Stiffness and Buckling Behavior of Woven Columns

TL;DR

This work develops purely analytical expressions for the buckling load and stiffness of dense woven columns, integrating vertical and horizontal weaver geometry with material properties. The total buckling load is expressed as $P_{cr,total}=P_{cr,h}+P_{cr,v}$, with explicit forms for $P_{cr,v}$ and $P_{cr,h}$ that incorporate Koiter's knockdown factor and imperfection size, and the column stiffness as $k_{total}=k_h+k_v$ with detailed forms for $k_v$ and $k_h$. Experimental validation across varied weaver dimensions demonstrates that buckling load scales as $P_{cr}\propto t_v^3$ (and with $t_h$) and that stiffness scales with $t_v^3$ and $t_h^2$, while the buckling mode can be steered by the relative widths of vertical and horizontal weavers. The study also classifies buckling into local and global modes, providing a linear boundary in the $(w_v,w_h)$ space, and offers design guidelines for optimizing performance of hierarchical 3D woven structures in applications such as soft robotics, wearable devices, metamaterials, and aerospace systems.

Abstract

Woven shell structures are beneficial for applications requiring lightweight, damage resilience, and design tunability, such as in wearable devices, soft robotics, and aerospace systems. A fundamental component of woven structures is the woven column. While the mechanical properties of a woven column can be determined using sophisticated finite element (FE) simulations, these FE models are computationally expensive and do not explain the underlying mechanics behind scaling relationships. In this work, we derive purely analytical models for the buckling load and stiffness of woven columns, and discuss the criteria that lead to different buckling modes of the woven columns. The simulated results based on our models closely match experimental data across various weave design parameters. This work advances our understanding of the mechanics of woven systems and serves as a baseline for the design of next-generation hierarchical structures and materials.

Figures (6)

• Figure 1: Overview of the woven columns and their buckling behaviors which are explored in this work. a. Fabrication of a plain woven column, which is a fundamental unit for 3D woven shell structures. We assemble the vertical and horizontal weavers (left) into a woven column (right) using a plain weave pattern. b. A comparison shows that a woven column (top) does not experience permanent damage after buckling, whereas a continuous column (bottom) made with the same amount of material experiences plastic deformation and fracture. c. Buckling modes of twelve different woven columns, where localized out-of-plane deformations of weavers contribute to buckling of the columns. The buckling pattern is dependent on the parameters of the vertical and horizontal weavers. Scale bars are 5 cm.
• Figure 2: Derivation of the stiffness and the buckling load of a woven column.a. Schematic of the vertical weavers. Each segment of a vertical weaver is modeled as an independent column undergoing buckling. To derive stiffness, the inner force and bending moment are analyzed at locations of maximum curvature. b. Schematic of the horizontal weavers. Horizontal weavers are modeled as axisymmetric sinusoidally perturbed cylindrical shells stacked on top of one another. The horizontal cross-section is approximated as a polygon with $n_v$ sides. Given a cross-section, the perturbation $\delta_h$ is derived by averaging $R_{max}$ and $R_{min}$. The contact area $A$ for deriving the horizontal stiffness is taken as the total contact area between adjacent horizontal weavers.
• Figure 3: Fabrication of woven columns and the test setup. a. Assembly of a column where vertical weavers are connected at the top and bottom. This design maintains consistent spacing in samples, but can only be used for testing in which the thickness of vertical weavers remains constant. Scale bars are 4 cm. b. Assembly of a column where vertical weavers are mechanically joined using split pins at the top and bottom of horizontal weavers. This design maintains consistent end effects, and is only used in a few of our tests where vertical thickness varies. Scale bars are 4 cm. c. Test setup using Mark-10® ESM 1500 single-column tabletop testing system. d. Schematic for plate-to-plate compression loading of woven columns. Vertical and horizontal weavers are both assumed to be deformed with sinusoidal deflections. e. A typical force--displacement curve obtained from an experiment with the buckling force and stiffness measured.
• Figure 4: Comparison of buckling modes. a. Depending on the widths of horizontal and vertical weavers in a woven column, the buckling mode of the column switches between global, local, and combination modes. b. Typical force--displacement curve for a column that experiences global buckling into a diamond pattern, with no pre-buckling behavior observed. c. Typical force--displacement curve for a column that experiences local buckling, where pre-buckling is observed, and local force maxima occur before a peak force is obtained.
• Figure 5: Influence of design parameters on buckling forces of woven columns. Buckling force with respect to: a. Vertical weaver thickness, b. Horizontal weaver thickness, c. Column height, d. Vertical weaver width, e. Horizontal weaver width. f. Schematic of all design parameters.