Numerical Analysis of Reinforcing Effect for Scissors-Type Bridge with Strut Members

Abstract

Recently, a scissor mechanism was efficiently applied in the safety engineering field as an emergency structure owing to the advantages of mobility, transformability, and re-usability. This paper focuses on the advantages of this mechanism and puts forward a deployable emergency bridge called Mobile Bridge as a smart bridge. To deploy this bridge in an emergency situation, the structural safety, such as strength and stiffness, must be ensured through proper reinforcing methods. Several research studies concerning the reinforcing effect to scissors structures have been conducted using a cable and/or strut. However, the reinforcing situation was limited, and it is not clear where and how much reinforcement should be introduced. In this paper, we discuss the reinforcing effect of simple struts through a theoretical and numerical approach. Then, we evaluate their applicability to the Mobile Bridge based on numerical simulation. The advantage of the proposed reinforcing method is evaluated, focusing on the reduction of the bending moment which is the dominant sectional force in the scissor structure. We found the reinforcing effect has a nonlinear relationship between the stress and ratio of extension rigidity. The most effective reinforcing configuration was a double warren truss with the vertical element in a two-unit scissors-type bridge and a double warren truss without the vertical element in a three-unit scissors-type bridge. The necessary sectional area of the strut elements was more than 0.2 times that of the scissors member. These results imply that the smart bridge can enhance its performance by using proper reinforcement of the struts.

1. Introduction

Climate change caused by global warming has a wide range of effects, including an increase in the frequency of heavy rains and the growing typhoons’ power. In Japan, the annual number of short-time heavy rainfall over 50 mm per hour has been increasing [1], and there are concerns about the damage to bridges [2,3]. After a disaster, rapid transportable and constructable bridges are essential for saving human lives and supporting restoration activities. Most of those structures are modular type bridges so as to reduce the number of workers and construction time on site. For example, the Bailey bridge designed by D. Bailey consists of a modular bridge and deck parts. The usability of this bridge is found in its extensibility by adding a variety of modular parts both vertically and horizontally to increase the load capacity, and it is possible to use in a wide range of post-disasters [4,5]. In some cases, the bridges have continued to be used as permanent bridges without being removed [6]. The Bailey bridge has evolved through a series of improvements as the Mabey and Acrow bridges (e.g., [7,8]).

On the other hand, this type of bridges usually requires heavy machinery to bring and construct the bridge parts on site. There is, however, another type of rapid bridge system using a deployable structure that is more suitable for use in severe situations with time limitations. Typical deployable bridges are two-fold or three-fold girder bridges such as the Portable bridge by ERE Logistics [9,10] and the Armored Vehicle-Launched Bridge [11,12]. The parts of the prefabricated bridge are equipped on a trailer and tank, and it is installed by working using an oil pressure system. Besides these, Lederman et al. proposed a deployable tied-arch bridge. Unlike the above deployable bridges, the entire bridge is folded into a roll to enable simple deployment [13]. Because of the prefabricated bridge parts, these bridges can be installed on-site more quickly than modular bridges despite the limitation of the utilization area.

This paper focuses on the advantages of a deployable structure and proposes a foldable emergency bridge, called the Mobile Bridge (herein called MB), with a scissor mechanism [14,15,16]. The basic scissor unit consists of two linear elements joined at a pivot, providing a hinge-connection at their centres [17]. In the fully deployed state, two members are in the shape of the character ‘X’ creating a single scissor unit, and it is connected to a next unit by hinges. These features allow one or two dimensional folding [18,19,20,21]. It is also possible to fold and/or deploy the whole system easily and quickly with a few workers. Up to now, we have demonstrated the design performance using a prototype of a full-scaled MB (herein called MB version 1.0 with two scissors units [22] and MB version 1.1 with three scissors units [23]). These articles indicated the basically analytical method of the scissor structure based on an equilibrium equation and proved its experimental validation. The latest MB version 4.0 can be refferenced in [24].

On the other hand, the structural safety of the MB must be ensured for practical use. The scissors-type bridge is typically weak at its pivot due to the effect on high bending moments (e.g., [16,25]). Studies on the functionality of recent scissors structures (e.g., [26,27]) have been presented; however, most of the structures are not bridges but relatively lightweight structures for roofing materials. Heavy temporary structures, such as bridges, need not only to support moving objects, but also a combination of reinforcing elements. Hence, some research was conducted to improve the performance of the scissors-type bridges, focusing on the type of the cross sections and decks [28,29]. However, since the basic structural form remains the same, it is difficult to expect significant changes. On another front, modular bridges are applying additional reinforcement parts and cable before/after installation aimed at improving the load capacity and stability [30]. These reinforcing elements should be designed to be lightweight so they can be installed without heavy machinery. The reinforcing techniques of the scissor structure before/after installation are discussed in a few research studies. For example, Kokawa [31,32] used a cable and Saitoh [33,34] used a combination of a cable and a strut member. The former study arranged the reinforcing cable starting from a central hinged point and zigzagged it through pulleys to each supporting strut before returning to the starting point. The latter research set the cable and the strut member between each hinge point. From the point of application on the bridge, the latter reinforcing method is expected to reduce the high sectional force in the scissors structure and increase the structural stiffness. Indeed, Yu et al. performed a static loading test using a laboratory scale experimental bridge with cable reinforcement [35]. The experimental result based on the strain and displacement measurement shows the reinforcing effects in comparison with a nonreinforced bridge. In the authors’ previous studies, the reinforcing effect of the MB was theoretically and numerically investigated using strut reinforcing members to resist both compressive and tensile forces [36,37,38]. However, considering the reinforcing situation was limited, it is not clear where and how much reinforcement should be introduced in the MB.

