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Sutural Pattern and Shell Stress in Baculites with Implications for Other Cephalopod ShellMorphologiesAuthor(s): David K. JacobsSource: Paleobiology, Vol. 16, No. 3 (Summer, 1990), pp. 336-348Published by: Paleontological SocietyStable URL: http://www.jstor.org/stable/2400792Accessed: 23/02/2010 16:21

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Paleobiology, 16(3), 1990, pp. 336-348

Sutural pattern and shell stress in Baculites with implications for other cephalopod shell morphologies

David K. Jacobs

Abstract.-In Baculites, a straight shelled ammonite, the constructional limits on shell shape resulting from the limited strength of nacre in tension are circumvented by a system of vaults in the phrag- mocone. Vaults bridge between regions of the phragmocone supported by the complex ammonite septal suture, and maintain the shell wall in compression when hydrostatic load induces bending moments. To determine how these vaults interact in the phragmocone to resist hydrostatic loading, measurements were made on a suite of Baculites specimens. In Baculites there is a statistically sig- nificant inverse relationship between circumferential curvature (radius of curvature) of the vaulted shell surface and the size of vaults spanning between sutural elements supporting the phragmocone. The inverse relationship between radius of curvature and the sizes of spans in this system of vaults results in the generation of comparable reactive forces at the ends of the vault spans where adjacent vaults interact. The equivalence of these reactive forces prevents the lateral displacement of the vault ends. Consequently, compressive stresses from adjacent vaults are superimposed on, and reduce, the tensional stress component of bending. Limiting tensile stress is of utmost importance in a lightweight shell composed of a brittle material such as nacre, which is strong in compression but weak in tension.

Baculites is particularly appropriate for this study because its straighlt shell is curved only in the circumferential direction, thus simplifying the problem. However, sutural patterns in coiled am- monites appear to be similarly constrained to produce vaults in the phragmocone which vary inversely in curvature and span size.

David K. Jacobs. Department of Geological Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0796. Current address: Museum of Paleontology, University of California, Berkeley, California 94720

Accepted: March 21, 1990

Introduction

The structural implications of sutural com- plexity in ammonites have long been a subject of scientific interest. Buckland (1836) argued that the complex septal sutures of ammonites served to support the flattened flanks of am- monite phragmocone against hydrostatic pressure.This idea was also advocated by Pfaff (1911), Spath (1919), and Westermann (1956). However, conclusive evidence of the pres- sure difference across the shell wall of ceph- alopods was not available to these authors (for a historical review of functional analyses of shelled cephalopods see Jacobs 1990). Fi- nally, Denton and Gilpin-Brown (1961, 1966) and Denton et al. (1967) unequivocally dem- onstrated that modern shelled cephalopods contain gas at or below atmospheric pressure while submerged. Consequently, modern shelled cephalopods and, by analogy, fossil shelled cephalopods support the entire hy- drostatic load of the overlying water column

? 1990 The Paleontological Society. All rights reserved.

in the structure of the shell itself. This un- derstanding of the hydrostatic load borne on the shell led to the application of membrane stress formulae for hollow thin-walled spheres and cylinders to the concave portions of nautiloid septa and the tubular siphuncle (e.g., Westermann 1973, 1977, 1982). In these thin-walled structures, hydrostatic load pro- duces a purely tensional or compressional stress, termed a "membrane stress." Mem- brane stresses are proportional to the radius of curvature and the pressure difference across the concave nautiloid septum and tubular si- phuncle; they are inversely proportional to the septal or siphuncular thickness.

The distribution and magnitude of the stresses in the ammonite phragmocone are less easily assessed than the stresses in con- cave septa or tubular siphuncles. There are two reasons for this. First, ammonite phrag- mocones do not conform to simple shells of revolution for which equations of membrane

0094-8373/90/ 1603-0006/$1.00

SUTURAL PATTERN AND SHELL WALL STRESS 337

stress are readily derived. The radius of cur- vature around the whorl in the ammonite phragmocone is not constant; consequently, one would expect differential stress genera- tion in different regions of the phragmocone (Westermann 1971). Secondly, the phrag- mocone is supported at intervals by the septa. This support results in a flexural or bending problem as the shell wall, bearing hydrostatic load, passes over and between the supports provided by the septal suture.

