Design of Manual Handling Carts: A Novel Approach Combining Corrective Forces and Modelling to Prevent Injuries

Abstract

Design standards for manual handling equipment tend to measure maximal loads and moving forces using a smooth, flat, horizontal steel plate; yet, in everyday use, such equipment is used on floor coverings. Such test methods therefore overestimate the maximal loads acceptable for operators, which increases the risk of injury including the development of musculoskeletal disorders. This study presents a new approach for calculating the pushing force for manually handled equipment moving longitudinally on resilient floor coverings from the pushing force measured on a steel plate. This method combines corrective forces with the pushing force model presented in this study. Corrective force abaci, which describe corrective forces as functions of the hardness of the floor covering’s base foam, are provided for each type of tread and bearing in the cart’s wheels. These abaci have been elaborated from pushing force measurements obtained with 44 wheel designs (of varying diameters, treads and bearings) tested on five different floors on a custom-built test bench. A mean deviation between experimental results and model predictions of 5.1% is obtained for pushing forces. These results permit us to account for the real conditions in which manual handling equipment is used and help in reducing the incidence of musculoskeletal disorders.

1. Introduction

Manual handling tasks are still prevalent in Europe, as over 30% of European workers are concerned [1,2]. Moreover, across occupational settings, a number of tasks involve manually handled equipment. Some authors report that 50% of manual handling tasks consist of pushing and pulling loads [3,4]. Regular pulling and pushing tasks involving manually propelled wheeled equipment, such as trolleys, mobile hoists, hospital beds, roll cages, and meal delivery carts, often demand considerable effort from operators and can result in musculoskeletal disorders [5,6,7,8,9,10]. The European Union information agency for occupational safety and health (EU-OSHA) reports that work-related musculoskeletal disorders are the most prevalent work-related health issue (60%) in the European Union [11]. Additionally, the National Institute for Occupational Safety and Health (NIOSH) specifically reported that approximately 20% of these injuries were linked to push/pull tasks [12]. A literature review conducted by Hoozemans et al. highlighted pushing and pulling tasks as significant contributors to low back pain, with associated risk factors explored through epidemiological, psychophysical, physiological, and biomechanical studies [5]. More specifically, 9–20% of low back injuries have been linked to pushing and pulling tasks [13,14]. Additionally, a cross-sectional survey involving 434 workers revealed a strong association between such tasks and an increased risk of low back and shoulder complaints [6]. The prevalence of low back disability was more than double in a high-exposure group compared to a reference group, with a prevalence rate ratio of 2.15. For shoulder disability, the prevalence rate ratio was even higher, at 3.70. In addition, musculoskeletal disorders similarly affect the neck and forearms [5,15,16,17,18]. All of these disorders are among the most commonly reported workplace disabilities [14,19,20,21,22].

To reduce workplace injuries and ensure tasks align with human physical capabilities, various standards [23,24,25,26] provide guidelines derived from the Snook and Ciriello push/pull tables [27]. These tables offer ergonomic guidelines, including recommended force values, for pushing and pulling tasks to evaluate and design safe manual handling activities. However, pushing and pulling tasks performed using wheeled equipment often require higher forces than those recommended by these standards when nominal loads defined by manufacturers are applied [28,29]. It has been reported that the load carried by the equipment is too heavy, thereby putting strain on the musculoskeletal system [7,30,31,32,33].

Moreover, the design standards for manual handling equipment, such as mobile hoists or manual industrial trucks [34,35], describe test methods for discerning moving forces and maximal loads. Traditional methods consist of moving the equipment bearing a maximum load in a straight line and on a smooth, flat, horizontal steel plate. The initial force (defined when the equipment starts moving and also known as the starting force) and the sustained force (defined when the equipment is moving at constant speed and also known as the driving force) are measured using a dynamometer. When testing commences, a force is applied to the handle of the cart until it moves (the highest force recorded is the initial force); the sustained force is then measured at a constant velocity. Regulatory limit values of initial and sustained maximal forces are defined so that, if the force required for moving the cart is greater than the limit value, the load of the cart must be reduced. Such equipment, however, is only rarely used on a smooth, flat steel floor; instead, in everyday use it is moved on various types of adapted floors (e.g., tiles, carpet, resilient floor covering). Indeed, due to the benefits of resilient floor coverings (e.g., design, sustainability, cheap, easy to install, waterproofing), they are increasingly common in various occupational sectors, including in hospitals, clinics, care homes and commercial and educational settings. Previous studies have shown that the effort required to move wheel-mounted equipment depends on a number of factors, such as load distribution, inertia force, wheel diameter, the tread and bearings in the wheel and the air pressure in the tyres [36,37,38,39,40,41]. In a previous study, we demonstrated that resistive forces were higher for these floors than for steel plates [42]. And as a result of our earlier findings, we discovered that traditional test methods overestimate the maximal load of manual handling equipment. As occupational settings avail themselves of a large number of possible combinations of wheels and floorings, manufacturers face a challenge if they seek to propose an optimal combination that will reduce the likelihood of musculoskeletal disorders developing in the workforce. Therefore, a new approach is proposed in this article to enhance the test methods described in the design standards of manual handling equipment.

The main contributions of this work are:

(1)

A novel method is proposed for changing traditional test methods in manual handling equipment design standards. This new method retains the steel plate as a reference floor while incorporating corrective forces in order to take into account the real conditions of use, notably the type of wheel and the type of floor. This method consists of combining a pushing force model of an equipment with corrective forces;

(2)

The definition and assessment of corrective forces, based on resistive forces measured for different types of wheels in contact with various floor types, in order to correct the pushing force measured on the steel plate;

(3)

The provision of different abaci taking into account the type of wheel (its tread and bearing), the load carried by the wheel and the hardness of the base foam of floor coverings;

(4)

A basic mathematical model of the pushing force required to move a four-wheeled item of equipment, which combined with different corrective force abaci, allows us to assess the moving force (as a function of floor and wheel combinations), and thus to estimate the maximum load in real usage conditions.

To our knowledge, this present study is the first of its kind on this subject; indeed, our model combined with these abaci will be the first to estimate the real pushing force of any cart being moved on different floor coverings. This work may be used to change the design standards to take into account the real conditions of use in the workplace, and thus preventing the occupational risks associated with pulling and pushing tasks.