In this paper, we make the reinforcing effect for the scissors structure with strut members through a theoretical and numerical approach clear. When the scissors structure is reinforced by the strut members, the rotation of the scissors member is constrained, and the sectional force distribution significantly changes compared to the nonreinforced state. However, the change in sectional force depends on the conditions, such as the deployment angle of the scissors structure and the cross section of the scissors and strut members. Hence, first, we evaluate the advantages of strut reinforcement by theoretically using a single scissors structure model with different reinforcing positions, focusing on the reduction of the bending moment which is the dominant sectional force. We found that the reinforcing effect indicates a nonlinear solution depending on the deployment angle and cross section of the scissors and strut members. Then, the applicability of the strut reinforcement to the MB with two and three scissors units was evaluated with a numerical simulation. We considered several reinforcing patterns, that is, additional horizontal, vertical, or a combination of horizontal and vertical reinforcing members, and tried to seek optimal reinforcing layout and its sectional area. These results implied that the scissors-type bridge, including the MB, can enhance its performance by using proper reinforcement of the strut members.

2. Theoretical Evaluation for Strut Reinforcement

2.1. Basic Theory Based on Equilibrium Equations

Let us consider the reinforcing effect of the strut member from a theoretical approach based on the equilibrium equations [39]. This method can easily obtain nodal forces acting on hinges and pivots in the scissors structure. The authors confirmed usability of this method in our previous works through numerical and experimental validation [22,23]. Here, we expand the equilibrium equations as a statically indeterminate problem.

2.1.1. A Unit Model in a Statically Determinate Problem

A free-body diagram (FBD) for a scissor structure is shown in Figure 1. It is assumed that the length for each member is $L_0$, and the inclination angle is $\theta$ measured from the vertical direction; the panel length $\lambda$ and the height $\eta$ are related by $\lambda =L_{0}sin\theta$ and $\eta =L_{0}cos\theta$. Basically, the nodal forces of the scissor structure in Figure 1 can be solved by using the simple equilibrium equations. According to the equilibrium equations for the external forces in each x and y direction and the moments at point C of the intersecting members $\overline{B^L{A}^R}$ and $\overline{B^R{A}^L}$, it can express the matrix in the case of a cantilever model that has pinned support at points $A^L$ and $B^L$ shown below in Equation (1).

$$\begin{array}{ccc}\hfill L\left\{\begin{array}{c}{\left(B^L\right)}_x\\ {\left(B^L\right)}_y\\ {\left(A^L\right)}_x\\ {\left(A^L\right)}_y\end{array}\right\}& =& -R\left\{\begin{array}{c}{\left(B^R\right)}_x\\ {\left(B^R\right)}_y\\ {\left(A^R\right)}_x\\ {\left(A^R\right)}_y\end{array}\right\}-\left\{\begin{array}{c}{\left(C\right)}_x\\ {\left(C\right)}_y\\ 0\\ 0\end{array}\right\}\hfill \end{array}$$

(1)

where the symbols L and R are expressed as the following matrices:

$$\begin{array}{c}\hfill L=\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ -\eta & \lambda & 0& 0\\ 0& 0& \eta & \lambda \end{array}\right],R=\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 0& \eta & -\lambda \\ -\eta & -\lambda & 0& 0\end{array}\right]\end{array}$$

(2)

From for paragraph under the formulas, please confirm the similar changed in full text. CH: OK Equations (1) and (2), once the loads in the structure are clear, the unknown reaction forces can be obtained. Similarly, this method can be applied in the case of a simple supported model that has pinned support at points $B^L$ and $B^R$ shown below in the equation.

$$\begin{array}{ccc}\hfill L\left\{\begin{array}{c}{\left(B^L\right)}_x\\ {\left(B^L\right)}_y\\ {\left(B^R\right)}_x\\ {\left(B^R\right)}_y\end{array}\right\}& =& -R\left\{\begin{array}{c}{\left(A^L\right)}_x\\ {\left(A^L\right)}_y\\ {\left(A^R\right)}_x\\ {\left(A^R\right)}_y\end{array}\right\}-\left\{\begin{array}{c}{\left(C\right)}_x\\ {\left(C\right)}_y\\ 0\\ 0\end{array}\right\}\hfill \end{array}$$

(3)

When the nodal forces are clear, the sectional forces in each member can be calculated by coordinate-transforming for the obtained nodal forces.