Hewitt and Westermann (1986) attempted to model the bending stresses in spaces be- tween sutural support in the relatively flat flanks of a Calliphylloceras shell by analogy to a flat plate. However, even the flanks of oxy- conic ammonites have some curvature; they are rarely, if ever, exactly flat. Due to their curvature, portions of the phragmocone bridging between sutural elements form a se- ries of vaults rather than flat plates. In a tra- verse around the non-circular whorl section of a compressed ammonite morph, the radius of curvature necessarily changes (Fig. 1). In addition, spacing of sutural support of the shell wall is not uniform. The complexity of the suture and coiling geometry results in variation in the size of spans of shell wall between sutural elements in the sutural pat- tern. This variation in sutural spacing appears to relate to the local curvature of the shell around the whorl. The larger radius of cur- vature, "flatter," portions of the whorl have more closely spaced sutural support than the more tightly curved (smaller radius of cur- vature) portions of the shell (Buckland 1836; Spath 1919; Westermann 1956, 1971, 1975). Due to the complex interdigitation of adja- cent ammonoid sutures, septal support sur- rounds the vault spaces in the sutural pattern. A greater proportion of the load on a doubly curved vault surface will be transferred in the direction of the smaller radius of curvature. The smaller radius of curvature necessarily occurs around the whorl rather than around the coil of a planispiral shell. Consequently, vaults in the phragmocone will bear more of the load around the whorl rather than around the coil, and it is this circumferential direc- tion around the whorl that is of interest.

In this work, the relationship between shell

RADIUS OF CURVATURE

Ri-Flank Radius

R2-Dorsal Radius

R3-Ventral Radius

R2

Ri

R3

FIGURE 1. The thick curve represents a cross-section through a Baculites shell. Three radii of curvature de- scribe the major regions of curvature around the circum- ference of a Baculites shell. The radius of curvature of the broad flanks of the shell, Rl, is many times greater than that of the intermediate radius of curvature of the dorsal shell surface, R2, which is in turn substantially greater than the radius of curvature of the ventral region, R3.

curvature and the spacing of support is ex- amined to determine if it relates to the func- tion of the ammonite shell as a series of vaults supporting hydrostatic load. This will address the long standing question of the function of the ammonite suture and how the ammonite suture may be functionally constrained.

The Phragmocone Conceived of as a Series of Vaults. -Vaults combine the axial or mem- brane stresses, associated with thin-walled pressurized spheres and cylinders, with the bending stresses normally associated with loaded beams or flat plates (Salvadori 1971). Unlike plates, vaults produce outward thrust- ing at their ends when loaded. This outward thrusting is a consequence of bending mo-

338 DAVID K. JACOBS

INTERACTION OF THRUSTS IN A SERIES OF VAULTS

c

A -t B TA TB

TA' TB'

S S

FIGURE 2. In a series of uniformly loaded vaults which meet at supports, SA and SB, the outward thrusts of vault AB, TA and TB must be met by thrusts of similar magnitude, TA' and TB', generated by adjacent vaults if vault AB is to remain in compression. If TA' and TB' are not as large as TA and TB then the vault ends will displace outward and tensile stress will develop on the interior of the vault at C.

ments in the vault. Unless this outward thrusting is met by an opposing reaction force, bending moments in the vault will result in a large tensile stress component on the in- terior of the vault (Fig. 2). Ammonite shells are composed largely of nacre, a brittle ma- terial; nacre from Nautilus is reported to be two to three times stronger in compression than tension (Currey and Taylor 1974; Currey 1976). In addition, cracks or imperfections in the material will limit tensile strength to a greater extent than it will limit compressive strength (Hewitt and Westermann 1986). As a consequence of these properties of nacre, tensile stress due to bending must be limited to a small fraction of compressive stress in an optimally designed cephalopod shell.

One strategy for reducing the tensile com- ponent of bending in a beam or vault is to superimpose a compressive axial stress on the whole structure. In concrete beams this is done by application of a stress prior to loading in what is termed "prestressing." In vaults, a compressive stress is superimposed on a bending stress with the same result; tensile bending stresses are avoided in a brittle ma- terial weak in tension, but strong in com- pression. The superimposed compressive stress is supplied by the outward thrusting of the vault itself. To generate a compressive stress effective in opposing tension due to bending, the outward thrusting of the vault must be met by an equal and opposite force. The problem can also be expressed in terms of strain or displacement in the vault; if out- ward displacement of the vault ends is pre-

vented by equivalent reaction forces, then no dilation of the material on the interior of the vault can occur. If there is no dilation then there must be no tensional stress.

In a Roman aqueduct each vault has an ad- jacent vault available to supply an opposing force maintaining the series of vaults in com- pression. Similarly, around the whorl of the ammonite phragmocone there are adjacent spaces in the sutural pattern. These spaces should function as vault structures in the phragmocone providing reactive forces to the vault ends in a manner similar to adjacent vaults in an aqueduct (Fig. 2).

Hydrostatic pressure uniformly loads the vaults comprising the ammonite phragmo- cone. These vaults vary in radius of curvature and in span length. Theoretically, thickness of the shell wall (vault thickness) could also vary. Variation in vault thickness would have consequences for the stresses resulting from the reaction forces and bending moments in the vault. However, variation in shell thick- ness around the ammonite whorl is not large (Westermann 1971). In this study, no system- atic variation around the whorl of Baculites was observed. A series of measurements at various points around the whorl of two Bac- ulites specimens reveals that the minor vari- ation in thickness observed was not associ- ated with a particular region of the shell (Table 1).

