Multi-point support reaction forces calculation based on unilateral constraint

Abstract

Preventing overturning of construction machinery necessitates precise calculation of support reaction forces. The primary challenges reside in: (1) statically indeterminate force distributions, (2) indeterminacy of ground-contact states at support points. While conventional engineering methods yield ambiguous solutions, finite element method (FEM) proves computationally intensive for this purpose. Crucially, the relationship between reaction force and penetration depth embodies a unilateral constraint mechanism. A method based on unilateral constraint that correlates micro-displacements of support structures with reaction forces is proposed in this study. By formulating equilibrium equations in terms of micro-displacements and providing initial values and the corresponding Jacobian matrix, the accuracy and computational efficiency are significantly enhanced. Several examples are provided to verify the correctness of the proposed method. The proposed method offers an innovative idea for resolving statically indeterminate problems while simultaneously determining contact states—a critical factor in construction machinery overturn analysis.

Introduction

Accurate acquisition of support reaction forces is critical for various construction machinery such as cranes, as inaccurate estimation constitutes the primary cause of overturning accidents in practice. ¹ With the advancement of the crane industry, loading conditions of support structures have garnered extensive research attention. Current crane design standards prioritize estimating support reaction forces via simplified formulations, primarily for overturning safety verification. The principal challenges in resolving these forces are twofold: (1) support structures are always statically indeterminate, as the number of support points is always larger than three, and multi-point contact leads to high-order statically indeterminate issues; (2) the contact status at each support point should be determined, which is essential in defining loading configuration. Current study primarily employs virtual stiffness models and the finite element method to calculate support reaction forces.

The introduction of virtual stiffness is aimed at establishing an approximate relationship between the displacement of the support point and the corresponding support reaction force. Romanello ² treated the support structure as rigid body, and assumed that reaction forces had a linear relationship with displacement of outriggers, then calculated the reaction forces of cranes with four outriggers. Furthermore, Romanello ³ presented a graphical approach for reaction force based on his previous work, which is more efficient. Lee et al. ⁴ studied the details of the outrigger plates and proposed a method for selecting candidate cranes under different working conditions by the examples stored in the database provided by their users. Chang et al. ⁵ applied principle of virtual displacement to solve statically indeterminate problem, and deal with the case when outrigger expands arbitrarily. Jeng et al. ⁶ utilized two-phase simplex algorithm and minimum/maximum pruning operation to solve equations of support reaction forces. Engineering practices prove that assuming that the ground is elastic is more reasonable than assuming the support structure is flexible. However, existing researches focus on support structures with four outriggers, while multi-point support contact and the determining of contact status still remain to be studied.

Another way to obtain support reaction force is to take advantage of finite element method, which is widely applied in engineering. Teng and Wang ⁷ calculated support reaction forces of truck cranes based on moment equilibrium, and verified the results with ANSYS. Karaman and Öztürk ⁸ obtained the reaction force of hydraulic outrigger and stabilizer via NASTRAN, and studied the displacement and stress of the support structure. Ali et al. ⁹ applied FEM and determined the reaction forces of a crane with 16 support points. Suh and Yoon ¹⁰ computed the contact reaction forces using ADAMS to estimate the maximum lifting capacity. Based on certain load configuration, FEM assumes that support structure is flexible, but also fails to determine the contact status between the support point and the ground.

Studies on other mechanisms such as feeding mechanism ¹¹ or crankshaft^12–14 also inspires us that reaction forces vary from different numbers and arrangements of support points. Besides, the lateral force of the crane should be taken into consideration, and numerous studies have been performed on wind loads,^15,16 which is important to the overturn analysis.

Actually, the key to deal with the high-order statically indeterminate issue and determine the contact status lies in the application of the unilateral constraint model, the effect of which is overlooked in prior studies. Unilateral constraint model is employed in this paper to analyze overturning stability. By exploiting the mathematical properties of unilateral constraints, the contact force under varying contact states can be determined. Based on unilateral constraint, a method to solve support reaction force utilizing micro-displacement and virtual stiffness is proposed in this paper, where the penetration depth of the support points is represented by the micro-displacements of the structure. A smoothing method is implemented to ensure the continuity of the relationship between the penetration depth and reaction force, which guarantees the equation is solvable.