2.1.2. A Unit Model in a Statically Indeterminate Problem

An FBD for a scissor structure with an upper strut member is shown in Figure 2a. This model has one degree of redundancy in comparison with the model in Figure 1. The statically indeterminate force X in the scissor structure can be led by the unit load method. It is necessary to think the two conditions depend on the number of the degree of redundancy as shown in Figure 2b,c. For instance, the statically indeterminate force X in the model Figure 2a can be solved by use of generated sectional forces in the models of Figure 2b,c as follows:

$$\begin{array}{ccc}\hfill X& =& -\frac{{\delta }_{00}}{{\delta }_{01}}\hfill \\ & =& -\frac{{\displaystyle \sum _{j=1}^4{\int }_0^{L_0/2}\frac{M_{0j}{M}_{1j}}{EI}d\chi +\sum _{j=1}^4\frac{N_{0j}{N}_{1j}}{EA}\frac{L_0}{2}}}{{\displaystyle \sum _{j=1}^4{\int }_0^{L_0/2}\frac{M_{1j}^2}{EI}d\chi +\sum _{j=1}^4\frac{N_{1j}^2}{EA}\frac{L_0}{2}+\frac{N_{15}^2}{EB}\frac{L_{0}sin\theta }{2}}}\hfill \end{array}$$

(4)

where $\chi$ is an arbitrary distance from each nodal point in axial direction, $L_0/2$ is a half length of member, $\theta$ is an expanding angle, E is elastic modulus, A is a sectional area of scissor member, B is a sectional area of the strut member, $N_{0j}$ is an axis force of the basic system, $M_{0j}$ is the bending moment of the basic system, $N_{1j}$ is an axis force of the first system, and $M_{1j}$ is the bending stress of the first system. This equation can be rewritten as follows:

$$\begin{array}{ccc}\hfill X& =& -\frac{{\displaystyle \frac{2A}{I{L}_0}\sum _{j=1}^4{\int }_0^{L_0/2}{M}_{0j}{M}_{1j}d\chi +\sum _{j=1}^4{N}_{0j}{N}_{1j}}}{{\displaystyle \frac{2A}{I{L}_0}\sum _{j=1}^4{\int }_0^{L_0/2}{M}_{1j}^{2}d\chi +\sum _{j=1}^4{N}_{1j}^2+\frac{sin\theta }{\alpha }}}\hfill \end{array}$$

(5)

where $\alpha =EB/EA$ which is the ratio of extension rigidity between the strut and the scissor member. It is understood that X becomes large according to an increase of the parameter $\alpha$.

Of course, it is also possible to determine the unknown reinforced force X by considering the strain energy. The strain energy in this structure is given by the following equation:

$$\begin{array}{ccc}\hfill \mathcal{U}& =& \sum _{j=1}^4\left(\frac{N_{\left(j\right)}^2}{2EA}\frac{L_0}{2}+{\int }_0^{L_0/2}\frac{M_{\left(j\right)}^2}{2EI}d\chi \right)+\frac{X^2{L}_{0}sin\theta }{2EB}\hfill \end{array}$$

(6)

Now, unknown internal force X can be calculated using the principle of strain energy minimization:

$$\begin{array}{ccc}\hfill \frac{\partial \mathcal{U}}{\partial X}& =& 0\hfill \\ & =& \frac{1}{2EA}\frac{L_0}{2}\frac{\partial }{\partial X}\left(\sum _{j=1}^4{N}_{\left(j\right)}^2\right)+\frac{1}{2EI}\frac{\partial }{\partial X}\left(\sum _{j=1}^4{\int }_0^{L_0/2}{M}_{\left(j\right)}^{2}d\chi \right)+\frac{2X{L}_{0}sin\theta }{2EB}\hfill \\ & =& \frac{L_0}{4EA}\sum _{j=1}^4\frac{\partial {\left(N_{\left(j\right)}\left(X\right)\right)}^2}{\partial X}+\frac{1}{2EI}\sum _{j=1}^4{\int }_0^{L_0/2}\frac{\partial {\left(M_{\left(j\right)}\left(X\right)\right)}^2}{\partial X}d\chi +\frac{X{L}_{0}sin\theta }{EB}\hfill \\ & =& \frac{1}{4}\sum _{j=1}^4\frac{\partial {\left(N_{\left(j\right)}\left(X\right)\right)}^2}{\partial X}+\frac{A}{2I{L}_0}\sum _{j=1}^4{\int }_0^{L_0/2}\frac{\partial {\left(M_{\left(j\right)}\left(X\right)\right)}^2}{\partial X}d\chi +\frac{Xsin\theta }{\alpha }\hfill \end{array}$$

(7)

