Engineering beam theory has been used to analyse the ways in which body shape and elastic modulus of two species of sea anemones affect their mechanical responses to flow.

  1. Anthopleura xanthogrammica is exposed to wave action, but because it is short, wide, and thick-walled, maximum tensile stresses in its body walls due to flow forces are an order of magnitude lower than those in the tall, slim, thin-walled, calm-water sea anemone Metridium senile.

  2. The elastic modulus of M. senile body wall is more dependent on extension rate than is that of A. xanthogrammica. Because the extension rate of M. senile body wall in tidal currents is higher than that of A. xanthogrammica in wave surge, the moduli of walls from these species when exposed to such flow conditions are similar, between 0·1 and 0·3 MN.m-2.

  3. The flexural stiffness of M. senile is lowest in the upper column where the anemones bend in currents: this orients their filter-feeding oral discs normal to the currents. The flexural stiffness of A. xanthogrammica is one to two orders of magnitude higher than that of M. senile;, A. xanthogrammica remain upright in wave surge and feed on mussels that fall on their oral discs.

  4. The deflexions of these anemones predicted using beam theory are consistent with those observed in nature.

  5. The critical stress to produce local buckling is an order of magnitude lower for M. senile than for A. xanthogrammica.

  6. Several general principles of the organization of cantilever-like sessile organisms are revealed by this study of sea anemones.

The shape of a structure determines the distribution of mechanical stress (force per cross-sectional area) within the structure for a given load distribution. The rigidity of the materials comprising a structure determines how much it will deform in response to stresses. Thus, the ability of an organism to resist deformation by an imposed load depends upon both the shape of its body and the mechanical properties of its tissues. This paper describes how aspects of body shape and material can maximize or minimize the deformation of an organism when it is loaded.

Sea anemones are basically hydrostatically supported cylinders attached at one end to the substratum. Large Metridium senile fimbriatum (Verrill) are subtidal anemones that are bent over by slow tidal currents and filter-feed through their lobed discs (Koehl, 1976). In contrast, Anthopleura xanthogrammica (Brandt) are interial anemones that remain upright when exposed to wave action (Koehl, 1976) and feed on mussels that fall on their oral discs (Dayton, 1973). Since the same flow force that causes M. senile to bend over does not noticeably deform an A. xanthogrammica (Koehl, 1976), I expected that the latter would have a more stress-minimizing shape and/or a stiffer body wall than the former.

A sessile anemone can be considered as a cantilevered beam supporting a feeding apparatus in flowing water in a suitable orientation for food capture. A cantilevered beam (sea anemone) subjected to a load (flow force) undergoes shearing (Fig. 1 A) and bending (Fig. 1B). The oral disc of an anemone deformed in shearing remains parallel to the substratum and the flow direction, whereas the oral disc of an anemone deformed in bending does not.

Fig. 1.

Diagram of a cantilevered beam supporting a load (F). (A) Deformation in shearing, (B) deformation in bending (the arrows in (A) and (B) indicate the directions in which molecules of the beam are tending to be moved), (C) side view of the beam and cross-section of the beam at distance x from the free end of the beam. The unit of area dA is at distance y from the axis of bending of the beam.

Fig. 1.

Diagram of a cantilevered beam supporting a load (F). (A) Deformation in shearing, (B) deformation in bending (the arrows in (A) and (B) indicate the directions in which molecules of the beam are tending to be moved), (C) side view of the beam and cross-section of the beam at distance x from the free end of the beam. The unit of area dA is at distance y from the axis of bending of the beam.

I used beam theory (which is discussed in Faupel, 1964; Alexander, 1968; and Wainwright et al. 1976) to analyse the effects of body shape and materials on the mechanical responses of M. senile and A. xanthogrammica to flow forces.

M. senile along the Pacific coast of North America range from small, shallow-water anemones with large tentacles to large, deeper-water animals with lobed oral discs covered with numerous small tentacles (Hand, 1955b). The present study is concerned only with the latter form of M. senile. Although a few individuals of A. xanthogrammica can be found in the more protected coastal sites, they are most abundant in exposed surge channels (Koehl, 1976), and it is these animals that were used in the present study.

