Analysis and optimization of brace system in scaffold

Scaffold consists of horizontal components, vertical components and lateral braces, while the lateral stiffness of structure is mainly provided by brace system. Brace system can be classified into three groups: concentric brace, eccentric brace and knee brace. Concentric brace enables to offer greater stiffness, for providing direct load transfer. Eccentric brace and knee brace belong to ductile system, maintaining stability by energy consumption braces. Eccentric brace dissipates energy by deforming and buckling, while knee brace disposes energy by form a plastic hinges. Concentric brace is a common way for enhancing lateral stiffness in scaffold. Current studies aimed to investigate detail parameter effects in one brace pattern, while few comparative studies were carried on multiple brace pattern. In this paper, a simplified simulation method was proposed. Throughout simulation analysis, some regulations of brace arrangement were summarized, which provides a guidance during the scaffolding design and optimization process.


Introduction
As a temporary structure, scaffold benefits from light self-weight and easier fabrication, which is broadly applied in some accidental scenarios, such as construction platform and festival meeting.[1] Scaffold consists of horizontal components, vertical components and lateral braces.According to study of Brito and Pimentel, the stiffness of scaffold is affected by connection and brace.[2] The connections of scaffold impact the way of load transfer.Brace system in scaffold is able to resist lateral deformation, enhancing structural stiffness dramatically.[3] Ji and Ellis focus the brace arrangement to set up a series of layout in a certain scaffold, which found that efficient brace arrangement can increase stiffness 284%.[4] As for load resistance, vertical loads is transferred from horizontal components to vertical components, while the horizontal load is primarily carried by braces.The brace connects the horizontal and vertical members to form a complete truss, which can efficiently increase the lateral stiffness and limit the story drifts.There are three brace mode: concentric brace, eccentric brace, knee-brace.[5] Concentric brace is most commonly used, which is easiest to install.Maheri and Sahebi experimentally tested the shear, tension and compression in frame and braced frame.The results indicated that the stiffness of frame with concentric brace is 2.4 to 4.0 times that of the frame without brace.[6] Maheri explored the pushover-load displacement of frame without brace, frame with concentric brace and frame with knee brace, which showed that concentric brace offered greater stiffness.[7] Turker and Bayrakstar carried on theoretical and experimental analysis about brace type.The result shows that the frame with cross brace had the highest stiffness configuration.Concentric braces also have better performance under earthquake load.[8] Yang figured out the natural period and fragility data in multifarious brace configuration, which found that the concentric braces are more efficient to offer the stiffness of structure.[9] Eccentric brace refers to the brace intersection of structure is a certain distance from the connection of the horizontal and vertical members.At intersection of eccentric brace, bending moment and shear force would be generated, which forms ductile connection and dissipates energy by deforming and buckling.[10] Adil and Esra compared the free vibration period of different brace type, which shows V brace is stiffer than K brace, then followed by D brace.[11] Knee brace is another kind of disposable component, which is able to form a plastic hinges to dissipate energy.The pushover test carried by Maheri also showed that X-brace and knee-brace enable to increase yield capacity and strength capacity.Compared with X brace, the failure pattern of knee brace was bending failure before shear failure, which offer a more effective collapse-level earthquake resistance.
Although plenty research had been conducted on detail parameter effects in one brace pattern, there is few comparative study on multiple brace pattern.In this paper, a simplified simulation method was proposed.Throughout simulation analysis, some regulations of brace arrangement were summarized, which provides a guidance during the scaffolding design and optimization process.

The eigenvalue of analysis
The research focus on the relationship between structure behavior and brace layout.The stiffness is regarded as the characteristic parameter of structure behavior.The relationship of the stiffness (k), the natural frequency (w) and self-weight (m), can be expressed as: Then it can be derived as: k = w 2 m (2) According to the equation above, the stiffness (k) is proportional to the square of the natural frequency (k).Therefore, natural frequency (w), obtained in finite element simulation, is regarded as the eigenvalue of analysis.