Once statically indeterminate force X is clear, it can provide feedback as the live loads to the original model. That is, the following equations are obtained for a cantilever and a simple supported condition based on Equations (1) and (3):

$$\begin{array}{ccc}\hfill L\left\{\begin{array}{c}{\left(B^L\right)}_x\\ {\left(B^L\right)}_y\\ {\left(A^L\right)}_x+X\\ {\left(A^L\right)}_y\end{array}\right\}& =& -R\left\{\begin{array}{c}{\left(B^R\right)}_x\\ {\left(B^R\right)}_y\\ {\left(A^R\right)}_x-X\\ {\left(A^R\right)}_y\end{array}\right\}-\left\{\begin{array}{c}{\left(C\right)}_x\\ {\left(C\right)}_y\\ 0\\ 0\end{array}\right\},\hfill \\ \hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill L\left\{\begin{array}{c}{\left(B^L\right)}_x\\ {\left(B^L\right)}_y\\ {\left(B^R\right)}_x\\ {\left(B^R\right)}_y\end{array}\right\}& =& -R\left\{\begin{array}{c}{\left(A^L\right)}_x+X\\ {\left(A^L\right)}_y\\ {\left(A^R\right)}_x-X\\ {\left(A^R\right)}_y\end{array}\right\}-\left\{\begin{array}{c}{\left(C\right)}_x\\ {\left(C\right)}_y\\ 0\\ 0\end{array}\right\}\hfill \end{array}$$

(9)

In addition, the sectional forces $N_i$ and $M_i$ in the reinforced scissor structure can be calculated using statically indeterminate force X as follows:

$$\begin{array}{ccc}\hfill N_i& =& N_{0j}+N_{1j}X,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill M_i& =& M_{0j}+M_{1j}X\hfill \end{array}$$

(11)

2.2. Example Calculation for a Unit Model

2.2.1. Calculation Models

Using the above theoretical concepts of the scissor structure, three different cantilever models for bridge deployment are considered; these are shown in Figure 3. We consider a unit scissor structure under the cantilever condition, which includes pinned support at two nodal points $B_1^L$ and $A_1^L$ and a load P at point $A_1^R$.

Each statically indeterminate force X of Figure 3b,c is obtained by equilibrium equations as follows:

$$\begin{array}{c}\hfill X\left\{\begin{array}{cc}=\frac{\alpha P\left(cot\theta +(\beta -1)sin2\theta +tan\theta \right)}{2\left(sin\theta +\alpha (\beta {cos}^2\theta +cot\theta +sin\theta )\right)}\hfill & \cdots \mathrm{for the upper horizontal strut in Figure 3b}\hfill \\ \hfill \\ =\frac{P}{2}\left(\frac{1}{1+\frac{\alpha }{cos\theta }\left(1+\beta +(1-\beta )cos\frac{\theta }{2}\right)}-1\right)\hfill & \cdots \mathrm{for the right vertical strut in Figure 3c}\hfill \end{array}\right.\end{array}$$

(12)

where P is load, $L_0$ is length, $\theta$ is expanding angle, $\alpha$ is $EB/EA$ which is the ratio of extension rigidity between the strut and scissor member, $\beta$ is $A{L}_0^2/\left(12I\right)$. The bending moment of the dominant sectional force in every model is obtained using Equation (10).

2.2.2. Calculation Results

Let us consider the reinforcing effect using specifically sectional properties of $L_0$ = 1000 mm; A = 564 mm²; and I = 3.3 × 10${}^5$ mm⁴ based on the pedestrian type of the MB [23].

The change in the bending stress depends on the angle $\theta$, which acts on the concentrated load (P = 1 kN) at the end of the system, as shown in Figure 4. The abscissa of the graph shows the angle $\theta$ of the scissor structure, and the ordinate shows the absolute values of the maximum bending stresses in the model. In both results, the maximum bending stress is obtained at the pivot in the member $\overline{B_1^L{A}_1^R}$, and the stress values are increasing according to expansion of the scissor structure. When we focus on the design angle of 60 degree of the MB in Figure 4a, the maximum bending stress without reinforcement, that is $\alpha =0$, is over 45 MPa. However, it can be seen that the maximum bending stress at the pivot is largely decreased by applying the upper reinforcement in comparison with the initial state. Even if the value of $\alpha$ is small, e.g., $\alpha$ = 0.1 and 0.2, we can obtain a good reinforcing effect. In other words, there is an optimal size of the reinforcing member which shows the stiffening effect with light weight. Similarly, Figure 4b shows the reduction of the maximum bending stress approximately half from the initial state, when the value of $\alpha$ is over 0.05. Thus, it is considered that the vertical reinforcing member equally distributes the force to each scissor member if it has enough extensional rigidity.

The change in the maximum bending stress in the scissor member $\overline{B_1^L{A}_1^R}$ and the axial stress in the upper horizontal strut member $\overline{A_1^L{A}_1^R}$ and in the right vertical strut member $\overline{A_1^R{B}_1^R}$ depends on the ratio of extensional rigidity $\alpha$, as shown in Figure 5. The abscissa of the graph shows $\alpha$, and the ordinate shows the absolute values of stresses in the design angle of 60 degrees. From Figure 5a, both stresses in the scissor member and the strut member depend on $\alpha$. When $\alpha$ is smaller than 0.1, the bending stress in the scissor member increases suddenly. On the other hand, the bending stress is almost constant after $\alpha$ is 0.3. From Figure 5b, the bending stress in the scissor member is decreased by the value of $\alpha$. The stress value is almost constant when the value of $\alpha$ is over 0.05. As well as the result in Figure 4b, it is thought that the vertical member is capable of dispersing the force.