In contrast to the minimal variation in shell thickness and hydrostatic loading, there is considerable variation in span size and radius of curvature in the series of vaults comprising

SUTURAL PATTERN AND SHELL WALL STRESS 339

the ammonite phragmocone. If this system of vaults is to perform optimally when loaded, span size and radius of curvature must covary in a manner that produces equivalent reac- tion forces at the vault ends. Equivalence of reaction forces would limit the tensional stresses due to bending so that the brittle na- creous material could be used to advantage.

Testing the Interactive Vault Hypothesis. -The vaults comprising the ammonite phragmo- cone are complex in shape, and accurate cal- culations of reaction forces cannot be readily obtained by conventional engineering pro- cedures. Consequently, an expected general relationship characteristic of vaults is used to test the interactive vault hypothesis. Engi- neering equations can be used to calculate thrusts at the ends of vaults of uniform width that lack lateral support (Roark and Young 1975); no such equations exist for vaults of the complex shape defined by the spaces be- tween the sutural elements of the ammonite phragmocone. Perhaps more importantly, en- gineering equations assume precise end con- ditions; vaults are either pinned, that is, free to rotate in the vertical direction, or fixed, where no end rotation is possible. Because the septum is thin and curved where it meets the shell, sutural support of the phragmocone is likely to be strongly elastic or spring-like (Hewitt and Westermann 1986). In addition, the vaults end in regions of more closely spaced sutural elements. The combination of proximity of sutural support at the vault ends, and the elasticity of the septa allows septal support of the shell to be diffused over a re- gion of the shell wall. These regions of su- tural support occur between, and determine the end conditions of, the vault spans (Fig. 3). The diffuse support between vaults is nec- essary to limit bending stresses over the sup- port. A single point of support would con- centrate stress and result in bending in the shell wall as it passed over the localized sup- port, resulting in tension on the exterior of the shell. Such bending is observed over the simple strut-like septal support of the flank of Nautilus (Hewitt and Westermann 1987b). Diffuse support limits bending and the pro- duction of tensile stress in the nacreous shell wall. Although structurally advantageous, the

TABLE 1. Measurements of shell thickness around the whorl of two Baculites specimens.

Specimen K L Species B. sp. B. sp. Repository Peabody Mus. AMNH Specimen No. A361 Am 29545 Whorl height 3.4 cm 5.6 cm Whorl width 1.5 cm 3.7 cm

Data (shell thickness mm) K L

Dor- Flank Flank Dor- Venter sum 1 2 Venter sum Flank

0.36 0.36 0.38 0.36 1.30 1.33 1.33 0.34 0.35 0.33 0.36 1.28 1.31 1.40 0.35 0.36 0.34 1.39 1.41 1.51

0.39 0.37 1.54 1.48

diffuse end conditions provided by the com- plex ammonite sutural pattern make it diffi- cult to determine a precise location where the vaults end, much less how to characterize them in engineering terms. Due to the com- plex shape of the vaults comprising the Bac- ulites phragmocone, and the difficulties in as- sessing the exact nature of the end conditions of these vaults, direct calculation of the re- action forces at the vault ends is not under- taken here.

On a more general level, vaults combine the structural attributes of cylindrical shells, where the radius of curvature is the critical variable, with a bending problem where the length of the span is critical (Salvadori 1971). This is illustrated by an examination of the simplest vault equation, that of a statically determined vault or arch of unit width; this equation suggests that an inverse relation- ship between vault size and radius of cur- vature would be expected if reaction forces are to be equivalent at the vault ends. The outward horizontal thrusts at the supports at the ends of statically determined arches or vaults of unit width can be readily calculated (Salvadori 1971):

T = M/h = 0.5w12/h (1)

The horizontal thrust T at the vault ends is a function of the bending moment M, which is in turn a function of the load w and the moment arm 1 (Fig. 4). The bending moment at the vault end will be given by the formula

340 DAVID K. JACOBS

SUTURAL PATTERN

DORSAL

CROSS NK , INTERSUTURAL

VENTRAL

FIGURE 3. The sizes of the vaults spanning between septal sutures in the shell wall can be approximated by the sizes of circles inscribed between the sutures in the complex sutural pattern. Note that the largest circles are in the ventral region, the region of the shell with the smallest radius of curvature, and small circles are present in the large radius of curvature flanks of the shell. Circles of intermediate size fit within the sutural pattern in the dorsal region of the shell of intermediate curvature.