This article is structured as follows. The definition of the corrective forces, the objectives of the mathematical model for the pushing force of a four-wheeled cart, and the experimental method to assess the overall effectiveness of the combination of corrective forces and pushing force model, are described in Section 2. The way to assess the corrective forces from resistive forces data, the deduced corrective force abaci, the pushing force model and the results of comparison between the experimental measurement of the pushing force and its modelled value are presented in Section 3. This Section is followed by a Discussion where the results are analysed and interpreted. Perspectives on this issue are presented at the end.

2. Materials and Methods

This section describes the method used to predict the pushing force of a manually propelled four-wheeled item of equipment on a floor covering, where this value is known for a steel plate. We sought to evaluate the maximal permissible load that could be applied to the equipment on a floor covering, without exceeding the recommended exposure limit for a worker in push/pull tasks as provided in ergonomic standards [23,24,25].

2.1. Definition of Corrective Forces for a Single-Wheel

To successfully complete this goal, we propose that pushing force should still be evaluated on a smooth, flat, horizontal steel plate (it is then considered as the reference floor), and that it should be subsequently corrected taking into account a corrective force. Indeed, a steel plate is considered to be the standard for all such tests. We then combine these corrective forces with a model of pushing force to evaluate the corresponding effort required for floor coverings, given the pushing force of the equipment moving on a steel plate.

Corrective forces can be derived from the minimum forces required to push a wheel to overcome the forces that oppose that movement, such as, for example, the force linked to the deformation of the wheel and the ground. These forces, called resistive forces, have been investigated in a previous publication [42]. These resistive forces were obtained for 44 different wheels (of varying wheel diameters, treads and bearings) moving on a steel plate and on four resilient floor coverings (three PVC multilayer floor coverings, PVC.Fl.1, PVC.Fl.2, PVC.Fl.3, and one monolayer rubber floor covering, Rub.Fl.). Resistive forces, obtained for a single wheel in a uniform rectilinear motion (i.e., without swivelling) and progressively loaded with 5 different charges (from 13 to 62 kg in accordance with the static and dynamic load capacities of the wheel), were measured using a custom-built test bench (Figure 1). Each wheel was mounted on a universal stiffness clevis (developed to accommodate all the tested wheels) and moved across each selected floors (i.e., the steel plate and the floor coverings glued to a steel plate). This motorized test bench is fully described in our previous work [40]. The wheel diameters tested were 80, 100, 160 and 200 mm; the treads tested were solid rubber, elastic tyre, thermoplastic rubber, synthetic rubber, injected polyurethane, casted polyurethane and polyamide; and the wheel bearings were sleeve bearings (S.B.), cone ball bearings (C.B.B.) and precision ball bearings (P.B.B.). Some results are given in Section 3.1.

As a result of this earlier study, we defined corrective force ξ (Equation (1)) as being the difference between the floor resistive force F_R,flo (obtained for a given couple wheel/floor covering) and the steel resistive force F_R,ste (obtained for the same wheel on the smooth, flat, horizontal steel plate). At this stage, since the resistive forces were obtained for a single wheel moving on a floor (to eliminate all energy losses not related to the contact between the wheel and the floor), the corrective forces are also being determined for a single wheel.

$$\mathsf{\xi }={\mathrm{F}}_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{o}}-{\mathrm{F}}_{\mathrm{R},\mathrm{s}\mathrm{t}\mathrm{e}}$$

(1)

2.2. Transition from a Single-Wheel to a Four-Wheeled Cart System to Estimate the Pushing Force

The pushing force required to move the equipment in a straight line on a floor covering must be estimated for a four-wheeled cart, as used in the workplace, whereas the corrective forces were obtained for a single wheel. Therefore, it is necessary to transition from a single-wheel configuration to a four-wheeled cart system. The cart configuration studied here, which is the most common, consists of two castors at the rear and two at the front. Although different wheel configurations are possible, the rear and front wheels may differ in certain cases. Taking these considerations into account, the pushing force can be estimated using a model. Moreover, to ensure its practical application, the model must be user-friendly and easily adopted into the methods for evaluating forces in the design standards for manual handling equipment. This simple mechanical model can be based on Newtonian mechanics and derived from the forces and torques acting on the equipment. Furthermore, this model can adjust the pushing force measurement obtained from the conventional method, which uses a steel plate covering as a reference, by considering different characteristics of the floors and wheels, in order to better reflect the real usage conditions of the equipment.

To achieve this goal, the model must be able to:

-

calculate the pushing force for any floor coverings in the knowledge of different pieces of information: the pushing force measured on a steel plate covering and the corrective forces corresponding to the type of rear and front wheels fitted the four-wheeled cart (these forces depend on the floor coverings under consideration, and the load carried by the rear and front wheels);

-

predict the pushing force of any cart in the knowledge of the acceleration applied to the equipment and the resistive forces of any wheels in contact with the floor coverings under consideration;

-

take into account the differences between front and rear wheels (diameter, load distribution, etc.) as well as the acceleration of the equipment.

The definition and application of the model are described in detail in Section 3.4.

2.3. Experimental Validation of Combination of Corrective Forces and Pushing Force Model

Experimental trials were conducted with a four-wheeled manual handling cart to validate the combination of corrective forces (obtained for a single wheel) and pushing force model (describing the pushing force for a complete four-wheeled equipment). Thereby, the experimental results of the pushing force required to move the four-wheeled item of equipment on a floor covering were compared with the pushing force curve determined by the model. More specifically this modelled curve of pushing force is obtained by combining the corrective force values at the loads carried by the rear and front wheels, provided in the corrective force abaci (selected according to the type of rear and front wheels, and to the type of floor covering) and the pushing force equation given by the model. To carry out these comparisons, the manual handling cart was equipped with various combinations of wheels, in which the diameter, tread and bearings of the wheels varied. Different combinations of rear and front wheels were also used to test the performance of the model. This cart was equipped with instruments to measure both the pushing force and the cart’s acceleration. Force was measured using a six-axis force sensor FTS-Delta SI-660-60 (SCHUNK, Lauffen/Neckar, Germany). This sensor was attached on an additional handle fixed to the tray of the cart. An accelerometer 3711D1FA3G (PCB Piezotronics, Depew, NY, USA) was also attached to the cart to measure longitudinal acceleration. Force and acceleration were recorded by a DEWE-43A data acquisition system (DewesoftX Version: 2022.4) with a sample rate of 100 Hz. Figure 2 shows the instrumented cart used in the present study.

