Inextensibility and Its Effect on the Number of Equilibria of Shallow Buckled Beams

Shallow arches are known to exhibit complex non-linear behavior when subjected to transverse loads. These relatively simple structures—which underlie more complicated civil, mechanical, aerospace, and biological systems—have a rich history of research (e.g., Refs. [1–6]), yet continue to yield new and exciting results for a variety of boundary and loading conditions (e.g., Refs. [7–15]). Recent interest has stemmed in part from “smart” devices, energy harvesting, and morphing systems that harness buckling and bistable post-buckling behavior [16–18].

In a recent study [19], a relationship was discovered between the Euler buckling loads of an initially straight beam and the lateral force–displacement curve of a buckled beam under lateral loading. In particular, it was shown that the number of unstable equilibria could be determined purely based on the buckled beam’s geometry. The number of unstable equilibria was related to the number of “loops” in the lateral load–deflection relation, quantified by the number of equilibria where the transverse force was zero (called “zero-transverse-load crossings”). This number was determined from the “squash” load at which the extensible beam would flatten to conform to the prescribed end displacement. This paper seeks to prove that the number of loops, and correspondingly the number of zero-transverse-load crossings, is infinite for the inextensible beam. Furthermore, the previous study [19] relied on numerical (finite element method (FEM)) results to demonstrate the hypothesized relationship, whereas this paper derives the result analytically.

The following work considers an elastic, inextensible, initially straight beam that is buckled to give a pre-stressed, shallow arch [11], as opposed to a curved, stress-free arch [20]. It is hoped that this work can provide insight on the behavior post-buckled structures, approximate preliminary design equations, and motivate extensions for other boundary conditions. The analysis of the buckled beam is formulated in Sec. 2. Using a constrained energy minimization approach, the lateral load–deflection relation is derived, leading to a proof of the infinite number of zero lateral-load equilibria for the archetypal case of the pin-ended, inextensible, buckled beam. Some results are presented and discussed in Sec. 3, followed by a comparison to the extensible case in Sec. 4. Section 5 gives some concluding remarks.

Consider the initially straight, inextensible beam of length L, shown in Fig. 1(a). It is assumed homogeneous and uniform with flexural rigidity EI. When subjected to a prescribed end displacement of Δ₀ (Fig. 1(b)), a shallow lateral deflection y₀(x) results, with an associated rise of H at midspan. Now a lateral constraint is imposed at a point $x¯$ along the length of the buckled beam such that the lateral deflection $y(x¯)=A$ (Fig. 1(c)), where y(x) is the general deflected shape and A is the amplitude of the deflected shape at $x¯$⁠. This constraint is taken here to be “knife-edge” (Fig. 1(c)), where the rotation at $x¯$ is not hindered.

Finally, note that the physical interpretations of μ and λ are the axial load required to induce Δ = Δ₀ and the lateral load required to induce $y(x¯)=A$⁠, respectively. Because of the way the constraints were augmented to U (see Eq. (11)), the axial load μ is positive in tension and the lateral load λ is positive upward (i.e., opposite as drawn in Fig. 1(c)).

In this section, some results for the pin-ended beam are shown to demonstrate the theory developed in Sec 2. The results are presented primarily in the form of normalized lateral load–deflection (λ–A) relations. Additionally, the influence of increasing the number of modes retained in the expansion [Eq. (15)] on the lateral load λ and axial load μ is presented.