The paper is organized as follows. In Section “Support reaction force calculation,” considering unilateral constraint, the relationship between the micro-displacements and support reaction forces is studied, which provides supplementary equation to the statically indeterminate issue. In Section “Equilibrium equations of the support structure,” equilibrium equations of the system are established, and initial values of the variables and Jacobian matrix of the equations are provided. In Section “Verification example,” several examples with different working conditions are provided to verify the proposed method. The research methodology is drawn in Figure 1.

Figure 1.

Flowchart of the research methodology.

Support reaction force calculation

Structure of support system and basic assumptions

The configuration of support structures depends on the working conditions, as illustrated in Figure 2. The number of support points and their arrangement modes vary significantly.

Figure 2.

Common support structures of crane: (a) crane with four outriggers, (b) crane with five outriggers, (c) crawler crane, and (d) marine crane.

The support reaction force on each support point can be decomposed into the following form

$${\mathit{f}}_g=f_x{\mathit{g}}_x+f_y{\mathit{g}}_y+f_z{\mathit{g}}_z$$

(1)

where, ${\mathit{g}}_x$ , ${\mathit{g}}_y$ , ${\mathit{g}}_z$ are the base vectors of global coordinate system, and $f_x$ , $f_y$ , $f_z$ are the component product on these base vectors. It is acknowledged from equation (1) that the number of unknown component forces depends on the quantity of support points. However, only six equilibrium equations are available, resulting in statically indeterminate problem. Supplementary equations must be introduced to establish relationships among the variables. A methodology employing micro-displacements and virtual stiffness to determine support reaction forces is studied in this paper, and two assumptions are proposed:

1. The support structure is modeled as a rigid body which can penetrate into the ground according to Figure 3, and the penetration depth of each support point can be solved by $[u_z,\alpha ,\beta ,\gamma ]$ , the micro-displacements of the support structure under the external force ${\mathit{f}}_e$ , where $u_z$ is the displacement along axis ${\mathit{g}}_z$ and $[\alpha ,\beta ,\gamma ]$ are cardan angles which describe the rotation of the support structure. All the micro-displacements are close to zero, so that the position of centroid and point of force application are not influenced by micro-displacements.

2. Owing to the unilateral constraint, the support reaction forces at each point are functions of both the penetration depth and virtual stiffness. The virtual stiffness serves as an intermediate variable; however, it does not represent the actual contact stiffness.

Figure 3.

Micro-displacements of the support structure.

The relationship between penetration depth and micro-displacement of the support structure

The support structure is constrained to move solely along axis ${\mathit{g}}_z$ , while the displacements along axes  ${\mathit{g}}_x$  and  ${\mathit{g}}_y$  are restricted by structural limitations such as slide rails. Assuming the origin of support structure’s center coincides with the global coordinate origin before penetration, its post-penetration position can be expressed as:

$${\mathit{r}}_c=u_z{\mathit{g}}_z$$

(2)

The base vectors of the local coordinate system of the support structure can be expressed by cardan angles.

$${\mathit{e}}_x=c_2{c}_3{\mathit{g}}_x+(s_1{s}_2{c}_3+c_1{s}_3){\mathit{g}}_y+(s_1{s}_3-c_1{s}_2{c}_3){\mathit{g}}_z$$

(3)

$${\mathit{e}}_y=-c_2{s}_3{\mathit{g}}_x+(c_1{c}_3-s_1{s}_2{s}_3){\mathit{g}}_y+(c_1{s}_2{s}_3+s_1{c}_3){\mathit{g}}_z$$

(4)

$${\mathit{e}}_z=s_2{\mathit{g}}_x-s_1{c}_2{\mathit{g}}_y+c_1{c}_2{\mathit{g}}_z$$

(5)

where $s_1=\sin \alpha$ , $c_1=\cos \alpha$ , $s_2=\sin \beta$ , $c_2=\cos \beta$ , $s_3=\sin \gamma$ , $c_3=\cos \gamma$ .