Pushing force was applied to the cart by an experimenter starting from a static location, with different loads deposited on the tray of the cart, resulting in total loads of 87, 126, 164 and 202 kg. The load distribution between rear and front wheels was measured using a portable wheel scale. The cart was moved with different accelerations (for also test the performance of the model) on three PVC multilayer floor coverings (referred to hereafter as PVC.Fl.1, PVC.Fl.2 and PVC.Fl.3) and on the smooth, flat, horizontal steel plate. Thus 14 configurations of wheels were tested with various accelerations and conditions of loads and floors, in order to compare experimental results of pushing force of the cart and the value calculated by the model (with the combination of the correctives forces). These configurations gave us 731 comparisons between experimental and modelled pushing forces. The 14 configurations of front and rear wheels (diameter, wheel tread and wheel bearing) are described on

Table 1. Description of the 14 configurations of front and rear wheels used for experimental validation (inj. polyur. = injected polyurethane, C.B.B. = cone ball bearing, S.B. = sleeve bearing).

N°	Front Wheels (Diameter/Wheel Tread/Wheel Bearing)	Rear Wheels (Diameter/Wheel Tread/Wheel Bearing)
1	160 mm/ inj. polyur./C.B.B.	160 mm/ solid rubber/S.B.
2	160 mm/ inj. polyur./C.B.B.	80 mm/ solid rubber/C.B.B.
3	80 mm/ solid rubber/S.B.	80 mm/ solid rubber/S.B.
4	160 mm/ solid rubber/S.B.	160 mm/ solid rubber/S.B.
5	160 mm/ solid rubber/C.B.B.	160 mm/ solid rubber/C.B.B.
6	80 mm/ solid rubber/C.B.B.	80 mm/ solid rubber/C.B.B.
7	80 mm/ inj. polyur./C.B.B.	80 mm/ inj. polyur./C.B.B.
8	160 mm/ inj. polyur./C.B.B.	160 mm/ inj. polyur./C.B.B.
9	160 mm/ solid rubber/S.B.	160 mm/ solid rubber/C.B.B.
10	160 mm/ solid rubber/S.B.	80 mm/ solid rubber/C.B.B.
11	160 mm/ solid rubber/S.B.	160 mm/ inj. polyur./C.B.B.
12	160 mm/ solid rubber/C.B.B.	160 mm/ inj. polyur./C.B.B.
13	80 mm/ inj. polyur./C.B.B.	80 mm/ solid rubber/C.B.B.
14	80 mm/ inj. polyur./C.B.B.	160 mm/ inj. polyur./C.B.B.

.

3. Results

3.1. Resistive Forces

As mentioned above, corrective forces were derived from the measurements of resistive forces (which are the minimum pushing forces needed to move the wheel on the floor).

Our previous study has shown [42] that for all configurations of wheels and floors, resistive forces increase linearly with the load; these resistive forces are significantly influenced by the wheels’ diameter (resistive forces sensitively decreased for a diameter above 100 mm), the nature of the tread (but not by its hardness) and the type of bearing in the wheels. In addition, the lowest resistive forces are those for the steel plate (therefore, using a steel plate to measure the pushing forces, without correcting them, leads to overestimate the maximum load of the equipment). Moreover, the resistive forces are heavily influenced by the type of floor and, in particular, by the hardness of floor covering’s base foam. The harder the base foam (i.e., higher hardness value), the lower the effort required. In our previous study, in compliance with the requirements of EN ISO 868 [43], hardness values A/15 were measured at different points in the floor covering specimen in a vertical direction using a shore-A durometer coupled with a Bareiss BS 61 II test stand at a temperature of 25 °C. The values were read 15 s after the specimen was pressed by the indenter. The hardness values of the base foams of the four resilient floor coverings are given in

Table 2. Hardness values (in Shore-A/15) of the four floor coverings tested.

	Rub.Fl.	PVC.Fl.1	PVC.Fl.2	PVC.Fl.3
Base foam	A/15: 95	A/15: 73	A/15: 52	A/15: 49

. Rub.Fl. is hard, whereas PVC.Fl.2 and PVC.Fl.3 are softer materials.

As an example, Figure 3a–f displays the scatter plots of the mean values overlaid with the standard deviation (error bars) and the least-squares regression lines of the resistive forces plotted according to the load for six different wheels on the steel plate with PVC.Fl.1, PVC.Fl.2, PVC.Fl.3 and Rub.Fl., two treads (injected polyurethane and elastic tyre), for two diameters (100 mm and 200 mm) and two bearings (C.B.B. and P.B.B.). Through our analysis of all of the trials conducted (not detailed here), we have observed that floor covering material and wheel tread are the most important parameters governing resistive forces, while wheel diameter and wheel bearing have less influence. To reduce the effort required to move equipment in a straight line, it is recommended to choose large-diameter wheels with treads, made for example, of injected or casted polyurethane, and equipped with P.B.B. or C.B.B. bearings. Additionally, the load distribution should be balanced to prevent overloading one axle of the equipment, as this would generate high resistive forces that could hinder its movement. If the equipment moves on an uphill slope, the slope will contribute to increasing both the initial force and the required sustained force. Furthermore, the acceleration applied to the equipment, which is not studied in this work, contributes significantly to the initial effort required to set the equipment in motion.

3.2. Evaluation of Corrective Forces

In accordance with the definition of corrective forces (see Equation (1)), we have used the values of resistive forces, measured using the custom-built test bench on the different floors, to estimate the corrective forces (for a single wheel) for different configurations (as were described in Section 3.1). Figure 4a–f corresponds to the scatterplot of the experimental data of the corrective forces. As we can observe, there is a linear relationship between corrective forces and loads on the three PVC multilayer floor coverings (PVC.FL.1 to PVC.Fl.3) and on the monolayer rubber covering (Rub.Fl.).

