# Mechanical Properties of New Composite Wood-Plastic Formworks with Aluminum Alloy Frame

Authors: Kairong Shi; Jiebin Xu; Zhengrong Jiang; Junfeng Lv; Yonghui Lu
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Formwork engineering plays a crucial role in cost, efficiency, quality, and schedule in civil engineering. Currently, wood-plastic formwork, which has favorable mechanical properties such as wood and excellent stability, formability, ease of demolding, and time-saving as plastic, is earning its increasing reputation in construction. This work focuses on mechanical properties of two types of new composite wood-plastic formworks with aluminum alloy frame used for construction, that is, single-span simply supported and three-span continuous formworks. Experimental investigation shows that the two types of wood-plastic formworks demonstrate favorable performance, and the deflections and stresses are within the allowable range, thereby satisfying the structural bearing requirements. Numerical analyses confirm that the results of the refined and general finite element models are consistent with the experimental results, but the former has a higher accuracy. When the requirement of accuracy is not too strict, the general model is preferred, given the modeling convenience and high efficiency. On the basis of experimental and numerical investigations, practical simplified formulas are proposed to facilitate rapid calculation and evaluation considering transverse deflection and inconsistency of two materials. Therefore, the results in this work can provide a theoretical basis for developing and applying the new formworks.


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## Body

## 1. Introduction

Cast-in-place reinforced concrete structures play a crucial role in civil engineering and comprise a large part of an infrastructure. Many experiments and simulation analyses are conducted to ensure that the mechanical properties of concrete are within limits considering mix design and curing condition [1–4]. Concrete cannot be cast without formwork, which is the foundation for concrete structure forming. Formwork engineering is an important component and the core process of concrete structure construction. Various formworks result in different costs, efficiencies, qualities, and schedules [5–7].Several formworks are commonly used in construction. The first one is wooden formwork, which is gradually being replaced by plywood formwork given a small dimension, poor bearing capacity, large density, and poor water resistance of wooden formwork. The second one is steel formwork. In comparison with wooden formworks, steel formworks raise the rate of use and reduce water pressure. However, the large density of steel complicates transportation and installation. Moreover, steel is easily corroded without careful maintenance. The third one is aluminum alloy formwork, which has a lighter weight and higher corrosion resistance than a steel formwork. However, its Young’s module is only approximately one-third of the steel template, thus leading to poor behavior in deflection control in formworks. The last one is plastic formwork. In contrast to traditional formworks, plastic formworks exhibit certain unique characteristics. Plastic formworks are easy to demold given the smooth surface and its considerable water resistance capacity. In terms of malleability, plastic formworks are easy to process, manufacture, and recycle. However, plastic formworks are prone to aging and warping, and the white waste formed by plastic formworks causes serious environmental pollution. In this work, a new composite wood-plastic formwork is introduced. It is a new polymer formwork, which comprises wood and plastic in a certain manner. It has favorable mechanical properties such as wood and excellent stability, formability, advantage of reuse, and high efficiency of demolding as plastic.Several research works have been conducted to assist in selecting appropriate formwork systems. Tam et al. [8] and Hanna and Senouci [9] developed neural network models for selecting vertical and horizontal formwork systems, respectively. Nikhil and Patil [10] used fuzzy logic based on comparative analysis to automate the formwork knowledge base, which aids in formwork selection. Martinez et al. [11] developed the choosing by advantages decision-making method to lead decision makers in selecting the suitable formwork. Other research works have focused on the mechanical analyses of formworks with different materials. Meng et al. [12] presented experimental and mechanical studies on flexural behaviors of fiber-reinforced polymer fabric-reinforced ultrahigh-performance concrete panels. Carrillo et al. [13] used finite element (FE) software to assess the mechanical performance of different sizes of steel formwork. Gallego et al. [14] applied the ANSYS FE program to simulate wooden and steel formworks in calculating the lateral pressures of a complex-shaped formwork. Wang et al. [15] conducted numerous comparative analyses of bearing capacity and stability for plywood formwork. Wang et al. [16] investigated the effect of internal curing with presoaked lightweight aggregate and engineered cementitious composite permanent formwork on moisture distribution in a drying environment. However, an analysis of references shows that few researchers have concentrated on the mechanical analysis of the new composite wood-plastic formwork.The use of concrete as a structural material in civil engineering requires adopting formworks, which must remain in place until the concrete setting is completed [17]. Therefore, a formwork is an important tool for constructing concrete structures, especially for cast-in-place concrete. This work presents the results of experimental and numerical simulations of the two types of new composite wood-plastic formworks (i.e., single-span simply supported and three-span continuous formworks). Single-span simply supported formwork only can be used in concrete slab less than 6000 mm. Here, the two types of formworks perform well and can, therefore, be used.


## 2. Geometric Parameters and Material Properties

### 2.1. Geometric Parameters

The structure of the proposed new formworks consists of two parts, namely, wood-plastic template (surface slab) and aluminum alloy frame. Rivets are used to tie a template and a frame together. This work targets two types of specimens, namely, Type I (single-span simply supported formwork; Figure1) and Type II (three-span continuous formwork; Figure 2). The dimension of Type I is 1500 mm × 600 mm × 65 mm, whereas that of Type II is 2700 mm × 600 mm × 65 mm. The template thickness of both specimens is 10 mm.Figure 1
Geometric model of a single-span simply supported formwork (Type I). (a) Model of a frame without a template. (b) Model with a template. (c) Dimensions of the formwork. (d) Sections of ① and ②. (e) Sections of ③ and ⑤. (f) Section of ④.Figure 2
Geometric model of a three-span continuous formwork (Type II). (a) Model of a frame without a template. (b) Model with a template. (c) Dimensions of the formwork. (d) Sections of ① and ②. (e) Sections of ③ and ⑤. (f) Section of ④.
(a)(b)(c)(d)(e)(f)


### 2.2. Material Properties

The template of the formwork is made of a new kind of environmentally friendly composite material called wood-plastic. All other members in the formwork are made of 6061-T6 aluminum alloy in addition to steel rivets. The detailed material data are listed in Table1.Table 1
Material properties of the composite wood-plastic formwork.
MemberMaterialDensity (kg/m3)Young’s modulus (N/mm2)Poisson’s ratioTemplateWood-plastic1.26 × 1033.84 × 103—Frame6061-T6 aluminum alloy2.70 × 1030.70 × 1050.3RivetSteel7.85 × 1032.06 × 1050.2


## 2.1. Geometric Parameters

The structure of the proposed new formworks consists of two parts, namely, wood-plastic template (surface slab) and aluminum alloy frame. Rivets are used to tie a template and a frame together. This work targets two types of specimens, namely, Type I (single-span simply supported formwork; Figure1) and Type II (three-span continuous formwork; Figure 2). The dimension of Type I is 1500 mm × 600 mm × 65 mm, whereas that of Type II is 2700 mm × 600 mm × 65 mm. The template thickness of both specimens is 10 mm.Figure 1
Geometric model of a single-span simply supported formwork (Type I). (a) Model of a frame without a template. (b) Model with a template. (c) Dimensions of the formwork. (d) Sections of ① and ②. (e) Sections of ③ and ⑤. (f) Section of ④.Figure 2
Geometric model of a three-span continuous formwork (Type II). (a) Model of a frame without a template. (b) Model with a template. (c) Dimensions of the formwork. (d) Sections of ① and ②. (e) Sections of ③ and ⑤. (f) Section of ④.
(a)(b)(c)(d)(e)(f)


## 2.2. Material Properties

The template of the formwork is made of a new kind of environmentally friendly composite material called wood-plastic. All other members in the formwork are made of 6061-T6 aluminum alloy in addition to steel rivets. The detailed material data are listed in Table1.Table 1
Material properties of the composite wood-plastic formwork.
MemberMaterialDensity (kg/m3)Young’s modulus (N/mm2)Poisson’s ratioTemplateWood-plastic1.26 × 1033.84 × 103—Frame6061-T6 aluminum alloy2.70 × 1030.70 × 1050.3RivetSteel7.85 × 1032.06 × 1050.2


## 3. Experimental Investigation

### 3.1. General Test and Loading Scheme

To investigate the mechanical properties of formworks, three specimens are fabricated for each type of formwork. Table2 summarizes the general information of the formwork experiments. Steel blocks (2.5 kg) are used to simulate the uniform load. The measurement contains the following main parts:(1)
Deflection in the middle of the span(2)
Stress and strain in the middle of the span and supportTable 2
Loading program of the formwork.
Specimen numberDimension of the formworkType of supportSpan (mm)Length of the cantilever (mm)Maximum test load (kN/m2)1, 2, 31500 mm × 600 mm × 65 mmSingle-span simply supported900300154, 5, 62700 mm × 600 mm × 65 mmThree-span continuous750 × 322530


### 3.2. Single-Span Simply Supported Formwork

#### 3.2.1. Arrangement of the Measuring Points

(1) Arrangement of the Measuring Point of Deflection. Figure 3 illustrates five measuring points (A–E) to monitor the vertical deflection in the middle of the span.Figure 3
Arrangement of the measuring points of deflection in the single-span simply supported formwork.(2) Arrangement of the Measuring Point of Strain. Figure 4 depicts four measuring points (1–4) placed on the interesting points of the frame to monitor the strain. The measuring points are separately distributed in the middle of four longitudinal beams of the frame.Figure 4
Arrangement of the measuring points of strain in the single-span simply supported formwork.