The heights above the substratum and the diameters at various points along the bodies of expanded A. xanthogrammica were measured to the nearest 5 mm in the field. To minimize sampling bias, I measured every anemone encountered in a tide pool or surge channel. The heights and diameters of expanded M. senile were measured to the nearest 5 mm on photographs of the anemones with a centimetre grid held next to them. To minimize sampling bias, the groups of anemones to be photographed were chosen in a haphazard manner and all expanded individuals whose entire bodies were visible in a photograph were measured. Nine M. senile and 17 A. xanthogrammica were anaesthetized in solutions of 20% MgSO4.7H2O one to one with Instant Ocean (Pantin, 1964) and strips were cut from the columns of relaxed anemones with a razor blade. The thickness of such strips was measured to the nearest 0·1 mm.

Anemones of both species were collected and kept in refrigerated aquaria containing Instant Ocean. Every fourth day, M. senile were fed brine shrimp nauplii and A. xanthogrammica were fed mussels or snails. Only the body walls of actively posturing animals that responded to food were used for mechanical testing. Chapman (1953) observed that the passive mechanical behaviour of anemone body wall depends on the connective tissue but not the muscles of the body wall. I therefore used relaxed strips of body wall for mechanical tests and did not risk damaging the delicate strips by attempting to remove the muscle fibres. The strips were pulled at various rates in an Instron Universal Testing Instrument (Model TT-C) that simultaneously recorded force (standard error, S.E. = ±0·5%) and extension (S.E. = ±0·25%). The strips were attached with contact cement (Eastman 9−10) to two stainless-steel spring clamps which were gripped in the Instron. The length between the grips, the width, and the thickness of each strip was measured to the nearest 0·1 mm using callipers. The strips being pulled were immersed in a bath of the anaesthetic solution kept at 10°C by an ice jacket (water temperatures in areas along the coast of Washington where the anemones are found range from 5 to 16°C (Connell, 1970)). Plots of stress versus extension (the ratio of change in length of the specimen to its original unstressed length) were made. The slope of the straight portion of each such stressextension curve was taken to be the modulus of elasticity (E) for that specimen.

Body shape

M. senile are tall (mean height above the substratum = 38·0 cm, standard deviation, S.D. = 9·0, number of measurements, n = 28) and slim (expanded M. senile are roughly three times taller than they are wide) (Table 1). M. senile columns taper, being narrowest about three-quarters of the column height above the substratum. In contrast A. xanthogrammica are short (mean height = 2·5 cm, S.D. = 0·9, n = 71) and squat (expanded A. xanthogrammica in surge channels are roughly three times wider than they are tall) (Table 2).

Table 1.

Body proportions of expanded Metridium senile

Body proportions of expanded Metridium senile
Body proportions of expanded Metridium senile
Table 2.

Body proportions of expanded Anthopleura xanthogrammica

Body proportions of expanded Anthopleura xanthogrammica
Body proportions of expanded Anthopleura xanthogrammica

A. xanthogrammica body walls are twice as thick as those of M. senile ; the mean body-wall thickness measured at mid-column height of A. xanthogrammica is 2·2 mm and of M. senile is 1·1 mm. This difference in thickness is significant = 228·7, P < 0·001). The body walls of both species are thicker towards the base (Fig. 2) where the bending moments caused by flow forces are greatest.

Fig. 2.

Graph of the thickness of body wall at various heights along the columns of an A. xanthogrammica (△) and a M. senile (▼). Twelve strips the length of the column were cut from each anemone and the thickness at 1-cm intervals along the strips were measured. Error bars indicate standard deviation.

Fig. 2.

Graph of the thickness of body wall at various heights along the columns of an A. xanthogrammica (△) and a M. senile (▼). Twelve strips the length of the column were cut from each anemone and the thickness at 1-cm intervals along the strips were measured. Error bars indicate standard deviation.

The second moments of area (I, a measure of the distribution of material around the axis of bending) of typical M. senile and A. xanthogrammica are calculated in the Appendix and listed in Table 3. The I of the lower column of M. senile is of the same order of magnitude as that of A. xanthogrammica; although the body wall of A. xanthogrammica is thicker, the diameter of the M. senile lower column is greater. The I of the thin-walled, narrow upper column of a typical M. senile is an order of magnitude lower, however, than that of the lower column.

Table 3.