In order to analyse the linear fit, we used adjusted coefficients of determination (r²). Considering all of the wheels and floor coverings tested, it appears that r² is greater than 0.8 in 81% of cases. More precisely, in 72% of cases, r² exceeds 0.9. Lower values of r² are obtained when corrective forces are close to zero (i.e., for floor coverings with a resistive force similar to that of steel under the same wheel and load conditions). Linear regression therefore fits the data well. Normalised-root-mean-square error (NRMSE) is complementary to the adjusted coefficients of determination r² because it evaluates the accuracy of the linear regression model’s predictions by comparing the predicted values to the experimental data.

Table 3. NRMSE repartition for the different wheels and floor coverings tested.

NRMSE	% of Predictions with This NRMSE
NRMSE ≤ 10%	53%
10% < NRMSE ≤ 30%	21%
30% < NRMSE ≤ 50%	3%
NRMSE > 50%	23%

summarises NRMSE percentage values obtained for all of the wheels and floor coverings tested. These results show that the model accurately predicts the data. Indeed, NRMSE is less than 30% in 74% of cases. Higher NRMSE (>50%) is obtained, for floor coverings with high base foam hardness value, when corrective forces are close to zero.

3.3. Corrective Forces Abacus

The corrective forces are close enough when it comes to wheels with the same tread and the same bearings. Thus, it is possible to consider an average corrective force for a specific tread and wheel bearing and as a function of the base foam hardness of the floor covering. For instance, Figure 5 and Figure 6 present two different results. Figure 5 shows the dispersion diagram and fitting of the data points of the corrective forces obtained for wheels of four diameters (80, 100, 160 and 200 mm) with an injected polyurethane tread and a C.B.B. wheel bearing on the PVC.Fl.3 floor covering (with a base foam with a hardness of A/15: 49). Figure 6 shows the dispersion diagram and fitting of the data points of the corrective forces obtained for wheels of three diameters (100, 160 and 200 mm) with an elastic tyre tread and a C.B.B. wheel bearing on the PVC.Fl.2 floor covering (with a base foam with a hardness of A/15: 52).

In the light of these results for the different wheels and floor coverings, we propose a number of corrective forces abaci. These abaci describe the evolution of corrective forces for loads carried by a single wheel, ranging from 10 to 150 kg, as a function of the hardness of a floor covering’s base foam, for each type of tread and wheel bearing. Figure 7 shows a corrective forces abacus for wheels with injected polyurethane treads and C.B.B. moving on floor coverings of varying hardness (A/15: 49, A/15: 52, A/15: 73 and A/15: 95). Figure 8 shows a corrective forces abacus for wheels with elastic tyre tread and C.B.B. moving on A/15: 49, A/15: 52, A/15: 73 and A/15: 95. A corrective forces abacus for the other types of wheel tread and bearing are presented in Appendix A.

3.4. Pushing Force Model

A simple mechanical model, based on Newtonian mechanics, was developed in order to evaluate the pushing force required to move a rigid four-wheeled manually handled item of equipment longitudinally. We assume that there is no difference between wheels of the same train (the equipment is symmetrical along the longitudinal centreline), and that it is therefore reasonable to combine them as if they were a single wheel. Thus, the four-wheeled item of equipment is described using a two-wheeled model, as shown at Scheme 1, which illustrates the forces acting on the equipment. The equipment moves at a velocity V without slipping on a two-dimensional inclined plane with a slope angle α. Air resistance is neglected. The longitudinal pushing force F_T along is applied to the handle on point D at height h_D from the floor. The total mass (equipment and load carried) is M. The radii of the front and rear wheels are R_f and R_r, respectively. The centres of the front and rear wheels are O_f and O_r, respectively. The wheelbase (distance between the rear and the front wheel axles) is L. The centre of mass G, where the total weight $\overrightarrow{\mathrm{W}}$ is applied, is positioned at height h_G from the floor. The distances between the centre of mass and the rear and front wheel axles are L_b and L_a, respectively. $\overrightarrow{\mathrm{F}}$_inertia is the inertia force when the equipment is submitted to acceleration γ. Aerodynamic drag is not considered [44]. ${\overrightarrow{\mathrm{N}}}_{\mathrm{f},\mathrm{x}}$ and ${\overrightarrow{\mathrm{N}}}_{\mathrm{f},\mathrm{y}}$ are, respectively, the tangential and the normal components of the ground reaction force $\overrightarrow{{\mathrm{N}}_{\mathrm{f}}}$ exerted on the front wheel at point A_f. ${\overrightarrow{\mathrm{N}}}_{\mathrm{r},\mathrm{x}}$ and ${\overrightarrow{\mathrm{N}}}_{\mathrm{r},\mathrm{y}}$ are, respectively, the tangential and the normal components of the ground reaction force $\overrightarrow{{\mathrm{N}}_{\mathrm{r}}}$ exerted on the rear wheel at point A_r. Motion resistance due to contact between wheels and floor is represented by rolling resistance parameters (also called rolling friction parameters in the literature) δ_f and δ_r for front and rear wheels, respectively. Rolling parameter reflects the shift of the point of application of the ground reaction (point A_f for the front wheel and point A_r for the rear wheel) from the centre of the wheel projected on the floor (point B_f for the front wheel and point B_r for the rear wheel). This shift is due to higher pressure at the front of the wheels when the pushing force is applied.

A glossary of the different parameters used in the model with their definition and unit is given in

Table 4. Glossary of all the parameters used to establish the mechanical model.

