Marmo F et al. (2022), Flexible sutures reduce bending moments in shel...

ECB-ART-50321

R Soc Open Sci 2022 May 04;95:211972. doi: 10.1098/rsos.211972.

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Flexible sutures reduce bending moments in shells: from the echinoid test to tessellated shell structures.

Marmo F , Perricone V , Cutolo A , Daniela Candia Carnevali M , Langella C , Rosati L .

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In the field of structural engineering, lightweight and resistant shell structures can be designed by efficiently integrating and optimizing form, structure and function to achieve the capability to sustain a variety of loading conditions with a reduced use of resources. Interestingly, a limitless variety of high-performance shell structures can be found in nature. Their study can lead to the acquisition of new functional solutions that can be employed to design innovative bioinspired constructions. In this framework, the present study aimed to illustrate the main results obtained in the mechanical analysis of the echinoid test in the common sea urchin Paracentrotus lividus (Lamarck, 1816) and to employ its principles to design lightweight shell structures. For this purpose, visual survey, photogrammetry, three-dimensional modelling, three-point bending tests and finite-element modelling were used to interpret the mechanical behaviour of the tessellated structure that characterize the echinoid test. The results achieved demonstrated that this structural topology, consisting of rigid plates joined by flexible sutures, allows for a significant reduction of bending moments. This strategy was generalized and applied to design both free-form and form-found shell structures for architecture exhibiting improved structural efficiency.

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	Figure 1. . Regular echinoid. Schematic reconstruction of a regular echinoid, its skeletal test in aboral view and functional characteristic of a tessellated shell structure. The aboral view shows the subdivision in ambulacral (I–V) and interambulacral (1–5) zones along with respective plates and the symmetry plan. aa = adapical suture; ao = adoral sutures; ad = adradial suture: ii = interradial suture.
	Figure 2. . Internal force components in shells and the corresponding Cauchy stresses: (a) membrane forces, (b) bending moments and out-of-plane shear forces, (c) Cauchy stresses associated with membrane forces, and (d) Cauchy stresses associated with bending moments and out-of-plane shear forces.
	Figure 3. . Paracentrorus lividus test: functional details from macro to microscale. Test with plate diversity in (a) ambulacral plates with adapical and adradial sutures and (b) interambulacral plates with adradrial and interratial sutures; (c) plates with curved edges indicated in (a); (d) trivalent vertex arrangement magnified from (b); (e) knob-like protrusions; and (f) collagen fibres magnified from (d). Asterisk = functional features. aa = adapical suture; ao = adoral sutures; ad = adradial suture: ii = interradial suture.
	Figure 4. . Macro and microstructure of an interambulacral plate. (a) Interambulacral plate with detail of knob-like protrusions, extracted from P. lividus's test. SEM micrographs showing: (b) plate transversal section with microstructural variability (a, imperforated; b, tubercle galleried; c, labyrinthic; d, perforated; e, suture galleried) identified by a dashed line; suture area of (c) joined and (d) divided plates.
	Figure 5. . Mechanical behaviour of a curved plate. (a) Flat plates can undergo relative rotation by keeping adjacent edges in contact. (b) Joined echinoid plates with curved geometry. (c,d) Relative rotation between curved plates causes separation of adjacent edges.
	Figure 6. . Experimental set-up. Three-point bending test executed by a TA Instruments ElectroForce 200 N - 4 motor Planar Biaxial Test Bench on ad-hoc supports manufactured in thermoplastic material (ABS) via a three-dimensional printing system (Stratasys Object 30 Pro). Plate–plate sample before (a) and after (b) testing.
	Figure 7. . Experimental curves for single plates: force–displacement curves (a) are interpreted by the scheme (b) to obtain the stress–strain curves (c).
	Figure 8. . Experimental curves for plate–plate pairs: force–displacements curves (a) are interpreted by the scheme (b) to obtain the secant stiffness–relative rotation (c) and the tangent stiffness–relative rotation (d) curves. Solid curves refer to plate–plate pairs in natural states while dashed curves refer to plate–plate pairs treated with K+ solution. Red dotted curves are relevant to a parametric description of the two curves by the analytical models for the secant stiffness (equation (1.3)) and the tangential stiffness (equation (1.4)).
	Figure 9. . Three-dimensional reconstruction of the P. lividus's test. (a) Three-dimensional model obtained by a photogrammetric reconstruction. (b) Parameterized geometry model with visible plates and sutures. (c) Three-dimensional mesh used for finite-element analyses. (d) Application of pressures on a rectangular region of the test, a, and spring hinges at the base, b.
	Figure 10. . FEA of the P. lividus's test. (a) FEA and response of the monolithic model with homogeneous flexural resistance with a magnified detail in (d). (b) FEA and response of the tessellated model are characterized by reduced flexural stiffness of the elements corresponding to collagenous sutures with a magnified detail in (e). Values represented by the chromatic scale are in N mm mm−1 and increase from blue to red. (c) Diagram showing the relationship between the value of the maximum principal bending moment Mmax computed at the quadrature points of the FE model and the number of quadrature points where Mmax is attained.
	Figure 11. . Maximum principal bending moments in funicular concrete shells: monolithic (a,c) versus tessellated (b,d); effect of vertical (e) and horizontal (f) loads. Values in the colour bars are expressed in N m m−1. Diagrams showing the relationship between the value of the maximum principal bending moment Mmax computed at the quadrature points of the FE model and the number of quadrature points where Mmax is attained.
	Figure 12. . Maximum principal bending moments in concrete arch shells: monolithic (a,c) versus tessellated (b,d); effect of vertical (e) and horizontal (f) loads. Values in the colour bars are expressed in N m m−1. Diagrams showing the relationship between the value of the maximum principal bending moment Mmax computed at the quadrature points of the FE model and the number of quadrature points where Mmax is attained.

References [+] :

Abou Chakra, Holotestoid: a computational model for testing hypotheses about echinoid skeleton form and growth. 2011, Pubmed, Echinobase