Based on the above examples, we found the reduction of the stress in scissor member by applying the reinforcing member between each hinge part. The reinforcing effect showed the nonlinear solution depending on the deployment angle and cross section of the scissors and strut members. In addition, this tendency differed according to the reinforcing layout. Hence, in order to provide feedback to the design of the reinforced scissor structure, it is important to seek the optimal reinforcing layout and allowable size of the reinforcing member.

3. Application of Strut Reinforcement

Based on the previous section, the reinforcing method by the strut member is applied to the scissors-type bridge. It is considered that the reinforcing effect depends on its reinforcing layout, specification of the strut member, configuration of the bridge, and so on. As a first trial, this paper mainly focuses on the reinforcing pattern and makes the change of stress and displacement clear.

3.1. Numerical Model Based on Experimental Bridge

An FE numerical model based on the full-scaled MB having two and three scissors units is created by ABAQUS 6.12 as shown in Figure 6. Each nodal point is defined as $A_1,\cdots ,A_n$ at the upper hinge parts, $B_1,\cdots ,B_n$ at bottom hinge parts, and $C_1,\cdots ,C_n$ at pivot parts, respectively. This simulation model uses linear beam elements and considers a simplified one-side model due to the symmetry of the MB. In order to model pivots and hinges, each scissor member is connected by the joint elements which do not transfer any bearing. Similarly, each reinforcing member is also connected with a scissor member by the joint elements without any friction.

At full extension, the total length of the span is 7.0 m in the two units model and 10.5 m in the three units model, respectively. The height in both models is 2.0 m. Each model is supported by pins in supported boundary conditions because it is assumed that we will apply the reinforcing member after the bridge is deployed. Now, the vertical and horizontal reinforcing members are assumed to have the same specifications for scissor member and deck boards. Detailed sectional and material properties of structural members are provided in our previous paper [22].

3.2. Reinforcing Pattern

Reinforcing effects of the strut member are examined numerically on the basis of the nonreinforcing state without any strut members.

Table 1. Numerical cases with different reinforcing patterns.

Case	Horizontal mem.	Vertical mem.	Configuration
0	No	No
1-1		No
1-2	Only	Center place
1-3	Lower place	Both side place
1-4		All place
2-1	Lower place	No	-
2-2	and	Center place	-
2-3	Central	Both side place	-
2-4	upper place	All place	-
3-1	All	No
3-2	Lower place	Center place
3-3	and	Both side place
3-4	Upper place	All place

shows the detail of reinforcing patterns. We focus on the number of horizontal and vertical reinforcing members and their setting position. The point is to find key horizontal and vertical reinforcing members.

In the case of the two scissors units model, there are two different setting places for the horizontal reinforcing members. First, it is considered that there are only deck boards on the frame member. Hence, the horizontal members are applied to the bottom hinge parts (Case 1) and the upper and bottom hinge parts (Case 3). Then, the vertical members are given four patterns as: (1) nonmember, (2) center place, (3) both side places and (4) all hinge places.

In the case of the three scissors units model, there are three different setting places for the horizontal reinforcing members. The horizontal members are applied to the bottom hinge parts (Case 1), the central upper and bottom hinge parts (Case 2), and the upper and bottom hinge parts (Case 3). Then, the vertical members are given four patterns as well as in the case of the two scissors units model. According to these different configurations, the reductions of the sectional forces and displacement are evaluated.

3.3. Loading Condition

Each model considers only deadweight as the loading condition. The deadweight for structural members is given by the distribution load based on its sectional and material properties. The weight of shafts connecting two scissor planes and pivots connecting scissor members are given by concentrated load, and each value of shaft and pin are 10 N and 322 N, respectively.

4. Numerical Results

This section describes the numerical results for the basic frame model in Case 0, not including the strut member at first. Then, results from Case 1 assuming a through bridge are shown, and its reinforcing effect is evaluated as an example. Thereafter, the whole of numerical cases are discussed, focusing on the change of physical quantity with regards to reinforcing patterns.

4.1. Two Scissors Unit Model

4.1.1. In the Case of Non-Reinforcing Pattern (Case 0)

Figure 7 shows the numerical results for Case 0 which is only the frame model without the strut member. Figure 7a shows the bending moment distribution (BMD), and Figure 7b shows the axial force distribution (AFD). The units of these sectional forces are mN*mm and mN, respectively.

From both figures, the maximum bending moment occurs at pivots of ‘Λ’-shaped members. and the maximum axial force is present in the bottom parts of the ‘V’-shaped members of the bridge. From these maps of sectional forces, we can see that they are symmetrically distributed. These results allow us to see that the maximum bending moment is due to pulling the ‘Λ’-shaped members by the maximum axial forces in the bottom parts. When each sectional force converts to stress, $|{\sigma }_{M0}|$ = 6.3 MPa and $|{\sigma }_{N0}|$ = 0.7 MPa are obtained. It is seen that in the nonreinforced state the effect of the bending stress is more than 10 times compared with the axial stress. Furthermore, the maximum displacement of 2.6 mm is obtained at the central bottom hinge (Point $B_2$ in Figure 6a).