#### 3.2.2. Loading Process

The loading process of the single-span simply supported formwork is demonstrated as follows: first, the proposed formwork specimens are placed on a special experimental bench. Second, the deflection meters, resistance strain gauges, and other devices are installed. To ensure uniformity of loading with steel blocks, the top surface of the wood-plastic template is covered with a thin layer of fine sand. Then, the steel blocks (2.5 kg) are applied uniformly on the formwork step by step to achieve a uniform load. The loading region is demonstrated in Figure5. The strain data are collected in the computer transmitted by strain gauges. The deflection data can be read directly from the meters.Figure 5
Loading diagram of the simply supported formwork.Before the formal experiment, the specimens are preloaded to check whether the specimens are placed stably, and the test instruments are set up correctly, thus ensuring that the experimental devices can work accurately and reduce the test error during the whole experiment. The abovementioned preloading reaches 5 kN and is held for 3 min.During the formal loading, 2.5 kN/m2 at each step is loaded until it reaches 15 kN/m2 (self-weight of 600 mm concrete). All data are recorded after loading for more than 30 s. The loading procedure is presented as follows: 0 ⟶ 2.5 kN/m2 ⟶ 5 kN/m2 ⟶ 7.5 kN/m2 ⟶ 10 kN/m2 ⟶ 12.5 kN/m2 ⟶ 15 kN/m2 ⟶ 0.During the actual process of loading step by step, the deformation of the formwork and the corresponding stress and strain increase with the uniform load. The experimental data collected are acceptable, and no apparent abnormality is found. The entire process is completed smoothly. The details of the loading process are exhibited in Figure6.Figure 6
Loading process of the single-span simply supported formwork.


#### 3.2.3. Results of the Experiments

Figures7 and 8 display the average experimental results of deflections and stresses of the formwork specimens, respectively.Figure 7
Load-deflection curves of the single-span simply supported formwork specimens.Figure 8
Load-stress curves of the single-span simply supported formwork specimens.The load-deflection curves of the specimens are nearly linear. The largest deflection is located in point C in the middle of the span and reaches 1.954 mm when subjected to the load of 15 kN/m2, which is smaller than the allowable value of 2.25 mm (1/400 of span) [18].Figure8 presents that the stress at the critical points increases linearly with the load, and the average value of the maximum tensile stress is approximately 60 N/mm2, while the average value of the maximum compressive stress is approximately −30 N/mm2, which is smaller than the allowable value of 200 N/mm2 (yield strength).In addition, during the loading process and after unloading, no apparent damage or bending buckling occurs.


### 3.3. Three-Span Continuous Formwork

#### 3.3.1. Arrangement of the Measuring Points

(1) Arrangement of the Measuring Points of Deflection. Figure 9 illustrates the five measuring points (A–E) to monitor the vertical deflection along with the three spans.Figure 9
Arrangement of the measuring points of deflection on the three-span continuous formwork.(2) Arrangement of the Measuring Points of Strain. Figure 10 depicts the five measuring points (1–5) placed on the interesting points of the frame to monitor the strain. They are separately distributed along a longitudinal beam of the frame.Figure 10
Arrangement of the measuring points of strain on the three-span continuous formwork.


#### 3.3.2. Loading Process

The loading process of the three-span continuous formwork is similar to that of the single-span simply supported formwork. The loads are applied between the outermost supports (Figure11). Moreover, the maximum value of preloading reaches 10 kN/m2. For the formal loading, the maximum load reaches 30 kN/m2. The loading procedure is demonstrated as follows: 0 ⟶ 5 kN/m2 ⟶ 10 kN/m2 ⟶ 15 kN/m2 ⟶ 20 kN/m2 ⟶ 25 kN/m2 ⟶ 30 kN/m2 ⟶ 0.Figure 11
Loading diagram of the three continuous supported formworks.The details of the loading process are exhibited in Figure12.Figure 12
Loading process of the three-span continuous formwork.


#### 3.3.3. Results of the Experiments

Figures13 and 14 display the average experimental results of deflections and stresses of the formwork specimens, correspondingly.Figure 13
Load-deflection curves of the three-span continuous formwork specimens.Figure 14
Load-stress curves of the three-span continuous formwork specimens.Similarly, the load-deflection curves of the specimens are nearly linear. Moreover, the results of the three tests are nearly consistent with one another. The largest deflection is located in points B and E and reaches approximately 1.336 mm when subjected to the load of 30 kN/m2, which is smaller than the allowable value of 1.875 mm (1/400 of span) [18].The stress at the critical point increases linearly with the load, and the average value of the maximum tensile stress is approximately 60 N/mm2, whereas the average value of the maximum compression stress is approximately −60 N/mm2, which is much smaller than the allowable value of 200 N/mm2 (yield strength).Furthermore, during the loading process and after unloading, no damage, bending, or buckling occurs.


## 3.1. General Test and Loading Scheme

To investigate the mechanical properties of formworks, three specimens are fabricated for each type of formwork. Table2 summarizes the general information of the formwork experiments. Steel blocks (2.5 kg) are used to simulate the uniform load. The measurement contains the following main parts:(1)
Deflection in the middle of the span(2)
Stress and strain in the middle of the span and supportTable 2
Loading program of the formwork.
Specimen numberDimension of the formworkType of supportSpan (mm)Length of the cantilever (mm)Maximum test load (kN/m2)1, 2, 31500 mm × 600 mm × 65 mmSingle-span simply supported900300154, 5, 62700 mm × 600 mm × 65 mmThree-span continuous750 × 322530


## 3.2. Single-Span Simply Supported Formwork

### 3.2.1. Arrangement of the Measuring Points

(1) Arrangement of the Measuring Point of Deflection. Figure 3 illustrates five measuring points (A–E) to monitor the vertical deflection in the middle of the span.Figure 3
Arrangement of the measuring points of deflection in the single-span simply supported formwork.(2) Arrangement of the Measuring Point of Strain. Figure 4 depicts four measuring points (1–4) placed on the interesting points of the frame to monitor the strain. The measuring points are separately distributed in the middle of four longitudinal beams of the frame.Figure 4
Arrangement of the measuring points of strain in the single-span simply supported formwork.


### 3.2.2. Loading Process

The loading process of the single-span simply supported formwork is demonstrated as follows: first, the proposed formwork specimens are placed on a special experimental bench. Second, the deflection meters, resistance strain gauges, and other devices are installed. To ensure uniformity of loading with steel blocks, the top surface of the wood-plastic template is covered with a thin layer of fine sand. Then, the steel blocks (2.5 kg) are applied uniformly on the formwork step by step to achieve a uniform load. The loading region is demonstrated in Figure5. The strain data are collected in the computer transmitted by strain gauges. The deflection data can be read directly from the meters.Figure 5
Loading diagram of the simply supported formwork.Before the formal experiment, the specimens are preloaded to check whether the specimens are placed stably, and the test instruments are set up correctly, thus ensuring that the experimental devices can work accurately and reduce the test error during the whole experiment. The abovementioned preloading reaches 5 kN and is held for 3 min.During the formal loading, 2.5 kN/m2 at each step is loaded until it reaches 15 kN/m2 (self-weight of 600 mm concrete). All data are recorded after loading for more than 30 s. The loading procedure is presented as follows: 0 ⟶ 2.5 kN/m2 ⟶ 5 kN/m2 ⟶ 7.5 kN/m2 ⟶ 10 kN/m2 ⟶ 12.5 kN/m2 ⟶ 15 kN/m2 ⟶ 0.During the actual process of loading step by step, the deformation of the formwork and the corresponding stress and strain increase with the uniform load. The experimental data collected are acceptable, and no apparent abnormality is found. The entire process is completed smoothly. The details of the loading process are exhibited in Figure6.Figure 6
Loading process of the single-span simply supported formwork.


### 3.2.3. Results of the Experiments

Figures7 and 8 display the average experimental results of deflections and stresses of the formwork specimens, respectively.Figure 7
Load-deflection curves of the single-span simply supported formwork specimens.Figure 8
Load-stress curves of the single-span simply supported formwork specimens.The load-deflection curves of the specimens are nearly linear. The largest deflection is located in point C in the middle of the span and reaches 1.954 mm when subjected to the load of 15 kN/m2, which is smaller than the allowable value of 2.25 mm (1/400 of span) [18].Figure8 presents that the stress at the critical points increases linearly with the load, and the average value of the maximum tensile stress is approximately 60 N/mm2, while the average value of the maximum compressive stress is approximately −30 N/mm2, which is smaller than the allowable value of 200 N/mm2 (yield strength).In addition, during the loading process and after unloading, no apparent damage or bending buckling occurs.


## 3.2.1. Arrangement of the Measuring Points

(1) Arrangement of the Measuring Point of Deflection. Figure 3 illustrates five measuring points (A–E) to monitor the vertical deflection in the middle of the span.Figure 3
Arrangement of the measuring points of deflection in the single-span simply supported formwork.(2) Arrangement of the Measuring Point of Strain. Figure 4 depicts four measuring points (1–4) placed on the interesting points of the frame to monitor the strain. The measuring points are separately distributed in the middle of four longitudinal beams of the frame.Figure 4
Arrangement of the measuring points of strain in the single-span simply supported formwork.


## 3.2.2. Loading Process

The loading process of the single-span simply supported formwork is demonstrated as follows: first, the proposed formwork specimens are placed on a special experimental bench. Second, the deflection meters, resistance strain gauges, and other devices are installed. To ensure uniformity of loading with steel blocks, the top surface of the wood-plastic template is covered with a thin layer of fine sand. Then, the steel blocks (2.5 kg) are applied uniformly on the formwork step by step to achieve a uniform load. The loading region is demonstrated in Figure5. The strain data are collected in the computer transmitted by strain gauges. The deflection data can be read directly from the meters.Figure 5
Loading diagram of the simply supported formwork.Before the formal experiment, the specimens are preloaded to check whether the specimens are placed stably, and the test instruments are set up correctly, thus ensuring that the experimental devices can work accurately and reduce the test error during the whole experiment. The abovementioned preloading reaches 5 kN and is held for 3 min.During the formal loading, 2.5 kN/m2 at each step is loaded until it reaches 15 kN/m2 (self-weight of 600 mm concrete). All data are recorded after loading for more than 30 s. The loading procedure is presented as follows: 0 ⟶ 2.5 kN/m2 ⟶ 5 kN/m2 ⟶ 7.5 kN/m2 ⟶ 10 kN/m2 ⟶ 12.5 kN/m2 ⟶ 15 kN/m2 ⟶ 0.During the actual process of loading step by step, the deformation of the formwork and the corresponding stress and strain increase with the uniform load. The experimental data collected are acceptable, and no apparent abnormality is found. The entire process is completed smoothly. The details of the loading process are exhibited in Figure6.Figure 6
Loading process of the single-span simply supported formwork.


