Dynamic Behavior of an Arch Structure Under Moving Point Loads

An arch structure under single moving point load had been previously studied (see Fig. 1). This project was intended to understand the dynamic response, especially the snap-through behavior over time, of such structure under two moving point loads toward each other (see dashed line in Fig. 1). Using Mathematica software package, it is possible to study the characteristics and determine the dangerous zone where snap-through would occur under certain combination of different load magnitudes and load speeds.

Fig. 1 Problem description of an arch under moving point load(s)

(courtesy of Solid Mechanics Laboratory, NTU)

Initial shape of the arch is assumed to be u₀(ξ)=h sin(ξ), using expansion solution the shape of the loaded arch can be expanded as u(ξ)=Σ α_n(τ)sin(ξ). Due to symmetry, the solution for the latter case retains only odd-integer mode expansion.

After obtaining the governing equation of α_n, we first consider the quasi-static configurations, that is, cases when the moving speed c is small and the acceleration term can be neglected. Fig. 2 shows α₁ as a function of load position e=cτ when h=8. 4 modes are used in the expansion, and there are 9 equilibrium positions while only 7 of them are real.

Fig. 2 α₁ as a function of load position e with h=8: (a) Q=18, (b) Q=20, and (c) Q=18.16

(courtesy of Solid Mechanics Laboratory, NTU)

It demonstrates that as Q decreases from 20 the saddle-node bifurcation points in Fig. 2(b) approach each other and eventually merge into transcritical bifurcation points when Q=Qcr=18.16 shown in Fig. 2(c). The saddle-node bifurcation points are responsible for dynamic snap-through.

Fig. 3 further compares quasi-static solutions denoted by thin lines and dynamic simulation results marked by thick lines. It is shown that when the speed c is increased from zero, snap-through may occur even when the load Q is smaller than Qcr, and the quasi-static solutions are no longer valid. Also note that when c is increased to 1.5, the arch doesn't respond fast enough so the snap-through will not occur when the load reaches the other end, and we can deduce that there are two critical speeds c_cr^-and c_cr⁺, and that speed within these two limits may lead to dynamic snap-through.

Fig. 3 Dynamic and quasi-static solutions for an arch with h=8, μ=0.001, and Q=18

(courtesy of Solid Mechanics Laboratory, NTU)

In some cases it is also possible that the arch may continue to deform and snap after the moving point load leaves the arch, this is demonstrates by Fig. 4 when c=2.4. To determine whether this will happen or not one needs to check that energy gained by external force is smaller than the energy barrier between two distant stable equilibrium positions, and that α₁ remains greater than zero while the load is on the arch.

Fig. 4 α₁ v.s. load position e for an arch with h=8, μ=0.001, and Q=25. α₁ history of c=2.4 and c=2.6 after the load leaves the arch are shown in 4(a) and 4(b) respectively.

(courtesy of Solid Mechanics Laboratory)

In the final analysis, the dangerous speed zone in Q-c space can be determined for an arch specified as shown in Fig. 5.

Fig. 5 Dangerous speed zone for an arch h=8, μ=0.001 in Q-c space. Solid circle indicates the arch will snap before settling to a final stable equilibrium position.

(courtesy of Solid Mechanics Laboratory)