Tendon constrained inflatable architecture: rigid axial load bearing design case

The bladder material stiffness determines the maximum internal pressure, while its geometry impacts the rigid load-bearing capacity. A stiff bladder material is capable of withstanding high internal pressure due to reduced bladder wall deformation, in comparison to a soft bladder material. Thus, a stiff bladder material is capable of higher rigid load-bearing capacity, but the TCI would be adversely affected in its stowability. While soft bladders experience increased bladder tension and larger deformation for a given pressure, this can be mitigated by circumferential constraint rings. In addition, durability concerns such as fatigue or wear resistance must be considered (although are beyond the scope of this paper) and the selection of tougher, but less soft materials as well as protective layers can be employed to strike a balance between durability and performance. The geometric dimensions of the bladder also impact the performance of the TCI. The radius of the undeformed bladder determines the maximum load-bearing capacity with deformation for a given pressure. Additionally, the undeformed bladder radius, thickness, and height contribute to the TCI's rigid load-bearing capacity. A long, thick, and small bladder experiences reduced bladder tension and can withstand high internal pressure, but at the cost of stow capability. The optimal bladder geometry in relation to constraint geometries for a specified rigid load-bearing capacity must be selected by considering the bladder deformation due to pressure and load.

The interplay between the tendons' and the bladder's geometry and material determines the rigid load-bearing capacity and the deployed height. An inflatable without any constraints is synonymous with an inflatable constrained by tendons of infinite length. With infinite tendon length, the inflatable's load-bearing functionality and deploy-and-stow capability are determined solely by the bladder material and geometry. The bladder's deformation due to external load and maximum deployed size both depend on the bladder's material and geometry. On the other hand, with finite tendon length, the TCI's rigid load-bearing functionality (including the capacity and the threshold point) and deploy-and-stow capability are limited by the tendon length and material. The maximum rigid load-bearing capacity depends on the tendon's tensile strength because the TCI's rigid load-bearing capacity is equal to the tension of active tendons for a given pressure. The rigid load-bearing threshold point depends on the ratio between the tendon length and bladder height because not only does this ratio determine when the tendons will become active due to tension but it also impacts the TCI's deployment as ${l_{{\text{deployed}}}}$ depends purely on the tendon length assuming the soft bladder is capable of stretching to tendon length. These dependencies are only applicable for tendons with high stiffness and strength since low stiffness tendons will deform with loading and allow TCI axial displacement even when the tendons are active, while low strength tendons will not allow a high pre-tension. The internal axial constraint tendon properties must be selected based on desired rigid load-bearing functionality and the tendons' interaction with pressure and bladder.

The circumferential constraint rings' material and geometry as well as the number of constraint rings influence the bladder tension that determines the maximum applicable internal pressure. A ring is placed where the bladder experiences maximum circumferential bladder stretch to reduce the bladder wall area over which the pressure can act. To do so, the rings must have high stiffness to restrain inflation. Furthermore, if the ring radius is equal to the undeformed bladder radius, the ring becomes active at any nonzero pressure, and if the ring thickness is increased, it decreases bladder tension but reduces TCI's stow capability. Similarly, an increase in the number of constraint rings reduces the overall TCI's bladder tension but also reduces the TCI's stow capability. The selection of the number of constraint rings is a careful balance between stow capability and maximum allowable internal pressure.

To be able to accurately design and gain quantified insight into the relationship between all of these drivers, an analytical model is called for that relates the rigid axial load-bearing capacity as a function of the geometric variables and material properties, internally applied pressure, and external axial load. An unconstrained inflation model analyzes the bladder's deformation due to unloaded and loaded unconstrained inflation. Since the constraints depend on the bladder geometry, the unconstrained inflation model is further developed by incorporating the axial and circumferential constraints in the boundary conditions to derive the axial and circumferential constraint models. The two constraint models are combined to create the axial rigid load-bearing TCI performance model, which can be used to predict and design the axially rigid load-bearing capacity of a TCI. This model establishes the axially rigid structural support design space and provides the overall rigid axial load-bearing capacity of a TCI as a function of bladder and constraint properties to set the foundation for the analysis of tendon constraints on TCIs with multi-axial rigid structural support, controlled compliance, and posable functionalities.

The inflation of an elastic cylinder of finite length with end caps on each side can be predicted using different approaches. Kydoniefs and Spencer (1969) produced an exact solution for an initially cylindrical membrane deforming into an axisymmetric membrane. However, it is modeled specifically for one type of hyperelastic material model, Mooney-Rivlin, and has restrictions on the shape of the deformed membrane. Another solution was posed by Pamplona et al (2006), who use finite elements, but is computationally intensive for design. Instead, a promising approach by Yang and Feng (1970) that numerically solves the general axisymmetric nonlinear deformation is considered. Unfortunately, it does not account for the boundary conditions and geometry specific to a cylindrical bladder with end caps. This paper extends Yang and Feng's method by imposing TCI's boundary condition. A summary of Yang and Feng's method is discussed in sections 3.1.1.1 and 3.1.1.2, while a boundary condition specific to the cylindrical bladder with end cap, not presented in Yang and Feng's work, is developed in section 3.1.1.3.

3.1.1.1. Stretch kinematics.

The inflatable bladder depicted in figure 4 is a cylindrical tube of radius $\,r$, thickness $\,h$, and height ${l_{\text{B}}}$. The geometry of the nonlinear axisymmetric membrane is represented in polar coordinates. The cylindrical bladder axisymmetrically deforms with an increase in pressure as shown in figure 4. Throughout the deformation, the polar angle ${{\Theta }}\,$ of the differential element is constant, but the azimuthal angle $\theta$ is variable. The undeformed membrane is defined by coordinates $\left( {r,z,{{\Theta }}} \right)$ and the deformed membrane is defined by coordinates$\,\left( {\rho ,\,\eta ,\,{{\Theta }}} \right)$. The deformation of the membrane is defined by the stretch ratios ${\lambda _1},\,{\lambda _2},\,{\lambda _3}$ in the meridional, circumferential, and radial direction that relate the undeformed configuration to deformed configuration where stretch is related to strain by$\,\left( {1 + \varepsilon } \right)$. For the rest of this analysis, subscripts 1, 2, and 3 will denote meridional, circumferential, and radial directions. Each of the stretch ratios is defined as

