroids, such as E. tribuloides (Fig. 2 A, B) have massive, dense stereom, and relatively prominent sutures with collagen fibers equally distributed throughout the thickness of radial and circumferential sutures. In contrast, the perradial and interradial sutures of D. antillarum (Fig. 2 C - F ) differ. Interradial sutures (Fig. 2C) show no collagen at the coelomic surface but it is present within the suture. The perradial sutures (Fig. 2D) are not as close-fitting and are b o u n d with more numerous collagen fibers. As in the interambulacra, circumferential ambulacral sutures (Fig. 2 E) have less collagen than radial sutures (Fig. 2F). The phymosomatoid A. punctulata (Fig. 3 A, B) has close-fitting joints with moderate amounts of collagen, especially in radial sutures (Fig. 3 B). A m o n g the Echinoida, P. miliaris (Fig. 3 C, D) likewise has more numerous collagen fibers in radial sutures (Fig. 3D). The same distribution of collagen was seen in E. gracilis, S. droebachiensis and E. lucunter. However, in the latter species, which has a relatively thicker test, there

Table 7. Ambital test thicknesses for irregular echinoids. Thickness at anterior ambulacrum given in gm, other measurements as percent

differences from the reference thickness. A = ambulacrum, I = interambulacrum Species

Anterior

I

A

I

A

1

Posterior

Meomaventricosa Apatopygusrecens Echinoneus cyclostomus Clypeaster rosaceus

2900 600 800 6800

+ 3 + 8 0 - 18

-3 +8 0 - 3

- 7 +17 + 6 - 15

-17 + 8 + 13 + 3

--31 + 4 + 25 - 20

+ 3 + 31 - 25

119

Fig. 2. A Eucidaris tribuloides: circumferential (compression) suture near the ambitus. B E. tribuloides: radial (tension) suture. Note dense stereom and equal amounts of fibers binding both sutures. C-F Diadema antillarum. C Junction between tension (transverse) and compression (vertical) sutures in interambulacral region. D Compression joint between ambulacral plates, bound by collagen fibers. E Compression joint between ambulacral plates broken open to show collagen fibers throughout thickness of joint. F Tension suture between ambulacral plates broken open: note greater numbers of collagen fibers. (All scale bars 50 gm)

is much less collagen and in the circumferential ambulacral sutures it is almost absent. Several species, such as P. miliaris (Fig. 3 E, F) have greatly thickened trabeculae at the coelomic (Fig. 3 E) and external surfaces of the plates. Quite fortuitously, the preparation used hefe (Fig. 3F) clearly shows the polycrystalline structure. Similar trabecular thickening was seen in E. gracilis and, on the outer surface only, in the spatangoid P. cristatus. The cassiduloid, A. recens (Fig. 4A) has very tightly interlocking sutures with collagen reduced to a few strands at the sutural junctions. The sutures are reinforced at intervals by fusion o f surface Echinoneus cyclostomus (Holectypoida) trabeculae.

(Fig. 4 B, C) has more extensive fusion of trabeculae across the sutures than any other echinoid seen in this study. In the interambulacra, the radial sutures (Fig. 4B) are bordered by highly modified, dense stereom, with numerous points o f fusion. In the ambulacra (Fig. 4C), the sutures are largely obscured by the extent of fusion, especially between adjacent ambulacral plates. Considerable variation was found within the spatangoids (Fig. 4 D - F ) . The stereom of P. cristatus is a delicate looking, open meshwork and the sutures (Fig. 4D, E) are all tied with collagen throughout their thickness. In M . ventricosa (not illustrated) the plates are very thick, the stereom dense and the sutures

120

Fig. 3. A Arbacia punctulata." Tightly fitting trabeculae and sparse collagen fibers along compression suture between interambulacral plates. B A. punctulata: Collagen fibers in tension suture between interambulacral plates. C-F Psammechinus miliaris. C Collagen fibers, compression suture in interambulacral region (specimen slightly etched in formalin). D Mat of collagen fibers binding tension suture. E Thickened trabeculae at coelomic surface. F Polycrystalline structure of trabeculae. (All scale bars 50 ~tm)

strongly reinforced by collagen fibers. The tiny spatangoid, A. excentricus (Fig. 4F) has very thin, tightly locked plates with sparse, delicate fibrils of collagen, not visible from the coelomic surface. Most clypeasteroids have extensive collagen binding in the sutures but this has been dealt with separately, elsewhere (Harold 1985; Telford 1984). D. Discussion