The establishment of simplified modelling
In the process of model establishment, the type of connection and load transmission should be taken into account.The connection in finite element model includes beam element and truss element.The beam elements are connected rigidly.The connection between truss elements is pinned connection, and load transfer is mainly through axial force.Practically, the connection between the members is a semi-rigid rather than an idealized pinned, so the simulation of the connection with the truss element is not accurate enough.
Firstly, some models (shown in figure 1) were established respectively with beam element and truss element to explore the differences between rigid and pinned connections.As shown in figure 1, The length of vertical and horizontal member is 1000mm.Except the different brace layout.all the details (such as material property, section and boundary condition) of members are exactly same, which are listed in table1.

Beam element Truss element
Finite element analysis was carried on, modifying the type of element.The natural frequency of these frame layout with beam and truss elements are listed in table 2. The result shows that the influence of brace layout on stiffness is greater than that of connection.Therefore, it can be considered that the simplification of connection does not affect the analysis of brace layout.Besides, the scaffold is a 3D resistance system, but 3D model analysis would require much time on model establishment and analysis.To explore the effect of model dimension, frame A and frame B with pin connection are established in 2D and 3D model respectively, and the nature frequency of them are listed in table 3. Compared with the layout of frame, the 3D model is slightly stiffer than 2D model.For frame under lateral force, the lateral resistance is primarily offered by the plane frame parallel to the force.4.
The stiffness comparison is also based on simplified pin connected frame plane.The structural stiffness (k) can be obtained by external force (F) and corresponding maximum displacement ( Δ) , which expressed as: In the model of simulation, the external force (F=5kN) is applied equally as nodal force (P=1kN) to top 5 points.Calculating stiffness by lateral force and corresponding displacement, the stiffness of 11 brace arrangements is shown in table 5. Side-span centralized brace arrangement (7) (8) (9) (10) Weak story brace arrangement (11) From table 5 the following regulation can be acquired: (a) The top 4 ranking belong to global arrangement, indicating that the global arrangement is better than any other arrangement.The stiffness rank of brace arrangement: global braced arrangement is larger than Mid-span centralized brace arrangement, followed by side-span centralized brace arrangement and weak story brace arrangement.

Conclusion
Scaffold consists of horizontal components, vertical components and lateral braces, while the lateral stiffness of structure is mainly provided by brace system.This paper reviews tested and simulated result of three brace pattern commonly used (concentric brace, eccentric brace and knee brace).Based on current finite element simulation and theoretical formula, a simplified simulation analysis method for stiffness comparison is proposed.By using this simplified method of simulation, a simplified model is established to compare the stiffness with different brace type and arrangement.Though finite element analysis, some regulation of brace arrangement can be summarized: (a) The stiffness rank of brace arrangement: global braced arrangement is larger than Mid-span centralized brace arrangement, followed by side-span centralized brace arrangement and weak story brace arrangement.(b) Braces in different bay and same layer should be arranged in multiple directions.(c) Braces through layers should be connected as continuously as possible.(d) The weak layers should be avoided in practice.These rules could offer a reasonable prediction of lateral stiffness for scaffold, which provides a guidance during the scaffolding design and optimization process.

Figure 1 .
Figure 1.layout of frame model

3 .
Finite Element (FE) Simulation of brace type and arrangement For the purpose of exploring the stiffness of different brace type and arrangement, a simplified model (shown in figure 2) was established.All members adopt scaffold pipe with same material S355 [E = 210×10 9 N/m 2 , ρ = 7847 kg/m 3 ] and same section 48.3×3.0mm[A = 4.3cm 2 , I = 21.9 cm 4 ].

Figure 2 .
Figure 2. The layout of the frame and applied nodal load

Table 5 .
(d) Model(11) is the least stiff structure, which illustrates that weak story of bracing would reduce the overall stiffness of structure.The stiffness of 11 brace arrangements (the stiffness rank from high to low)

Table 2 .
Natural frequency of braced frame with different connection.
* Relative difference of Layout=

Table 3 .
Natural frequency of 2D and 3D frame with pin connection * Relative difference of Layout=

Table 4 .
Classification of brace type and arrangement