The origin of the contact point before penetration ( ${\overline{\mathit{r}}}_{\mathrm{gi}}$ ) and after penetration ( ${\mathit{r}}_{\mathrm{gi}}$ ) can be written as

$${\overline{\mathit{r}}}_{\mathrm{gi}}={\mathit{r}}_c+x_i{\mathit{g}}_x+y_i{\mathit{g}}_y+z_i{\mathit{g}}_z$$

(6)

$${\mathit{r}}_{\mathrm{gi}}={\mathit{r}}_c+x_i{\mathit{e}}_x+y_i{\mathit{e}}_y+z_i{\mathit{e}}_z$$

(7)

where $(x_i,y_i,z_i)$ are the components of origin of the contact point before penetration, which are design-determined constants. Thus, the penetration depths of the support point $i$ along the base vectors of global coordinates has the form

$${\epsilon }_{\mathrm{zi}}=z_i-{\mathit{g}}_z^T({\mathit{r}}_c+x_i{\mathit{e}}_x+y_i{\mathit{e}}_y+z_i{\mathit{e}}_z)$$

(8)

$${\epsilon }_{\mathrm{xi}}={\mathit{g}}_x^T({\mathit{r}}_c+x_i{\mathit{e}}_x+y_i{\mathit{e}}_y+z_i{\mathit{e}}_z)-x_i$$

(9)

$${\epsilon }_{\mathrm{yi}}={\mathit{g}}_y^T({\mathit{r}}_c+x_i{\mathit{e}}_x+y_i{\mathit{e}}_y+z_i{\mathit{e}}_z)-y_i$$

(10)

Through the derivation above, the relationship between penetration depth and the micro-displacements $[u_z,\alpha ,\beta ,\gamma ]$ is derived.

Unilateral constraint between support reaction force and penetration depth

In fact, the relationship between penetration depth and support reaction force exhibits unilateral constraint behavior, formally expressed by Signorini’s condition.^17–20

$${\epsilon }_n\ge 0,f_n\ge 0,{\epsilon }_n{f}_n=0$$

(11)

where, ${\epsilon }_n$ is the penetration depth obtained in Section “The relationship between penetration depth and micro-displacement of the support structure,” and $f_n$ is support reaction force. Taking the assumption 2 in Section “Structure of support system and basic assumptions” into consideration, the relationship between the penetration depth and support reaction force is illustrated in Figure 4. Only when the support point penetrates into the ground ( ${\epsilon }_n>0$ ), is the value of support reaction force positive.

Figure 4.

Penetration depth of the outrigger: (a) ${\epsilon }_n<0,f_n=0$ , (b) ${\epsilon }_n=0,f_n=0$ , and (c) ${\epsilon }_n>0,f_n>0$ .

To simulate unilateral constraint mentioned above, a penalty method is applied, as Figure 5(a) shows.

$$f_z=k_z{\epsilon }_z$$

(12)

$$k_z=\{\begin{array}{c}0\mathrm{when}{\epsilon }_z\le 0\hfill \\ \multicolumn{1}{c}{k_0\mathrm{when}{\epsilon }_z>\delta }\end{array}$$

(13)

where, $k_z$ is the virtual stiffness of the ground, $k_0$ is the value of and virtual stiffness, and ${\epsilon }_z$ is the penetration depth along axis ${\mathit{g}}_z$ .

Figure 5.

Value of the ground stiffness before and after correction: (a) $k_z$ before correction and (b) $k_z$ after correction.

From Figure 5(a), we can find there is a breaking point of $k_z$ when ${\epsilon }_z$ is close to zero, so the value of $k_z$ is not continuous, which is known as switch effect that leads to failure in solving nonlinear equations. In order to deal with numerical singularity, a smoothing method based on polynomial interpolation is applied.

A polynomial function $\lambda (\chi )$ is constructed in a correction interval near the singular point to ensure that the function is first order continuous. The boundary conditions for the value of $\lambda (\chi )$ are

$$\lambda =\{\begin{array}{c}\lambda =0,\frac{\partial \lambda }{\partial {\epsilon }_z}=0\mathrm{when}\chi =0({\epsilon }_z=0)\hfill \\ \multicolumn{1}{c}{\lambda =1,\frac{\partial \lambda }{\partial {\epsilon }_z}=0\mathrm{when}\chi =1({\epsilon }_z=\delta )}\end{array}$$

(14)

where, $\chi ={\epsilon }_z{\delta }^{-1}$ , and $\delta$ is the length of correction interval. According to the continuity requirements, cubic polynomial equations are utilized. Combining with equation (14), $\lambda (\chi )$ can be expressed as

$$\lambda =\{\begin{array}{cc}0\hfill & \mathrm{when}{\epsilon }_{\mathrm{zi}}\le 0\hfill \\ \multicolumn{1}{c}{3{\chi }^2-2{\chi }^3}& \mathrm{when}0<{\epsilon }_{\mathrm{zi}}\le \delta \hfill \\ \multicolumn{1}{c}{{\delta }^2{(\chi -1)}^2+1}& \mathrm{when}{\epsilon }_{\mathrm{zi}}>\delta \hfill \end{array}$$