Force or Quantity	Definition	Unit
$\overrightarrow{\mathrm{F}}$T	Longitudinal pushing force applied to the handle	N
$\overrightarrow{\mathrm{W}}$	Total weight applied at the centre of mass G	N
$\overrightarrow{\mathrm{F}}$inertia	Inertia force for an acceleration γ	N
${\overrightarrow{\mathrm{N}}}_{\mathrm{f}}$	Ground reaction force exerted on the front wheel	N
${\overrightarrow{\mathrm{N}}}_{\mathrm{r}}$	Ground reaction force exerted on the rear wheel	N
${\overrightarrow{\mathrm{N}}}_{\mathrm{f},\mathrm{x}}$	Tangential component of the ground reaction force $\overrightarrow{{\mathrm{N}}_{\mathrm{f}}}$ exerted on the front wheel	N
${\overrightarrow{\mathrm{N}}}_{\mathrm{f},\mathrm{y}}$	Normal component of the ground reaction force $\overrightarrow{{\mathrm{N}}_{\mathrm{f}}}$ exerted on the front wheel	N
${\overrightarrow{\mathrm{N}}}_{\mathrm{r},\mathrm{x}}$	Tangential component of the ground reaction force $\overrightarrow{{\mathrm{N}}_{\mathrm{r}}}$ exerted on the rear wheel	N
${\overrightarrow{\mathrm{N}}}_{\mathrm{r},\mathrm{y}}$	Normal component of the ground reaction force $\overrightarrow{{\mathrm{N}}_{\mathrm{r}}}$ exerted on the rear wheel	N
V	Velocity of the equipment	m/s
α	Angle of slope	rad
hD	Height from the floor at which the pushing force is applied to the handle	m
hG	Height between the floor and the centre of mass G	m
M	Total mass (four-wheeled manual handling equipment and load carried)	kg
Rf	Front wheel radius	m
Rr	Rear wheel radius	m
L	Wheelbase (distance between the rear wheel axle and the front wheel axle)	m
LD	Distance between the front wheel axle and the point D where the pushing force is applied to the handle	m
Lb	Distance between the centre of mass and the rear wheel axle	m
	Distance between the centre of mass and the front wheel axle	m
γ	Acceleration of the equipment	m/s2
δf	Rolling resistance parameter for front wheels	m
δr	Rolling resistance parameter for rear wheels	m
g	Acceleration of gravity	m/s2

.

The model is based on a simple representation of a four-wheeled cart that is broken down into three solids (equipment’s frame loaded, front wheels, rear wheels) which are successively isolated. For each solid, the mechanical action tensors to which they are subjected are identified before applying the fundamental principles of dynamics. The pushing force F_T applied at point D is then given by:

$${\mathrm{F}}_{\mathrm{T}}=\mathrm{M}\left[1+{\displaystyle \frac{{\mathrm{h}}_{\mathrm{G}}}{\mathrm{L}}}\left({\displaystyle \frac{{\mathsf{\delta }}_{\mathrm{r}}}{{\mathrm{R}}_{\mathrm{r}}}}-{\displaystyle \frac{{\mathsf{\delta }}_{\mathrm{f}}}{{\mathrm{R}}_{\mathrm{f}}}}\right)\right]\left(\mathsf{\gamma }+\mathrm{g}\sin \mathsf{\alpha }\right)+{\displaystyle \frac{\mathrm{M}\mathrm{g}\cos \mathsf{\alpha }}{\mathrm{L}}}\left({\displaystyle \frac{{{\mathrm{L}}_{\mathrm{b}}\mathsf{\delta }}_{\mathrm{f}}}{{\mathrm{R}}_{\mathrm{f}}}}+{\displaystyle \frac{{{\mathrm{L}}_{\mathrm{a}}\mathsf{\delta }}_{\mathrm{r}}}{{\mathrm{R}}_{\mathrm{r}}}}\right)$$

(2)

Then, considering resistive forces F_R,r and F_R,f for front and rear wheels:

$${\mathrm{F}}_{\mathrm{R},\mathrm{f}}={\displaystyle \frac{\mathrm{M}\mathrm{g}{\mathsf{\delta }}_{\mathrm{f}}}{{\mathrm{R}}_{\mathrm{f}}}}$$

(3)

$${\mathrm{F}}_{\mathrm{R},\mathrm{r}}={\displaystyle \frac{\mathrm{M}\mathrm{g}{\mathsf{\delta }}_{\mathrm{r}}}{{\mathrm{R}}_{\mathrm{r}}}}$$

(4)

Thereby, the total pushing force F_T for the four-wheeled manual handling cart is given by:

$${\mathrm{F}}_{\mathrm{T}}=\mathrm{M}\mathsf{\gamma }+{\displaystyle \frac{{\mathrm{h}}_{\mathrm{G}}\mathsf{\gamma }}{\mathrm{L}\mathrm{g}}}\left({\mathrm{F}}_{\mathrm{R},\mathrm{r}}-{\mathrm{F}}_{\mathrm{R},\mathrm{f}}\right)+\mathrm{M}\mathrm{g}\sin \mathsf{\alpha }+{\displaystyle \frac{{\mathrm{h}}_{\mathrm{G}}}{\mathrm{L}}}\sin \mathsf{\alpha }\left({\mathrm{F}}_{\mathrm{R},\mathrm{r}}-{\mathrm{F}}_{\mathrm{R},\mathrm{f}}\right)+{\displaystyle \frac{\cos \mathsf{\alpha }}{\mathrm{L}}}\left({\mathrm{L}}_{\mathrm{b}}{\mathrm{F}}_{\mathrm{R},\mathrm{f}}+{\mathrm{L}}_{\mathrm{a}}{\mathrm{F}}_{\mathrm{R},\mathrm{r}}\right)$$

(5)

For a horizontal and flat ground, pushing force is expressed by:

$${\mathrm{F}}_{\mathrm{T}}=\mathrm{M}\mathsf{\gamma }+{\displaystyle \frac{{\mathrm{h}}_{\mathrm{G}}\mathsf{\gamma }}{\mathrm{L}\mathrm{g}}}\left({\mathrm{F}}_{\mathrm{R},\mathrm{r}}-{\mathrm{F}}_{\mathrm{R},\mathrm{f}}\right)+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{b}}}{\mathrm{L}}}{\mathrm{F}}_{\mathrm{R},\mathrm{f}}+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{a}}}{\mathrm{L}}}{\mathrm{F}}_{\mathrm{R},\mathrm{r}}$$

(6)

The full method for obtaining Equation (6) for pushing force is provided in Appendix B.