## 3.2.3. Results of the Experiments

Figures7 and 8 display the average experimental results of deflections and stresses of the formwork specimens, respectively.Figure 7
Load-deflection curves of the single-span simply supported formwork specimens.Figure 8
Load-stress curves of the single-span simply supported formwork specimens.The load-deflection curves of the specimens are nearly linear. The largest deflection is located in point C in the middle of the span and reaches 1.954 mm when subjected to the load of 15 kN/m2, which is smaller than the allowable value of 2.25 mm (1/400 of span) [18].Figure8 presents that the stress at the critical points increases linearly with the load, and the average value of the maximum tensile stress is approximately 60 N/mm2, while the average value of the maximum compressive stress is approximately −30 N/mm2, which is smaller than the allowable value of 200 N/mm2 (yield strength).In addition, during the loading process and after unloading, no apparent damage or bending buckling occurs.


## 3.3. Three-Span Continuous Formwork

### 3.3.1. Arrangement of the Measuring Points

(1) Arrangement of the Measuring Points of Deflection. Figure 9 illustrates the five measuring points (A–E) to monitor the vertical deflection along with the three spans.Figure 9
Arrangement of the measuring points of deflection on the three-span continuous formwork.(2) Arrangement of the Measuring Points of Strain. Figure 10 depicts the five measuring points (1–5) placed on the interesting points of the frame to monitor the strain. They are separately distributed along a longitudinal beam of the frame.Figure 10
Arrangement of the measuring points of strain on the three-span continuous formwork.


### 3.3.2. Loading Process

The loading process of the three-span continuous formwork is similar to that of the single-span simply supported formwork. The loads are applied between the outermost supports (Figure11). Moreover, the maximum value of preloading reaches 10 kN/m2. For the formal loading, the maximum load reaches 30 kN/m2. The loading procedure is demonstrated as follows: 0 ⟶ 5 kN/m2 ⟶ 10 kN/m2 ⟶ 15 kN/m2 ⟶ 20 kN/m2 ⟶ 25 kN/m2 ⟶ 30 kN/m2 ⟶ 0.Figure 11
Loading diagram of the three continuous supported formworks.The details of the loading process are exhibited in Figure12.Figure 12
Loading process of the three-span continuous formwork.


### 3.3.3. Results of the Experiments

Figures13 and 14 display the average experimental results of deflections and stresses of the formwork specimens, correspondingly.Figure 13
Load-deflection curves of the three-span continuous formwork specimens.Figure 14
Load-stress curves of the three-span continuous formwork specimens.Similarly, the load-deflection curves of the specimens are nearly linear. Moreover, the results of the three tests are nearly consistent with one another. The largest deflection is located in points B and E and reaches approximately 1.336 mm when subjected to the load of 30 kN/m2, which is smaller than the allowable value of 1.875 mm (1/400 of span) [18].The stress at the critical point increases linearly with the load, and the average value of the maximum tensile stress is approximately 60 N/mm2, whereas the average value of the maximum compression stress is approximately −60 N/mm2, which is much smaller than the allowable value of 200 N/mm2 (yield strength).Furthermore, during the loading process and after unloading, no damage, bending, or buckling occurs.


## 3.3.1. Arrangement of the Measuring Points

(1) Arrangement of the Measuring Points of Deflection. Figure 9 illustrates the five measuring points (A–E) to monitor the vertical deflection along with the three spans.Figure 9
Arrangement of the measuring points of deflection on the three-span continuous formwork.(2) Arrangement of the Measuring Points of Strain. Figure 10 depicts the five measuring points (1–5) placed on the interesting points of the frame to monitor the strain. They are separately distributed along a longitudinal beam of the frame.Figure 10
Arrangement of the measuring points of strain on the three-span continuous formwork.


## 3.3.2. Loading Process

The loading process of the three-span continuous formwork is similar to that of the single-span simply supported formwork. The loads are applied between the outermost supports (Figure11). Moreover, the maximum value of preloading reaches 10 kN/m2. For the formal loading, the maximum load reaches 30 kN/m2. The loading procedure is demonstrated as follows: 0 ⟶ 5 kN/m2 ⟶ 10 kN/m2 ⟶ 15 kN/m2 ⟶ 20 kN/m2 ⟶ 25 kN/m2 ⟶ 30 kN/m2 ⟶ 0.Figure 11
Loading diagram of the three continuous supported formworks.The details of the loading process are exhibited in Figure12.Figure 12
Loading process of the three-span continuous formwork.


## 3.3.3. Results of the Experiments

Figures13 and 14 display the average experimental results of deflections and stresses of the formwork specimens, correspondingly.Figure 13
Load-deflection curves of the three-span continuous formwork specimens.Figure 14
Load-stress curves of the three-span continuous formwork specimens.Similarly, the load-deflection curves of the specimens are nearly linear. Moreover, the results of the three tests are nearly consistent with one another. The largest deflection is located in points B and E and reaches approximately 1.336 mm when subjected to the load of 30 kN/m2, which is smaller than the allowable value of 1.875 mm (1/400 of span) [18].The stress at the critical point increases linearly with the load, and the average value of the maximum tensile stress is approximately 60 N/mm2, whereas the average value of the maximum compression stress is approximately −60 N/mm2, which is much smaller than the allowable value of 200 N/mm2 (yield strength).Furthermore, during the loading process and after unloading, no damage, bending, or buckling occurs.


## 4. FE Analysis

### 4.1. Methodology of ABAQUS Modeling

#### 4.1.1. Principle of Finite Element Analysis

The basic step of finite element analysis is to discrete the structure. In ABAQUS, one of the discretization methods used in contact pair is the discretization method of surface-to-surface. In the physical sense, the pressure will transmit between two objects that contact each other at the same time, and there exists shearing force, to be precise, friction to prevent the tangential movement of objects. In the numerical calculation, the contact is extremely discontinuous boundary conditions of nonlinearity; that is, the contact constraints exist only if the contact planes contact each other without gap. If the contact planes separate, the contact constraints will be invalid. Generally, the type of contact can be divided into two kinds. One is surface-based contact while the other is element-based contact. We will discuss surface-based contact, which will be used in the following finite element analysis. According to Figure15, the discretization of surface-to-surface is that the node on the average region of the slave surface discretizes normally into several continuous small planes. The contact is built on the discrete surface of the slave surface and the master surface adjacent to it. In contact pair, both the slave surface and the master surface are considered as continuous faces, and the contact direction is the normal direction of slave surface. The node from the master surface is not allowed to penetrate the slave surface, but instead, the node of the slave surface can penetrate through the master surface.Figure 15
Discretization method of node to the surface.As for the tracking method of contact, there are two ways to calculate the relative slip between the contact surfaces: finite sliding and small sliding. As shown in Figure16, the position of the finite slip node is updated depending on the actual shape of the slave surface, and when the contact constraint is generated, the node 1 of slave surface will be constrained to slip along the path of the master surface, regardless of its direction and deformation. It is used when the instances contacted with each other have large deformation, rotation, and large displacement sliding. The tracking method of the small sliding, assuming that the nodal displacement on the slave surface is relatively small compared with the local curvature of the master surface, updates the location of slave node along with the tangent direction of the master surface, as shown Figure 17.Figure 16
Path of finite sliding.Figure 17
Path of small sliding.In addition, the initial geometrical imperfection [19] also has a certain effect on the mechanical properties of the structure, especially for pressure bar and shell structure, but this factor can be ignored for the bending formwork under normal construction load in this paper.


#### 4.1.2. Modeling

In this work, the FE analysis software ABAQUS is used to conduct the numerical simulation of this experiment. Two types of FE models, namely, refined and general, are established.In the first FE model (refined model), several contact pairs are set, including the relationship between the template and frame, the interaction between the template and rivet, and the interaction between the frame and rivet. The interactions of all the contact pairs are simulated by the contact function in ABAQUS. All contacts are defined to be surface-to-surface contacts where a “hard contact” is the pressure-overclosure on normal behavior. However, the contact pair between the formwork and support assumes that the friction formulation on tangential behavior is a penalty function, whereas the friction formulation on the tangential behavior of the three other contact pairs is rough, thereby indicating that no slip will occur. Small sliding is used for all the abovementioned contacts. The density of the seeds on frame, template, and supports is all set to 5 mm with the element type assigned as C3D8R (eight-node linear brick elements with reduced integration). The density of mesh seeds on the rivet is small, given the small dimension, which is 1 mm (Figure18). For the boundary condition, pinned supports with a linear displacement of all its nodes impeded in all directions are assumed for the models.Figure 18
First FE model (refined models). (a) Single-span simply supported formwork. (b) Three-span continuous formwork.
(a)(b)In the second FE model (general model), the rivets are ignored for simplified modeling, but a tie constraint exists between the template and frame, thereby indicating that the elements on the contact surface will deform simultaneously. The cross sections of a fringe frame are simplified as rectangular shaped in accordance with the stiffness equivalence principle, and the mesh seed density of all the parts is equal to 10 mm (Figure19). Furthermore, the boundary condition is the same as the first model (refined models).Figure 19
Second FE model (general models). (a) Single-span simply supported formwork. (b) Three-span continuous formwork.
(a)(b)


### 4.2. Single-Span Simply Supported Formwork

#### 4.2.1. FE Analysis Result

Figures20–22 illustrate that two models are consistent with each other in terms of deformation shape and stress distribution.Figure 20
Deflections of the template (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 21
Deflections of the frame (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 22
Longitudinal stresses of the frame (unit: N/mm2). (a) Refined model. (b) General model.
(a)(b)From the deflection contours of the formwork, under the maximum uniform load with 15 kN/m2, the layout of the maximum deflection of the template is located at the centroid, thus accounting for 2.014 mm (Figure 20(a)) and 2.009 mm (Figure 20(b)). In addition, it appears as a concave phenomenon in the middle of the span on the template because two transverse beams are supported under the formwork in the middle of the span, thereby increasing the local stiffness and helping reduce the deflection. Moreover, the deflections at the two cantilevered ends of formwork are positive because the loads are only applied between the supports such that both cantilevered ends will be upturned. The deflection contours of the frame show that the negative extreme value appears in the middle span, whereas the value of the two ends is positive.The longitudinal stress contour of the frame shows that the maximum longitudinal stress (55 and 60.07 N/mm2 in Figures 22(a) and 22(b), respectively) appears in the middle of the span, which is far less than the allowable value (200 N/mm2). The bending moment is largest at the midpoint of the beam, such that the stress, which is positively correlated with the moment, is also the largest.