(15)

So vertical support reaction force is corrected to

$$f_{\mathrm{zi}}=\lambda k_z{\epsilon }_{\mathrm{zi}}$$

(16)

The function graph of $k_z$ after correction is illustrated in Figure 5(b) for comparison, and we can find that the value of $k_z$ is continuous after smoothing. The relationship between horizontal reaction forces and horizontal penetration depths is also assumed as linear. When the support points penetrate into the ground, the horizontal reaction forces have the form

$$f_x=-\sqrt{k_x^2}{\epsilon }_x$$

(17)

$$f_y=-\sqrt{k_y^2}{\epsilon }_y$$

(18)

where, ${\epsilon }_x$ and ${\epsilon }_y$ are the penetration depth along axis ${\mathit{g}}_x$ and ${\mathit{g}}_y$ , while $k_x$ and $k_y$ are the virtual stiffness along axis ${\mathit{g}}_x$ and ${\mathit{g}}_y$ . $k_x$ and $k_y$ cannot be predetermined due to various contact situations, so they are treated as the unknown variables of equations.

In this section, to address the statically indeterminate problem, two assumptions are first proposed, which are the foundation of complementary equations. The unilateral contact model is then employed to establish the relationship between the penetration depth and the support reaction force.

Equilibrium equations of the support structure

Equilibrium equations

According to Section “Introduction,” the number of equilibrium equations is six, so the number of unknown variables should be six correspondingly. The relationship between penetration depths and micro-displacements has been derived in Section “The relationship between penetration depth and micro-displacement of the support structure,” and the virtual stiffness is introduced to establish the connection between penetration depth and support reaction force. Thus, micro-displacements and virtual stiffness mentioned in Section “Support reaction force calculation” are treated as unknown variables, which yields

$$\mathit{q}=[u_z,\alpha ,\beta ,\gamma ,k_x,k_y]$$

(19)

The support reaction forces can be expressed as the function of $\mathit{q}$ , and the number of the unknown variables is equal to the number of equations, so equilibrium equations can be solved. An example of the support reaction force analysis of a truck crane has been conducted, which is equipped with four outriggers as Figure 6 shows, where $\alpha$ is the rotation angle of the crane. Assuming that the mass of structure is $m_c$ , correspondingly the origin of the centroid can be expressed as

$${\mathit{r}}_c=x_c{\mathit{e}}_x+y_c{\mathit{e}}_y+z_c{\mathit{e}}_z$$

(20)

Figure 6.

Force analysis of a truck crane.

Assuming that the resultant external force of the support structure is ${\mathit{f}}_e$ , which is composed of support reaction forces of boom and cylinder. In engineering practice, loads such as wind and laterally loads can affect the overturning stability of cranes, and they are collectively treated as a part of external force ${\mathit{f}}_e$ in this paper, and the variation in wind load is directly manifested through changes in the external force. The origin of the point of force application has the form

$${\mathit{p}}_e={\mathit{r}}_c+x_e{\mathit{e}}_x+y_e{\mathit{e}}_y+z_e{\mathit{e}}_z$$

(21)

In equations (20) and (21), $[x_c,y_c,z_c]$ and $[x_e,y_e,z_e]$ are constants determined by structure design. Through the derivations above, the resultant force and moment of the support structure can be obtained

$${\mathit{F}}_c={\mathit{f}}_{g1}+{\mathit{f}}_{g2}+{\mathit{f}}_{g3}+{\mathit{f}}_{g4}+m_{c,1}\mathit{g}+{\mathit{f}}_e$$

(22)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{M}}_c={\mathit{r}}_{g1}\times {\mathit{f}}_{g1}+{\mathit{r}}_{g2}\times {\mathit{f}}_{g2}+{\mathit{r}}_{g3}\times {\mathit{f}}_{g3}}\\ \multicolumn{1}{c}{+{\mathit{r}}_{g4}\times {\mathit{f}}_{g4}+m_{c,1}{\mathit{r}}_{c,1}\times \mathit{g}+{\mathit{p}}_e\times {\mathit{f}}_e}\end{array}$$