4.1.2. In the Case of Lower Reinforcing Patterns (Case 1)

Figure 8 shows numerical results which are reinforced in bottom parts. The left and right figures show BMD and AFD for each numerical case. It is found from Case 1-2 and 1-4 that the maximum bending moment decreases greatly due to restraining the deformation of the bridge by applying the vertical member at the central part. On the other hand, they do not made a dramatic difference in Case 1-1 which has only the horizontal member at the lower hinge place and Case 1-3 which sets on the vertical member at both ends because these reinforcing patterns cannot reduce the deflection of the MB1.0 directly.

Now, let us consider the reinforcing effects by focusing on the high bending members (‘Λ’-shaped members) in Case 0. Figure 9 shows the sectional distribution of member $\overline{B_1{A}_2}$. The longitudinal axis depicts the change in sectional forces in each case, and the abscissa axis depicts the attention point along with the member axis direction. Point $B_1$ and central pivot show x = 0 mm and 2150 mm in the figure.

When looking at Figure 9a, it is found that the maximum axial force of 2 kN in Case 0 was increased by the strut member except Case 1-3. However, these increments of the axial force are not a big problem because the effect of the bending moment is decreased greatly from the initial state. When looking at Figure 9b, Case 1-1 is unable to decrease the bending moment, but the maximum value increases 1.5 times from Case 0 according to the increment of self-weight. However, it is understood that the bending moments are basically decreased by applying a vertical member at least. Figure 9c shows detail of the bending moment distribution for each case. It is found that the minimum bending moment is obtained in Case 1-4.

From the above, it is clear that the vertical reinforcing members can reduce the maximum bending moment approximately 1/20–1/30 times in comparison to the initial state.

4.1.3. Effects of Decreasing the Displacement and Sectional Forces Depending on the Reinforcing Layout

Effects of decreasing the displacement and sectional forces are evaluated by Case 1 with the bottom horizontal members and Case 3 with the upper and bottom horizontal members. Figure 10 and Figure 11 show the nondimensional value of displacement and sectional force for each numerical case divided by ones in Case 0 without any strut members. Lower and Lower + Upper in the abscissa axis in both figures show the numerical results of Case 1 and 3. Moreover, in Figure 11a,b, bar graphs colored in blue and purple depict the maximum value in the whole of the model and ‘Λ’-shaped members.

It is understood from Figure 10a that of every reinforcing pattern, only the horizontal member without the vertical member (Case 1-1 and Case 3-1) and Case 1-3 increase the maximum displacement more than the result of Case 0. It is seen that reinforcing effects of the lower horizontal member are few. However, the remaining patterns show decreasing one by setting on the vertical member. Figure 10b depicts the detail of displacement in a small range, and it shows almost a displacement below 1/10 times from the initial state. A high effect of decreasing is obtained in Case 3-4.

Figure 11a shows that of every reinforcing pattern, only the horizontal member without the vertical member (Case 1-1 and Case 3-1) and Case 1-3 increase the maximum bending moment approximately 1.2–1.5 times more than Case 0. However, the remaining patterns clearly show decreasing one under 1/10 times by applying the vertical member. Furthermore, when looking at the whole of the bridge colored in blue, the maximum bending moment also becomes small due to transmitting forces to another member by the vertical member. At this time, the minimum reduction is obtained in Case 3-4. Figure 11b shows that the maximum axial force generates near the supporting points in the whole case. These values colored in purple are approximately two times higher than the nonreinforcement state at maximum. That is to say, the axial force does not increase extremely by reinforcing, and its merits are quite good given the effects of decreasing the displacement and bending moment.

4.2. Three Scissors Unit Model

As in the previous section, the numerical results of the basic Case 0 without any reinforcement and Case 1-1 to Case 1-4 with the horizontal members on the bottom parts are shown first. Then, the overall evaluation of the three scissors unit models is presented, focusing on the change in sectional forces and displacement according to the placement pattern of each reinforcement.

4.2.1. In the Case of Nonreinforcing Pattern (Case 0)

Figure 12 shows the numerical results for Case 0 without any reinforcement. Figure 12a shows the bending moment distribution (BMD), and Figure 12b shows the axial force distribution (AFD). The units of each result are the same as in Figure 7. From Figure 12a, the maximum bending moment is at the central scissors unit. As can be seen from the BMD, no bending moment is generated in the outermost member whose end is not fixed, similar to the results in the two scissors unit model. Figure 12b shows that the maximum tensile force is between the member point $\overline{B_2{C}_1}$, and the maximum compressive force is the point $\overline{B_2{C}_1}$. It can also be seen that the tensile forces are acting on the lower half of the scissors members except for the members connecting to the supports, and the compressive forces are acting on the upper half of the scissors members. The obtained maximum stresses are $|{\sigma }_{M_0}|$ = 8.5 MPa and $|{\sigma }_{N_0}|$ = 0.4 MPa. Compared with the maximum value in the two scissors unit model, the maximum bending stress increases by 1.4 times and the maximum axial stress decreases by 0.6 times due to the increment of the scissors unit. These results indicate that the maximum bending moment increases with the number of scissors units, but the maximum axial force does not necessarily increase with its number.