Calculated stresses in and deflexions of sea anemones in flowing water

Calculated stresses in and deflexions of sea anemones in flowing water
Calculated stresses in and deflexions of sea anemones in flowing water

Shear stresses and tensile stresses

The shear stresses and the tensile stresses associated with bending are calculated in the Appendix and listed in Table 3 for typical M. senile in tidal current and A. xanthogrammica in wave surge. Shear stresses are minimized in beams with large cross-sectional areas, but are independent of cross-sectional shape. In contrast, tensile stresses associated with bending are minimized when the most material is the greatest possible distance from the axis of bending (i.e. when I is high). The shear stresses in M. senile are of the order of 103N.m-2 whereas the maximum tensile stresses are of the order of 104 N.m-2. If a cantilever has a uniform cross-sectional shape, maximum stresses occur near the base of the beam; in M. senile, however maximum stresses occur at the narrow region of the upper column where I is lowest. The shear and maximum tensile stresses in A. xanthogrammica are only between 102 N.m-2 and 103 N.m-2, with shear stresses being greater. Thus, although A. xanthogrammica is exposed to wave action, because it is short and wide, the maximum tensile stresses in its body wall due to flow forces are an order of magnitude lower than those in a tall, calm-water M. senile. Note that although the I of the lower column of the typical M. senile is slightly greater than that of the A. xanthogrammica, the tensile stresses in the former are nearly 20 times larger than in the latter, because M. senile is taller.

M. senile and A. xanthogrammica illustrate the profound effect that body height and cross-sectional dimensions can have on the magnitude and distribution of stresses in beam-like organisms subjected to loads such as flow forces.

Material rigidity

Representative graphs of the stress in strips of body wall from M. senile and A. xanthogrammica at various extensions are shown in Fig. 3. Note that the stress-extension curves for both species have steeper slopes when extension rate (change in length per original length per s) is higher. This indicates that the faster the anemone body wall is stretched, the higher is its modulus of elasticity (E). This increase rigidity with increased extension rate is more pronounced for M. senile body wall than for A. xanthogrammica body wall.

Fig. 3.

Graphs of stress (MN.m-2) against extension (the ratio of the increase in length to the original length of the specimen) for strips of body wall from a M. senile (◯) and an A. xanthogrammica (▲) pulled along their longitudinal axes in an Instron testing machine at a rapid (dashed line) and a slow (solid line) extension rate (

ε·
⁠).

Fig. 3.

Graphs of stress (MN.m-2) against extension (the ratio of the increase in length to the original length of the specimen) for strips of body wall from a M. senile (◯) and an A. xanthogrammica (▲) pulled along their longitudinal axes in an Instron testing machine at a rapid (dashed line) and a slow (solid line) extension rate (

ε·
⁠).

Analyses of variance reveal that for an individual anemone at each of the extension rates used there is no significant difference between the elastic moduli of strips of body wall from the same or from different regions of the column (upper, middle, or lower column), or between moduli of strips of body wall pulled along different axes (longitudinal or circumferential axes of the anemone). Moduli of strips of body wall from different individuals of the same species pulled at a given strain rate are in some instances significantly different (0·025 < P < 0·05), but the coefficients of determination are low (0·01 or less). All elastic modulus data is pooled for each species to graph modulus against extension rate. Such graphs indicate that modulus has a power law relationship to extension rate, hence a log-log plot of elastic modulus against extension rate for each species should yield a straight line (Fig. 4). Analysis of covariance reveals that the regression line fitted to the data for A. xanthogrammica and that for M. senile are significantly different in slope (F[1,182]= 6·80, 0·01 < P < 0·025) and in elevation (F[1,182]= 7 ·40, 0 ·005 < P < 0 ·01). By converting the equations from their logarithmic form (Fig. 4) to their power law form, the relationship between modulus and extension rate for M. senile body wall becomes

Fig. 4.

Graph of the log of elastic modulus (N.m-2) against the log of extension rate (s-1) for M. senile body wall (▼) and A. xanthogrammica body wall (△). The dashed line represents the regression line fitted by the least squares method to the set of data points for M. senile, and the solid line represents the regression line fitted to the data points for A. xanthogrammica.

Fig. 4.

