Experiments and Numerical Analysis of Crane Toppling Using Ultra-Small Model

A significant number of toppling accidents in various structures are thought to involve structural instability. Examples include bridge girder drops caused by jack overturning [1,2], as well as heavy machinery toppling accidents involving cranes and pile drivers [3]. These incidents suggest that not only overturning moments but also structural instability play a role, motivating theoretical studies based on structural stability theory [4,5]. In these studies, in addition to conventional static analyses, the influence of dynamic inertial forces in facilitating toppling has also been highlighted [6,7]. Recent research has examined crane operational parameters, such as working height and radius, and evaluated toppling safety criteria under these conditions, considering factors such as suspended loads and supporting ground stiffness [8]. Numerical analyses incorporating structural instability have further explored the relationships among load magnitude, load height, ground stiffness, and toppling conditions [9]. The present study aims to experimentally validate the toppling conditions obtained from these numerical analyses. To achieve this, various experiments were conducted using an ultra-small-scale crane model. This paper first presents an overview of the experimental setup, including the miniature model used, followed by an analysis and discussion of the toppling test results from multiple perspectives. The study provides empirical data on toppling behavior and clarifies the role of structural instability in crane and pile driver toppling accidents, thereby contributing to future accident prevention measures.

Ultra-Small-Scale Experimental Model

The ultra-small-scale crane model used in the experiments is shown in Figure 1, with its specifications summarized in Table 1. The commercially available miniature model is based on a largescale building demolition machine originally designed for structures approximately 65 m in height (equivalent to 21 stories). With a scale ratio of 1:228, the model has a boom length of approximately 30 cm, a total mass of 65.5 g, and a crawler center-to-center distance of 3.3 cm. Due to its extremely small size, accuracy compared to full-scale behavior and reproducibility on soft ground may be poor. Nevertheless, the simplicity and compactness of the model allow for experiments under a variety of conditions, facilitating the evaluation of the effects of changes in experimental parameters. Figure 1 also indicates the locations of the suspended loads and the corresponding basic toppling test cases. For reference, the structural appearance of the machine’s lower section as seen from the front is shown in Photograph 2 in Appendix. Although the original machine was a building demolition machine and thus has a slightly different boom tip configuration compared to conventional cranes, the fundamental structural characteristics are preserved. Therefore, the influence of this difference on the primary objective of understanding toppling behavior is expected to be minimal. Additionally, as shown in Figure 1, the experimental model features a non-telescoping boom, so the weight of the crane itself remains constant and does not vary with the position of the suspended load.

Table:1Ultra-Small Model.

Types of Supporting Foundations and Load Application

Four types of supporting foundations were used to support the crane model, as summarized in Table 2. These were selected in order of hardness:

A. on laptop computer – hard
B. on urethane mattress – slightly hard
C. on towel (folded four times) – soft
D. on cushion – very soft

Table:2Basic Toppling Experiment Conditions.

Photographs of the actual supporting foundations are provided in Appendix. For the C. on towel configuration, the towel rests on a hard computer which likely influenced the experimental results. For the D. on cushion, the deformation changes over time due to surface fabric tension and internal air compression. Consequently, load history affects the C. on towel and D. on cushion conditions, resulting in reduced reproducibility. Nevertheless, the experiments are considered sufficiently reliable and provide valuable data on toppling behavior that would be difficult to obtain in full-scale testing. Loads were applied by suspending very lightweight bags from the boom, as shown in Figure 1. Three types of weights were used: glass beads (≒1.5 g), plastic Othello pieces (≒1.2 g), and aluminum 1-yen coins (≒1 g). Suspended loads were applied at three positions along the boom: 1 upper, 2 middle, and 3 lower, as illustrated in Figures 1 and 3. For all tests, the boom rotation angle was fixed at 90° (horizontal direction) to simplify analysis.

Measurement Instruments and Methods

Mass measurements were conducted using a kitchen scale (ELECOM, maximum capacity 2 kg, minimum display 0.1 g) as shown in Figure 2. For cases in which the smallest weight (1- yen coin, 1 g) caused toppling, the pre- and post-toppling results were used to estimate the critical load with 0.1 g resolution. Boom height and working radius were measured using a tape measure; digital measurements were not performed. In the Basic Toppling Experiments, the initial boom inclination angle was set using a protractor (Photograph 4 in Appendix), and the working radius was calculated from the measured load height. Due to the extremely small scale of the model and the use of analog measurement methods, the absolute precision of the experiments is limited. However, the primary objective of this study—to analyze crane overturning behavior from multiple perspectives using a simple experimental setup—has been largely achieved.

A total of five experimental categories were conducted using the ultra-small-scale crane model, each designed to address specific research objectives.