4.2.2. In the Case of Lower Reinforcing Patterns (Case 1)

Figure 13 shows the numerical results which are reinforced in the bottom parts. The left and right figures show BMD and AFD for each numerical case. The maximum and minimum values of contour figures are based on the unreinforced condition (Case 0).

It is found from Case 1-2 and 1-4 that the deformation of the bridge is decreased greatly by applying the vertical member at the central part. This is because the presence of the vertical members between $\overline{B_2{A}_2}$ and $\overline{B_3{A}_3}$ reduce the bending moment acting on the scissors members. On the other hand, they do not make a dramatic difference in Case 1-1 which has only the horizontal member at a lower hinge place, similar to the two scissors unit model.

4.2.3. Effects of Decreasing the Displacement and Sectional Forces Depending on the Reinforcing Layout

The effects of decreasing the displacement and sectional forces are evaluated by Case 1 to Case 3 according to the position of the reinforcing members. Figure 14 and Figure 15 show the nondimensional value of the central displacement and sectional force for each numerical case divided by the ones in Case 0 without any strut members. Lower, Lower +1, and Lower + Upper in the abscissa axis in both figures show the numerical results of Case 1, 2, and 3. Moreover, in Figure 15a,b, bar graphs colored in blue and purple depict the maximum value in the whole of the model and member $\overline{B_2{A}_3}$.

It is understood from Figure 14a that Case 1-1 with only the horizontal members and Case 1-3 with the horizontal members and side vertical members increase maximum displacement more than the result of Case 0 without any reinforcement. However, remaining patterns show a high reduction of central displacement. Figure 14b depicts the detail of displacement in small range, and it shows almost a displacement below 1/20 times from the initial state. The highest decrease is obtained in Case 3-3 similar to a double warren truss (DWT) configuration, and its value is smaller than that of Case 3-4 in a fully reinforcement pattern. This result implies that side vertical members on the supports are more important in the case of Case 3 with both upper and horizontal members.

Figure 15a shows that Case 1-1 with only the horizontal members and Case 1-3 with the horizontal members and side vertical members increase the maximum bending moment approximately 1.4 times more than Case 0. However, remaining patterns clearly show a decrease under 1/10 times colored in purple by applying reinforcing members. Furthermore, when looking at the whole of the bridge colored in blue, the maximum bending moment also becomes small due to transmitting forces to another member by the vertical and horizontal member. At this time, the minimum reduction is obtained in Case 3-4.

Figure 15b shows that the maximum axial force generates near the supporting points in the whole case. These values colored in purple are approximately two times higher than the nonreinforcement state at maximum. That is to say, the axial force does not increase extremely by reinforcing as well as in the two scissors unit model.

5. Discussion

According to the numerical results, this section discusses the necessary sectional area of the reinforcing area and the optimal reinforcing process.

5.1. Necessary Sectional Area

Figure 16, Figure 17 and Figure 18 show the relationship between the nondimensional parameter and the ratio of extensional rigidity $\alpha$ focusing on the effective reinforcing layout of Case 1-2, Case 1-4, and Case 3-4 of the MB1.0 based on Table 2. The left figure of the longitudinal axis shows the nondimensional maximum bending moment $M/M_0$ in the member $\overline{B_1{A}_2}$, and the right figure shows the nondimensional displacement $\delta /{\delta }_0$ at the center of the bridge. The abscissa axis shows the ratio of extensional rigidity between the scissor member and the reinforcing member. In each result, the $\alpha$ was given as between 0 and 1. Thus, when the $\alpha$ is equal to 0, the numerical model is the same as Case 1-1. The numerical result in Figure 16 indicates that the nondimensional value becomes smaller rapidly by application of the reinforcing members. The significant decrement shows until $\alpha =0.1$, and then the decrease gradually becomes smaller. The nondimensional value is almost constant after $\alpha =0.2$. In other words, a high value of $\alpha$ is not always necessary to reinforce the bridge, and a too strong reinforcing member has the possibility to increase the deadweight uselessly. These tendencies are also found in Figure 17 and Figure 18.

From above results, when we reinforce the MB1.0, it is enough to apply central vertical reinforcement like a Case 1-2 with $\alpha >0.2$. This result was slightly smaller than the case study using single scissors unit in Section 2. When we refer the result on three scissors unit with central upper reinforcement in [16], the required $\alpha$ is larger than 0.1, even the strut member is made of the steel or aluminum material. Therefore, the engineering needs to decide carefully the design of the reinforcing member because of the variation in the minimum value of $\alpha$ according to loading situation and number of scissors units.

5.2. Reinforcing Process

Effective reinforcing procedures to reduce central displacement and bending moment are considered based on numerical results in last section. Table 2 shows optimal reinforcing steps in order to minimize the displacement and bending moment occurred in ‘Λ’-shaped members from the model without reinforced member (Case 0) to the model with the bottom horizontal members (Case 1). Where the symbol H is horizontal member, the symbol ‘V’ is vertical member, the values are decreasing rate of the displacement and bending moment from Case 0. According to progress of step 1 and 2 in the table, both displacement and bending moment become small and minimize in step 3. Both displacement and bending moment decrease by setting the reinforcing member as center vertical member → side vertical member → upper horizontal member.