M = 0.5w12. This moment, when divided by the maximum height or rise of the arch, h, above the horizontal span, provides the for- mula for the horizontal thrust T generated at the end of the arch. Note that the length term is squared in this expression as a consequence of uniform load along the length of the beam; the load itself as well as the moment arm are functions of beam length. Hydrostatic load will also act horizontally on the height of the vault, but this term is a negligible contribu- tion to the outward thrust of the vault. From this reaction equation it is apparent that if T, the horizontal force at the ends of the vault, is to remain constant, the square of the half- span length, 12, will have to vary in concert with the rise of the vault, h. The height h varies inversely with radius of curvature (if length of the vault is held constant then the height of the vault depends on its radius of curvature alone); therefore, 12 is expected to vary inversely with the radius of curvature R. The static vault equation above applies only to vaults which are pinned and free to rotate at their ends and midpoint. The radius of cur- vature and bending terms related to span length are also critical in calculating the out- ward thrusts produced by indeterminant vaults, vaults that are fixed at their ends (Roark and Young 1975). Thus regardless of the de-

tails of the type of vault, a general expectation of an inverse relationship between a squared span length term and radius of curvature is predicted if the thrusts at the ends of the vaults are to be comparable in magnitude.

Methods

Measurements. -An uncoiled ammonite, Baculites, was chosen because the straight shell provides a variety of curvatures around the whorl without the complexity associated with curvature normal to the whorl section that results from coiling. Three criteria were used to select specimens for measurement. Only specimens lacking lateral ornamentation in the form of ribs or bullae were used, since such ornament would complicate measure- ment and analysis. Only specimens that showed no obvious asymmetry or deforma- tion were selected. Lastly, only specimens with at least three chambers exposed were used. This allowed measurement of the su- tural pattern at a sufficient number of loca- tions for a statistical test. For each of the ex- posed chambers, measurements of radius of curvature and span length were made for the three regions of uniform curvature (venter, flank, and dorsal) in ten specimens of Bacu- lites.

Measurements of radius of curvature were

SUTURAL PATTERN AND SHELL WALL STRESS 341

Vault Equation

~ w

T

R ~~~T= Al21 R ~~TW2h

FIGURE 4. The thrust T at the ends of a uniformly loaded vault is proportional to the load per unit length, w, multiplied by the square of the halfspan length 1. This thrust is inversely related to the height or rise of the vault, h; the rise of the vault is inversely related to the radius of curvature, R. It follows that the thrust at vault ends will be proportional to 12 and R. If the thrust produced by the vaults adjacent regions of the shell are to be equal, 12 must vary inversely with R.

made by molding the specimen with mod- eling clay and cutting away strips of the clay. The strips of clay were then placed on paper and the curvature of the mold drafted on the paper. The radius of this curvature was then determined using a compass. This procedure was repeated for the three distinct curvatures of the shell, the ventral, flank, and dorsal curvature for every chamber measured in the ten specimens (Fig. 1).

To obtain estimates of the vault span lengths between sutural elements supporting the shell, the sutural pattern was drafted on my- lar; a compass was then used to find the larg- est circle which would just fit within the spaces in the sutural pattern (Fig. 3). These circles provide an approximation of the spacing of sutural support in the circumferential or hoop direction around the Baculites. Although lon- ger transects that do not touch sutures may be found in the circumferential direction, these transects pass through regions where the shell is closely supported by adjacent su- tures. This use of the largest inscribed circle provides an arbitrary but consistent means of estimating the size of vaults in the circum- ferential direction given the complex and variable interdigitations of the suture pattern

encountered in a transect around the shell. The length of the radius of each of these cir- cles was recorded for the ventral, flank, and dorsal curvatures characteristic of the shell. These measurements were made for each chamber where the sutural line was ade- quately exposed on the ten specimens. The radius of the largest circle inscribed in the sutural pattern consistently provides a rough measure of the effective halfspan length in the circumferential direction. The largest vaults in a region of constant shell curvature are the vaults of interest since they will pro- vide the largest outward thrusts which must be met by thrust of vaults in adjacent regions of the shell. These vaults are separated by one or several closely spaced septal elements. These regions where one or a number of sep- tal elements meet the shell in the septal su- ture support the ends of, and provide the end conditions for, adjacent vaults (Fig. 3).

The Statistical Test. -The general expecta- tion of an inverse relationship between vault span size between sutures, 12, and the radius of curvature, R, of the shell was tested using regression analyses. Because an exponential rather than a linear function was expected, a natural logarithm transformation of both

342 DAVID K. JACOBS

variables was employed. The exponential re- lationship in question was then tested by the slope and significance of the regression line produced by the two log-transformed vari- ables. A slope of -1 would indicate a simple inverse relationship. Least squares regres- sions were performed and subsequently con- verted to reduced major axis regressions through division of the regression slope by the correlation coefficient, as suggested by LaBarbera (1986). The reduced major axis form of regression was preferred here because nei- ther variable was "independent." Regres- sions were performed for each of the ten spec- imens measured (Table 2).