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{\lambda _1} = \frac{{{\text{d}}S}}{{{\text{d}}s}} = {{\left( {{{\left( {\frac{{{\text{d}}\rho }}{{{\text{d}}z}}} \right)}^2} + {{\left( {\frac{{{\text{d}}\eta }}{{{\text{d}}z}}} \right)}^2}} \right)}^{\frac{1}{2}}}{\text{ and}}} \end{array}\end{equation} \tag{ 1 }$$

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{\lambda _2} = \frac{\rho }{r},\,} \end{array}\end{equation} \tag{ 2 }$$

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where ${\text{d}}S$ and ${\text{d}}s$ are the deformed and undeformed meridional arc lengths shown in figure 4. The inflatable bladder is assumed to be incompressible, and thus, the radial stretch ratio is defined by the constraint

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{\lambda _1}{\lambda _2}{\lambda _3} = 1.\,} \end{array}\end{equation} \tag{ 3 }$$

3.1.1.2. Bladder equilibrium.

The differential equilibrium in the meridional–tangential and the normal directions are expressed as

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{\frac{{{\text{d}}{T_{{\text{B1}}}}}}{{{\text{d}}\rho }} + \frac{1}{\rho }\left( {{T_{{\text{B}}1}} - {T_{{\text{B2}}}}} \right) = {\text{0 and}}} \end{array}\end{equation} \tag{ 4 }$$

$$\begin{equation}\begin{array}{*{20}{c}} {{K_1}{T_{{\text{B}}1}} + {K_2}{T_{{\text{B}}2}} = P,} \end{array}\end{equation} \tag{ 5 }$$

where ${T_{{\text{B}}1}}$ and ${T_{{\text{B2}}}}$ are the resultant bladder stresses in their respective directions with a dimension of force per unit membrane edge length; ${K_1}$ and ${K_2}$ are principal curvatures in their respective directions; and $\,P$ is the pressure acting in the normal direction of deformed membrane. The resultant bladder stresses are related to stretch ratios, ${\lambda _1}$ and ${\lambda _2}$ by a material constitutive model.

As the cylindrical bladder inflates, it takes on an oblate spheroid shape with its ends confined by the end caps. To predict the bladder shape, the deformation on the meridional and circumferential curvatures are expressed in terms of the deformed geometry $\rho$ and $\eta$ as

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{ }{K_1} = - \frac{{{\text{d}}\theta }}{{{\text{d}}S}} = { }\frac{{ - 1}}{{{{\left( {{\rho ^{\prime \,2}} + {\eta ^{\prime \,2}}} \right)}^{\frac{1}{2}}}}}\frac{{\text{d}}}{{{\text{d}}z}}\left[ {{{\cos }^{ - 1}}\frac{{ {- \rho}^{\prime} \,}}{{{{\left( {\rho^{\prime} {\,^2} + {\eta ^{\prime \,2}}} \right)}^{\frac{1}{2}}}}}} \right]{\text{ and}}} \end{array}\end{equation} \tag{ 6 }$$

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{K_2} = \frac{{{\text{sin}}\theta }}{\rho } = \frac{{\eta^{\prime}}}{{\rho {{\left( {{\rho ^{\prime \,2}} + {\eta ^{\prime \,2}}} \right)}^{\frac{1}{2}}}}},{ }} \end{array}\end{equation} \tag{ 7 }$$

where the prime denotes differentiation with respect to $z$. Using stretch equations (1) and (2), ${\eta}^{\prime} \,$ is expressed in terms of the stretch ratios,

$$\begin{equation}\begin{array}{*{20}{c}} {\eta^{\prime} {\,^2} = \lambda _1^2 - {r^2}\lambda _2^{\prime \,2}.{ }} \end{array}\end{equation} \tag{ 8 }$$

The deformed curvatures are defined in relation to stretch ratios by substituting stretch equations (1), (2), and (8) into the curvature equations (6) and (7), resulting in

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{K_1} = \frac{{r\lambda _2^{\prime} \,\lambda _1^{\prime} - r{\lambda _1}\lambda _2^{\prime\prime}}}{{\lambda _1^2{{\left( {\lambda _1^2 - {r^2}\lambda _2^{\prime \,2}} \right)}^{\frac{1}{2}}}}}{\text{ and}}} \end{array}\end{equation} \tag{ 9 }$$

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{K_2} = \frac{{{{\left( {\lambda _1^2 - {r^2}{\lambda}_{2}^{\prime \,2}} \right)}^{\frac{1}{2}}}}}{{r{\lambda _1}{\lambda _2}}}.} \end{array}\end{equation} \tag{ 10 }$$

Combining the differential equilibrium equations (4) and (5) with curvature equations (9) and (10) produces a set of three first-order ordinary differential equations in ${\lambda _1}^{\prime} \,,{ }{\lambda _2}^{\prime} \,,$ and ${\lambda _2}^{\prime\prime}$. Thus, the relationship between applied internal pressure and stretch ratios are defined.

3.1.1.3. Unloaded unconstrained inflation boundary conditions.

Yang and Feng (1970) had applied their axisymmetric nonlinear model to predict the deformation of an initially cylindrical model with rigid end caps due to external tension only and set one of the boundary conditions as the axial displacement. To apply the model to an inflatable with rigid end caps undergoing pressure inflation, boundary conditions were selected at the end cap and the equator. At the equator, the differential bladder element is vertical due to symmetrical inflation such that

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{\lambda _2^{\prime} = 0{\text{ at }}z = \frac{{{l_{\text{B}}}}}{2}.} \end{array}\end{equation} \tag{ 11 }$$

At an end cap, the bladder is held fixed to the end cap restricting circumferential stretch such that

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{\lambda _2} = 1{\text{ at }}z = 0.{ }} \end{array}\end{equation} \tag{ 12 }$$

The second boundary condition at the end cap is given by ${\lambda _1}$. During free inflation, pressure and the resultant bladder stress act on the end cap. The angle of the bladder tension with respect to the horizontal $\left( {\phi = \frac{\pi }{2} - \theta } \right)$ acting on the end cap is given by

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle {{\text{sin}}\phi = \frac{{{\text{d}}\eta }}{{{\text{d}}S}} = \frac{{{{\left( {\lambda _1^2 - {r^2}\lambda _2^{\prime \,2}} \right)}^{\frac{1}{2}}}}}{{{\lambda _1}}}.} \end{array}\end{equation} \tag{ 13 }$$

The equilibrium on the end cap is then defined as

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{\frac{{Pr}}{2} = {T_{{\text{B}}1}}{\text{sin}}\phi .} \end{array}\end{equation} \tag{ 14 }$$

Each of the first-order differential equations requires a boundary condition; however, they are not co-located, requiring a shooting method, a numerical analysis that solves the system of differential equations by reducing a boundary value problem to a system of initial value problems. The third boundary condition required at the end cap is determined by guessing ${ }{\lambda _2}^{\prime} \,$, applying an integration method such as Runge–Kutta up to the equator and checking whether the shooting target, i.e. equatorial boundary condition (equation (11)), has been satisfied.