Representative equilibrium configurations are shown for seven equilibria (i–vii) along one of the detached equilibrium paths (gray) for $x¯=0.45L$ (Fig. 2(c)). The initial equilibrium configuration (ii) corresponds to the first Euler buckling mode, for which λ = 0 and A/H is slightly less than 1 because A is measured at $x¯=0.45L$⁠, whereas H is at x = L/2. Configuration (i) corresponds to a slight increase in A accompanied by a sharp increase in load, which continues to infinity. Alternatively, if A is decreased starting from configuration (ii), the lateral load λ decreases (positive stiffness) until a minimum is reached at configuration (iii)—a limit point, corresponding to snap-through under load-control—at which point the load begins to increase (negative stiffness) until reaching zero at configuration (iv). Configuration (iv) corresponds to the second Euler buckling mode, with A slightly less than zero (for similar reason as before); configuration (iv) also corresponds to the point of snap-through under unilateral displacement control [9,11,21]. Continuing to decrease A, the load increases up to a critical point—corresponding to configuration (v)—at which stability is lost. This critical point coincides with a vertical tangent. Following the equilibrium path further along the unstable branch, with increasing A, the load increases and then decreases, returning to zero load at configuration (vi). Configuration (vi) corresponds to the third Euler buckling mode. Finally, if A is further increased, passing through configuration (vii), the load blows up to negative infinity.

It is worth noting that, experimentally, the loss of stability at configuration (v) would propagate a jump—sometimes called “snap-back” [9,11]—to a neighboring stable equilibrium, i.e., the black line in Fig. 2(c). Similar equilibrium configurations exist on this path (black line), but are not shown because they are simply mirror images of those shown for the other path (gray line). As discussed in Sec. 3.2, retaining more terms in the expansion of y(x) results in more “loops” in the equilibrium paths and more ZLLCs, but the two detached equilibrium paths will never connect for the case of the inextensible beam.

With only three terms, there are six equilibria with λ = 0 (i.e., ZLLCs): $A/H≃$ ±1, 0 (double root), and ±1/3. The ZLLCs at A/H ≃ ±1 and 0 (e.g., configurations (ii) and (iv), respectively, in Fig. 2(c)) correspond to stable solutions, whereas the ZLLCs at A/H ≃ ±1/3 (e.g., configuration (vi) in Fig. 2(c)) are unstable. As the lateral constraint moves away from midspan (⁠$x¯=L/2$⁠), the stable ZLLCs near A/H ≃ 0 (i.e., close to the horizontal, the chord connecting the arch ends) progressively move away from A/H = 0, corresponding to a more gradual transition from positive to negative stiffnesses. This positive-to-negative transition is not pronounced for large eccentricities, such as $x¯=0.33L$ (Fig. 2(d)). Furthermore, the critical points at which stability is lost (e.g., configuration (v) in Fig. 2(c)) progressively move away from the nearby stable branch, resulting in larger jumps when stability is lost. There is a corresponding increase in the critical lateral load at which this jump occurs, discussed in Sec. 3.1.2.

It should be noted that not all parts of the equilibrium paths, such as those in Fig. 2, will be experimentally observable, at least not readily. Of course, at each value of λ, the equilibria occur where the total potential has zero-valued gradient, but these equilibria may be valleys (stable), hilltops (unstable), or saddles (unstable) in the potential energy hypersurface. The unstable equilibria will both be difficult to “find” experimentally [22], and it would also be difficult to hold the structures in such configurations under the unavoidable perturbations and imperfections that occur in experiments. Furthermore, many regions of the potential energy landscape are “steep” (i.e., this is a stiff system), meaning that divergence away from unstable equilibria after a small perturbation may be quite violent. Another subtle issue is that displacement control provides a constraint that restricts certain routes through the potential energy landscape; this may stabilize some, but not all unstable equilibria. The simplest example of this is that displacement control at or near the midpoint stabilizes the primary bifurcation path, but not paths with higher spatial wave numbers. Additional constraints could potentially be applied to stabilize more of the equilibrium path [23]; however, without feedback control these constraints may inadvertently change the force pattern to be something other than what is intended.