Given that we know the pushing force F_T,ste of the equipment moving on the smooth, flat, horizontal steel plate and the acceleration γ, by using Equation (6) the pushing force F_T,flo on the floor covering can be evaluated using formula:

$$\begin{array}{c}{\mathrm{F}}_{\mathrm{T},\mathrm{f}\mathrm{l}\mathrm{o}}={\mathrm{F}}_{\mathrm{T},\mathrm{s}\mathrm{t}\mathrm{e}}+{\displaystyle \frac{\mathsf{\gamma }{\mathrm{h}}_{\mathrm{G}}}{\mathrm{L}\mathrm{g}}}\left({\mathrm{F}}_{\mathrm{R},\mathrm{r},\mathrm{f}\mathrm{l}\mathrm{o}}-{\mathrm{F}}_{\mathrm{R},\mathrm{r},\mathrm{s}\mathrm{t}\mathrm{e}}+{\mathrm{F}}_{\mathrm{R},\mathrm{f},\mathrm{s}\mathrm{t}\mathrm{e}}-{\mathrm{F}}_{\mathrm{R},\mathrm{f},\mathrm{f}\mathrm{l}\mathrm{o}}\right)+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{b}}}{\mathrm{L}}}\left({\mathrm{F}}_{\mathrm{R},\mathrm{f},\mathrm{f}\mathrm{l}\mathrm{o}}-{\mathrm{F}}_{\mathrm{R},\mathrm{f},\mathrm{s}\mathrm{t}\mathrm{e}}\right)+\\ {\displaystyle \frac{{\mathrm{L}}_{\mathrm{a}}}{\mathrm{L}}}\left({\mathrm{F}}_{\mathrm{R},\mathrm{r},\mathrm{f}\mathrm{l}\mathrm{o}}-{\mathrm{F}}_{\mathrm{R},\mathrm{r},\mathrm{s}\mathrm{t}\mathrm{e}}\right)\end{array}$$

(7)

where:

F_R,r,flo and F_R,r,ste are the pushing forces for the rear wheel on the floor covering and the steel plate, respectively;

F_R,f,flo and F_R,f,ste are the pushing forces for the front wheel on the floor covering and the steel plate, respectively.

Using our definition of corrective force given in Equation (1), the corrective forces ξ_r and ξ_f for the rear and front wheels, respectively, are as follows:

$${\mathsf{\xi }}_{\mathrm{r}}={\mathrm{F}}_{\mathrm{R},\mathrm{r},\mathrm{f}\mathrm{l}\mathrm{o}}-{\mathrm{F}}_{\mathrm{R},\mathrm{r},\mathrm{s}\mathrm{t}\mathrm{e}}$$

(8)

$${\mathsf{\xi }}_{\mathrm{f}}={\mathrm{F}}_{\mathrm{R},\mathrm{f},\mathrm{f}\mathrm{l}\mathrm{o}}-{\mathrm{F}}_{\mathrm{R},\mathrm{f},\mathrm{s}\mathrm{t}\mathrm{e}}$$

(9)

Thus, using Equations (8) and (9), pushing force F_T,flo (Equation (7)) on the floor covering becomes:

$${\mathrm{F}}_{\mathrm{T},\mathrm{f}\mathrm{l}\mathrm{o}}={\mathrm{F}}_{\mathrm{T},\mathrm{s}\mathrm{t}\mathrm{e}}+{\displaystyle \frac{\mathsf{\gamma }{\mathrm{h}}_{\mathrm{G}}}{\mathrm{L}\mathrm{g}}}\left({\mathsf{\xi }}_{\mathrm{r}}-{\mathsf{\xi }}_{\mathrm{f}}\right)+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{b}}}{\mathrm{L}}}{\mathsf{\xi }}_{\mathrm{f}}+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{a}}}{\mathrm{L}}}{\mathsf{\xi }}_{\mathrm{r}}$$

(10)

Given this equation, in addition to the result of the corrective forces, it is necessary to know the equipment’s acceleration, its wheelbase L, the height h_G of the centre of mass, and the distance between the centre of mass and the axles of the front L_a and rear wheels L_b. Except for acceleration, which depends on the operator and requires the use of an accelerometer (low-cost accelerometers are sufficient for this purpose), the other information is easily accessible to designers and users of carts.

This model is intended to be applied to equipment carrying stable loads. Indeed, if the load distribution of the equipment varies during movement (as is the case when moving liquids in tanks, for example), the contact properties between the wheels and the floor vary over time. In such cases, the model cannot accurately evaluate the pushing force required.

3.5. Comparison of Experimental and Modelled Pushing Forces

Given that the pushing force of the cart on the steel plate, its acceleration γ and the load distribution between front and rear wheels are all known, the pushing force on floor coverings as well as the values of corrective force for rear and front wheels (respectively ξ_r and ξ_f) can be estimated using Equation (10). These values of corrective force are provided by the abaci (chosen according to the type of tread and bearing of the front and rear wheels fitted on the four wheels equipment) at load carried by the front and rear wheels and as a function of the floor covering where the equipment is used (defined by the hardness of the base foam for the covering in question). A set of experiments consisting of 168 different combinations (14 cart configurations as listed in Table 1, using four loads and three floor coverings) was designed to assess the robustness of the model incorporating different wheels at the front and rear of the cart. A total of 731 trials were conducted with the same operator, applying different accelerations. The experimental results are compared to the values obtained from the model. First of all, an example of the experimental pushing force obtained from two trials of the cart under the same configuration (wheels, load, load distribution between the rear and front wheels, floor covering), but with two different accelerations (arbitrarily called low and high accelerations) applied by the same operator, is illustrated in Figure 9. This figure shows the two pushing forces and the corresponding inertia forces for both trials. We observed that the gap between the pushing forces and the corresponding inertia forces is similar for both applied accelerations. This gap depends on the resistive force associated with the type of wheels fitted to the cart and the type of floor over which it moves.

Then, out of the 731 comparisons conducted, Figure 10a–c represents 3 of them. Figure 10a shows a comparison of the cart fitted with wheel configuration N°4, bearing a load of 202 kg and moving on the floor covering PVC.Fl.3. A deviation of 1.8% from the mean is observed between this experimental peak value of the pushing force and the peak value obtained with the predictive model. Figure 10b depicts wheel configuration N°12 with a load of 126 kg and for PVC.Fl.1 floor covering. Here, a deviation of 1.6% from the mean is observed. Configuration N°10 with a load of 126 kg and for PVC.Fl.2 floor covering is applied in Figure 10c, where a deviation of 4.3% from the mean is observed. The mean deviation, calculated from the 731 trials, is 5.1%. Thus, we consider the model to be well adapted for describing the pushing force for carts that move on different floor coverings, as it evaluates the actual resistive forces associated with the contact between each wheel of the cart and the floor. Of course, additional experiments on other types of floor coverings (linoleum, etc.) need to be performed in order to extend the use of this model to any kind of configuration. In particular, it would be necessary to have a broader range of hardness for floor covering’s base foam to complete the x-axis of the corrective forces’ abaci.