#### 4.2.2. Comparison of the FE Analysis and Experimental Results

Figures23 and 24 depict the comparisons between the FE and experimental results. The prediction of FE is consistent with the corresponding experimental result, thereby denoting that the proposed FE models can reflect the mechanical behavior of formworks well.Figure 23
Comparison of the deflections between the experiment and FE models.Figure 24
Comparison of the stresses between the experiment and FE models. (a) Top surface. (b) Bottom surface.
(a)(b)Tables3 and 4 summarize the comparisons of deflection and stress between the experiment and FE models. The maximum error of deflection between the two types of FE models is 8.94% and is located at points A and E. The error of deflection between the refined model and the experiment is within 5.5%. Furthermore, the error of stress between the two models and experiment is nearly less than 10%.Table 3
Comparison of the deflections between the experiment and FE models.
PointDeflectionRefined model (mm)General modelAverage value of the experimentResult (mm)ErrorResult (mm)ErrorPoint A1.1091.2088.941.1675.19Point B1.6731.7263.151.7313.47Point C2.0142.0100.201.9533.00Point D1.6711.7263.271.7011.77Point E1.1091.2088.941.1695.43Table 4
Comparison of the stresses between the experiment and FE models.
PointStressRefined model (mm)General modelAverage value of the experimentResult (N/mm2)ErrorResult (N/mm2)ErrorPoint 1 (top)−25.68−23.229.57−25.371.23Point 2 (top)−26.60−25.364.68−28.205.98Point 3 (top)−26.49−25.364.28−29.9713.10Point 4 (top)−26.78−23.2213.30−29.439.88Point 1 (bottom)31.1929.664.9230.552.08Point 2 (bottom)52.8053.711.7252.950.28Point 3 (bottom)52.8053.711.7152.450.68Point 4 (bottom)31.5929.666.1033.475.93Overall, the results between the numerical simulation and the experiment are consistent. By contrast, the refined model has high accuracy, which is consistent with the experimental results. When the requirement of accuracy is not too strict, the general model is preferred, given modeling convenience and high efficiency.


### 4.3. Three-Span Continuous Formwork

#### 4.3.1. FE Analysis Result

Similar to the results of the single-span simply supported formwork, both models for the three-span continuous formwork are consistent in terms of deformation shapes and stress distributions.The deflection contours of the template (Figure25) exhibit that extremes appear in the middle of the two side spans, thus accounting for 1.277 and 1.322 mm (Figures 25(a) and 25(b), correspondingly). Figure 26 displays that the extreme values appear in the middle of the side span. The longitudinal middle beams deform more clearly than the longitudinal side beams because they bear additional loads.Figure 25
Deflections of the template (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 26
Deflections of the frame (unit: mm). (a) Refined model. (b) General model.
(a)(b)In Figure27, the extreme values of longitudinal stress appear in the middle of each span and on top of the support because the maximum positive moments are located in the middle of the span, and the minimum negative moments are located on top of the support. Accordingly, the maximum longitudinal stresses of the two models (90.96 and 81.00 N/mm2) are within the limit (200 N/mm2).Figure 27
Longitudinal stresses of the frame (unit: N/mm2). (a) Refined model. (b) General model.
(a)(b)


#### 4.3.2. Comparison of the FE Analysis and Experimental Results

Figures28 and 29 illustrate the comparisons between the experimental and FE results. The prediction of FE is consistent with the corresponding experimental result, thus indicating that the proposed FE models can reflect the mechanical behavior of formworks well.Figure 28
Comparison of the deflections between the experiment and FE models.Figure 29
Comparison of the stresses between the experiment and FE models.
(a)(b)Table5 lists that the maximum error between the first model and the experiment is located at point A with 6.1%, thus denoting that the refined model can reflect the mechanical properties of formworks well. Moreover, the error between the two FE models is less than 6%. Table 6 summarizes that the errors of the stresses between the two FE models and experiment are nearly less than 10%.Table 5
Comparison of the deflections between the experiment and refined FE models.
PointDeflectionRefined modelGeneral modelAverage value of the experimentResult (mm)ErrorResult (mm)ErrorPoint A0.8250.8003.070.8766.10Point B1.2761.2095.291.3324.34Point C0.5870.6083.560.5683.32Point D0.8420.8222.390.8222.40Point E1.2901.2155.831.3363.55Table 6
Comparison of the stresses between the experiment and FE models.
PointStressRefined modelGeneral modelAverage value of the experimentResult (N/mm2)ErrorResult (N/mm2)Error (%)Point 1 (top)−20.26−20.772.51−20.501.14Point 2 (top)37.0135.115.1339.807.54Point 3 (top)−12.41−12.712.41−12.601.50Point 4 (top)36.0633.038.3841.6015.36Point 5 (top)−19.68−21.479.07−18.904.00Point 1 (bottom)39.8636.737.8644.2010.86Point 2 (bottom)−57.76−56.681.86−60.905.44Point 3 (bottom)19.8420.965.6319.203.26Point 4 (bottom)−46.10−46.020.17−47.803.67Point 5 (bottom)41.8638.168.8349.7018.71In conclusion, the two proposed FE models can reflect the mechanical behavior of formworks well, and the refined model is more accurate than the general model. Similarly, when the requirement of accuracy is not too strict, the general model is suggested.


## 4.1. Methodology of ABAQUS Modeling

### 4.1.1. Principle of Finite Element Analysis

The basic step of finite element analysis is to discrete the structure. In ABAQUS, one of the discretization methods used in contact pair is the discretization method of surface-to-surface. In the physical sense, the pressure will transmit between two objects that contact each other at the same time, and there exists shearing force, to be precise, friction to prevent the tangential movement of objects. In the numerical calculation, the contact is extremely discontinuous boundary conditions of nonlinearity; that is, the contact constraints exist only if the contact planes contact each other without gap. If the contact planes separate, the contact constraints will be invalid. Generally, the type of contact can be divided into two kinds. One is surface-based contact while the other is element-based contact. We will discuss surface-based contact, which will be used in the following finite element analysis. According to Figure15, the discretization of surface-to-surface is that the node on the average region of the slave surface discretizes normally into several continuous small planes. The contact is built on the discrete surface of the slave surface and the master surface adjacent to it. In contact pair, both the slave surface and the master surface are considered as continuous faces, and the contact direction is the normal direction of slave surface. The node from the master surface is not allowed to penetrate the slave surface, but instead, the node of the slave surface can penetrate through the master surface.Figure 15
Discretization method of node to the surface.As for the tracking method of contact, there are two ways to calculate the relative slip between the contact surfaces: finite sliding and small sliding. As shown in Figure16, the position of the finite slip node is updated depending on the actual shape of the slave surface, and when the contact constraint is generated, the node 1 of slave surface will be constrained to slip along the path of the master surface, regardless of its direction and deformation. It is used when the instances contacted with each other have large deformation, rotation, and large displacement sliding. The tracking method of the small sliding, assuming that the nodal displacement on the slave surface is relatively small compared with the local curvature of the master surface, updates the location of slave node along with the tangent direction of the master surface, as shown Figure 17.Figure 16
Path of finite sliding.Figure 17
Path of small sliding.In addition, the initial geometrical imperfection [19] also has a certain effect on the mechanical properties of the structure, especially for pressure bar and shell structure, but this factor can be ignored for the bending formwork under normal construction load in this paper.


### 4.1.2. Modeling

In this work, the FE analysis software ABAQUS is used to conduct the numerical simulation of this experiment. Two types of FE models, namely, refined and general, are established.In the first FE model (refined model), several contact pairs are set, including the relationship between the template and frame, the interaction between the template and rivet, and the interaction between the frame and rivet. The interactions of all the contact pairs are simulated by the contact function in ABAQUS. All contacts are defined to be surface-to-surface contacts where a “hard contact” is the pressure-overclosure on normal behavior. However, the contact pair between the formwork and support assumes that the friction formulation on tangential behavior is a penalty function, whereas the friction formulation on the tangential behavior of the three other contact pairs is rough, thereby indicating that no slip will occur. Small sliding is used for all the abovementioned contacts. The density of the seeds on frame, template, and supports is all set to 5 mm with the element type assigned as C3D8R (eight-node linear brick elements with reduced integration). The density of mesh seeds on the rivet is small, given the small dimension, which is 1 mm (Figure18). For the boundary condition, pinned supports with a linear displacement of all its nodes impeded in all directions are assumed for the models.Figure 18
First FE model (refined models). (a) Single-span simply supported formwork. (b) Three-span continuous formwork.
(a)(b)In the second FE model (general model), the rivets are ignored for simplified modeling, but a tie constraint exists between the template and frame, thereby indicating that the elements on the contact surface will deform simultaneously. The cross sections of a fringe frame are simplified as rectangular shaped in accordance with the stiffness equivalence principle, and the mesh seed density of all the parts is equal to 10 mm (Figure19). Furthermore, the boundary condition is the same as the first model (refined models).Figure 19
Second FE model (general models). (a) Single-span simply supported formwork. (b) Three-span continuous formwork.
(a)(b)