Regression analyses of height and diameter for several speeies of regular echinoids (Table 1) provided slopes > 1.00. Analysis of covariance showed no significant differences

in slopes for the individual species, but highly significant differences in intercept values. Echinometra lucunter, unequal in length and width, was initially ommitted from the analysis of covariance because use of the mean diameter could alter the variance structure of the data, reducing its comparability with the other species. However, when E. lueunter was included in a separate analysis of covariance, no difference in slope was found ( P > 0 . 0 5 ) but there was a difference among intercepts (P<0.01). With this one slight reservation about the data for E. lucunter, I have included the species in this discussion. The data for all species show that height increases relatively faster than diame-

121

Fig. 4. A Apatopygus recens: Junction of compression (left) and tension suture (top to bottom). Collagen is confined to these points of convergence. B, C Eehinoneus cyclostomus. B Greatly modified stereom bordering radial (tension) suture in interambulacral region. C Adradial and ambulacral sutures, partially obscured by fusion of trabeculae. Large apertures are podial pores. D, E Paleopneustes cristatus. D Interambulacral tension suture swathed in collagen fibers. E Interambulacral compression suture. F Agassizia excentricua: Close fitting sutures with some trabecular fusion, between interambulacral and ambulacral plates. (All scale bars 50 ~tm)

ter (slopes > 1.00) and that the rate of increase is essentially the same in all species (no difference in slopes). The significantly different intercepts indicate that, at equal diameters, the h/d ratios for these species are different. The final h/d ratio depends on the adult diameter. With a single exception (A. magnifica), the h/d ratio in all of the species examined (Table 2) was greater than 0.5, which is the ratio for a hemisphere. The ner result o f growth in all of these species is that the ratio height/diameter increases with size: the dornes become relatively taller as the urchins get bigger. Amongst those for which adequate allometric data are available, Stylocidaris affinis has the highest h/d ratio and

S. droebachiensis the lowest. Circumferential tensile forces in domes, as noted in the introduction, are inversely proportional to the h/d ratio. Thus, these forces are effectively reduced or minimized by this growth pattern. Differences in h/d ratios for regular echinoids appear to be related to other structural features and probably also to life style. The thickness ratio o f engineered structures (R/t) is provided by the radius of curvature/minimum thickness (Heyman 1977). Structures in which R/t > 20 are called thin shells. All of the echinoids considered here qualify as thin shell structures (Table 5). Other things being equal, the load bearing capacity of a dome increases with thick-

122 ness. Stylocidaris affinis has the lowest R/t ratio and the thickness increases uniformly from near the apex to the edge of the peristome. It has already been shown that expected stresses are greatest towards the margin of a dome (or the ambitus of an urchin). The thickening of S. affinis towards the ambitus and mouth increases its strength. Burkhardt et al. (1983), Gordon (1978) and many others, have remarked that in engineering, elegant and cost-efficient design uses the minimum amount of material possible, concentrated where the stresses are highest. Echinoids are no exception to this generality. Apart from the cidaroid already discussed, in all of the species examined, thickness increases towards the ambital region by 50-150%, and usually decreases again toward the peristome (Table 5). In general, the higher the R/t ratio (ie the thinner the shell), the greater the percent thickening toward the ambitus and the higher and h/d ratio. For example, the rather thin-shelled T. edulis with R/t=36 is thicker at the ambitus by 134% and is relatively high domed, with h/d= 0.62. In constrast, S. droebachiensis, which is relatively low domed (h/d= 0.53), has a thicker shell (R/t = 29) and a smaller percent thickening at the ambitus (47%). These three features, then, are clearly inter-related: shape of dome, amount of building material and its precise distribution. An alternative model might be to consider echinoids as geodesic structures rather than as simple domes. However, geodesics are structures held together by tensile forces (Kenner 1976). Stresses are carried longitudinally in the components of a space frame, skirting around the surface planes. In an echinoid, stresses are carried throughout the surface material and transmitted across the suture lines. Again, in a geodesic stresses are reacted by the entire structure whereas echinoids react forces locally and require local differences in the quantity of skeletal material. At present it seems that the simple dome model is most suitable. Thus far, only the membrane characteristics of urchins have been considered. Dornes are sometimes reinforced with radial ribs and, as noted by Lin and Stotesbury (1981), if the ribs are deep enough they will resist bending. In such dornes it is not necessary to apply hoop components nor to assume pure shell action: the design is exactly analogous to an arch (Lin and Stotesbury 1981). Plate edges along radially oriented sutures are thickened in many echinoids, and form more or less distinct ribs tapering into the upper reaches of the dome. Such ribs have a two-fold function. First, they provide increased flexural stiffness (ie stiffness of the dome structure itself) which means that the ribs will carry a greater proportion of the stress (Lin and Stotesbury 1981). For this reason, the intervening parts of the dome can be correspondingly thinner. Secondly, the greater thickness reinforces the radial sutures, strengthening them in resistance to tensile forces, by increasing the area of contact between plates. Circumferential tensile forces within the shell are themselves decreased by the action of the ribs. The ribs within a dome can be treated as arches (Heyman 1977) carrying stress primarily in compression, but subject also to bending moments. For purposes of calculation, an arch can be replaced by an imaginary beam and the bending moments determined along its length (Mukhin et al. 1983). Increasing the thickness of a beam decreases surface stress during bending (Gordon 1968) and thus increases the load which it can sustain. For example, increasing beam thickness by 40% approximately doubles its strength. Taking the center of the interambulacral plates at the ambitus as