Graph of the log of elastic modulus (N.m-2) against the log of extension rate (s-1) for M. senile body wall (▼) and A. xanthogrammica body wall (△). The dashed line represents the regression line fitted by the least squares method to the set of data points for M. senile, and the solid line represents the regression line fitted to the data points for A. xanthogrammica.

formula
and for A. xanthogrammica body wall,
formula
where E is elastic modulus in MN. m-2 and ε is extension rate in s-1. Note that because the modulus of M. senile wall is more dependent on extension rate than that of A. xanthogrammica wall, at high extension rates (between 0·1 s-1 and 1·0 s-1) M. senile is as rigid as A. xanthogrammica, but at slower extension rates (lower than 0·001 s-1) A. xanthogrammica is significantly more rigid than M. senile (F[1,182]= 25·62, P <0·001)

The deformations of these two species of anemones depend upon the elastic moduli of their body walls, which in turn depend upon the rates at which they are stretched. Therefore, the extension rates of the animals’ body walls when subjected to flow forces must be assessed before their mechanical responses to those forces can be predicted. These extension rates can be estimated by doing ‘creep’ tests (Koehl, 1977b). In a creep test, a relaxed strip of M. senile body wall can be subjected to longitudinal stress of the order of 104 N. m-2 to simulate the maximum tensile stresses it experiences in tidal currents. The slope of the straight portion of a plot of extension of the strip against time should be a reasonable estimate of the extension rate of the anemone wall. The mean extension rate of M. senile body wall subjected to stresses between 3·0 × 104 N.m-2 and 7·5 × 104 N.m-2 is 0·0016 s-1 (S.D. = 0·0004, n =11). Therefore, by equation (1) an estimate of the elastic modulus of the body wall of a M. senile in a tidal current is 0·17 MN.m-2. When A. xanthogrammica body wall is subjected to longitudinal stresses between 0·6 × 103 N.m-2 and 1·7 × 103 N.m-2 to simulate stresses due to wave surge, the mean extension rate is only 0·00025 s-1 (S.D. = 0·00024, n = 21) and the calculated modulus (by equation 2) is 0·25 MN.m-2. A. xanthogrammica walls in surge extend at a rate roughly ten times lower than do M. senile walls in tidal currents; hence, because the elastic modulus of anemone body wall depends upon extension rate, the walls of M. senile in tidal currents are as rigid as those of A. xanthogrammica in surge, even though A. xanthogrammica walls are stiffer at a given extension rate (when < 0·01 s-1).

Flexural stiffness

An index for the resistance of a beam to being bent by a load is flexural stiffness, the product of E (a material property) and I (a shape property). The El of a typical M. senile and a typical A. xanthogrammica are calculated in the Appendix and listed in Table 3. Note that the flexural stiffness of the upper column of M. senile is roughly 20 times lower than that of the lower column. Since body wall from any location in the M. senile column has the same modulus, the local increase in flexibility at the upper column is due to shape. It is not surprising that M. senile in a current is bent at this upper region of its column; the anemone’s filter-feeding oral disc is thus oriented normal to the direction of flow and is held out in the more rapidly flowing water away from the substream.

The flexural stiffness of a typical A. xanthogrammica in surge is an order of magnitude greater than that of the upper column of a M. senile in a tidal current, again due to a difference in cross-sectional shape. In particular cases where the extension rate on the wall of A. xanthogrammica is greater than that mentioned above, the elastic modulus of the A. xanthogrammica body wall is higher than that of M. senile wall; in such cases the difference between the flexural stiffnesses of the two species becomes greater and is Mue to differences in both shape and material rigidity (Koehl, 1976). A. xanthogrammica do not bend visibly in moving water ; their oral discs remain upright where mussels are more likely to fall on them.

Deflexion

The linear deflexion of the free end (oral disc) of a typical M. senile deformed by a tidal current and of a typical A. xanthogrammica deformed by surge can be predicted using beam theory (Appendix). The predicted deflexions (Table 3) are consistent with deflexions of the anemones observed in the field.

Local buckling

A beam may undergo local buckling, i.e. may kink as a drinking straw does when bent too far. Such buckling reduces the size of the lumen of a hollow beam and thus is resisted if the beam is filled with fluid under pressure. However, because the internal pressures of sea anemones are only of the order of 0·1−10 cm H2O (Chapman, 1974), I have considered anemones to be open tubes as a first approximation to calculate the critical stress to produce local buckling (Appendix).