## 4.1.1. Principle of Finite Element Analysis

The basic step of finite element analysis is to discrete the structure. In ABAQUS, one of the discretization methods used in contact pair is the discretization method of surface-to-surface. In the physical sense, the pressure will transmit between two objects that contact each other at the same time, and there exists shearing force, to be precise, friction to prevent the tangential movement of objects. In the numerical calculation, the contact is extremely discontinuous boundary conditions of nonlinearity; that is, the contact constraints exist only if the contact planes contact each other without gap. If the contact planes separate, the contact constraints will be invalid. Generally, the type of contact can be divided into two kinds. One is surface-based contact while the other is element-based contact. We will discuss surface-based contact, which will be used in the following finite element analysis. According to Figure15, the discretization of surface-to-surface is that the node on the average region of the slave surface discretizes normally into several continuous small planes. The contact is built on the discrete surface of the slave surface and the master surface adjacent to it. In contact pair, both the slave surface and the master surface are considered as continuous faces, and the contact direction is the normal direction of slave surface. The node from the master surface is not allowed to penetrate the slave surface, but instead, the node of the slave surface can penetrate through the master surface.Figure 15
Discretization method of node to the surface.As for the tracking method of contact, there are two ways to calculate the relative slip between the contact surfaces: finite sliding and small sliding. As shown in Figure16, the position of the finite slip node is updated depending on the actual shape of the slave surface, and when the contact constraint is generated, the node 1 of slave surface will be constrained to slip along the path of the master surface, regardless of its direction and deformation. It is used when the instances contacted with each other have large deformation, rotation, and large displacement sliding. The tracking method of the small sliding, assuming that the nodal displacement on the slave surface is relatively small compared with the local curvature of the master surface, updates the location of slave node along with the tangent direction of the master surface, as shown Figure 17.Figure 16
Path of finite sliding.Figure 17
Path of small sliding.In addition, the initial geometrical imperfection [19] also has a certain effect on the mechanical properties of the structure, especially for pressure bar and shell structure, but this factor can be ignored for the bending formwork under normal construction load in this paper.


## 4.1.2. Modeling

In this work, the FE analysis software ABAQUS is used to conduct the numerical simulation of this experiment. Two types of FE models, namely, refined and general, are established.In the first FE model (refined model), several contact pairs are set, including the relationship between the template and frame, the interaction between the template and rivet, and the interaction between the frame and rivet. The interactions of all the contact pairs are simulated by the contact function in ABAQUS. All contacts are defined to be surface-to-surface contacts where a “hard contact” is the pressure-overclosure on normal behavior. However, the contact pair between the formwork and support assumes that the friction formulation on tangential behavior is a penalty function, whereas the friction formulation on the tangential behavior of the three other contact pairs is rough, thereby indicating that no slip will occur. Small sliding is used for all the abovementioned contacts. The density of the seeds on frame, template, and supports is all set to 5 mm with the element type assigned as C3D8R (eight-node linear brick elements with reduced integration). The density of mesh seeds on the rivet is small, given the small dimension, which is 1 mm (Figure18). For the boundary condition, pinned supports with a linear displacement of all its nodes impeded in all directions are assumed for the models.Figure 18
First FE model (refined models). (a) Single-span simply supported formwork. (b) Three-span continuous formwork.
(a)(b)In the second FE model (general model), the rivets are ignored for simplified modeling, but a tie constraint exists between the template and frame, thereby indicating that the elements on the contact surface will deform simultaneously. The cross sections of a fringe frame are simplified as rectangular shaped in accordance with the stiffness equivalence principle, and the mesh seed density of all the parts is equal to 10 mm (Figure19). Furthermore, the boundary condition is the same as the first model (refined models).Figure 19
Second FE model (general models). (a) Single-span simply supported formwork. (b) Three-span continuous formwork.
(a)(b)


## 4.2. Single-Span Simply Supported Formwork

### 4.2.1. FE Analysis Result

Figures20–22 illustrate that two models are consistent with each other in terms of deformation shape and stress distribution.Figure 20
Deflections of the template (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 21
Deflections of the frame (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 22
Longitudinal stresses of the frame (unit: N/mm2). (a) Refined model. (b) General model.
(a)(b)From the deflection contours of the formwork, under the maximum uniform load with 15 kN/m2, the layout of the maximum deflection of the template is located at the centroid, thus accounting for 2.014 mm (Figure 20(a)) and 2.009 mm (Figure 20(b)). In addition, it appears as a concave phenomenon in the middle of the span on the template because two transverse beams are supported under the formwork in the middle of the span, thereby increasing the local stiffness and helping reduce the deflection. Moreover, the deflections at the two cantilevered ends of formwork are positive because the loads are only applied between the supports such that both cantilevered ends will be upturned. The deflection contours of the frame show that the negative extreme value appears in the middle span, whereas the value of the two ends is positive.The longitudinal stress contour of the frame shows that the maximum longitudinal stress (55 and 60.07 N/mm2 in Figures 22(a) and 22(b), respectively) appears in the middle of the span, which is far less than the allowable value (200 N/mm2). The bending moment is largest at the midpoint of the beam, such that the stress, which is positively correlated with the moment, is also the largest.


### 4.2.2. Comparison of the FE Analysis and Experimental Results

Figures23 and 24 depict the comparisons between the FE and experimental results. The prediction of FE is consistent with the corresponding experimental result, thereby denoting that the proposed FE models can reflect the mechanical behavior of formworks well.Figure 23
Comparison of the deflections between the experiment and FE models.Figure 24
Comparison of the stresses between the experiment and FE models. (a) Top surface. (b) Bottom surface.
(a)(b)Tables3 and 4 summarize the comparisons of deflection and stress between the experiment and FE models. The maximum error of deflection between the two types of FE models is 8.94% and is located at points A and E. The error of deflection between the refined model and the experiment is within 5.5%. Furthermore, the error of stress between the two models and experiment is nearly less than 10%.Table 3
Comparison of the deflections between the experiment and FE models.
PointDeflectionRefined model (mm)General modelAverage value of the experimentResult (mm)ErrorResult (mm)ErrorPoint A1.1091.2088.941.1675.19Point B1.6731.7263.151.7313.47Point C2.0142.0100.201.9533.00Point D1.6711.7263.271.7011.77Point E1.1091.2088.941.1695.43Table 4
Comparison of the stresses between the experiment and FE models.
PointStressRefined model (mm)General modelAverage value of the experimentResult (N/mm2)ErrorResult (N/mm2)ErrorPoint 1 (top)−25.68−23.229.57−25.371.23Point 2 (top)−26.60−25.364.68−28.205.98Point 3 (top)−26.49−25.364.28−29.9713.10Point 4 (top)−26.78−23.2213.30−29.439.88Point 1 (bottom)31.1929.664.9230.552.08Point 2 (bottom)52.8053.711.7252.950.28Point 3 (bottom)52.8053.711.7152.450.68Point 4 (bottom)31.5929.666.1033.475.93Overall, the results between the numerical simulation and the experiment are consistent. By contrast, the refined model has high accuracy, which is consistent with the experimental results. When the requirement of accuracy is not too strict, the general model is preferred, given modeling convenience and high efficiency.


## 4.2.1. FE Analysis Result

Figures20–22 illustrate that two models are consistent with each other in terms of deformation shape and stress distribution.Figure 20
Deflections of the template (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 21
Deflections of the frame (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 22
Longitudinal stresses of the frame (unit: N/mm2). (a) Refined model. (b) General model.
(a)(b)From the deflection contours of the formwork, under the maximum uniform load with 15 kN/m2, the layout of the maximum deflection of the template is located at the centroid, thus accounting for 2.014 mm (Figure 20(a)) and 2.009 mm (Figure 20(b)). In addition, it appears as a concave phenomenon in the middle of the span on the template because two transverse beams are supported under the formwork in the middle of the span, thereby increasing the local stiffness and helping reduce the deflection. Moreover, the deflections at the two cantilevered ends of formwork are positive because the loads are only applied between the supports such that both cantilevered ends will be upturned. The deflection contours of the frame show that the negative extreme value appears in the middle span, whereas the value of the two ends is positive.The longitudinal stress contour of the frame shows that the maximum longitudinal stress (55 and 60.07 N/mm2 in Figures 22(a) and 22(b), respectively) appears in the middle of the span, which is far less than the allowable value (200 N/mm2). The bending moment is largest at the midpoint of the beam, such that the stress, which is positively correlated with the moment, is also the largest.


## 4.2.2. Comparison of the FE Analysis and Experimental Results

Figures23 and 24 depict the comparisons between the FE and experimental results. The prediction of FE is consistent with the corresponding experimental result, thereby denoting that the proposed FE models can reflect the mechanical behavior of formworks well.Figure 23
Comparison of the deflections between the experiment and FE models.Figure 24
Comparison of the stresses between the experiment and FE models. (a) Top surface. (b) Bottom surface.
(a)(b)Tables3 and 4 summarize the comparisons of deflection and stress between the experiment and FE models. The maximum error of deflection between the two types of FE models is 8.94% and is located at points A and E. The error of deflection between the refined model and the experiment is within 5.5%. Furthermore, the error of stress between the two models and experiment is nearly less than 10%.Table 3
Comparison of the deflections between the experiment and FE models.
PointDeflectionRefined model (mm)General modelAverage value of the experimentResult (mm)ErrorResult (mm)ErrorPoint A1.1091.2088.941.1675.19Point B1.6731.7263.151.7313.47Point C2.0142.0100.201.9533.00Point D1.6711.7263.271.7011.77Point E1.1091.2088.941.1695.43Table 4
Comparison of the stresses between the experiment and FE models.
PointStressRefined model (mm)General modelAverage value of the experimentResult (N/mm2)ErrorResult (N/mm2)ErrorPoint 1 (top)−25.68−23.229.57−25.371.23Point 2 (top)−26.60−25.364.68−28.205.98Point 3 (top)−26.49−25.364.28−29.9713.10Point 4 (top)−26.78−23.2213.30−29.439.88Point 1 (bottom)31.1929.664.9230.552.08Point 2 (bottom)52.8053.711.7252.950.28Point 3 (bottom)52.8053.711.7152.450.68Point 4 (bottom)31.5929.666.1033.475.93Overall, the results between the numerical simulation and the experiment are consistent. By contrast, the refined model has high accuracy, which is consistent with the experimental results. When the requirement of accuracy is not too strict, the general model is preferred, given modeling convenience and high efficiency.