From the preceding discussion, three critical equilibria can be identified from the lateral load–deflection curves: (1) a limit point resulting in snap-through under load-control at a critical lateral load, denoted $λLP*$⁠; (2) a snap-through instability under unilateral displacement-control at a critical displacement, denoted $AUL*$⁠; and (3) a snap-back instability at a critical lateral load, denoted $λSB*$⁠. These three critical conditions are depicted in Fig. 3 and are recognized as corresponding to configurations of the type (iii), (iv), and (v), respectively, illustrated in Fig. 2(c). All three of these critical equilibria correspond to the onset of snapping behavior depending on the experimental loading protocol used; namely, types (1) and (2) for load-control and unilateral displacement-control, respectively, whereas type (3) occurs for both load- and (bilateral) displacement-control tests.

Figure 4 portrays the effect of the lateral constraint location $x¯$ on these three critical values when three terms are used in the approximation of y(x). The values reported are for equilibrium paths corresponding to the gray lines in Fig. 2. For a midpoint lateral constraint (⁠$x¯=L/2$⁠), the critical values are $λLP*$ = P^C, $AUL*$ = 0, and $λSB*$ = −P^C. As the lateral constraint moves away from midspan, the critical lateral loads $λLP*$ and $λSB*$ both increase, whereas the critical displacement $AUL*$ decreases as previously discussed. The critical limit-point lateral load $λLP*$ increases to about −P^C/2 near $x¯=L/3$ and 2L/3, before decreasing and going to negative infinite as $x¯→0$ and L; therefore, it is physically the easiest (i.e., requiring the smallest absolute load) to initiate snap-through under load-control for a constraint applied close to the third points of the buckled beam. The critical displacement $AUL*$ decreases to about −H/2 at $x¯=L/4$ and 3L/4, before returning to 0 at $x¯=0$ and L; therefore, under unilateral (and bilateral) displacement-control, the constraint location point is able to pass below the horizontal (⁠$AUL*$ < 0) before the arch snaps downward for all constraint locations except $x¯=0,$L/2, and L. The critical snap-back lateral load $λSB*$ peaks at $x¯=L/3$ (i.e., the node line of the third buckling mode), reaching nearly 2.5 times P^C; this large load corresponds to a bifurcation, which can be seen in Fig. 2(d) for $x¯=0.33L$⁠. The lowest $λSB*$ is half of P^C for a constraint location close to $x¯=L/9$ and 8L/9, but this minimum critical lateral load is dependent on the number of terms used to approximate y(x) (see Sec. 3.2).

To better understand these results, Fig. 5 portrays the axial load μ versus deflection A, lateral load λ, and coefficients Q_i for the same cases considered in Fig. 2. The axial load μ is normalized by the first Euler buckling load for a pin-ended beam (i.e., P_e = π²EI/L²); note that, as defined, μ is negative for compression, so the sign is reversed in Fig. 5. For the nearly midspan load [Fig. 5(a)], the deflection A, lateral load λ, and coefficients Q_i (i = 1, 2, 3) gradually vary up to an axial load of −4π²EI/L² (i.e., the second Euler buckling load), at which point the deflection A and lateral load λ abruptly change signs. This transition is realized through the growth of the second coefficient (Q₂), while the first and third coefficients (Q₁ and Q₃, respectively) go to zero. This is more clearly shown by the coefficient for $x¯=0.45L$ in Fig. 5(c), where the coefficients are distinguished by the line weight—Q₁ (thickest) to Q₃ (thinnest)—and are normalized by $∑j=13|Qj|$⁠. A similar effect is observed at an axial load of −9π²EI/L² (i.e., the third Euler buckling load), where only the third coefficient (Q₃) is present. Finally, it should be noted that the ZLLCs directly correspond to the Euler buckling loads [i.e., −μ/(π²EI/L²) = 1, 4, and 9].