Results

The regressions produced highly signifi- cant negative slopes for all ten specimens, confirming the inverse relationship between vault size and radius of curvature. The re- duced major axis slopes ranged from -0.865 to -1.270, with an average slope of -0.996, trivially different from -1. All regression slopes were significant at the P < 0.0005 level, confirming an excellent fit of the regression lines to the data. These results indicate that in a series of vaults determined by the suture pattern of Baculites, vault area and radius of curvature covary in an inverse manner. The conformity to a slope of negative one suggests that a simple inverse or hollow curve type of relationship exists between the two variables. This simple inverse relationship indicates that the series of spans between sutural elements that comprise the ammonite shell, despite the complexities of their shape and end condi- tions, interact in the predicted manner, func- tioning as a series of vaults.

The inverse covariation between halfspan length squared and radius of curvature re- sults in relative equality of the reaction force of the vaults around the phragmocone. This would minimize the tensile forces resulting from the bending inherent to vault action. This minimization of tensile stress is advan- tageous, and would be expected, in a shell composed of brittle material such as nacre.

Discussion

Application of the Model to Other Shell Mor- phologies. -The work presented here was per-

formed on Baculites because it is straight and lacks a secondary curvature associated with coiling. However, theoretical considerations and a number of observations of the spacing of septal elements on coiled shells suggest that the vault model applies to coiled am- monites as well.

A straight phragmocone can be analogized to a thin-walled cylindrical surface in the sense that the circumferential stress around the whorl will be the primary stress; on a cylindrical surface the stress is borne pref- erentially in the circumferential rather than the longitudinal direction. This is not the case in a thin-walled sphere, where all directions of curvature are comparable and the stress generated in all directions is half that borne in the circumferential direction of a cylinder of equal radius of curvature and thickness supporting the same pressure difference across the wall. The cylindrical and spherical shapes represent end points in a continuum of dou- bly curved surfaces. In a cylinder, the radius of curvature in the longitudinal direction is infinite and only the smaller direction of cur- vature need be considered. If the radius of curvature around the coil greatly exceeds that around the whorl, the situation approaches the cylindrical condition and the larger stress will be borne in the direction of smaller cur- vature. We would therefore expect that in those ammonites where the ventral curvature around the whorl is much smaller than around the coil the interactive vault model would apply and the spacing of sutural elements would depend on the radius of curvature around the whorl.

In a transit around the whorl of an oxyconic form such as Placenticeras, the ventral region has a small radius of curvature in the direc- tion around the whorl, whereas the radius of curvature in the direction of coiling is much larger (Fig. 5, A). Assuming that the larger direction of curvature around the coil can be neglected, Placenticeras should be analogous to Baculites, where a venter of small radius of curvature interacts with the broadly curved flanks of the whorl. As in Baculites, vaults in the ventral region supply compressive reac- tion forces to the flanks of the ammonite. One can therefore predict that the sutural pattern in Placenticeras should produce small vault

SUTURAL PATTERN AND SHELL WALL STRESS 343

FIGUREs 5. A, on the broadly curved flanks of Placenticeras, the sutures are closely spaced and therefore the vault spans are short. At the venter, where there is a smaller radius of curvature, the sutural spacing is much wider resulting in larger vault spans. Also note that on the extensive flank region of the compressed involute ammonite an increased number of sutural lobes, marked P, are required to achieve the close sutural spacing. B, in a globose pachydiscid ammonite, curvature around the venter and flanks is relatively constant. Note that the spacing between the elements of the complex suture is also relatively constant.

spans on the flattened flanks of the shell and large spaces between sutures on the venter, similar to those produced by the spacing of sutural elements in Baculites. Examination of a Placenticeras reveals ornate closely spaced sutures on the flanks and widely spaced su- tures on the venter, thus confirming the pre- dictions of the model (Fig. 5, A).

In an ammonite lacking variation in radius of curvature around the shell, minimal vari- ation in sutural spacing would be predicted by the model. On globose ammonites, cur- vatures in the coiling direction and around the whorl are uniform and nearly equal over the exterior portion of the phragmocone (Fig. 5, B). In globose forms such as Stephanoceras or Macrocephalites, sutural spacing varies min- imally over the uniformly curved portions of the phragmocone. In scaphites there are lat- erally compressed forms and globose forms. Laterally compressed forms such as Hoplo- scaphites show close sutural spacing on the flanks whereas globose forms such as some Scaphites show even septal spacing around the phragmocone.