Basic Toppling Experiments

The first set of experiments, referred to as the basic toppling experiments, examined the influence of supporting foundation, initial boom inclination angle, boom length, working radius, and other parameters on the critical overturning load (toppling load). Four types of supporting foundations, arranged in order of increasing compliance, were used: A. on laptop computer, B. on urethane mattress, C. on towel, and D. on cushion. The initial boom inclination angles were set at [1] θ₀ = 30° and [2] θ₀ = 45° with respect to the vertical axis. Suspended loads were applied at three positions along the boom, as shown in Figure 1 and Figure 3: the uppermost point (load point 1), intermediate point (load point 2), and lowest point (load point 3). Table 2 summarizes the conditions for the basic toppling experiments, and photographs in Appendix illustrate each experimental configuration. The results of the basic overturning experiments provided the basis for four additional experimental categories, each designed to analyze specific aspects of crane toppling behavior, as explained in the following.

Experiment for Influence of Initial Boom Inclination Angle

The second experimental category investigated the effect of initial boom inclination angle θ₀ on the critical load and other overturning characteristics. These tests were conducted on the C. on towel supporting foundation, with θ₀ varied across five values: 0°, 10°, 20°, 30°, and 45°. In addition to the 30° and 45° cases tested in the basic experiments, supplemental tests were conducted for 0°, 10°, and 20° to enable a comprehensive comparison across all five inclination angles.

Rotational Spring Stiffness Investigation (Fixed Working Radius)

The third experiment measured displacements under a fixed working radius on the C. on towel foundation while varying the initial boom inclination angle for a given load. From these measurements, the rotational spring stiffness of the foundation was estimated and compared, as the stiffness should theoretically be the same for identical supporting conditions. The calculated stiffness values were then used to generate load–displacement angle curves, allowing validation of the experimental methodology. It should be noted that this experiment does not involve toppling; rather, it focuses on the estimation of rotational spring stiffness.

Experiment for Influence of Supporting Foundation (θ₀ = 10°)

The fourth experimental category examined the influence of supporting foundation stiffness at a relatively small initial boom inclination angle (θ₀ = 10°), where structural instability effects are more pronounced. Supporting foundations were varied across four types: computer desk, four-layer towel, six-layer towel, and ten-layer towel. For each foundation type, suspended loads were applied at the three load points (1, 2, and 3), resulting in different working radii due to the constant inclination angle. The objective was to evaluate how foundation compliance affects toppling behavior under small initial tilt conditions.

Structural Instability Influence (Equilibrium-Transition Toppling)

The fifth experimental category focused on toppling behavior influenced by structural instability, specifically equilibriumtransition- type toppling. Experiments were conducted on a very stiff supporting foundation (wooden board) representing the overturning-moment-dominated case and on a very soft foundation (futon) representing the equilibrium-transition case. The suspended load was applied at the uppermost point (load point 1), where structural instability effects are most pronounced. The working radius was set to a = 8 cm with an initial boom inclination angle of θ₀ = 15.7° for both cases. The magnitude and height of the load at toppling were measured, enabling a direct comparison between overturning-moment-dominated and structuralinstability- dominated behaviors. The following sections present and discuss the results of each experimental category.

Analysis of Overturning Stability

The results of the basic toppling experiments summarized in Table 2 are presented by supporting foundation type in Tables 3-1 through 3-4: A. on laptop computer, B. on urethane mattress, C. on towel, and D. on cushion. In these tables, the upper-left (shaded blue) values correspond to measured data, while the remaining entries are calculated values derived from a combination of measured data, initial settings (e.g., initial inclination angle), and structural parameters of the model. In each table, the experimental columns [1] (initial inclination angle θ₀₁ = 30°) and [2] (θ₀₂ = 45°) correspond to the toppling experiment results. The columns on the right, [1′] (toppling) and [1″] (non-toppling), show numerical values calculated from simulations based on these experimental results. The following discussion focuses first on the experimental results in columns [1] and [2].

The first four rows of Table 3 (a–d) present measured quantities as follows:
P: critical toppling load (suspended load)
t _W : total weight (suspended load plus crane weight)
0 _h : suspended load height before testing
1 _h : suspended load height immediately before toppling

Details of the calculation methods are described in Reference 8; a brief summary is provided here. The structural model used in the stability analysis is a simple rigid-bar–rotational-spring system, as illustrated in Figure 4 [10]. The toppling condition is expressed by the following equation [8]:

Table:3-1Results of Basic Toppling Experiments: A. on PC.

Table:3-2Results of Basic Toppling Experiments: B. on Mattress.

Table:3-3Results of Basic Toppling Experiments: D. on Cushion.