Figure 2(b) showed the normalized lateral load–deflection relation for a three-term approximation of the deflected shape y(x) for a constraint location $x¯=0.49L$⁠. Figure 6 shows the same relation, but for varying numbers of terms used in the approximation: (a) 5, (b) 8, (c) 11, and (d) 22 terms. As more terms are retained in the expansion of y(x), additional “loops” in the load–deflection curves are observed. A similar effect has been observed when the rise of an extensible arch is increased [19], resulting in progressively more ZLLCs. For the system considered in Ref. [19], however, the number of loops was determined to be limited based on the axial load required to “flatten” the beam. In the formulation presented here, progressively more “loops” and ZLLCs are recovered as more terms are retained in the expansion, with the relationship holding up to an infinite number of ZLLCs for the infinite series (Eq. (15)). The ZLLCs for an approximately midspan constraint (⁠$x¯≈0.5L$⁠) correspond to A/H ≃ ±1, 0, ±1/3, 0, ±1/5, …, where each 0 is a double root.

It is worth noting that the additional loops all correspond to unstable branches of the load–deflection relations, yet the stable branches remain largely unchanged with more terms being retained. As a result, the limit-point lateral load $λLP*$ (not shown) was unaffected (within numerical precision) by the number of terms retained in the approximation of y(x), whereas the snap-back lateral load $λSB*$ (Fig. 7(a)) and critical displacement $AUL*$ (Fig. 7(b)) showed varying degrees of sensitivity to the number of terms retained. In particular, for constraint locations near midspan (⁠$0.45L⩽x¯⩽0.55L$⁠), the snap-back lateral load $λSB*$ (Fig. 7(a)) is relatively insensitive to the number of terms. This critical load is more sensitive to the number of terms when the constraint is located near the beam’s third points (⁠$x¯≃L/3$ and 2L/3) or ends (⁠$0<x¯⩽L/6$ and $5L/6⩽x¯<L$⁠) where the critical load peaks. In the case of the former (third points), retaining more terms reduces the peak critical load, with less effect when more than four terms are retained. In the case of the latter (close to beam ends), the effect of the number of terms is more dramatic, resulting in a wide range of critical loads, including both positive and negative loads. For three, four, or five terms, $λSB*$ has a distinct minimum that approaches $x¯=0$ and L as more terms are retained, but these minimum critical loads never go below zero in these cases. Conversely, for six or more terms, $λSB*$ crosses zero for load locations close to 0.05L and 0.95L and asymptotes to negative infinity as $x¯→0$ and L. If $λSB*$ is less than zero, then the stable branch of the lateral load–deflection path (e.g., solid lines in Figs. 2 and 6) would not reach zero load before going unstable, and hence instability under unilateral displacement-control testing would occur at a lateral load of $λSB*$ instead of at the critical displacement $AUL*$ which corresponds to a ZLLC (i.e., λ = 0). Therefore, two of the ZLLCs that are typically stable would be unstable in this case. This behavior was experimentally observed in Ref. [21]; the portion of the stable branch between configurations (iv) and (v) in Fig. 2(c), called the “secondary branch” in Ref. [21], could not be found experimentally for constraint locations near the supports (⁠$x¯<0.152L$ and $x¯>0.848L$⁠). Furthermore, the critical displacement $AUL*$ as defined in Sec. 3.1.2 is not well defined in this case (i.e., it is unstable), indicated in Fig. 7(b) by the termination of the lines. Otherwise, the critical displacement $AUL*$ is independent of the number of terms retained in the approximation of y(x). Note that, in Fig. 7(b), there are in fact seven lines, but they fall on top of each other, obstructing the cases with fewer terms.

Figure 8(a) shows the full range of the lateral load–deflection relation for the case of an 11-term approximation of y(x), corresponding to Fig. 6(c). It can be observed that the lateral load λ increases with each successive loop that is added. This is illustrated in Fig. 8(b); as the axial load μ increases, the amplitude of the lateral load λ increases with each successive ZLLC. Note that in Fig. 8(b) the square root of the normalized axial load is plotted to stretch the μ-axis. In doing so, the ZLLCs occur at integer values (n = 1, 2, …), corresponding to the Euler buckling loads n²π²EI/L². At each buckling load, the corresponding coefficient Q_n is present with all other coefficient going to zero.