Buckland (1836) and Westermann (1956, 1971, 1975) have similarly commented on the

closer spacing of support on the flanks of am- monites and have related this close spacing of sutural elements to the need for greater support against pressure in the flatter regions of the shell, although not to the vault model of support presented here. These authors ex- plicitly recognized a functional role in the relationship between shell curvature and proximity of sutural support. In addition, oth- er authors have reported the related empirical relationship between shell curvature and the number of sutural elements. Spath (1919) ob- served that increased numbers of sutural el- ements were found on the flanks of phyllo- ceratids and ammonites that had evolved broad curvature of the flank region. He sim- ilarly observed that the number of lobes in- creased in the broadly curved dorsal region of depressed ammonites and that ammonites with round whorl sections had a relatively even distribution of lobes around the whorl. Spath also cited examples of evolutionary re- sponse of the suture line to change in whorl shape within lineages. Checa (1986) and Hewitt and Westermann (1987a) also ob- served the relationship between involute compressed morphologies and the develop-

344 DAVID K. JACOBS

ment of additional lobes in the region of the suture supporting the flank. On the flat flanks of increasingly involute or oxyconic lineages, the sutures extend across the breadth of the flank and converge toward the diminishing umbilicus (Fig. 5, A). The sutures approach each other as they approach the umbilicus much as the radial spokes of a wheel approach each other adjacent to the hub. Due to the elongation toward the umbilicus and the lat- eral compression of the whorl, additional su- tural complexity on preexisting lobes of the suture can not readily control or contribute to sutural proximity in the region of the flank approaching the umbilicus. To control sutur- al spacing in this region in lineages devel- oping this compressed involute oxyconic morphology, new umbilicular lobes evolved.

Workers have tended to concentrate on the addition of sutural elements with increasing radius of shell curvature such as in the broad flanks of oxycones. A concomitant decrease in proximity of sutural elements in those regions of the shell simultaneously under- going a reduction in radius of curvature also occurs in the scaphites and in cardioceratid and macrocephalitid lineages observed by Spath (1919). This suggests that selection on the suture operated to create a series of vaults of comparable reactive force, rather than just adding support on the flanks as curvature changed.

Observations of the spacing of sutural el- ements and their relationship to shell cur- vature support predictions of vault spacing generated by the interactive vault hypothesis. The vault hypothesis predicts sutural spacing and constrains the evolution of greater su- tural complexity to particular regions of the shell in a wide range of coiled as well as straight ammonites. Consequently, the exter- nal shape of the shell functionally and adap- tively controlled the evolution of sutural pat- tern in ammonites.

Functional Constraint. -If the sutural pat- tern in ammonites is functionally constrained by the external shape of the shell, there are a number of implications for the study of am- monite evolution. This functional constraint contributes to the frequent incidence of par- allel evolution involving both the suture line

and the outer shape of the shell in ammonites. Consequently, the structural relationship ex- pressed here should be important in con- structing phylogenies. As previously noted (Westermann 1956, 1971), heavy reliance on the sutural morphology of an ammonite may be poor taxonomic practice in light of the potential for convergent evolution.

The frequency of involute compressed oxy- cones with similar sutural development in the flank regions is an enlightening example. Oxyconic homeomorphs with similar exter- nal and sutural morphology occur at widely spaced intervals throughout the Mesozoic (Spath 1919). Repeated evolution of involute compressed morphologies with similar su- tural patterns also occurs within individual evolutionary lineages (Bayer and McGhee 1984). This repetition of oxyconic morphol- ogy may relate to adaptive influences on shell shape which in turn constrain sutural pat- tern.

Hydrodynamically Efficient Shell Shapes and Sutural Complexity. -Sutural complexity may have evolved in response to selective forces driving the ammonoid morphology away from circular whorl sections. Circular whorls are structurally advantageous; they do not re- quire vault support to avoid bending stresses. Consequently, departure from a circular whorl section would require some alternative or competing selective influence. Selection for hydrodynamically efficient morphology may provide such a selective advantage favoring non-circular whorl sections and resulting in selection for sutural complexity to accom- modate these shapes.

Oxyconic morphologies confer hydrody- namic efficiency (Chamberlain 1976, 1981). Oxyconic shapes require the greatest varia- tion in surface curvature, from the venter to the flank, of any planispiral ammonoid whorl shape. Without complex sutures dividing the flanks of an oxycone into a series of smaller vaults, outward thrusting of the flanks could not be balanced by the vault force produced in the ventral region. Sutural complexity al- lows the attainment of the large variation in shell curvature observed in oxycones.

The argument for the adaptive influence of hydrodynamics on shell shape and its rela-

SUTURAL PATTERN AND SHELL WALL STRESS 345

tionship to sutural pattern is supported by the shapes attained by ammonites and nau- tiloids in the Mesozoic and Cenozoic. During the Mesozoic, ammonoids occupy, and appear to competitively exclude nautilids from, the morphospace of hydrodynamically advanta- geous laterally compressed forms that mini- mize drag (Ward 1980; Chamberlain 1981). In the Tertiary, after the demise of the ammo- nites, nautilids developed a certain amount of sutural complexity in conjunction with laterally compressed morphologies (e.g., Atu- ria). These observations suggest that evolu- tion of laterally compressed hydrodynamical- ly efficient cephalopod shell morphologies is associated with the evolution of sutural com- plexity. The possibility that sutural complex- ity evolved as a consequence of selection for hydrodynamically efficient shell shapes mer- its further investigation.