## 4.3. Three-Span Continuous Formwork

### 4.3.1. FE Analysis Result

Similar to the results of the single-span simply supported formwork, both models for the three-span continuous formwork are consistent in terms of deformation shapes and stress distributions.The deflection contours of the template (Figure25) exhibit that extremes appear in the middle of the two side spans, thus accounting for 1.277 and 1.322 mm (Figures 25(a) and 25(b), correspondingly). Figure 26 displays that the extreme values appear in the middle of the side span. The longitudinal middle beams deform more clearly than the longitudinal side beams because they bear additional loads.Figure 25
Deflections of the template (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 26
Deflections of the frame (unit: mm). (a) Refined model. (b) General model.
(a)(b)In Figure27, the extreme values of longitudinal stress appear in the middle of each span and on top of the support because the maximum positive moments are located in the middle of the span, and the minimum negative moments are located on top of the support. Accordingly, the maximum longitudinal stresses of the two models (90.96 and 81.00 N/mm2) are within the limit (200 N/mm2).Figure 27
Longitudinal stresses of the frame (unit: N/mm2). (a) Refined model. (b) General model.
(a)(b)


### 4.3.2. Comparison of the FE Analysis and Experimental Results

Figures28 and 29 illustrate the comparisons between the experimental and FE results. The prediction of FE is consistent with the corresponding experimental result, thus indicating that the proposed FE models can reflect the mechanical behavior of formworks well.Figure 28
Comparison of the deflections between the experiment and FE models.Figure 29
Comparison of the stresses between the experiment and FE models.
(a)(b)Table5 lists that the maximum error between the first model and the experiment is located at point A with 6.1%, thus denoting that the refined model can reflect the mechanical properties of formworks well. Moreover, the error between the two FE models is less than 6%. Table 6 summarizes that the errors of the stresses between the two FE models and experiment are nearly less than 10%.Table 5
Comparison of the deflections between the experiment and refined FE models.
PointDeflectionRefined modelGeneral modelAverage value of the experimentResult (mm)ErrorResult (mm)ErrorPoint A0.8250.8003.070.8766.10Point B1.2761.2095.291.3324.34Point C0.5870.6083.560.5683.32Point D0.8420.8222.390.8222.40Point E1.2901.2155.831.3363.55Table 6
Comparison of the stresses between the experiment and FE models.
PointStressRefined modelGeneral modelAverage value of the experimentResult (N/mm2)ErrorResult (N/mm2)Error (%)Point 1 (top)−20.26−20.772.51−20.501.14Point 2 (top)37.0135.115.1339.807.54Point 3 (top)−12.41−12.712.41−12.601.50Point 4 (top)36.0633.038.3841.6015.36Point 5 (top)−19.68−21.479.07−18.904.00Point 1 (bottom)39.8636.737.8644.2010.86Point 2 (bottom)−57.76−56.681.86−60.905.44Point 3 (bottom)19.8420.965.6319.203.26Point 4 (bottom)−46.10−46.020.17−47.803.67Point 5 (bottom)41.8638.168.8349.7018.71In conclusion, the two proposed FE models can reflect the mechanical behavior of formworks well, and the refined model is more accurate than the general model. Similarly, when the requirement of accuracy is not too strict, the general model is suggested.


## 4.3.1. FE Analysis Result

Similar to the results of the single-span simply supported formwork, both models for the three-span continuous formwork are consistent in terms of deformation shapes and stress distributions.The deflection contours of the template (Figure25) exhibit that extremes appear in the middle of the two side spans, thus accounting for 1.277 and 1.322 mm (Figures 25(a) and 25(b), correspondingly). Figure 26 displays that the extreme values appear in the middle of the side span. The longitudinal middle beams deform more clearly than the longitudinal side beams because they bear additional loads.Figure 25
Deflections of the template (unit: mm). (a) Refined model. (b) General model.
(a)(b)Figure 26
Deflections of the frame (unit: mm). (a) Refined model. (b) General model.
(a)(b)In Figure27, the extreme values of longitudinal stress appear in the middle of each span and on top of the support because the maximum positive moments are located in the middle of the span, and the minimum negative moments are located on top of the support. Accordingly, the maximum longitudinal stresses of the two models (90.96 and 81.00 N/mm2) are within the limit (200 N/mm2).Figure 27
Longitudinal stresses of the frame (unit: N/mm2). (a) Refined model. (b) General model.
(a)(b)


## 4.3.2. Comparison of the FE Analysis and Experimental Results

Figures28 and 29 illustrate the comparisons between the experimental and FE results. The prediction of FE is consistent with the corresponding experimental result, thus indicating that the proposed FE models can reflect the mechanical behavior of formworks well.Figure 28
Comparison of the deflections between the experiment and FE models.Figure 29
Comparison of the stresses between the experiment and FE models.
(a)(b)Table5 lists that the maximum error between the first model and the experiment is located at point A with 6.1%, thus denoting that the refined model can reflect the mechanical properties of formworks well. Moreover, the error between the two FE models is less than 6%. Table 6 summarizes that the errors of the stresses between the two FE models and experiment are nearly less than 10%.Table 5
Comparison of the deflections between the experiment and refined FE models.
PointDeflectionRefined modelGeneral modelAverage value of the experimentResult (mm)ErrorResult (mm)ErrorPoint A0.8250.8003.070.8766.10Point B1.2761.2095.291.3324.34Point C0.5870.6083.560.5683.32Point D0.8420.8222.390.8222.40Point E1.2901.2155.831.3363.55Table 6
Comparison of the stresses between the experiment and FE models.
PointStressRefined modelGeneral modelAverage value of the experimentResult (N/mm2)ErrorResult (N/mm2)Error (%)Point 1 (top)−20.26−20.772.51−20.501.14Point 2 (top)37.0135.115.1339.807.54Point 3 (top)−12.41−12.712.41−12.601.50Point 4 (top)36.0633.038.3841.6015.36Point 5 (top)−19.68−21.479.07−18.904.00Point 1 (bottom)39.8636.737.8644.2010.86Point 2 (bottom)−57.76−56.681.86−60.905.44Point 3 (bottom)19.8420.965.6319.203.26Point 4 (bottom)−46.10−46.020.17−47.803.67Point 5 (bottom)41.8638.168.8349.7018.71In conclusion, the two proposed FE models can reflect the mechanical behavior of formworks well, and the refined model is more accurate than the general model. Similarly, when the requirement of accuracy is not too strict, the general model is suggested.


## 5. Practical Simplified Calculation Formula of the Formwork

To quickly and conveniently predict the deflection and evaluate the mechanical properties of the formwork in engineering practice, the practical simplified calculation formula must be deduced.


### 5.1. Section Conversion and Cross-Sectional Geometry

#### 5.1.1. Section Conversion

This new composite wood-plastic formwork with an aluminum alloy frame is assembled using different materials. Two kinds of materials must be converted into the same material through formula deduction. The equivalent principles of section conversion are presented as follows:(1)
The bending stiffness is consistent before and after the section conversion(2)
The deformation of the template on the contact surface is consistent with that of the frameFigure30 illustrates the composite cross section of the two materials. The lower part is the aluminum alloy frame, with the whole section area Af and the elastic modulus of Ef. The upper part is the wood-plastic template, with the section area of Aw and elastic modulus Ew. Figure 31 depicts the new cross section after section conversion, in which the wood-plastic template is replaced by an aluminum alloy template with the cross-sectional area Ae. The detailed deduction process of the section conversion is demonstrated as follows:where σw is the stress in the wood-plastic template, σe is the stress in the equivalent template, and ε is the strain in the template.Figure 30
Actual cross section of the formwork.Figure 31
Equivalent cross section of the formwork.(1)Aw⋅σw=Ae⋅σe,Ae=σwσe⋅Aw,∵σe=Ef⋅ε,σw=Ew⋅ε,Ae=EwEf⋅Aw,If(2)EfEw=n,then(3)Ae=1n⋅Aw.The sectional height remains unchanged, and only the width is changed, that is,(4)be=1n⋅bw.The physical meaning of equation (4) is that the width bw of the wood-plastic template is divided by the ratio n of the elastic modulus of the aluminum alloy and the wood-plastic material. Therefore, the equivalent width be and area Ae of the template are expressed as follows:(5)n=EfEw=0.7×1053.84×103=18.23,be=bwn=58.418.23=3.2cm,Ae=3.2×1=3.2cm2.


#### 5.1.2. Calculation of the Cross-Sectional Geometric Properties

The cross-sectional geometries of the longitudinal beams of the frame are presented in Table7.Table 7
Cross-sectional geometry of the longitudinal beams of the frame.
ShapeParameterArea (cm2)Distance between top and centroid (cm)Moment of inertia (cm4)4.583.0118.382.972.5315.27The distance from the neutral axis to the top surface of the equivalent section is as follows:(6)y0=4.58×3.01×2+2.97×3.53×2+3.20×0.54.58×2+2.97×2+3.20=2.74cm.The conversion moment of the inertia of the equivalent section is as follows:(7)Ie=18.38+4.58×3.01−2.742×2+15.27+2.97×3.53−2.742×2+3.20×2.74−0.052=37.43+34.25+23.16=94.84cm2.