4. Discussion

The main outcome of this study is that it proposes a new method for modelling and evaluating the effort required to move a four-wheeled manually handled cart on different floor coverings. In particular, it recommends that the effort required for specific floor coverings be evaluated on the basis of knowing the value required to move the same cart on a steel plate (in accordance with the protocol described in the design standards). This new method is based on the definition of a corrective force for a unique wheel and on a pushing force model that allows us to measure the effort required to move a full cart. The results obtained using a combination of corrective force abaci and the pushing force model as well as through experimental tests with a cart equipped with instruments accord well with one another. We have therefore shown that it is possible to predict the pushing forces required for different floor coverings depending on the existing configuration. The traditional test methods described in the design standards, which use a smooth, flat, horizontal steel plate to evaluate pushing force and maximal load of a piece of equipment, may, therefore, continue to be employed. The pushing force value is then adjusted, by means of this combination of corrective force abacus and of the pushing force model, to take into account the real conditions of use. More specifically, with the load carried by the rear and front wheel axles, the corrective forces corresponding to the type of wheel (defined by its tread and its bearing) and to the hardness values of the floor covering’s base foam, are given by the abacus. These values are then introduced into the Equation (10), thereby giving the adjusted pushing force. When the corrected pushing force is higher than the recommended limit values defined in the design standards, it is appropriate to decrease the value of the maximal load of the cart. Corrective forces are calculated by measuring the resistive forces for various wheels and floors, which have already been described in a previous article [42]. Just as is the case with resistive forces [36,42], a linear relationship is observed between the corrective force and the load carried by the wheel, for all floor coverings and types of wheels, as demonstrated in the previous section. Interestingly, while wheels with a solid rubber tread require a greater pushing force, the corrective forces are lower because there is only a minimal difference between the effort expended to push the same cart on floor coverings and to do the same on a steel plate. By contrast, when pushing forces are low (for polyurethane and polyamide treads), corrective forces are more important. These results show the importance of taking floor coverings into account when evaluating the effort required to push a cart. Moreover, corrective forces also vary with the type of wheel bearing used. Nevertheless, whether the wheel bearing has a greater or lesser influence depends, on the one hand, on the wheel’s tread and, on the other, on the floor covering in question. For example, a very low difference (1 N at 150 kg for A/15: 52 hardness) is observed between wheels with a solid rubber tread and sleeve bearings and wheels with the same tread but with cone ball bearings; this difference is, however, more significant (6 N at 150 kg for A/15: 52 hardness) in the case of polyamide tread. Nonetheless, for floor coverings with a harder base foam (A/15: 95), the difference is approximately 1 N. Using the pushing force model, the corrective force abacus and a given acceleration, a difference of 26 N (between the pushing forces on a steel plate and on a floor covering whose base foam has a hardness of A/15: 49) is obtained for a cart equipped with four thermoplastic rubber wheels with cone ball bearings and a total weight of 300 kg with a front/rear load distribution ratio of 40:60. In the same conditions, a difference of 36 N is observed for a cart equipped with four wheels with polyamide treads and cone ball bearings.

Ergonomic standard ISO 11228-2 [23,24] gives the recommended limits for pushing and pulling a cart by defining initial and sustained forces. For example, for a cart with a handle height of 135 cm, a travel distance of 8 m and a working frequency of 96 pushes/8 h (0.2/min), used by a mix of age and gender, the initial and sustained forces limits are 180 N and 80 N. If design standards for manual handling equipment do not consider the composition of the floor covering [34,35], these force limits are increased (of 26 N and 36 N respectively for this example). For a hoist, the design standard EN ISO 10535 [34] specifies that the maximum initial force must not exceed 160 N, and the maximum sustained force must not exceed 85 N. The tests consist of pushing the hoist (at its nominal load) onto a steel plate and ensuring that the measured forces do not exceed these maximum values. However, conducting these tests on the steel plate leads to an overestimation of the nominal load, as the pushing force is lower on steel than on resilient floor coverings. For example, these devices are generally fitted with 100 mm diameter wheels, both at the front and rear. Based on standard test method conducted on a steel plate and assuming a nominal load of 175 kg (with an empty weight of 50 kg and a front/rear load distribution ratio of 30:70), and considering wheels with casted polyurethane tread and P.B.B. bearing, the increase in the initial force required for the same acceleration reaches 19% when moving over a floor covering with a base foam hardness of A/15:49. The greater the load, the more significant this difference becomes. Moreover, in the absence of acceleration during the sustaining phase of movement, the sustained force remains within the maximum limit set by this standard (this observation is frequently made during tests conducted in companies). These observations highlight the potential for an increase in musculoskeletal disorders in the workforce. Indeed, the maximum effort values defined in the ISO 11228-2 [23,24] standard are based on recommendations established through the Snook and Ciriello psychophysical approach (this is based on individuals’ subjective evaluation of the maximum acceptable external forces) [27]. However, these recommendations do not systematically account for the actual biomechanical constraints imposed on the human body, particularly the spine, during pushing and pulling tasks. In fact, Weston et al. [45] conducted a biomechanical study on the maximum force limits to be respected for pulling and pushing tasks. The defined limit values can be up to 30% lower than those established by the Snook and Ciriello psychophysical approach [27]. Next, the differences in effort will accumulate. It has been shown that the cumulative effect of increased forces on operators over time leads to an increase in fatigue and, consequently, a higher risk of musculoskeletal disorders [46].

Thus, it is important to consider floor coverings when designing manual handling equipment in order to limit the resistive forces and thus the strength necessary to move them; this, in turn, will help to reduce the risk of musculoskeletal disorders.