the reference thickness (this is usually the thinnest point around the ambitus) it is apparent (Table 6) that almost all radially oriented sutures are thickened, most notably in the ambulacral region where the test is otherwise weakened by the presence of podial pores. By analogy with beams, it may be estimated that these ambulacral ribs locally increase load carrying capacity by 60-160%, depending on species. Another feature of echinoids is the continuity of the sides of the dome and oral surface, passing around the curve of the ambitus. Analysis of the forces transmitted to the oral surface is difficult. By comparison with an annular plate (here representing the oral surface of an echinoid) loaded around the outer edge (Ugural 1981), rotational forces which lift and, perhaps, bend the oral surface can be inferred. Thickening of the test around the ambitus diminishes the effect of such forces. In regular echinoids, rotational forces are probably not large, but in such specialised animals as E. pusillus they may be very significant (Telford 1985). The sutures themselves are not pure masonry structures, they are lashed together by varying amounts of collagen fibers (see SEM micrographs). It is well known that structures cannot be perfectly rigid (Gordon 1978). Applied loads are reacted by forces generated in deformations of the structural components. In the domes of urchins loaded at the apex, tensile circumferential forces especially are presumed to be accommodated by slight spreading of the radial sutures. Compression applied in a diameter through the ambitus, ie not near the apex, should put circumferential sutures into tension. The role of collagen fibers within the sutures is to react these forces. Although no data are available on the tensile strength of echinoderm calcite, it is thought, like masonry, to be stronger in compression than in tension. The radial sutures are most frequenctly subjected to tensile forces and these are most heavily bound by collagen (Fig. 2E, F; 3A, B, E, F; 4D, E). The amount of collagen within the suture might be indicated by collagen at the coelomic surface, as in E. tribuloides (Fig. 2 A) but often it is inconspicuous at the surface, as in the ambulacral sutures of D. antillarum (Fig. 2C). In this species there are pronounced differences in sutural collagen, depending on location. The interambulacral circumferential sutures have very little collagen and it remains sparse in the radial sutures. The ambulacral sutures are less tightly interlocking (Fig. 2C, D) and are generously supplied with collagen fibers (Fig. 2E, F). The ambulacra of the diadematoids are bowed outwards giving the dome a broadly corrugate surface. Both the magnitudes and directions of stresses in the ambulacra and interambulacra would differ for this reason, but its precise analysis would be very complex. In other echinoids the sutures become more closely interlocking and more rigid, as, for example, in A. punctulata (Fig. 3E, F) and many irregular forms (Fig. 4a-f). As Smith (1984) noted, rigid sutures allow ready transfer of stress. This is particularly true of compressive stress which will ultimately be transferred to the substrate. The circumferential sutures need to fit closely, without slippage, like the voussoirs of an arch. The increasing thickness of the plates ensures that strain remains within acceptable limits for the material. Smith (1980) has described the micro-architecture of sutures, especially the interlocking pegs and spikes of some clypeasteroids. Elsewhere, I have described the remarkable sutures of Echinocyamus, which are devoid of collagen and reinforced by interlocking knobs and hooks, as well as fused