(23)

The equilibrium equations of the support structure can be grouped as

$${\mathit{f}}_q=[{\mathit{F}}_c;{\mathit{M}}_c]=\mathit{0}$$

(24)

The initial value and Jacobian matrix of the equation

The accuracy of the solution of nonlinear equations depends critically on the initial value of the unknown variables, and different initial values may lead to different solutions. Reasonable initial values are essential to obtaining physically consistent solutions. The initial values of the variables are listed in Table 1. Especially, the initial value of $u_z$ must be positive, otherwise the solution of $u_z$ will be negative, which is inconsistent with the fact.

Table 1.

Initial value of the variables.

Variable	Value
uz (m)	0.05
α (rad)	0
β (rad)	0
γ (rad)	0
kx (N/m)	314,000
ky (N/m)	314,000

Newton’s method is a standard iterative technique for solving nonlinear equations, requiring explicit formulation of the Jacobian matrix. To enhance computational efficiency, analytical derivation of the Jacobian is strongly preferred. When closed-form expressions are unavailable, numerical solvers typically resort to finite-difference approximations, a computationally expensive alternative. Take the derivative of the equations with respect to time, we can observe that if the derivative of the equations can be written as the product of a transfer matrix ${\mathit{T}}_{f,q}$ and the derivative of variables $\stackrel{·}{\mathit{q}}$ , the Jacobian matrix is equal to ${\mathit{T}}_{f,q}$ by definition.

$${\stackrel{·}{\mathit{f}}}_q={\mathit{T}}_{f,q}\stackrel{·}{\mathit{q}}=\frac{\partial {\mathit{f}}_q}{\partial \mathit{q}}\stackrel{·}{\mathit{q}}$$

(25)

Accordingly, the changing rate of each item in equations (22) and (23) should be expressed as the product of a transfer matrix and $\stackrel{·}{\mathit{q}}$ . The time derivative of the displacement of origin of the support structure is equal to

$${\stackrel{·}{\mathit{r}}}_c={\stackrel{·}{u}}_z{\mathit{g}}_z={\mathit{T}}_{\mathrm{r}}\stackrel{·}{\mathit{q}}$$

(26)

The angular velocity of the structure under Cardan description has the form

$$\mathit{\omega }=\stackrel{·}{\alpha }{\mathit{p}}_1+\stackrel{·}{\beta }{\mathit{p}}_2+\stackrel{·}{\gamma }{\mathit{p}}_3={\mathit{T}}_{\omega }\stackrel{·}{\mathit{q}}$$

(27)

where, the rotation axes of the three rotations are

$${\mathit{p}}_1={\mathit{g}}_x$$

(28)

$${\mathit{p}}_2=\cos \alpha {\mathit{g}}_y+\sin \alpha {\mathit{g}}_z$$

(29)

$${\mathit{p}}_3=\sin \beta {\mathit{g}}_x-\sin \alpha \cos \beta {\mathit{g}}_y+\cos \alpha \cos \beta {\mathit{g}}_z$$

(30)

Thus, the time derivative of base vector of the local coordinate can be expressed as

$${\stackrel{·}{\mathit{e}}}_x=\mathit{\omega }\times {\mathit{e}}_x;{\stackrel{·}{\mathit{e}}}_y=\mathit{\omega }\times {\mathit{e}}_y;{\stackrel{·}{\mathit{e}}}_z=\mathit{\omega }\times {\mathit{e}}_z$$

(31)

Take the derivative of equations (1) and (7) with respect to time

$${\stackrel{·}{\mathit{f}}}_{g,i}={\stackrel{·}{\mathit{f}}}_{i,x}{\mathit{g}}_x+{\stackrel{·}{\mathit{f}}}_{i,y}{\mathit{g}}_y+{\stackrel{·}{\mathit{f}}}_{i,z}{\mathit{g}}_z+\mathit{\omega }\times {\mathit{f}}_{g,i}={\mathit{T}}_{g,i}\stackrel{·}{\mathit{q}}$$

(32)

$${\stackrel{·}{\mathit{r}}}_{\mathrm{gi}}={\stackrel{·}{u}}_z{\mathit{g}}_z+\mathit{\omega }\times ({\mathit{r}}_{\mathrm{gi}}-{\mathit{r}}_c)={\mathit{T}}_{\mathrm{rg},i}\stackrel{·}{\mathit{q}}$$