Instead of a stiff bladder membrane that would be capable of withstanding high pressure with minimal deformation at the cost of stow capability, a soft hyperelastic bladder membrane is considered so that the bladder can compactly fold when stowed. To predict the membrane's deformation due to pressurization, a constitutive hyperelastic model based on a strain energy density function is used to relate the bladder stress resultant to the stretch ratios. There have been many efforts in modeling the behavior of a nonlinear membrane; however, since there is such a large variety of nonlinear materials, there are various classes of hyperelastic constitutive models. Each represents different properties that capture different material behaviors; however, Yang and Feng (1970) have only applied the Mooney–Rivlin model for their model demonstrations. A general procedure to capture the behavior of a hyperelastic bladder is to apply multiple hyperelastic material models. Three popular hyperelastic material models are considered for this paper to characterize a silicone bladder. Two material models are based on phenomenological descriptions of observed behavior, Mooney–Rivlin model (Mooney 1940, Rivlin 1948) and Ogden model (Ogden 1972), and one material model is based on a hybrid of phenomenological and mechanistic models, Pucci–Saccomandi model (Pucci and Saccomandi 2002). One of the three models is chosen according to their fit to the experimental data for further analysis of TCIs. The use of a different hyperelastic material may require a different hyperelastic model to characterize its behavior.

3.1.2.1. Hyperelastic material models.

This section briefly summarizes the equations of incompressible isotropic non-linear elasticity (Ogden 1997). The hyperelastic material model is based on the strain energy density function, which can be represented by the stretch ratios, ${\lambda _1},\,{\lambda _2},\,{\lambda _3}$, and also by two independent invariants, ${I_1},\,{I_2}$, which themselves are functions of the stretch ratios. The invariants are defined in terms of stretch by

$$\begin{equation}\begin{array}{*{20}{c}} {{I_1} = \lambda _1^2 + \lambda _2^2 + \lambda _3^2\ {\text{and}}\ {I_2} = \lambda _1^{ - 2} + \lambda _2^{ - 2} + \lambda _3^{ - 2}.\,} \end{array}\end{equation} \tag{ 15 }$$

Thus, the strain-energy density function can be represented as functions of either the stretch ratios or the invariants

$$\begin{equation}\begin{array}{*{20}{c}} {W\left( {{\lambda _1},\,{\lambda _2},\,{\lambda _3}} \right) = \bar W\left( {{I_1},{I_2}} \right).\,} \end{array}\end{equation} \tag{ 16 }$$

The resultant bladder stress with a dimension of force per unit edge length of a membrane ${T_{{\text{B}}i}}$, is related to the strain energy density function through

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{T_{{\text{B}}i}} = h{\lambda _3}\left( {{\lambda _i}\frac{{\partial W}}{{\partial {\lambda _i}}} - q} \right),} \end{array}\end{equation} \tag{ 17 }$$

where $i \in \left( {1,2} \right)$ and $q$ is a Lagrange multiplier associated with incompressibility constraint equation (3).

The three hyperelastic material models considered to characterize the behavior of the hyperelastic bladder are Mooney–Rivlin, Ogden, and Pucci–Saccomandi models.

• Mooney–Rivlin model (Mooney 1940, Rivlin 1948) considers the strain-energy density function to be
  $$\begin{equation}\begin{array}{*{20}{c}} {\bar W = \mathop \sum \limits_{\left( {i,j} \right)} {C_{ij}}{{\left( {{I_1} - 3} \right)}^i}{{\left( {{I_2} - 3} \right)}^j},} \end{array}\end{equation} \tag{ 18 }$$
  where $\left( {i,\ j} \right) \in \left\{ {\left( {1,0} \right),\,\left( {0,1} \right),\,\left( {2,0} \right)} \right\}$, which are commonly used coefficients, and $C$ is a material constant determined through experimental curve-fitting.
• Ogden model (Ogden 1972) considers the strain-energy density function
  $$\begin{equation}\begin{array}{*{20}{c}} {W = \mathop \sum \limits_i^N \frac{{2{\mu _i}}}{{{\alpha _i}}}\left( {\lambda _1^{{\alpha _i}} + \lambda _2^{{\alpha _i}} + \lambda _3^{{\alpha _i}} - 3} \right),\,} \end{array}\end{equation} \tag{ 19 }$$
  where $\alpha$ and $\mu \,$are experimental material constants and $N$ is generally considered up to 3.
• Pucci–Saccomandi model (Pucci and Saccomandi 2002) considers the strain-energy density function as
  $$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{\bar W = - \frac{{\mu {J_{\text{m}}}}}{2}\ln \left( {1 - \frac{{{I_1} - 3}}{{{J_{\text{m}}}}}} \right) + {C_2}\ln \left( {\frac{{{I_2}}}{3}} \right),{ }} \end{array}\end{equation} \tag{ 20 }$$
  where $\mu$ is the shear modulus, ${J_{\text{m}}}$ is the material constant related to limited extensibility of a molecular chain, and ${C_2}$ is additional empirical material constant. The model is a modification of the Gent model (Gent 1996) by adding a logarithmic term. The original Gent model corresponds to ${C_2} = 0$. Thus, the Pucci–Saccomandi model is also commonly referred to as the Gent–Gent model.