The lower columns of M. senile in tides and A. xanthogrammica in surge have roughly the same flexural stiffness. However, because A. xanthogrammica columns are thicker-walled and M. senile lower columns are larger in diameter, the critical stress to produce local buckling is an order of magnitude greater for A. xanthogrammica than for M. senile (Table 3). This critical stress falls within the range of stresses M. senile encounter in the field, and in fact buckled M. senile are observed in the field (Fig. 5):

Fig. 5.

Tracing of a photograph of a M. senile that has undergone local buckling. The arrow indicates the direction of the tidal current.

Fig. 5.

Tracing of a photograph of a M. senile that has undergone local buckling. The arrow indicates the direction of the tidal current.

Mechanical organization of cantilever-like sessile organisms

By analysing the ways in which body shape and elastic modulus of two species of sea anemones affect their mechanical responses to flow, several general principles of the organization of cantiliver-like sessile organisms are revealed :

  1. Tall, slim shapes tend to maximize the stresses in a body for a given load distribution, whereas short, wide shapes tend to minimize the stresses for a given load distribution.

  2. The rigidity (elastic modulus) of the materials from which an organism is built can depend upon how fast the materials are stretched, which depends upon the magnitude of the stresses to which they are subjected, which in turn depend upon the shape of thè organism.

  3. Flexural stiffness is the ability of a beam-like organism to fesist bending. The higher the elastic modulus of the organism’s tissues and the greater the distance of those tissues from the axis of bending, the less the organism will bend when loaded

  4. A beam-like organism can be flexible at a specific region of its body by reducing the diameter (and/or wall thickness if it is hollow) of that region of the body, and/or by reducing the elastic modulus of the tissue there. Thus, a narrow region in a beamlike body can behave like a flexible joint (Wainwright & Koehl, 1976).

  5. If a hollow beam-like organism becomes too wide or thin-walled, it is liable to kink when loaded.

This work, which was supported by a Cocos Foundation Training Grant in Morphology, a Theodore Roosevelt Memorial Fund of the American Museum of Natural History Grant, and a Graduate Women in Science Grant, was the direct result of a most stimulating association with S. A. Wainwright. I gratefully acknowledge his advice and enthusiasm. I thank G. Pearsall for his advice and the Department of Mechanical Engineering, Duke University, for the use of their Instron. I am grateful to T. Suchanek for supplying me at Duke with freshly collected anemones from the Pacific, and to D. Crenshaw for his advice on statistical analyses.

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APPENDIX. SAMPLE CALCULATIONS OF STRESSES IN AND DEFLEXIONS OF SEA ANEMONES

The body dimensions of a ‘typical’ M. senile (Fig. 6b) were calculated from the mean height of M. senile measured in the field (, = 0·38 m, s.D. = 0·09, n — 28) using the body proportions listed in Table 1. The flow forces (heavy arrows) on the oral disc, upper column, and lower column were calculated using field measurements of flow velocities over M. senile (Koehl, 1977 a) and are assumed to act on the midpoint of each of the three portions of the body.

Fig. 6.

Diagrams of (a) M. senile (symbols used in Table 1), (b) ‘typical’ M. senile exposed to flow forces (see Appendix), (c) A. xanthogrammica (symbols used in Table 2), (d) ‘typical’ A. xanthogrammica exposed to flow forces (see Appendix).

Fig. 6.

Diagrams of (a) M. senile (symbols used in Table 1), (b) ‘typical’ M. senile exposed to flow forces (see Appendix), (c) A. xanthogrammica (symbols used in Table 2), (d) ‘typical’ A. xanthogrammica exposed to flow forces (see Appendix).

The body dimensions of a ‘typical’ A. xanthogrammica (Fig. 6d) were calculated from the mean height of A. xanthogrammica in exposed surge channels ( = 0·025 m, s.D. = 0·009, n = 71) using the proportions listed in Table 2. The flow forces (heavy arrows) on the oral disc and the column were calculated using field measurements of flow velocities over A. xanthogrammica in exposed surge channels (Koehl, 1977a) and are assumed to act on the midpoint of each of these two portions of the body.