(33)

Take the derivative of equations (17) and (18) and (16) with respect to time

$${\stackrel{·}{f}}_{\mathrm{kx}}=-(k_x^2{)}^{1/2}{\stackrel{·}{\epsilon }}_{\mathrm{kx}}-{\epsilon }_{\mathrm{kx}}(k_x^2{)}^{-1/2}{k}_x{\stackrel{·}{k}}_x={\mathit{T}}_{\mathrm{fkx}}\stackrel{·}{\mathit{q}}$$

(34)

$${\stackrel{·}{f}}_{\mathrm{ky}}=-(k_y^2{)}^{1/2}{\stackrel{·}{\epsilon }}_{\mathrm{ky}}-{\epsilon }_{\mathrm{ky}}(k_y^2{)}^{-1/2}{k}_y{\stackrel{·}{k}}_y={\mathit{T}}_{\mathrm{fky}}\stackrel{·}{\mathit{q}}$$

(35)

$${\stackrel{·}{f}}_{\mathrm{kz}}=k_z{\epsilon }_{\mathrm{kz}}{\stackrel{·}{\lambda }}_{\mathrm{kz}}+{\lambda }_{\mathrm{kz}}{k}_z{\stackrel{·}{\epsilon }}_{\mathrm{kz}}={\mathit{T}}_{\mathrm{fkz}}\stackrel{·}{\mathit{q}}$$

(36)

where, the time derivative of the penetration depth and ${\lambda }_{\mathrm{kz}}$ are functions of $\stackrel{·}{\mathit{q}}$ , which can be written as

$${\stackrel{·}{\epsilon }}_{\mathrm{kz}}=({\mathit{r}}_{\mathrm{gk}}-{\mathit{r}}_c{)}^T(\mathit{\omega }\times {\mathit{g}}_z)-{\stackrel{·}{u}}_z$$

(37)

$${\stackrel{·}{\epsilon }}_{\mathrm{kx}}=({\mathit{r}}_c-{\mathit{r}}_{\mathrm{gk}}{)}^T(\mathit{\omega }\times {\mathit{g}}_x)$$

(38)

$${\stackrel{·}{\epsilon }}_{\mathrm{ky}}=({\mathit{r}}_c-{\mathit{r}}_{\mathrm{gk}}{)}^T(\mathit{\omega }\times {\mathit{g}}_y)$$

(39)

$${\stackrel{·}{\lambda }}_{\mathrm{kz}}=\{\begin{array}{cc}0\hfill & \mathrm{when}{\epsilon }_{\mathrm{kz}}\le 0\hfill \\ \multicolumn{1}{c}{6{\delta }^{-1}(\chi -{\chi }^2){\stackrel{·}{\epsilon }}_{\mathrm{kz}}}& \mathrm{when}0<{\epsilon }_{\mathrm{kz}}\le \delta \hfill \\ \multicolumn{1}{c}{2\delta (\chi -1){\stackrel{·}{\epsilon }}_{\mathrm{kz}}}& \mathrm{when}{\epsilon }_{\mathrm{kz}}>\delta \hfill \end{array}$$

(40)

Take the derivative of equations (22) and (23) with respect to time

$${\stackrel{·}{\mathit{F}}}_c={\stackrel{·}{\mathit{f}}}_{g1}+{\stackrel{·}{\mathit{f}}}_{g2}+{\stackrel{·}{\mathit{f}}}_{g3}+{\stackrel{·}{\mathit{f}}}_{g4}+{\stackrel{·}{\mathit{f}}}_e$$

(41)

$$\begin{array}{c}{\stackrel{·}{\mathit{M}}}_c={\mathit{r}}_{g1}\times {\stackrel{·}{\mathit{f}}}_{g1}+{\mathit{r}}_{g2}\times {\stackrel{·}{\mathit{f}}}_{g2}+{\mathit{r}}_{g3}\times {\stackrel{·}{\mathit{f}}}_{g3}+{\mathit{r}}_{g4}\times {\stackrel{·}{\mathit{f}}}_{g4}\hfill \\ -{\mathit{f}}_{g1}\times {\stackrel{·}{\mathit{r}}}_{g1}-{\mathit{f}}_{g2}\times {\stackrel{·}{\mathit{r}}}_{g2}-{\mathit{f}}_{g3}\times {\stackrel{·}{\mathit{r}}}_{g3}-{\mathit{f}}_{g4}\times {\stackrel{·}{\mathit{r}}}_{g4}\hfill \\ +{\mathit{p}}_e\times {\stackrel{·}{\mathit{f}}}_e-{\mathit{m}}_{c,1}\mathit{g}\times {\stackrel{·}{\mathit{r}}}_{c,1}-{\mathit{f}}_e\times {\stackrel{·}{\mathit{p}}}_e\hfill \end{array}$$