3.1.2.2. Material coefficient characterization tests.

Material constants of the constitutive models are commonly found through experimental tests such as tensile and compression tests. Since the hyperelastic bladder undergoes multi-axial tension during inflation, uniaxial and biaxial tension tests are most applicable for determining the material constants. The uniaxial tension test defines one parameter and the biaxial tension test defines two parameters for the material constants in a hyperelastic model. These two tensile tests with a total of three dimensions can provide for all three constants of Mooney–Rivlin and Gent–Gent models; however, six independent mechanical tests are difficult to achieve for all Ogden's material constants. Therefore, for the given number of characterization tests, the Ogden model may be limited in providing accurate material constants to predict the bladder membrane's deformation. Each of the tensile tests, including fabrication and experiment, is discussed and applied to the hyperelastic models to yield material constants.

3.1.2.2.1. Uniaxial tension test.

The purpose of the uniaxial tension test is to produce a stress-stretch curve by applying load in one direction, resulting in ${ }{\lambda _2} = {\lambda _3}$ due to isotropy. The uniaxial true stress is calculated using the load applied to the specimen and the specimen's deformed cross-sectional area, while the uniaxial stretch is calculated using the change in length of the specimen due to load. To perform a uniaxial tension test, a dumbbell-shaped sample is required. To fabricate the sample, the two-part platinum curing silicone called DragonSkin10 Slow of durometer 10A from Smooth-On is mixed, degassed inside a vacuum chamber, and cured to shape using a 3D printed dumbbell shape mold (figure 5(a)). This results in a dumbbell-shaped silicone membrane of a total 6'' length, 1'' width, and 0.125'' thickness. Figure 5(b) shows the experimental set-up with one end of the dumbbell is held fixed, while the other end is loaded in the axial direction using weights. The dumbbell's sampled section is 1'' length, 0.5'' width, and 0.125'' thickness. The changes in length, height, and width of the sample's central section were measured using calipers as 20 g (0.04 lbs) weights were incrementally added until the sample stretched to 3.5 times its original length. These results are used to characterize one material constant for each hyperelastic model.

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3.1.2.2.2. Biaxial tension test.

The purpose of the biaxial tension test is to produce two stress-stretch curves by applying loads in two directions. A common method of applying biaxial tension on nonlinear membranes is through an inflation test (Li et al 2001, Mott et al 2003). Thus, a cylindrical bladder with end caps is inflated to measure the meridional and circumferential stretch of a differential element at its equator. In this biaxial test, the axial (meridional) stress seen by the differential element is determined by a force equilibrium in an axial direction at the equator,

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{{\sigma _{{\text{axial}}}} = \frac{{P{\lambda _2}r}}{{2{\lambda _3}h}},} \end{array}\end{equation} \tag{ 21 }$$

where ${\lambda _3}$ is calculated from incompressibility constraint equation (3) using the circumferential and meridional stretch. The circumferential stress ${\sigma _{{\text{circ}}}}$ is derived from the bladder differential equilibrium in the normal direction equation (5). To perform this biaxial tension test, a cylindrical bladder with end caps is required. The bladder is composed of the same silicone from the uniaxial tension test. The mixed silicone is degassed inside the vacuum chamber and poured between two acrylic cylinders of 4.75'' and 5.25'' outer diameter with 0.125'' thickness each that are held concentrically using spacers at the ends. The mixture is cured into a silicone bladder of 3.7'' length, 5'' outer diameter, and 0.125'' thickness (figure 5(c)). The two end caps are 3D printed using polylactic acid (PLA) of 5'' outer diameter and 0.5'' height. The bladder is attached to the end caps using hose clamps. For the biaxial tension test, one end cap is held fixed, while the other end cap is free to move with an increase in pressure. The cylindrical bladder's sampled section is a 0.20'' square marked at the equator in figure 5(d). The changes in horizontal and vertical dimensions of the sampled section and the meridional and circumferential curvatures were measured as the bladder was pressurized from 0 to 1.30 psi by 0.1 psi increments using photogrammetry. These results are used to characterize two material constants for each hyperelastic model.

3.1.2.2.3. Resulting hyperelastic model fit.

Figure 6 represents the least-squares curve fit for the Mooney–Rivlin, Ogden, and Gent–Gent model using the experimental uniaxial and biaxial tension test data set. The curve fit results in material parameters ${C_{1,0}} = 3.1,\,{C_{0,1}} = 0.4,$ and ${C_{2,0}} = 0.25$ for Mooney–Rivlin, ${\mu _1} = 373,\,{\mu _2} = 0,$ ${\mu _3} = 0.04,\,{\alpha _1} = 0.07,\,{\alpha _2} = 0,$ and ${\alpha _3} = \,7.4$ for Ogden and $\mu = 6.34,{ }{J_{\text{m}}} = 13.6,{ }$ and ${C_2} = 6.95$ for Gent–Gent. The average relative errors of the curve fit to the uniaxial tension and biaxial tension in axial and circumferential directions are 18%, 17%, and 20% for Mooney–Rivlin, 16%, 14%, and 16% for Ogden, and 7%, 8%, and 5% for Gent–Gent, respectively. The hyperelastic models fit the experimental data relatively well. However, it is evident in figure 6 that Mooney–Rivlin and Ogden models are unable to accommodate the rapid change in stress at large deformation, while Gent–Gent provides a close match with the lowest error. These differences in behaviors become more apparent when the models are applied to predict the deformation of the cylindrical bladder's differential element due to pressure.

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3.1.2.3. Hyperelastic material model comparison.

The stretch deformations with respect to pressure are found for each of the models in figure 7 from 0 to 1.3 psi with pressure increments of 0.1 psi. Of the classes of hyperelastic material models, the Gent–Gent model represents the best fit to experimentally measured stretch ratios with an average relative error of 4%. Mooney Rivlin and Ogden models match the experimental data with an average relative error of 9% and 13%. Although all the hyperelastic models predict small deformation well, as the pressure increases, the Mooney Rivlin model incorrectly predicts the inflation trend, while the Ogden model also incorrectly predicts the stretch at which the material becomes stiff from overstretching. For the rest of the paper, the Gent–Gent model with coefficients of $\mu = 6.34,{ }{J_{\text{m}}} = 13.6,{ }$ and ${C_2} = 6.95$ is considered for the silicone used in this experiment. Other bladder materials would result in different best fit hyperelastic material models, but the same approach can be applied to determine the best fit model.