Limitations of the Interactive Vault Model.- Not all cephalopod shells or even all portions of the ammonite phragmocone are equally constrained by the interactive vault model. Nautilids range from globose forms such as Eutrephoceras with concave septa unlikely to support the phragmocone to laterally com- pressed forms such as Aturia with a relatively complex suture supporting the flanks of the shell. Nautilus has a single septal strut sup- porting the slightly flattened flanks. Hewitt and Westermann (1987b) have observed that some bending of the shell does occur over this simple septal support. This suggests that nautilid shells do not function as a series of vaults in a manner here proposed for Bacu- lites. The failure of nautilids, or at least of Nautilus, to employ a series of vaults to min- imize tensional shell stress in bending may be related to the relative thickening of the shell wall in the flank region to accommodate bending (Westermann 1971). Thickening of the shell strengthens it in bending. However, stress is inversely related to the cross-section- al area acted on by a force; consequently, thickening to counteract bending will dimin- ish the compressive stress resulting from vault thrusting. It is this compressive stress which minimizes the tensile stresses resulting from bending. Local thickening as a solution to bending in the shell will tend to counteract

the advantageous use of the greater com- pressive strength of nacre allowed by vault action. Use of a series of vaults to avoid ten- sional bending stresses permits the construc- tion of a greater variety of shell shapes with less material. Nautiloid shells that counteract bending by local thickening will be more se- verely limited in the shapes they can attain than ammonites employing interactive vaults.

The relationship between the shell thick- ening on the flanks of nautilids and their lim- ited suite of shell morphologies relative to ammonites was pointed out by Westermann (1971) and shown empirically by Ward (1980). In the work presented in this paper, an in- teractive vault model provides a direct rela- tionship between external shell shape and optimal sutural pattern in ammonites. This model does not appear to describe the struc- tural interactions in nautiloid shells.

The Role of Ornament. -As observed by Buckland (1836), ornament can also have an important structural role in support of the shell. Ribbing reduces bending stress by plac- ing material farther away from the neutral axis, thereby increasing what is termed the "moment of inertia." The material displaced away from the neutral axis resists bending in a manner similar to the corrugations on a tin roof. Folding or "corrugation" of the shell wall produces the ribbing observed on the flanks of most ammonites. In contrast to ribs produced by thickening and thinning of the shell, folding of the shell wall does not great- ly increase the cross-sectional area of the shell. By minimizing the increase in cross-sectional area the axial compressive stress produced by vault action can be maintained to limit ten- sion due to bending. In addition, the ribs on the flanks of compressed ammonites are ori- ented in a nearly radial direction: ribs on the broad ventral region of depressed ammonites such as Cadoceras, Stephanoceras or Macroceph- alites are oriented transversely across the whorl. Ribs in these orientations will resist bending in the circumferential directions in regions of the shell where the greatest bend- ing stresses occur. Due to their orientation and the fact that they do not greatly increase cross-sectional area, most ribs facilitate struc- tural support in a manner consistent with the

346 DAVID K. JACOBS

TABLE 2. Specimen information and data on vault span and radius of curvature for 10 specimens of Baculites: B. comp. = Baculites compressus Say; R = radius of curvature; 12 = halfspan length squared.

Specimen A B C D Species B. sp. B. sp. B. comp. B. comp. Repository Raup, D. Field Mus. Field Mus. Field Mus. Specimen No. none 3544-1 3602 36390-a

Data (mm) R 12 R 12 R 12 R 12

5.0 18.14 18.5 2.05 27.0 14.32 20.0 0.94 5.0 16.55 8.4 5.86 13.0 19.58 7.0 1.59 6.0 16.55 4.5 14.99 9.0 30.08 3.2 3.90 7.0 14.17 20.0 1.83 31.0 12.57 21.0 0.76 7.0 15.76 8.5 6.34 18.0 14.96 6.5 1.35 7.0 14.96 4.5 19.83 8.5 36.13 3.5 3.76 7.0 14.96 19.0 2.38 30.0 11.30 18.0 0.94 7.0 8.12 7.0 9.60 16.0 15.12 6.0 1.94 8.5 8.12 4.3 17.99 11.5 26.26 3.0 4.38 9.0 8.75 21.0 1.83 32.0 13.85 18.0 0.94

10.0 8.12 6.8 7.35 15.0 21.17 5.8 1.78 10.0 9.23 3.0 14.26 6.0 40.59 2.8 3.76 11.0 8.12 21.5 1.83 34.0 7.96 16.5 0.70 35.0 7.48 7.0 7.35 19.0 12.57 5.5 1.40 36.0 2.71 4.2 14.99 9.5 31.99 3.0 3.60 37.0 2.07 23.0 1.83 17.0 0.76 39.0 2.39 7.4 7.32 5.2 2.96 42.0 2.39 4.0 13.53 3.5 3.39 41.0 2.39 22.0 1.83 45.0 2.07 6.0 6.84

4.0 13.53 Reduced major

axis -0.958 -1.270 -0.865 -0.919 Regression slopes (all slopes are significant at P < 0.0005)

interactive vault model of sutural support of the phragmocone.