### 5.2. Single-Span Simply Supported Formwork

#### 5.2.1. Calculation of Deflection

According to structural mechanics, the formula for calculating the maximum deflection of simply supported beams is as follows:(8)fmax=5384×ql4EI.The actual deflection is larger than the calculated deflection value because the composite formwork is incompletely integrated. To make the calculated deflection value reflect the actual deformation, the stiffness is suggested to be multiplied by the reduction factorη1 [20], and the maximum deflection value is revised as follows:(9)fmax=5384×ql4bEI×1η,where b is the width of the template. The maximum deflection of the template is as follows:(10)fmax1=5384×ql4bEI×1η=5384×0.015×9004×600−8−80.7×105×94.84×104×1η=1.1273ηmm.The abovementioned calculation assumes that the formwork is simplified as a beam. In comparison with the frame, the template has a relative deflection in a transverse direction:(11)fmax2=5384×ql4bEI=5384×0.015×200−404×5003.84×103×500×103/12=0.4mm.Thus, the total maximum deflection of formwork is as follows:(12)fmax1+fmax2=1.1273η+0.4mm.


#### 5.2.2. Stiffness Reduction Factor

The FE analysis results of the refined model are substituted into the total maximum deflection calculation formula as follows:(13)1.1273η1+0.4=2.014.The equation is solved, and the stiffness reduction factorη1=0.70 is obtained.


### 5.3. Three-Span Continuous Formwork

#### 5.3.1. Calculation of the Deflection

According to structural mechanics, the formula for calculating the maximum deflection of the three-span continuous beam is as follows:(14)fmax=6.77×10−3×ql4EI.As mentioned in Section 5.2, the stiffness must be revised by the reduction factorη2:(15)fmax=6.77×10−3×ql4EI×1η.The maximum deflection of the formwork is as follows:(16)fmax1=6.77×10−3×ql4bEI×1η=6.77×10−3×0.03×7504×5840.7×105×94.84×104×1η=0.5653ηmm.In the transverse direction, the formwork has a relative deflection. The difference between the transverse spans of the three continuous formworks is within 5%. Therefore, the formwork is simplified as an equal-span structure:(17)fmax2=6.77×10−3×ql4bEI=6.77×10−3×0.03×1604×7503.84×103×750×103/12=0.41mm.Thus, the total maximum deflection of formwork is as follows:(18)fmax1+fmax2=0.5653η+0.41mm.The FE analysis results are substituted to the total maximum deflection calculation formula as follows:(19)0.5653η2+0.41=1.277.The stiffness reduction factorη2=0.65 can be obtained.


### 5.4. Determination of a Unified Stiffness Reduction Factor

Tables8 and 9 summarize the comparisons of the different stiffness reduction factors (0.65 to 0.70 according to Sections 5.2.2 and 5.3.1) for the single-span simply supported and three-span continuous formworks.Table 8
Comparison of the deflection of different stiffness reduction factors for the single-span simply supported formwork.
Load (kN/m2)1.55852.33853.50705.26057.891511.836515.0000Analysis value of the refined model (mm)0.20760.31090.46610.69891.04801.57402.0140η=0.65Simplified calculation value (mm)0.22170.33270.49900.74851.12281.68412.1342Error6.81%7.02%7.06%7.09%7.14%7.00%5.97%η=0.66Simplified calculation value (mm)0.21900.32860.49280.73931.10901.66342.1080Error5.50%5.70%5.74%5.78%5.82%5.68%4.67%η=0.67Simplified calculation value (mm)0.21640.32470.48690.73031.09561.64332.0825Error4.22%4.43%4.46%4.50%4.54%4.40%3.40%η=0.68Simplified calculation value (mm)0.21380.32080.48110.72161.08261.62382.0577Error2.99%3.18%3.22%3.25%3.30%3.16%2.17%η=0.69Simplified calculation value (mm)0.21130.31710.47550.71321.06991.60482.0337Error1.78%1.98%2.01%2.05%2.09%1.96%0.98%η=0.7Simplified calculation value (mm)0.20890.31340.47000.70501.05771.58642.0104Error0.62%0.81%0.84%0.88%0.92%0.79%0.18%Table 9
Comparison of the deflections of different stiffness reduction factors for the three-span continuous formwork.
Load (kN/m2)3.00006.000010.500017.250027.375030.0000Analysis value of the refined model (mm)0.13490.26100.45210.73731.16501.2770η=0.65Simplified calculation value (mm)0.12860.25710.45000.73921.17311.2856Error4.70%1.48%0.47%0.26%0.70%0.68%η=0.66Simplified calculation value (mm)0.12720.25450.44540.73171.16111.2725Error5.67%2.49%1.49%0.76%0.33%0.36%η=0.67Simplified calculation value (mm)0.12600.25190.44090.72431.14951.2597Error6.62%3.47%2.48%1.76%1.33%1.36%η=0.68Simplified calculation value (mm)0.12470.24950.43650.71721.13811.2473Error7.54%4.42%3.44%2.73%2.31%2.33%η=0.69Simplified calculation value (mm)0.12350.24700.43230.71031.12711.2352Error8.43%5.35%4.37%3.67%3.25%3.27%η=0.7Simplified calculation value (mm)0.12240.24470.42820.70351.11651.2235Error9.30%6.24%5.28%4.58%4.17%4.19%Whenη=0.67, nearly all errors are less than 5%. Therefore, the unified stiffness reduction factor for the two types of the formworks is 0.67.


### 5.5. Practical Simplified Calculation Formula

The practical simplified calculation formulas of the single-span simple supported and three-span continuous formworks are expressed as follows:(20)fmax=5384×ql14b1EfIe×10.67+5384×ql24b2EwIw,fmax=6.77×10−3×ql14b1EfIe×10.67+6.77×10−3×ql24b2EwIw,where q is the load, l1 is the longitudinal span of the frame, l2 is the transverse length of the frame under loading, b1 is the transverse span of the formwork, b2 is the longitudinal length of the formwork under loading, Ef is the elastic module of the frame, Ew is the elastic module of the formwork, Ie is the conversion moment of the inertia of the section, and Iw is the conversion moment of the inertia of the formwork.


## 5.1. Section Conversion and Cross-Sectional Geometry

### 5.1.1. Section Conversion

This new composite wood-plastic formwork with an aluminum alloy frame is assembled using different materials. Two kinds of materials must be converted into the same material through formula deduction. The equivalent principles of section conversion are presented as follows:(1)
The bending stiffness is consistent before and after the section conversion(2)
The deformation of the template on the contact surface is consistent with that of the frameFigure30 illustrates the composite cross section of the two materials. The lower part is the aluminum alloy frame, with the whole section area Af and the elastic modulus of Ef. The upper part is the wood-plastic template, with the section area of Aw and elastic modulus Ew. Figure 31 depicts the new cross section after section conversion, in which the wood-plastic template is replaced by an aluminum alloy template with the cross-sectional area Ae. The detailed deduction process of the section conversion is demonstrated as follows:where σw is the stress in the wood-plastic template, σe is the stress in the equivalent template, and ε is the strain in the template.Figure 30
Actual cross section of the formwork.Figure 31
Equivalent cross section of the formwork.(1)Aw⋅σw=Ae⋅σe,Ae=σwσe⋅Aw,∵σe=Ef⋅ε,σw=Ew⋅ε,Ae=EwEf⋅Aw,If(2)EfEw=n,then(3)Ae=1n⋅Aw.The sectional height remains unchanged, and only the width is changed, that is,(4)be=1n⋅bw.The physical meaning of equation (4) is that the width bw of the wood-plastic template is divided by the ratio n of the elastic modulus of the aluminum alloy and the wood-plastic material. Therefore, the equivalent width be and area Ae of the template are expressed as follows:(5)n=EfEw=0.7×1053.84×103=18.23,be=bwn=58.418.23=3.2cm,Ae=3.2×1=3.2cm2.


### 5.1.2. Calculation of the Cross-Sectional Geometric Properties

The cross-sectional geometries of the longitudinal beams of the frame are presented in Table7.Table 7
Cross-sectional geometry of the longitudinal beams of the frame.
ShapeParameterArea (cm2)Distance between top and centroid (cm)Moment of inertia (cm4)4.583.0118.382.972.5315.27The distance from the neutral axis to the top surface of the equivalent section is as follows:(6)y0=4.58×3.01×2+2.97×3.53×2+3.20×0.54.58×2+2.97×2+3.20=2.74cm.The conversion moment of the inertia of the equivalent section is as follows:(7)Ie=18.38+4.58×3.01−2.742×2+15.27+2.97×3.53−2.742×2+3.20×2.74−0.052=37.43+34.25+23.16=94.84cm2.


## 5.1.1. Section Conversion

This new composite wood-plastic formwork with an aluminum alloy frame is assembled using different materials. Two kinds of materials must be converted into the same material through formula deduction. The equivalent principles of section conversion are presented as follows:(1)
The bending stiffness is consistent before and after the section conversion(2)
The deformation of the template on the contact surface is consistent with that of the frameFigure30 illustrates the composite cross section of the two materials. The lower part is the aluminum alloy frame, with the whole section area Af and the elastic modulus of Ef. The upper part is the wood-plastic template, with the section area of Aw and elastic modulus Ew. Figure 31 depicts the new cross section after section conversion, in which the wood-plastic template is replaced by an aluminum alloy template with the cross-sectional area Ae. The detailed deduction process of the section conversion is demonstrated as follows:where σw is the stress in the wood-plastic template, σe is the stress in the equivalent template, and ε is the strain in the template.Figure 30
Actual cross section of the formwork.Figure 31
Equivalent cross section of the formwork.(1)Aw⋅σw=Ae⋅σe,Ae=σwσe⋅Aw,∵σe=Ef⋅ε,σw=Ew⋅ε,Ae=EwEf⋅Aw,If(2)EfEw=n,then(3)Ae=1n⋅Aw.The sectional height remains unchanged, and only the width is changed, that is,(4)be=1n⋅bw.The physical meaning of equation (4) is that the width bw of the wood-plastic template is divided by the ratio n of the elastic modulus of the aluminum alloy and the wood-plastic material. Therefore, the equivalent width be and area Ae of the template are expressed as follows:(5)n=EfEw=0.7×1053.84×103=18.23,be=bwn=58.418.23=3.2cm,Ae=3.2×1=3.2cm2.