(42)

According to equations (26)–(36), equations (41) and (42) can be rewritten as the product of a transfer matrix and the derivative of variables $\stackrel{·}{\mathit{q}}$ .

$${\stackrel{·}{\mathit{F}}}_c={\mathit{T}}_{\mathrm{fc}}\stackrel{·}{\mathit{q}}$$

(43)

$${\stackrel{·}{\mathit{M}}}_c={\mathit{T}}_{\mathrm{mc}}\stackrel{·}{\mathit{q}}$$

(44)

Referring to equation (25), the Jacobian matrix of the equations is obtained

$$\mathit{G}=[{\mathit{T}}_{\mathrm{fc}};{\mathit{T}}_{\mathrm{mc}}]$$

(45)

In this section, equilibrium equations are established. During the solution of the equations, if a support point is off the ground, the corresponding support reaction force automatically vanishes. The proposed method enables the direct determination of the support reaction forces without requiring explicit prior judgment of the contact state.

Verification example

Accurate calculation of support reaction forces is critical for overturn analysis—a fundamental safety requirement in engineering that verifies lifting capacity. Common crane support configurations include: outriggers (truck cranes), crawler tracks (crawler cranes), and ring rails (marine cranes). The mechanical characteristics and load conditions differ significantly among support types. Figure 7 illustrates the support structures for crawler crane and marine crane.

Figure 7.

Arrangements of support points of crawler crane and marine crane: (a) crawler crane and (b) marine crane.

The crane’s primary load-bearing components rotate about their central axis, with different rotation angles (in subsequent text, this quantity is denoted as  $\alpha$ ) defining distinct operational scenarios, as visually mapped in Figure 6. For comprehensive safety verification, support reaction forces must be determined at critical rotation angles representative of maximum loading conditions. Three representative application cases of different cranes as verification examples are provided in this section.

Support reaction force of truck crane

Taking the structure in Figure 6 as an exemplar case, the four outriggers are positioned at the vertices of a rectangular configuration. Support reaction forces are computed under varying rotation angles. Values of parameters when the rotation angle is 0 are tabulated in Table 2.

Table 2.

Parameters of the truck crane.

Parameter	Value
Lateral span (m)	4.35
Longitudinal span (m)	4.8
Load of the car frame (t)	13.32
The distance between the center of gravity and the centroid of the car frame (m)	8.62
External load (t)	8.6
External moment (kN × m)	688.25
The distance between the rotation center and centroid of the car frame (m)	0.525

The results and comparisons with different methods applied in engineering are listed in Tables 3 and 4.

Table 3.

Support reaction forces (t) when the rotation angle is 90°.

Methods	$f_{g1,z}/\mathrm{t}$	$f_{g2,z}/\mathrm{t}$	$f_{g3,z}/\mathrm{t}$	$f_{g4,z}/\mathrm{t}$
Method in this paper	2.461	7.668	14.343	17.55
Method in Zhang et al. 17	2.711	6.097	16.235	17.817
FEM	0.887	5.924	18.058	18.144
Method in standard	2.534	4.282	17.148	18.896

Table 4.

Support reaction forces (t) when the rotation angle is 135°.

Methods	$f_{g1,z}/\mathrm{t}$	$f_{g2,z}/\mathrm{t}$	$f_{g3,z}/\mathrm{t}$	$f_{g4,z}/\mathrm{t}$
Method in this paper	8.314	0.76	21.392	11.557
Method in Zhang et al. 17	9.089	2.001	21.481	10.288
FEM	9.232	1.859	21.481	10.068
Method in standard	10.376	0.702	20.710	11.054

Through the aforementioned comparisons, the overall trends of the results are consistent, although the specific values exhibit discrepancies. These differences arise primarily from the application of distinct underlying assumptions. The results obtained with the proposed method show closer agreement with those of the standard method, which is also predicated on rigid-body motion. However, the standard method employs an empirical formula for the relationship of force and penetration depth, introducing approximations that limit its accuracy. In contrast, FEM treats the structure as a flexible body, while potentially more accurate for scenarios involving deformation, this flexibility consideration substantially increases computational cost. The comparative analysis demonstrates that the proposed method achieves comparable accuracy while offering superior computational efficiency due to its reduced number of degrees of freedom.