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With the adapted axisymmetric nonlinear deformation model and the Gent–Gent hyperelastic material model, the unloaded unconstrained inflation model can predict the deformation of the bladder with end caps due to pressure. To validate the unloaded unconstrained inflation model, the experiment uses the same prototype and experimental set-up as the biaxial tension test. However, instead of the differential element's change, the changes in global height and radius are considered. Thus, a cylindrical bladder with end caps of height 3.7'', diameter 5'', and thickness 0.125'' was pressurized from 0 to 1.1 psi and its change in height and radius were measured using calipers. The theoretical deformed bladder radius is defined by equation (2) and the theoretical deformed bladder height ${L_{\text{B}}}$ is defined by

$$\begin{equation}{L_{\text{B}}} = 2\int\limits_{z = 0}^{z = \frac{{{l_{\text{B}}}}}{2}} {} \eta^{\prime} {\text{d}}z\end{equation} \tag{ 22 }$$

where $\eta^{\prime} \,$ is defined by equation (8) and the integral interval is between the end cap and the undeformed equator. The test data is shown in figure 8 as data points and the lines represent the theoretical results using the Gent–Gent model with an average relative error of 2.5% with the correct inflation trend at small and large deformation. With this experimental validation, the unloaded unconstrained inflation model can accurately predict the inflation of hyperelastic cylindrical bladder with end caps and serves as the base model for the loaded unconstrained and constraint models.

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The boundary condition of the bladder with applied axial load is similar to that of the bladder at free inflation. The geometric boundary conditions in equations (11) and (12) remain the same and the end cap equilibrium boundary condition, equation (14), has an additional term related to the axial load

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{\frac{{Pr}}{2} = {T_{B1}}{\text{sin}}\phi + \frac{F}{{2\pi r}},} \end{array}\end{equation} \tag{ 23 }$$

where $F$ is the axial load applied to the end cap.

To experimentally validate the loaded unconstrained inflation model, the deformation of a pressurized bladder due to external load is measured. This experiment uses the same prototype as the biaxial tension test, a cylindrical silicone bladder of 3.7'' height, 5'' diameter, and 0.125'' thickness, with 3D printed PLA end caps of 5'' outer diameter and 0.5'' thickness. The inflated bladder with end caps in figure 5(d) is axially deformed using a micrometer (BM 25.40) with 0.1 mm resolution (0.0039'') attached to a miniature s-beam FUTEK load cell with 50 lbs capacity on one end cap, while the other is fixed as shown in figure 9. The prototype was inflated to three pressure values: 0.2 psi, 0.5 psi, and 0.7 psi, and the micrometer stage was moved down incrementally by approximately 1 lbf up to approximately 4 lbf, 10 lbf, and 14 lbf, the given pressure times the area of the end cap. Thus, the axial compliance of the bladder was experimentally validated as shown in figure 10 with an average error of 2.2% and 2.4% for radius and height in terms of undeformed bladder radius and height. The changes in height and radius were measured to decrease and increase respectively as the external load was incrementally increased. Thus, the loaded unconstrained inflation model can accurately predict the deformation of an unconstrained bladder due to external axial load and pressure. With the loaded unconstrained inflation model, the axial and circumferential constraints can be applied to predict the TCI rigid load-bearing capacity.

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To determine the rigid load-bearing capacity, the tendon tension ${T_{\text{T}}}$ for a given pressure with no applied external load $F = 0$ needs to be solved for. There are two main changes to the loaded unconstrained inflation model: the boundary condition and the numerical analysis procedure. The boundary condition at the end cap, equation (14), is expanded to include tendon tension ${T_{\text{T}}},$

$$\begin{equation}\begin{array}{*{20}{c}} \displaystyle{\frac{{Pr}}{2} = {T_{{\text{B}}1}}{\text{sin}}\phi + \frac{F}{{2\pi r}} + \frac{{{T_{\text{T}}}}}{{2\pi r}}.} \end{array}\end{equation} \tag{ 24 }$$

As external load is applied onto the end cap at constant pressure, the tendon tension decreases to maintain the boundary condition equilibrium in equation (24), while bladder tension remains constant because it is assumed in the model that the bladder deformation is negligible as long as the tendons are in tension.

Additionally, the numerical analysis procedure, applied to solve the unconstrained inflation model, must be adjusted to solve for the unknown ${T_{\text{T}}}$ added to the boundary condition using the axial constraint condition: the TCI's height ${L_{\text{B}}}$ must equal to tendon length ${l_T}$ when the tendons are active. Thus, in addition to the shooting method to find $\lambda _2^{\prime}$, a new shooting method is required to guess the tendon tension needed to maintain the deformed bladder height ${L_{\text{B}}}$ equal to the tendon vertical length ${l_{\text{T}}}$ (the shooting target), while assuming there is no applied external load $F = 0$. The first shooting method for $\lambda _2^{\prime}$ is nested inside the second shooting method and the second shooting method only becomes active after the deformed bladder height has reached the internal tendon length, the rigid load-bearing threshold point. These additions result in the internal axial constraint model.

The rigid load-bearing threshold point and capacity are determined by the ratio between the internal axial constraint length and the undeformed bladder height as a function of pressure. To understand and explore this relationship, a parametric study exploring different ratios of internal axial constraint length to undeformed bladder height was conducted theoretically and validated experimentally. The theoretical rigid load-bearing capacity was calculated by applying the internal axial constraint model and the experimental rigid load-bearing capacity was determined by measuring the TCI deformation due to load and calculating the rigid load-bearing threshold point, where TCI's stiffness drastically changes due to the tendons becoming inactive. The theoretical and experimental rigid load-bearing capacities were calculated and analyzed for three different tendon lengths less than (83%), equal to (100%), and greater than (103%) the undeformed bladder height at varying pressures.