The present findings, specifically directed toward designers of manually propelled wheeled equipment, were built upon previously obtained results [42]. These earlier findings provided guidelines for both users and designers to reduce the effort required to initiate and move such equipment. The results demonstrated that effort increases linearly with the load and is influenced by the wheel’s characteristics (diameter, tread type, and bearing type) as well as the floor type. Notably, effort decreases with larger wheel diameters. Using wheels with C.B.B. or P.B.B. bearings reduces effort, whereas S.B. bearings, which increase effort, should be avoided. Additionally, wheels with solid rubber treads increase effort, making elastic-tire or polyurethane treads preferable for reducing it. Furthermore, softer base foams in resilient floor coverings (low Shore-A hardness) increase the required effort. To minimize this, it is recommended to use hard monolayer floor coverings (>A75) or multilayer coverings with a hard base foam layer (>A75).

Although these results are intended to improve the design of the equipment. Then, it is necessary to question the influence of the operators on these results. Several studies address the influence of inter-operator variability on the pushing force. Kao et al. [33] showed that operators tend to bend their trunk forward to use their body weight to increase the pushing force. Hoffman et al. [47] observed two types of posture for pushing tasks. These are defined by the angular position of the elbows and the position of the legs. This study distinguishes between postures where the elbows are flexed and those where they are extended. Maras & Karwowski [48] demonstrated that the applied force depends on the position of several body segments, such as the height of the force application point (hand position), as well as the position of the trunk and feet. Regarding the influence of the operators’ sex, Van Der Beek et al. [49] showed, on one hand, that as the load increased the time gap between male and female subjects to perform the pushing tasks also increased (0.1, 0.3, 0.5, 1.6 s for loads of 130, 250, 400, and 550 kg, respectively). However, they also demonstrated that although male subjects exerted a greater initial force than female subjects, this difference disappeared when it was corrected by considering the subject’s weight, height, and maximum strength. However, Savin et al. [50] also showed that the same operator will not perform a task identically on two separate occasions, and also that two operators will not perform the task in the same way. Indeed, this variability in movements, both intra- and inter-individual, can be influenced by factors related to the nature of the task (such as mass, size, and geometry of the workstation) and the operators (such as age, sex, experience, and health status). Moreover, the corrective force abacus proposed in this study are based solely on the mechanical properties related to the contact between the wheels and the floor. Inter-operator differences are expressed in the acceleration applied to the equipment. Since we have proposed to measure the acceleration and then incorporating these data into the model to deduce the pushing force on a different type of floor covering, we take inter-individual differences into account.

However, this study is subject to a number of limitations. Only four floor coverings have been studied, and therefore only four hardness values pertaining to the floor coverings’ base foams have been considered. Indeed, if the current results were applied to floor coverings with floor covering’s base foam hardness different from those studied, the results would deviate from reality. However, the harder the base foam is, the smaller the corrective forces to be applied are. Thus, corrective forces may therefore be studied for a wider range of floor coverings, and especially for intermediate values of the hardness of the floor covering’s base foam in order to complete the corrective forces abacus:

o

between A/15: 52 and A/15: 73, the interval for which the variations in corrective forces may likely be the most significant due to the low hardness of the floor covering’s base foam (softness);

o

a study of corrective forces for floor covering’s base foam hardness between A/15: 73 and A/15: 95 (hardest floor coverings).

It will then be possible to explore the joint validation of the model and the corrective force abacus, for a wider range of hardness values of the floor covering’s base foam. This study must also be extended to other types of floor coverings (thick and thin), such as floorings made from natural (seagrass, wool, etc.) or synthetic fibres (polyamide, polypropylene, etc.), which can be found in hotels, retirement homes and planes as well as across a host of other settings. Indeed, some studies have reported that effort is higher for carpet than for other floor coverings, such as linoleum [51,52,53,54].

This model is intended to be applied to equipment carrying stable loads. Indeed, if the load distribution of the equipment varies during movement (as is the case when moving liquids in tanks, for example), the contact properties between the wheels and the floor vary over time. In such cases, it would then be necessary to determine this force for each time interval, when the load distribution between the rear and front wheel axles varies. Similarly, if the slope along the path followed by the equipment changes, the pushing force should be determined for each section where the slope varies.

Finally, the trials conducted to validate the combination of corrective forces and the pushing force model were performed with only one operator. Although we applied various levels of acceleration (from 1 m.s⁻² to 3 m.s⁻²) to simulate the variability induced by different operators, this comparison would deserve to be carried out with multiple operators.

Previous studies have analysed resistance during turning motions, especially for wheelchairs [55,56,57]. Future work may therefore study the influence of swivelling on pushing force, by isolating a caster using our test bench and studying various caster properties (tread, diameter and bearing of wheel and swivel offset) as well as various types of floor coverings. Swivelling corrective forces may be determined as a function of sensitive parameters of caster and floor.

5. Conclusions

This paper presents a new method for estimating the pushing force of a cart in a longitudinal motion on various floor coverings. This method combines corrective forces (obtained for a single wheel) with a model of the pushing force needed to move four-wheeled equipment. The corrective forces, which are defined as the difference between the resistive force obtained for a given wheel/floor coupling on a floor covering and the resistive force obtained for the same coupling on steel plate. The corrective force increases linearly with the load. Moreover, the present study demonstrates that the effort required depends on the hardness of the base foam under the floor coverings. In view of these results, an average corrective force is provided for a specific tread and wheel bearing and as a function of the floor covering’s base foam hardness (shore A/15) and of the load carried by the wheels. Based on these results, corrective forces’ abaci are proposed. The result of pushing force obtained from the combination of our pushing force model with these abaci is compared to the experimental pushing force measurements recorded using an instrumented cart. Several configurations of front and rear wheels (varying diameter, tread and bearing), combined with different loads and accelerations, were tested. These comparisons are in good accordance and demonstrate the applicability of our method. These results could be incorporated into the test methods, described in the design standards for assessing the pushing forces, in order to take into account the real conditions in which manual handling equipment is used. Thanks to these findings, the pushing force (through the total load carried by the equipment) of these manual handling equipments under real usage conditions can be reduced. Thus, the risk of musculoskeletal disorders developing in the workplace is reduced, as musculoskeletal disorders are one of the leading causes of disability worldwide.