Table:3-4Results of Basic Toppling Experiments: D. on Cushion.

where P_u =W_t is the total weight cr P is the elastic limit load (K_s/L ) , 0_θ is the initial boom inclination angle, θ_u is the overturning angle, K_s is the rotational spring stiffness, and L is the member length from the support point to the center of mass of P_u . In Table 3, g denotes the horizontal distance from the reaction point R_A to the total weight, and e denotes the horizontal distance from the rotation center to the total weight ( e = g − s / 2 ); see Figure 5 for geometric definition. Note that P_u in Equation (1) represents the total load used to evaluate the contribution of structural instability, which differs from the suspended load. For simplicity, the support point is assumed at the midpoint between the left and right crawlers, with the boom rotation center coincident with the support point.

According to the overturning-moment-dominated criterion, toppling occurs when the overturning moment exceeds the resisting moment [4]. In this case, as shown in Figure 5, the total weight W_t is aligned vertically with the crawler reaction R_B (R_B =W_t, R_A =0) . From the geometry, g=S and e=S/2 at toppling, which allows determination of the member length L and the toppling angle θ_u . In experimental columns [1] and [2] of Tables 3-1 to 3-4, the measured toppling load (row a) was used with Equation (1) to estimate the rotational spring stiffness K_s at toppling (row p; see Reference 9 for details).

The crane weight excluding the suspended load is denoted as W_M , and its center-of-mass position e_M is calculated from experimental measurements using the following equation:

where e_M is the horizontal distance from the midpoint between the crawlers to the crane center of mass, P_t is the suspended load, and a is the working radius of the suspended load. Other foundation-related parameters in Table 3 include k_v , the foundation reaction coefficient (load per unit settlement), and d = R_B/ k_v , the foundation settlement.

A comparison of experimental columns [1] (θ₀ = 30°) and [2] (θ₀ = 45°) shows that, as expected, the larger initial inclination in [2] results in a larger working radius and therefore a smaller critical toppling load. In Table 3-1 (A. on laptop computer), the contribution ratio of structural instability ( u cr P P , row n) is very small compared to the overturning-moment contribution ( 0 u θ θ , row m). This is consistent with the very stiff foundation, where cr P approaches infinity, making the overturning-moment effect dominant.

Concrete structures are often subjected to dynamic loads, ranging from moderate (e.g., wind) to extreme (e.g., explosions or vehicular impacts). Extensive research has shown that concrete responds differently under dynamic loading conditions compared to static ones. In particular, concrete exhibits a pronounced strainrate dependency, especially under tensile loading. This strain-rate effect has been shown to have a physical origin tied to the presence of free water within the hydrates of the cement paste [9, 10]. Experimental evidence indicates that when this physically bound water is removed, the strain-rate effect nearly disappears. This finding holds true across concretes with differing mix designs but similar levels of water retained in the nanoporosity of the hydrates. As such, they display a consistent absolute increase in tensile strength as the strain rate rises [9,10].

Unlike creep and quasi-static behavior—which are governed by larger pore structures and capillary water—strain-rate effects are governed by the nanometer-scale porosity of cement hydrates and involve a different type of water. The physical mechanism responsible has been hypothesized to resemble the Stefan effect, a well-known phenomenon in fluid mechanics. The Stefan effect occurs when a thin film of viscous fluid is confined between two parallel plates that are being pulled apart. The fluid resists this separation, generating a restoring force that is proportional to the rate of separation. In the case of concrete, this restoring force—produced by the viscous behavior of water in the hydrate nanopores—acts to resist the formation and growth of microcracks into macrocracks. Thus, the increased tensile strength of concrete at higher strain rates can be attributed to this Stefan-like resistance, which serves as a micro-scale damping mechanism opposing crack initiation and propagation.

In the following (Figure 1), a diagram is presented which synthesizes the different couplings mentioned previously as a function of the strain rate.

This article provides a comprehensive synthesis of previous and published research exploring the interactions between concrete cracking mechanisms and the physico-chemical processes associated with water at different scales of porosity within the material. The resulting synthesis offers a unified perspective on these interactions, encompassing the full spectrum of mechanical behaviors, from creep to dynamic loading. The key physicochemical mechanisms identified include:

a. The transfer of water and water vapor,
b. Capillary tension governed by Laplace’s law,
c. Self-healing of microcracks through continued hydration,
d. Stefan effect, which influences crack resistance at the nanoscale.

Beyond presenting a coherent framework linking these behaviors, the primary contribution of this work lies in its potential to inform the development of more accurate micromechanical and macroscopic models of concrete behavior. Accurate modeling requires a deep understanding of the underlying physical phenomena, something that purely empirical approaches cannot achieve. This unified view helps bridge that gap, providing essential insights for future research and modeling efforts in structural and materials engineering.

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