Features of the Shell Not Subject to the Inter- active Vault Mechanism. -The umbilical re- gion of coiled cephalopods presents the prob- lem of pressure inside rather than outside a cylindrical structure; therefore deep tubular umbilici will generate tensional rather than compressive stress. Avoiding a well defined umbilicus may be one mechanism of dealing with this potential problem. Shell thicken- ing, plugs and wrinkle layers in the umbili- cus may function to limit tensional stress in the umbilical wall. These features of the um- bilicus and possibly other structures such as keels represent structural solutions beyond the scope of the simple vault model.

Conclusions

In Baculites there is a statistically significant inverse relationship between the size of vault spans defined by the sutural pattern and the radius of curvature of these vaults which comprise the phragmocone. This relationship

confirms predictions based on an interactive vault model of the phragmocone. In this mod- el, the inverse relationship between the span size and radius of curvature results in equiv- alent reaction forces at the vault ends in the series of vaults comprising the phragmocone. The equivalence of the reaction forces at the vault ends provides a compressive stress that is superimposed on and limits tensional stress due to bending. Limitation of tensional stresses is necessary if the ammonite shell is to support hydrostatic load optimally with minimal nacreous shell material. This is a consequence of the greater strength of nacre in compression than in tension.

The interactive vault model is applicable to coiled as well as straight ammonite shells. It provides a major constraint on sutural spac- ing and therefore on sutural pattern and com- plexity. The vault model indicates a direct relationship between external shell mor- phology and sutural pattern. If sutural pat- tern and shell shape are linked in a functional sense, parallel evolution would be expected.

SUTURAL PATTERN AND SHELL WALL STRESS 347

TABLE 2. Extended.

E F G H I J B. sp. B. comp. B. comp. B. comp. B. sp. B. comp. UICC Col. Field Mus. UICC Col. UICC Col. UICC Col. UICC Col. Cm3 36390-b 29-i F24 Fm3 31

R 12 R 12 R 12 R 12 R 12 R 12

4.0 12.14 20.0 1.10 4.5 18.14 3.0 5.40 5.5 7.35 5.3 17.33 13.0 5.40 7.2 2.71 9.0 9.01 2.8 9.01 27.0 2.44 10.6 6.83 31.0 1.94 4.2 5.86 24.5 2.40 5.2 5.40 5.0 9.60 34.0 3.17

4.5 12.14 21.0 1.69 3.5 26.26 20.0 1.83 10.8 5.40 5.4 25.34 12.0 5.86 6.0 2.58 8.5 14.99 3.2 7.88 31.0 1.83 10.8 9.96 30.0 2.71 4.0 5.67 29.0 2.10 5.5 6.83 5.6 9.01 32.0 4.52

4.0 18.14 19.0 0.94 8.0 9.01 19.0 1.35 10.0 4.95 6.0 21.60 10.0 8.44 6.8 2.23 30.0 1.91 3.0 10.19 35.0 1.13 11.5 9.60 30.0 2.39 3.5 4.54 6.0 6.83 4.8 10.90 33.0 4.14

17.5 1.13 22.0 1.59 8.0 5.86 5.7 3.25 3.5 11.46 31.0 1.59 3.0 6.14 6.5 5.40

18.0 1.78 20.0 2.10 5.2 3.17 3.2 10.82 3.2 5.40 6.5 7.56

20.0 1.45

-0.924 -0.875 -1.248 -0.959 -0.980 -0.965 (all slopes are significant at P < 0.0005)

Homeomorphy (especially in lineages tend- ing toward compressed involute morpholo- gies where there is a tendency to generate new primary lobes on the flattened flanks of the whorl) may be explained by the con- straints imposed by this model. Recognition of the functional relationship between sutur- al pattern and shell shape should help dif- ferentiate between parallelism and homology in phylogenetic analyses of ammonites.

Nautiloids do not appear to employ the in- teractive model of shell support but locally thicken the shell instead. This would appear to explain the lower degree of sutural com- plexity and more limited suite of shell mor- phologies occupied by coiled nautiloids as opposed to ammonites.

The employment of a set of interactive vaults in the ammonite phragmocone capi- talizes on the greater strength of nacre in compression than tension. The complex am- monite septal suture serves to determine ap- propriate span size, and provides end sup- port, in this series of vaults.

Acknowledgments

I thank R. Hewitt, G. Westermann and J. Chamberlain for their critical reviews and R. Bambach, B. Bennington, R. Cowen, N. Gil- insky, A. Hubbard, and N. Landman for their assistance with the manuscript.

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