123 trabeculae (Telford 1985). Trabecular fusion across sutures is most strongly developed on the coelomic surface of the holectypoid, E. cyclostomus (Fig. 4 B, C). The internal sutute structure includes knobs, hooks and sparse collagen fibers. Similarly in the cassiduloid, A. recens (Fig. 4A) there is extensive fusion of trabeculae and collagen is restricted to points at which sutures converge. These urchins are rigid but extremely brittle. The design is economical of materials, compressive stress is transferred very efficiently, but tensile stress, which is accommodated by deformations of the calcite, must be minimised. Spatangoids also have tight-fitting sutures. In P. eristatus (Fig. 4D, E), the sutures are well supplied with collagen and in M. ventricosa, where the plates are relatively thick, there is also generous collagen binding, especially in the radial sutures. In A. exeentricus (Fig. 4F), sutures are very close fitting, rigid, and have sparse, delicate fibrils of collagen. The significance of general body shape in carrying stress in the various groups of irregular echinoids is not easy to assess. Agassizia excentrieus is almost circular around the ambitus, length and width are approximately equal, and the species could be regarded as a regular domed structure. The test is thin, rigid and brittle, the h/d ratio is high (Table 4). Many irregular echinoids have oval or elliptical outlines, longer in the anterior-posterior axis than in width. Analysis of length-width data is provided in Table 8. For these four species at least, the general form cannot be taken as a circular dome. Engineers frequently build elongated domes (using an orthogonal translation approach in which the curved arches forming the dome are matched to the required outline shape in plan) (Lin and Stotesbury 1981). As might be expected, calculation of forces in these dornes is complex and would be even more difficult in spatangoids with sunken ambulacra, mouth opening, anus and other irregularities. However, the gross distribution of forces can be approximated as inversely proportional to the square of the spans. An increasingly large proportion of the total load will be sustained in the width, as the difference between length and width increases. For example, at length 4.5 and width 4.0, only 44% of the total load will be carried in the longitudinal axis; at 5.0 and 4.0, it will be 39% and so forth. This relationship has been used to estimate the proportion of total load which might be carried in the longitudinal axis (Table 8). If the load is distributed in this way, the greater part carried across the width, one might expect differences in structure along the two axes. Ambital thicknesses of individual specimens, expressed as percent of anterior thickness, are given in Table 7. Only A. reeens shows any thickening in the transverse axis which might help to sustain a greater share of the load. Meoma ventrieosa is especially thick in the anterior region and around the periproctal area. It is worth noting that this species is also thicker in the roof of the dome, between the sunken ambulacra, than around the ambitus. This spatangoid is buried by day and emerges by night. During the day it must support a sediment load at least equivalent to its own weight, possibly greater. During the process of burrowing, whilst the animal is ploughing forward, considerable force would be acting in the longitudinal axis. Thickening of the test in C. rosaceus appears to be far greater than necessary to sustain simple loads or to react common external forces. This massive structure, which can support a man's weight, must be required for other reasons, such as resistance to predators, especially boring gastropods. And, of course, the

Table 8. Regression analyses of log-transformed data of length (y) and width (x) for four species of irregular echinoids and estimation of the proportion of total load carried in the longitudinal axis of the smallest and largest individuals in the regression series Species

a

b

r2

Meoma ventricosa Clypeaster rosaeeus Apatopygus reeens Echinoneus eyclostomus

1.93 1.10 1.25 1.27

0.89 1.03 0.96 1.00

0.99 1.00 0.99 1.00

%load : %load : small large 39 41 43 38

45 38 46 38

generally protective role of the test for all echinoids must not be overlooked. The foregoing analysis (which should be considered a first approximation) shows that in general body form, suture construction and details of structural reinforcement, echinoids are built to accommodate a variety of compressive (radial) and tensile (circumferential) stresses, by various combinations of arch and membrane action. It has been shown from arch and membrane theory that the magnitude of tensile forces depends partly on geometry and partly on loading. Self-weight is always part of total load but, clearly, echinoids are built to sustain rauch greater loads. The source of these loads is far from obvious. For the fibulariid, E. pusiIlus, I have suggested that crushing between rolling stones is probably a common occurrence (Telford 1985). All echinoids are subjected from time to time to what Smith (1984) calls impact loading, especially in hydrodynamically active environments. Strathmann (1981) has suggested that one role of echinoid spines is, in fact, to provide protection from impact loads, and this seems to be confirmed in experiments with E. pusillus (Telford 1985). The greater flexural stiffness provided by clypeasteroid supports (Telford 1985) is of obvious significance for this group which occupies highly active environments. By living in crevices, many echinoids avoid being dislodged by waves and currents. For example, E. tribuloides in coral reefs and S. droebachiensis in rocky outcrops, wedge into confined spaces, bracing themselves with the spines. This places relatively large loads on the test. Eehinometra lucunter often occurs on wave-swept rocks, migrating up and down individual channels into which they are braced by the spines. Indeed, the channels may themselves be scoured and enlarged by the constant abrasion of passing spines. Yet another example is the Indo-Pacific Colobocentrotus atratus, which receives constant battering from breaking waves. Dafni (1984) has suggested that muscular forces from the lantern, podia and mesenteries are also significant and might actually influence growth and shape. All such forces, constant, intermittent or occasional, must be successfully reacted if the urchin is to survive. Acknowledgements. This work has been supported by the Natural

Scienees and Engineering Research Council of Canada through Operating Grant # A4696. I am indebted to R.A. Collins, Department of Engineering, University of Toronto, for mueh valuable criticism and to E. Lin, Department of Zoology, for technical assistance in SEM. My colleagues, Rich Mooi and Tony Harold provided innumerable useful suggestions and constructive criticisms.

124

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Received November 26, 1984