The theoretical rigid load-bearing capacity was computed using the axial constraint model including the extended boundary condition in equation (24) and the adjusted numerical analysis procedure. The experimental rigid load-bearing capacity was determined by measuring the external load required for the tendons to just become slack. Thus, the threshold point is located where the change in the slope of the load–displacement curve occurs as the tendons become inactive. Experiments are performed using a TCI composed of the same cylindrical silicone bladder, 3D printed PLA end caps of 5'' outer diameter and 0.5'' thickness with four tendon attachment loops, and four polyethylene braided fishing lines of 0.4 mm diameter and 50lbs test from KastKing. Each end of the tendons is fastened to a loop at the upper and lower end caps to form four straight tendons of equal lengths connecting the upper and lower end caps. The bladder wraps around the tendon structure and the end caps and is attached to the end caps using hose clamps, forming a TCI prototype. Three TCI prototypes were fabricated of tendon lengths 83%, 100%, and 103% of the 3.7'' undeformed bladder height. These inflated TCI's were axially deformed using the same set-up for axially deforming an unconstrained inflatable in figure 9 with the addition of photogrammetry to measure the small deformation when tendons are active. The change in height was measured at various pressures of 0.2 to 0.8 psi range.

At each incremental pressure value, a TCI for a given ratio of tendon length to undeformed bladder height is axially loaded to measure its displacement to produce a force–displacement graph. One example of a force–displacement plot for a TCI with tendon length 83% of the undeformed bladder height at 0.5 psi is shown in comparison to an unconstrained inflatable (a TCI with infinite tendon length) with equal bladder and end cap geometry in figure 11. The deformation results of an unconstrained inflatable are used from the loaded and unconstrained inflation model validation case in figure 10. Figure 11 illustrates two key comparisons: (a) between active and inactive tendons in a constrained inflatable and (b) between a constrained and an unconstrained inflatable. As deformation is induced onto the TCI, the slope of the linear relationship between the applied load and the bladder height displacement drastically changes once the tendons become slack. For this particular TCI geometry and pressure, the rigid load-bearing capacity is defined by the intersection of the linear fits to active and inactive TCI shown as solid and dashed lines in figure 11. The capacity is represented as a red asterisk at 9.7 lbf with 0.06'' displacement. The TCI's initial stiffness is 179 lbf in⁻¹ when the tendons are taut but reduces to 5.3 lbf in⁻¹ when the tendons become slack; TCI with active tendons have a stiffness of 33.8 times that of a TCI with inactive tendons. Similar to the stiffness of TCI with slack tendons, the unconstrained inflatable's stiffness is 5 lbf in⁻¹, and at a force of 9.7 lbf, it experiences a total displacement of 1.1'' as shown in figure 11. The TCI with active tendons experiences only 5% of the axial displacement experienced by an unconstrained inflatable for an applied load of 9.7 lbf for this specific example. These comparisons demonstrate the TCI's rigid load-bearing functionality and improvement in stiffness compared to an unconstrained inflatable.

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The overall impact of tendons attached to the end caps inside an inflatable for different tendon length to undeformed bladder height ratio and pressure are shown with theoretical and experimental tendon tension in figure 12. The three sets of data demonstrate tendon length of 83%, 100%, and 103% of the undeformed bladder height. The theoretical results are shown as curves and experimental results from measuring rigid load-bearing capacity using the process shown in figure 11 are shown as circles. The internal axial constraint model is experimentally validated with a relative error of 1.4%, 19%, and 4.4% for tendon length of 83%, 100%, and 103% of the undeformed bladder height and total average relative error of 8.3%. The relative error for tendon length of 100% of undeformed bladder height is especially high due to small tendon tension values at low pressures although the absolute error remains small. The plot represents the rigid load-bearing capacity for given pressure input and tendon length. The current model for the TCI is only considered up to when the external load is equal to the force of the pressure, represented as the shaded region in figure 12 because when the bladder shape assumes a 'C' shape with excessive applied external force, it would begin to collide with itself when the constraint rings are implemented as well as collide with the external environment. The solid lines in the graph represent the theoretical tendon tension of the TCI designed for a given ratio of tendon length to undeformed bladder height from the internal axial constraint model and show that as pressure increases, the rigid load-bearing capacity increases as well. Thus, for a given pressure, a TCI can support a load with minimal deformation up to the rigid load-bearing capacity and will transition to a flexible unconstrained state for external loads greater than the rigid load-bearing capacity. A TCI at the overloaded deployed state or an inflated bladder of equal geometric dimensions can support a load with significant deformation up to the excessive force bound. The parametric study shows a general trend of increasing tendon tension for a given pressure as the tendon becomes shorter than the undeformed bladder height because the tendons are engaged from lower pressures. However, tendons shorter than the undeformed bladder height is not necessarily always advantageous as the TCI intersects the excessive force bound at lower pressures. When the tendon length equals the bladder height, the line intercepts with the origin. The x-axis intercept of the ratio for longer tendon than bladder height indicates that the tendons inside the TCI are only engaged from 0.38 psi. Thus, the TCI rigid load-bearing capacity is dependent on the tendon length and is a crucial design feature.

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To predict the deformation of the bladder with constraint rings, each ring is assumed to be inextensible and takes away a small finite height of the bladder. The model considers each ring to act as an end cap. Considering that the rings are equally spaced, each section between two circumferential constraints (the rings or the hose clamp at the end caps) is assumed to deform equally. The bladder tensions acting on the circumferential constraint rings by the resulting bladder sections are equal as shown in figure 13. As well, the bladder tension defined by the end cap equilibrium in equation (14) is equal to the bladder tension acting on the circumferential constraint rings. Thus, the same boundary condition for the end cap can be considered for the circumferential constraint rings. The deformation of a bladder section is calculated, and the height of each bladder section is summed to calculate the total change in height:

$$\begin{equation}{L_B} = 2n\int\limits_{z = 0}^{z = \frac{{{l_{\text{B}}}}}{{2n}}} {\eta^{\prime} {\text{d}}z} \end{equation} \tag{ 25 }$$

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where $n$ is the total number of bladder sections. The integral is calculated on the interval between an end cap and an equator for one bladder section since each bladder section is symmetric over the equator and has equal deformations.

The impact of number of constraint rings on the bladder's deformation due to pressure was explored by conducting a parametric study theoretically and validated experimentally. The theoretical and experimental bladder deformations were calculated and analyzed for a bladder with 0, 1, and 3 rings. Experiments are performed using the cylindrical silicone bladder of 4.3'' in height, 5'' in diameter, and 0.125'' in thickness and 3D printed PLA end caps of 5'' outer diameter and 0.5'' thickness. The rings are made of polyethylene braided fishing lines of 0.4 mm diameter with a 50 lbs test from KastKing. For each ring, the fishing line is wrapped around five times, tied to the bladder, and is fixed in place by coating the fishing line with the bladder material, DragonSkin10 SLOW. The change in height and radius were measured at 0.2 psi increments from 0 to 1.8 psi using photogrammetry.