Support reaction force of crawler crane

Unlike truck cranes, the support structure of a crawler crane comprises bilateral crawler assemblies, each consisting of support wheels distributed along both sides of the superstructure. This configuration yields significantly more ground contact points. Multiple loading conditions are analyzed using the proposed method, and the values of parameters when the rotation angle is 0 are listed in Table 5. The close agreement between results obtained from the proposed method and ANSYS finite element simulations are demonstrated in Figure 8.

Table 5.

Parameters of the crawler crane.

Parameter	Value
Load of turntable and parts above (t)	996
Moment on turntable and parts above (kN × m)	21,520
Load of the car frame (t)	194
Lateral span of the crawlers (m)	8.62
Longitudinal span of the crawlers (m)	8.6
Number of support wheels	32

Figure 8.

Comparison of crawler crane’s result of the proposed method and ANSYS.

To further evaluate performance across operational configurations, four loading conditions at distinct rotation angles are analyzed. The corresponding support reaction force at different support points are presented in Figure 9.

Figure 9.

Support reaction forces under different working conditions of crawler crane: (a) support reaction force when α = 0°, (b) support reaction force when α = 45°, (c) support reaction force when α = 90°, and (d) support reaction force when α = 135°.

As evidenced in Figures 8 and 9, the computational results obtained by the proposed method demonstrate a deviation of less than 5% comparing with the result in ANSYS, which verified the accuracy for the calculation of high-order statically indeterminate structures. Support reaction forces exhibit significant variation across rotation angles, and the support reaction force is greater at the contact point in closer proximity to the load position, which is consistent with the fact.

Support reaction force of marine crane

The marine crane’s support structure comprises multiple rollers arranged concentrically on a ring rail. Two distinct loading conditions are analyzed using the proposed method, with corresponding parameters documented in Table 6. Comparative validation against reference solutions is presented in Figure 10.

Table 6.

Parameters of the marine crane.

Parameter	Value
Load of turntable and parts above (t)	830, 880
Moment of turntable and parts above (kN × m)	17,610, 32,555
Rotation angle of the boom	0
Radius of the rail (m)	12
Number of support rollers	40

Figure 10.

Comparison of marine crane’s result of the proposed method and ANSYS.

Figure 10 validates the method’s accuracy across varying support point quantities and geometric configurations. The values of support reaction forces at different contact points are illustrated in Figure 11.

Figure 11.

Support reaction forces under different working conditions of mariner crane: (a) support reaction force under 830 t load and (b) support reaction force under 880 t load.

Figure 11 demonstrates the capability to detect ground contact status, evidenced by identifying elevated support points. As shown in Figure 11(b), support reaction forces of some contact points are zeros, which demonstrates the method’s applicability for determining multi-point support reaction forces under rigid-body motion, independent of support point quantity or spatial configuration.

Conclusion

A unilateral constraint-based method for analyzing multi-point support reaction forces in construction machinery is proposed in this study. By introducing micro-displacements and virtual stiffness, the approach resolves statically indeterminate problems while maintaining computational efficiency.

• (1) Unilateral constraint is introduced to describe the relationship between support force and penetration depth, where polynomial smoothing method eliminates singularities and analytical Jacobian matrices accelerate equation solving. The support reaction force can be directly determined without prior assessment of contact conditions.

• (2) The adoption of the rigid body motion assumption drastically reduces the number of degrees of freedom in the equations, thereby conferring a significant computational advantage over the finite element method in solving high-order statically indeterminate problems.

• (3) Comparative analysis of numerical examples and engineering tests are conducted to verify the correctness of the proposed method, which is proved to be a new approach for calculating support reaction forces in engineering structures.

• (4) The current analysis is limited to static support reaction force. Future investigations will explore the overturning stability under dynamic working conditions, such as crane luffing and slewing and time-varying loads. Due to experimental constraints, full-scale testing to validate the simulation results could not be conducted.