The formulation in Sec. 2 relies on the assumption of inextensibility, which is applied through the constraint in Eq. (5). This section explores the validity of this assumption through a comparison with an in-house co-rotational finite element model (cFEM). The details of the co-rotational model formulation can be found in Refs. [24,25]. The cFEM additionally provides a validation of the small deformation assumptions used in the formulation as it accurately captures not only the effect of extensibility, but also arbitrarily large global deformations, and is limited only by the fact that it assumes small strains.

The effect of the compression limit is demonstrated in Fig. 9, which shows a comparison of the results for extensible (cFEM) and inextensible (analytical) cases. The different results shown for extensible beams correspond to scenarios with identical normalized forces and deformations, but differing axial rigidities (see the simulation parameters in the figure caption). This is readily done in an FEM simulation, as the cross-sectional area can be increased without changing the bending stiffness. Figure 9(a) shows that the normalized lateral load–deflection curves approach the inextensible case as the axial stiffness is increased. The axial load plots shown in Fig. 9(b) for successively (axially) stiffer beams are perhaps more informative. They show how the squash limit (denoted by horizontal dash-dotted lines), from Eq. (35), serves as a “short circuit” that allows the equilibrium path to begin winding down in axial force toward the snapped-through configuration. The superimposed inextensible results, on the other hand, are unable to pass through this flattened configuration and hence two disconnected paths both extend as vertical asymptotes. While adding additional modes to the inextensible solution increases the number of loops that appear, the two paths will never connect.

The peak load (which is a ZLLC) corresponds with the completely flattened beam. This peak load is not (necessarily) identically equal to a buckling load, unless $n~$ in Eq. (35) is coincidentally an integer value. The approximate results for the four cases in Fig. 9(b), for example, are $n~=6.08,7.81,11.00$⁠, and 15.52, respectively. It should be noted that the curves in Fig. 9(b) come from the cFEM, which is exact, even for large deformation. Hence, the approximate expression for the squash load from Eq. (35) is surprisingly accurate at predicting both the squash load and the number of loops that will occur in the load–deflection curve. Additionally, $n~$ passing through increasing integer values corresponds with bifurcations resulting in additional loops in the force deformation curves. The transition through $n~=2$⁠, in particular, provides a useful estimate of the system parameters at which a bifurcation buckling may occur as opposed to limit point buckling, though the bifurcation path initially appears (generates) on the down-sloping part of the equilibrium paths.

An elastic, inextensible, initially straight beam that is buckled to give a pre-stressed, shallow arch was analyzed. The buckled beam, subject to both a prescribed end displacement and a lateral constraint, was modeled analytically using a constrained energy approach, and equilibria were characterized. These analytical results prove that, for inextensible beams, there are an infinite number of equilibrium configurations that require zero lateral load. Curves of the lateral constraint force versus the displacement at the constraint location were presented, which exhibited an increasing number of loops and zero lateral-load crossings with the number of terms retained in the expression for the deflected shape. Furthermore, it was shown that the zero lateral-load crossings corresponded to the Euler buckling modes for the flat beam. These results were verified numerically by adjusting the axial rigidity of a buckled beam modeled within a co-rotational finite element formulation. It was shown that the primary (stable) equilibrium path was insensitive to the number of terms retained in the analytical model, as well as the numerical model.  Appendix gives an extension of the energy minimization formulation presented in Sec. 2 to include extensibility. Other possible extensions of this work are the generalization to arbitrary boundary conditions and curved beams (i.e., stress-free arches).

This material is based upon work supported by the National Science Foundation under Grant Nos. NSF-CMMI-1663376 and NSF-CMMI-1943917 and the Air Force Office of Scientific Research Grant No. FA9550-19-1-0031. This support is greatly appreciated.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.