1. Second moment of area (I)

The anemone is considered to be a hollow cylindrical cantilever, and thus I is given by
formula
where r1 is the outer radius of the cylinder and r2 is the inner radius (r2 = r1t, where t is the thickness of the cylinder wall). In M. senile and A. xanthogrammica the contribution to I of the mesenteries is small because they are very thin relative to the body wall (see Hand, 1955a, b) and are closer to the axis of bending. Therefore, the mesenteries are neglected for simplicity’s sake in the following calculations.
M. senile. The mean wall thickness of M. senile body wall from the narrow region of the upper column (indicated by A in Fig. 6b) is 5 ·10−4 m (s.D. = 1×10−4, n = 45), and from the lower column (B in Fig. 6b) is 1·1 × 10−3 m (s.D. = 0·2 × 10−8, n = 45), hence
formula
and
formula
A. xanthogrammica. The mean wall thickness of A. xanthogrammica body wall near the base of the column (indicated by C in Fig. 6d) is 2·26 × 10−3 m (s.D. = 0·56 × 10−3 n = 53), hence
formula

2. Shear stress (σs)

Shear stress in a section of an anemone is given by
formula
where F is the load and A is the cross-sectional area of the beam. The cross-sectional area of a hollow cylinder is given by
formula
M. senile. The force on column section A is assumed to be the drag on the oral disc, and the force on column section B is assumed to be the drag on the entire anemone, hence
formula
A. xanthogrammica: The force on column section C is assumed to be the drag on the entire anemone, hence
formula

3. Maximum tensile stress associated with bending (σT)

The tensile stress (σT) in a bending cantilever is given by
formula
where x is the distance of the section being considered from the free end of the beam (hence Fx is the bending moment of the beam at that section), y is the distance of the area being considered from the centroid of the section (Fig. 1C), and I is the second moment of area of the section. Maximum tensile stress occurs at the upstream surface in a section of an anemone.
M. senile : the bending moment (Fx) to which section A is subjected is assumed to be due to the drag force acting on the midpoint of the oral disc. The bending moment to which section B is subjected is assumed to be due to the drag acting on the midpoint of the oral disc plus the drag acting on the midpoint of the upper column plus the drag acting on the midpoint of the lower column.
formula
and
formula
A. xanthogrammica. The bending moment to which section C is subjected is assumed to be due to the drag force acting on the midpoint of the oral disc plus the drag force acting on the midpoint of the column, hence
formula

4. Flexural stiffness (El)

M. senile. The E of M. senile body wall at an extension rate of 0·0016 s-1 is calculated using equation (1) to be 0·17 MN.m-2, hence
formula
and
formula
A. xanthogrammica. The E of A. xanthogrammica body wall at an extension rate of 0·00025 s-1calcuated using equation (2) to be 0·25 MN.m-2, hence
formula

5. Deflexion of the oral disc (δ)

The linear deflexion of the free end of a cantilevered beam is given by
formula
where L is the length of the beam.
M. senile. If the deflexion of the oral disc of the M. senile is assumed to be due to the drags acting at the midpoints of each of the three portions of the body diagrammed, and if the average of EIA and EIB is used as an estimate of El for the anemone, then by equation (7)
formula
A. xanthogrammica. If deflexion of the oral disc of A. xanthogrammica is assumed to be due to the flow forces acting at the midpoints of the oral disc and of the column, and if EIC is used as an estimate of El for the anemone, then
formula

6. Critical stress for local buckling (σL)

The stress required to produce local buckling in a hollow cylindrical beam is given approximately by the equation
formula
where t is the thickness of the beam’s wall, D is the diameter of the beam, and 0·5 is an empirically derived coefficient. Thus, although a very wide, thin-walled cylinder has a higher I than a narrower, thicker-walled cylinder of the same cross sectional area, the former would be more likely to undergo local buckling.
M. senile:
formula
and
formula
A. xanthogrammica ;
formula
Note ; The values for σs, σT, El and δ reported by Koehl (1976) for specific M. senile and A. xanthogrammica were calculated as described above: however, the values printed in Table 1 (Koehl, 1976) are in error and should be:

M. senile :

σs = 1739 N.m-2, σT= 15 450 N. m-2, El = 0·02 N.m2, δ5 = 6cm:

A. xanthogrammica-

σs = 1075 N.m-2, σT = 750 N.m-2, El = 1·5 N.m2, δ = 0·0003 cm.