The impact of rings on bladder deformation is shown with theoretical and experimental results in figure 14 for bladder with 0, 1, and 3 rings. Thus, the circumferential constraint model was experimentally validated with average errors of 2% and 3% for radius and height in terms of undeformed bladder radius and height. The model predicts that radial inflation decreases as the number of constraint rings increases as shown in figure 14(a). The bladder's height, on the other hand, is predicted to have a non-monotonic relationship with the number of rings depending on pressure. There are three operating regions shown in figure 14(b) and depending on the operating range, the optimal number of rings varies. At low pressures of 0 to 0.65 psi ①, a greater change in height is achieved by adding more rings onto the bladder because of the reduction in area for pressure to be applied. At intermediate pressures of 0.65 to 2.2 psi ②, the greatest change in height is achieved by no rings because the instability present for 0 rings drastically increases the bladder's height. At large pressure ③, the greatest change in height is achieved by increasing the number of rings with increasing pressure because of the residual impact from the change in height instability region. The constraint rings provide a valuable contribution to reducing the instability of change in height with respect to pressure by reducing the stretch in the material. Thus, for the same pressure, the bladder can inflate safely without a sudden change in the material stretch. To design, the number of rings must be considered with respect to the pressure for the desired operating range.

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The axial mode rigid load-bearing capacity is dependent on the bladder material parameter and the TCI's design parameters including the tendon length, the number of rings, the undeformed bladder radius, thickness, and height. A design plot is computed over one of these architectural parameters and a family of curves for varying values of the chosen geometric parameter is produced to represent the TCI's design space. Each of the curves is computed by predicting the deformation of TCI using the constraint models. The computation involves two shooting methods for the tendon tension and the differential of circumferential stretch. Depending on if the tendons are inactive or active, the tendon tension is initially assumed zero or a nonzero value, while the differential of circumferential stretch is assumed a value greater than or equal to 1. To predict the deformation for applied loads, the ordinary differential equations, based on the force equilibrium of the differential bladder membrane, are solved. To verify that the shooting targets have been met, the differential of the circumferential stretch at the equator is checked against its boundary condition and the axial constraint condition, requiring the total deformed bladder height is equal to or less than the tendon length for active and inactive tendons is checked. The ordinary differential equations are solved at incremental pressure until the TCI meets one of the two operating limits, the excessive force or a failure bound. The excessive force bound is when the tendon tension is greater than the load-bearing capacity, the net force due to pressure $P\pi {r^2}$. A failure bound is when the TCI breaks down due to causes such as mechanical failure, material failure, etc, and can be defined through experiments. A TCI's durability should also be considered for failure. If the bladder membrane fatigues or rubs against the external environment during operation or if the tendons wear due to continuous high tensile loads, different failure bounds should be considered based on a characterization of those failure modes as a function of inflation and loading compared to required lifetime. The resulting design space plot provides visualization for choosing a TCI design and operation parameters while staying safely within the operating bounds by visualizing the operation range and design margins.

An example design plot with a family of curves for varying undeformed bladder height is shown in figure 15 for a TCI composed of an undeformed silicone (DragonSkin10 Slow) bladder of 5'' diameter and 0.125'' thickness, inextensible tendons of 7'' length with 1 external circumferential ring. The design space is enclosed by the left diagonal excessive force bound and the right curved failure bound. For the tested prototype, the first mode of failure is due to the bladder-hose clamp failure. At experimental bladder tensions of 7 lbf in⁻¹ at the end cap, the bladder slips from the hose clamp holding the membrane and the end cap together. The shape of the curves and their intersection with the bounds depend on the TCI's undeformed bladder height. Undeformed bladder height longer than the tendon operates within or near the excessive force region, while heights equal to the tendon length, represented by the 7'' curve, inflates and supports loading but is limited by the excessive force region. Undeformed bladder heights shorter than the tendon length, on the other hand, require initial finite pressure for tendons to become active. The location of these intersections of the curves with excessive force bound occurs at larger rigid load-bearing capacity and internal pressure until a critical bladder height of 3.7'', shown as a red bold curve, is reached, at which the curve intersects both bounds in figure 15. If the undeformed bladder height is shorter than the critical bladder height, the curves begin to intersect the hose clamp-bladder slip bound because of an increase in bladder stretch required to activate the tendons and a decrease in its rigid load-bearing capacity. This design space plot illuminates the existence of a critical bladder height, which is important because it provides the largest achievable tendon tension for the given architecture.

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Additionally, the design plots are representations of the operation range and the design margins. The x-intercepts of the curves signify the tendons becoming active and the beginning of the TCI's rigid load-bearing operation range, while the curves' intersection with the bounds signifies the maximum rigid load-bearing capacity for a given architecture and the end of the TCI's rigid load-bearing operation range. The family of curves allows flexibility in the selection of bladder height and operating pressure in meeting design requirements. For example, the x-axis of a TCI with a bladder height of 3.7'' indicates the tendons engage at 1.4 psi and the TCI can support from 0 to 61 lbf from 1.4 to 3.2 psi. As well, for the same bladder height, if, for example, the TCI is operating at 2.5 psi to support 40 lbf, the plot shows the pressure margin and the bladder height tolerance. The TCI can operate up to a pressure of 3.2 psi without failure, while the undeformed bladder height can vary from approximately 3.4'' to 4'' to support the required loading as well. This analysis can be extended to other TCI's of different geometry to identify the operation range and design margins.

While the single design plot enables the analysis and selection of a single geometric parameter to achieve the best rigid load-bearing capacity; a series of design plots illustrate the rigid load-bearing capacity's dependence on multiple geometric parameters and provides flexibility in the selection of TCI's design parameters. For example, the undeformed bladder thickness and height are coupled in their impact on the rigid load-bearing capacity through the boundary conditions and the differential equations. A simple method of determining the individual and combined impact of these geometric parameters on rigid load-bearing capacity and their interaction with the operating bounds is by using a series of design plots over undeformed bladder thickness and height.