## 5.1.2. Calculation of the Cross-Sectional Geometric Properties

The cross-sectional geometries of the longitudinal beams of the frame are presented in Table7.Table 7
Cross-sectional geometry of the longitudinal beams of the frame.
ShapeParameterArea (cm2)Distance between top and centroid (cm)Moment of inertia (cm4)4.583.0118.382.972.5315.27The distance from the neutral axis to the top surface of the equivalent section is as follows:(6)y0=4.58×3.01×2+2.97×3.53×2+3.20×0.54.58×2+2.97×2+3.20=2.74cm.The conversion moment of the inertia of the equivalent section is as follows:(7)Ie=18.38+4.58×3.01−2.742×2+15.27+2.97×3.53−2.742×2+3.20×2.74−0.052=37.43+34.25+23.16=94.84cm2.


## 5.2. Single-Span Simply Supported Formwork

### 5.2.1. Calculation of Deflection

According to structural mechanics, the formula for calculating the maximum deflection of simply supported beams is as follows:(8)fmax=5384×ql4EI.The actual deflection is larger than the calculated deflection value because the composite formwork is incompletely integrated. To make the calculated deflection value reflect the actual deformation, the stiffness is suggested to be multiplied by the reduction factorη1 [20], and the maximum deflection value is revised as follows:(9)fmax=5384×ql4bEI×1η,where b is the width of the template. The maximum deflection of the template is as follows:(10)fmax1=5384×ql4bEI×1η=5384×0.015×9004×600−8−80.7×105×94.84×104×1η=1.1273ηmm.The abovementioned calculation assumes that the formwork is simplified as a beam. In comparison with the frame, the template has a relative deflection in a transverse direction:(11)fmax2=5384×ql4bEI=5384×0.015×200−404×5003.84×103×500×103/12=0.4mm.Thus, the total maximum deflection of formwork is as follows:(12)fmax1+fmax2=1.1273η+0.4mm.


### 5.2.2. Stiffness Reduction Factor

The FE analysis results of the refined model are substituted into the total maximum deflection calculation formula as follows:(13)1.1273η1+0.4=2.014.The equation is solved, and the stiffness reduction factorη1=0.70 is obtained.


## 5.2.1. Calculation of Deflection

According to structural mechanics, the formula for calculating the maximum deflection of simply supported beams is as follows:(8)fmax=5384×ql4EI.The actual deflection is larger than the calculated deflection value because the composite formwork is incompletely integrated. To make the calculated deflection value reflect the actual deformation, the stiffness is suggested to be multiplied by the reduction factorη1 [20], and the maximum deflection value is revised as follows:(9)fmax=5384×ql4bEI×1η,where b is the width of the template. The maximum deflection of the template is as follows:(10)fmax1=5384×ql4bEI×1η=5384×0.015×9004×600−8−80.7×105×94.84×104×1η=1.1273ηmm.The abovementioned calculation assumes that the formwork is simplified as a beam. In comparison with the frame, the template has a relative deflection in a transverse direction:(11)fmax2=5384×ql4bEI=5384×0.015×200−404×5003.84×103×500×103/12=0.4mm.Thus, the total maximum deflection of formwork is as follows:(12)fmax1+fmax2=1.1273η+0.4mm.


## 5.2.2. Stiffness Reduction Factor

The FE analysis results of the refined model are substituted into the total maximum deflection calculation formula as follows:(13)1.1273η1+0.4=2.014.The equation is solved, and the stiffness reduction factorη1=0.70 is obtained.


## 5.3. Three-Span Continuous Formwork

### 5.3.1. Calculation of the Deflection

According to structural mechanics, the formula for calculating the maximum deflection of the three-span continuous beam is as follows:(14)fmax=6.77×10−3×ql4EI.As mentioned in Section 5.2, the stiffness must be revised by the reduction factorη2:(15)fmax=6.77×10−3×ql4EI×1η.The maximum deflection of the formwork is as follows:(16)fmax1=6.77×10−3×ql4bEI×1η=6.77×10−3×0.03×7504×5840.7×105×94.84×104×1η=0.5653ηmm.In the transverse direction, the formwork has a relative deflection. The difference between the transverse spans of the three continuous formworks is within 5%. Therefore, the formwork is simplified as an equal-span structure:(17)fmax2=6.77×10−3×ql4bEI=6.77×10−3×0.03×1604×7503.84×103×750×103/12=0.41mm.Thus, the total maximum deflection of formwork is as follows:(18)fmax1+fmax2=0.5653η+0.41mm.The FE analysis results are substituted to the total maximum deflection calculation formula as follows:(19)0.5653η2+0.41=1.277.The stiffness reduction factorη2=0.65 can be obtained.


## 5.3.1. Calculation of the Deflection

According to structural mechanics, the formula for calculating the maximum deflection of the three-span continuous beam is as follows:(14)fmax=6.77×10−3×ql4EI.As mentioned in Section 5.2, the stiffness must be revised by the reduction factorη2:(15)fmax=6.77×10−3×ql4EI×1η.The maximum deflection of the formwork is as follows:(16)fmax1=6.77×10−3×ql4bEI×1η=6.77×10−3×0.03×7504×5840.7×105×94.84×104×1η=0.5653ηmm.In the transverse direction, the formwork has a relative deflection. The difference between the transverse spans of the three continuous formworks is within 5%. Therefore, the formwork is simplified as an equal-span structure:(17)fmax2=6.77×10−3×ql4bEI=6.77×10−3×0.03×1604×7503.84×103×750×103/12=0.41mm.Thus, the total maximum deflection of formwork is as follows:(18)fmax1+fmax2=0.5653η+0.41mm.The FE analysis results are substituted to the total maximum deflection calculation formula as follows:(19)0.5653η2+0.41=1.277.The stiffness reduction factorη2=0.65 can be obtained.


## 5.4. Determination of a Unified Stiffness Reduction Factor

Tables8 and 9 summarize the comparisons of the different stiffness reduction factors (0.65 to 0.70 according to Sections 5.2.2 and 5.3.1) for the single-span simply supported and three-span continuous formworks.Table 8
Comparison of the deflection of different stiffness reduction factors for the single-span simply supported formwork.
Load (kN/m2)1.55852.33853.50705.26057.891511.836515.0000Analysis value of the refined model (mm)0.20760.31090.46610.69891.04801.57402.0140η=0.65Simplified calculation value (mm)0.22170.33270.49900.74851.12281.68412.1342Error6.81%7.02%7.06%7.09%7.14%7.00%5.97%η=0.66Simplified calculation value (mm)0.21900.32860.49280.73931.10901.66342.1080Error5.50%5.70%5.74%5.78%5.82%5.68%4.67%η=0.67Simplified calculation value (mm)0.21640.32470.48690.73031.09561.64332.0825Error4.22%4.43%4.46%4.50%4.54%4.40%3.40%η=0.68Simplified calculation value (mm)0.21380.32080.48110.72161.08261.62382.0577Error2.99%3.18%3.22%3.25%3.30%3.16%2.17%η=0.69Simplified calculation value (mm)0.21130.31710.47550.71321.06991.60482.0337Error1.78%1.98%2.01%2.05%2.09%1.96%0.98%η=0.7Simplified calculation value (mm)0.20890.31340.47000.70501.05771.58642.0104Error0.62%0.81%0.84%0.88%0.92%0.79%0.18%Table 9
Comparison of the deflections of different stiffness reduction factors for the three-span continuous formwork.
Load (kN/m2)3.00006.000010.500017.250027.375030.0000Analysis value of the refined model (mm)0.13490.26100.45210.73731.16501.2770η=0.65Simplified calculation value (mm)0.12860.25710.45000.73921.17311.2856Error4.70%1.48%0.47%0.26%0.70%0.68%η=0.66Simplified calculation value (mm)0.12720.25450.44540.73171.16111.2725Error5.67%2.49%1.49%0.76%0.33%0.36%η=0.67Simplified calculation value (mm)0.12600.25190.44090.72431.14951.2597Error6.62%3.47%2.48%1.76%1.33%1.36%η=0.68Simplified calculation value (mm)0.12470.24950.43650.71721.13811.2473Error7.54%4.42%3.44%2.73%2.31%2.33%η=0.69Simplified calculation value (mm)0.12350.24700.43230.71031.12711.2352Error8.43%5.35%4.37%3.67%3.25%3.27%η=0.7Simplified calculation value (mm)0.12240.24470.42820.70351.11651.2235Error9.30%6.24%5.28%4.58%4.17%4.19%Whenη=0.67, nearly all errors are less than 5%. Therefore, the unified stiffness reduction factor for the two types of the formworks is 0.67.


## 5.5. Practical Simplified Calculation Formula

The practical simplified calculation formulas of the single-span simple supported and three-span continuous formworks are expressed as follows:(20)fmax=5384×ql14b1EfIe×10.67+5384×ql24b2EwIw,fmax=6.77×10−3×ql14b1EfIe×10.67+6.77×10−3×ql24b2EwIw,where q is the load, l1 is the longitudinal span of the frame, l2 is the transverse length of the frame under loading, b1 is the transverse span of the formwork, b2 is the longitudinal length of the formwork under loading, Ef is the elastic module of the frame, Ew is the elastic module of the formwork, Ie is the conversion moment of the inertia of the section, and Iw is the conversion moment of the inertia of the formwork.


## 6. Conclusion

(1)
During the loading process of the experiments, the deflections and stresses of the formworks increase linearly, and the test deflections and stresses are within the allowable values under the maximum experimental loads. Therefore, the new composite wood-plastic formworks perform well and can satisfy the structural bearing requirements.(2)
Numerical analyses confirm that the experimental considerations are accurate. The calculated values of the deflection and stress of the two FE models (refined and general) agree with the experimental results. Moreover, the general model can replace the refined model to simplify the problem when the accuracy requirement is not too strict.(3)
On the basis of the theoretical analysis and experimental research, the practical simplified formulas are proposed to facilitate the rapid calculation and evaluation for engineers considering the transverse deflection and inconsistency of two materials.



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*Source: 8831999-2020-12-07.xml*