Table 2. Optimal reinforcing procedure for two scissors unit model.

Step	Additional Member	Displacement	Bending Moment	Configuration
0	-	154.4%	156.4%
1	Center V mem.	5.2%	4.3%
2	Side V mem.	4.5%	3.8%
3	Upper H mem.	3.1%	2.9%

Table 2. Optimal reinforcing procedure for two scissors unit model.

Step	Additional Member	Displacement	Bending Moment	Configuration
0	-	154.4%	156.4%
1	Center V mem.	5.2%	4.3%
2	Side V mem.	4.5%	3.8%
3	Upper H mem.	3.1%	2.9%

Similarly,

Table 3. Optimal reinforcing procedure for three scissors unit model.

Step	Additional Member	Displacement	Bending Moment	Configuration
0	-	145.0%	137.6%
1	Upper H mem.	2.3%	1.4%
2	Center V mem.	2.2%	1.3%
3	Side V mem.	1.1%	1.3%

shows optimal reinforcing steps in order to minimize central displacement and bending moment occurred in member $\overline{B_2{A}_3}$. According to progress of step 1 and 2 in the table, both displacement and bending moment become small and minimize in step 3. Both displacement and bending moment decrease by setting the reinforcing member as central upper horizontal member → side vertical member → side upper horizontal member.

From the above results, it is understood that the displacement and bending moment can be reduced with a central vertical member in a two scissors unit model and by a central upper horizontal member in a three scissors unit model. According to the progress of the reinforcement, the displacement and bending moment are reduced less than 5% from a nonreinforced model. Especially, it is understood that first reinforcing member is quite important to improve the performance of a bridge.

5.3. Prediction of Optimal Reinforcing Layout

In the case of a two-unit scissors-type bridge, the maximum bending moment could be reduced dramatically by applying the vertical member at central parts similar to a king post in a truss bridge. In the case of a three-unit scissors-type bridge, the maximum bending moment could be reduced dramatically by applying a horizontal member at central parts. It can be seen that the horizontal reinforcement to the center is the key reinforcing member for the odd number of scissors unit, and the vertical reinforcement to the center is the key reinforcing member for the even number of scissors unit.

The most effective reinforcing configuration was DWT with the vertical member in a two-unit scissors-type bridge and DWT without the vertical member in a three-unit scissors-type bridge. When the bridge becomes a longer span, a DWT form is considered to be a good solution to improve the performance of a scissors-type bridge.

Based on the above, it is considered that the deflection and bending moment of scissors-type bridges with even and odd number of scissors unit can be efficiently reduced by adding reinforcement to fill the space between the hinges in the center of the bridge. Therefore, it can be safe to reinforce only central part of the bridge when it is a time-critical situation. After that, the bridge can be operated efficiently by installing reinforcement in the form of DWT according to the site conditions.

6. Conclusions

This paper discussed the reinforcing effect of the strut members to the scissors-type bridge through a theoretical and numerical approach. The advantages of this reinforcing method were evaluated by a statically indeterminate mechanics approach focusing on the reduction of the bending moment which is the dominant sectional force in the scissor structure. Then, we evaluated its applicability to the MB based on the numerical simulation. We considered several reinforcing patterns, which are additional horizontal, vertical, and a combination of horizontal and vertical reinforcing members and tried to seek optimal reinforcing layout and sectional area.

The following remarks can be made based on the results presented in this paper:

(1)

Reducing effect for the bending moment which is the dominant sectional force in the scissor structure was evaluated based on equilibrium equations as a statically indeterminate problem. Two examples indicated that sectional forces in a reinforced scissor structure changed depending on its reinforcing layout and expansion angle. Moreover, the reinforcing effect has a nonlinear relationship between the stress and ratio of extension rigidity (parameter $\alpha$).

(2)

Finite element analysis was carried out in order to evaluate the reinforcing effect according to different configurations based on the two-scissors and three-scissors unit model. It can be seen that the horizontal reinforcement to the center is the key reinforcing member for the even number of scissors unit, and the vertical reinforcement to the center is the key reinforcing member for the even number of scissors unit.

(3)

Although some members do not decrease the value of the axial force by reinforcement, the increment of the maximum axial force was approximately two times in comparison with the initial state at maximum. Thus, increase and decrease of the axial force is not big problem in this research.

(4)

The most effective reinforcing configuration was the double warren truss (DWT) with the vertical member in a two-unit scissors-type bridge and the DWT without the vertical member in a three-unit scissors-type bridge. Its necessary sectional area was $\alpha >0.2$ in this numerical discussion.

The obtained results imply that the scissors-type bridge, including the MB, can enhance its performance by using proper reinforcement of the strut members with necessary in the sectional area. We can design the effective reinforcement configuration for the scissors-type bridges with a small number of scissors units based on this research. We would like to consider further reinforcing effects, e.g., longer bridge spans, and optimization of cross-sectional dimensions in future research.