Figure 16 shows the rigid load-bearing capacity for a TCI of 5'' diameter, 7'' tendon length, and 2 rings at undeformed bladder height of 3.7'' and 5'' and varying undeformed bladder thickness. The impact of the first geometric parameter represented as a family of curves for varying undeformed bladder thickness can be determined through a single design plot. As shown in figure 16(a) or figure 16(b), a thin bladder requires only initial finite pressure to engage the tendons, but are limited by the excessive force bound, while a thick bladder requires additional pressure to engage the tendons, but are limited by the failure bound. The impact of the second geometric parameter, the undeformed bladder height, is determined by comparing the curves of the design plots. A curve of equal thickness in figures 16(a) and (b) show that as height is increased, the tendons engage at lower pressures due to a decrease in the bladder volume available to stretch. The impact of both geometric parameters is determined by comparing the most appropriate thickness for each of the design plots. In figure 16, the best rigid load-bearing capacity for an undeformed bladder height of 3.7'' is a thickness of 0.125'', while for an undeformed bladder height of 5'' is a thickness of 0.325''. With an increase in height, an increase in thickness is required to achieve the best rigid load-bearing capacity for a given architecture and the plots illustrate the balance required in geometric parameter values to provide good performance.

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The series of design plots in figure 16 enables the selection of either a thick and long bladder for high performance but at a cost of deploy-and-stow capability due to an increase in bladder material or a thin and short bladder with rings to achieve the same performance, while maintaining the deploy-and-stow capability. This analysis can be further extended to all design parameters. These plots simplify the understanding of the intertwined relationships between rigid load-bearing capacity and TCI's design parameters and enable a straightforward selection of appropriate TCI's design parameters by providing a new method of comparing the rigid load-bearing capacity's dependence on these parameters.

With the set geometric values for undeformed bladder radius of 5'', thickness of 0.125'' and tendon length of 7'', a series of design plots over undeformed bladder height and number of rings, figure 17, is generated. All of the design plots indicate that an undeformed bladder height of approximately 3.7'' is the most appropriate bladder height as the 3.7'' undeformed bladder height curve intersects both of the failure bounds. The most appropriate bladder height is drawn with the 100lbs load-bearing capacity requirement in figure 17 as red bold curved and horizontal lines, respectively. As the number of rings is increased from figures 17(a) to (d), the overall load-bearing capacity is increased as the TCI's design space expands with increasing applicable pressure. Both the design space plots of 0 and 1 ring, figures 17(a) and (b), show that they do not meet the specifications of the maximum supported load at the inflated bladder height because there is no intersection between the 3.7'' undeformed bladder height curve and the load-bearing capacity requirement. The design space plot of 2 rings, figure 17(c), indicates that the design specifications are met; however, the operation location is near the limits of the operating bounds of the design space. Instead, the design space plot of 3 rings, figure 17(d), demonstrates that the TCI can handle larger variances in undeformed bladder height, pressure, and load, while operating within the bounds, providing a higher safety margin. Using the intersection of the optimal tendon length curve and required 100 lbs rigid load-bearing capacity line, the design plots indicate that the TCI with 3 constraint rings should operate at 5.5 psi to stably support 80 lbf external load.

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To validate that the TCI with these design choices can rigidly support 80lbs, a sample prototype was fabricated and experimentally loaded to confirm the TCI's rigid load-bearing functionality. A TCI with the same silicone bladder, end cap, and tendon material from the axial constraint model validation was used along with the rings used in the experimental validation of the circumferential constraint model. The undeformed bladder was 3.7'' in height, 5'' in diameter, and 0.125'' in thickness, the end caps were 5'' in diameter and 0.5'' in thickness, while the tendons were 7'' in length. The tendons connecting the upper and lower end caps were configured to the truss design shown in figure 1 to provide axial rigidity and lateral stability when the load is applied. Thus, the TCI with three rings was fabricated to operate at a pressure of 5.5 psi to provide 100 lbf rigid load-bearing capacity to support 80 lbf external load with additional lateral stability.

Figure 18 shows one such TCI in the vacuumed stowed state, rigid constrained deployed state without load, and rigid constrained deployed state with a load of 80 lbs. In the vacuum stowed state, the TCI was reduced to the total end cap height of 1'' containing the bladder, tendons, and rings (figure 18(a)). In the rigid constrained deployed state without load, the TCI deploys by 7'' and reached the fully deployed height of 8'' with active tendons (figure 18(b)). Due to limitations of in-house manufacturing, the four tendons have slight variations in lengths resulting in a slightly slanted deployed TCI. In the rigid constrained deployed state with load, the TCI still maintains its height of 8'' (figure 18(c)). Note that the difference between the deployed unloaded and loaded state heights is indistinguishable as indicated by the red horizontal dashed line showing rigid axial load-bearing functionality with minimal axial compliance. These three images experimentally demonstrate the TCI's compact stow capability of 1'' height, deployability through deployment ratio of 8, and rigid load-bearing functionality in a relatively simple inflatable architecture. Despite the limitation of TCI's stow capability due to the rigid end caps, its stow package is only 1.1 times that of an inflatable structure of the same architecture but with flexible silicone end caps.

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Not only can the modules support 80lbs but also the same modules can operate at lower pressures to support smaller loads at the same deployed height. To support a person of 120lbs with 4 TCIs, each TCI needs to support 30lbs, and an extra 20lbs per TCI is considered for lateral stability. According to the design plot in figure 17(d), for a load-bearing capacity of 50lbs and undeformed 3.7'' bladder height, the TCI needs to be pressurized to 3.75 psi. Four TCI modules were assembled and pressurized to demonstrate their rigid load-bearing performance in figure 19. With the designed TCI platform, not only is the inflatable capable of withstanding axial load but also it can be considered as a structural element as the TCIs do not deform in height despite the person moving on the platform. These example studies demonstrate the usefulness of the TCI design space plots for designing an axial mode rigid load-bearing TCI to adjust performance to meet different design specifications and also the versatility of TCI design plot to consider different and multiple TCI design parameters to achieve the best performance. The TCI prototype demonstrated and validated that the trade-off between functionality and deploy-and-stow capability can be broken through the TCI's rigid load-bearing functionality and its expansive deployment with minimal volume stow using a simply constrained inflatable architecture.

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