Improved step-by-step modeling method for analyzing the mechanical behavior of steel structures during construction

Abstract

The step-by-step modeling method considering nonlinear effects is an effective method for analyzing the mechanical behavior of steel structures during construction. However, two problems have limited the widespread application of the traditional step-by-step modeling method: the positioning of newly assembled members and the transformation of the stiffness matrix and the nodal load and displacement vectors between different stages of construction. In this article, a new concept of “two moments of a construction stage” (the initial and final moments of each construction stage) is proposed to improve the step-by-step modeling method. Based on this concept, an improved step-by-step modeling method, which considers the modified design configuration positioning principle for newly assembled members and provides a method for modifying the structural stiffness matrix at the initial moment of any construction stage, is proposed. A calculation program block based on the proposed method is compiled to analyze the mechanical behavior of a plane frame and a plane shallow arch during construction. The mechanical behavior of an engineering application, the connective corridor of Shanghai International Financial Centre, is also analyzed using the proposed method, and the calculated results are compared with the monitoring results. The numerical results show that the modified design configuration positioning principle is applicable for the positioning of newly assembled members and that the modification method for the structural stiffness matrix and the nodal load and displacement vectors at the initial moment of a construction stage efficiently solves the problem of their transformation between different stages of construction. The improved step-by-step modeling method proposed in this article is more valid and accurate than other methods for analyzing the mechanical behavior of steel structures during construction.

Keywords

improved step-by-step modeling method modification of the structural stiffness matrix modified design configuration positioning principle transformation matrix two moments of a construction stage

Introduction

Complicated large-span structures are widely used in modern buildings. During construction, the geometry, structural system, and boundary conditions of a structure may change substantially, and these changes significantly influence the distribution of the member stresses and the deformations of the structure during construction (Choi et al., 1992; Cruz et al., 1998). Therefore, it is essential to analyze the mechanical behavior of a structure to ensure its safety during construction (Epaarachchi et al., 2002). Scholars and practitioners have drawn attention to such analysis in recent years, and the mechanical behavior of many structures during construction has been assessed, such as the Shanghai Center Tower (Lu et al., 2013), Shenzhen Universiade Sports Centre (Zhou et al., 2012), and Jinan Olympic Sports Centre (Li et al., 2012). However, no ideal general method has been developed for the construction analysis of steel structures.

Time-varying mechanics serves the basis of any mechanical behavior analysis of a structure during construction (Wang, 2000). Over several decades, a large number of scholars have made significant progress in construction mechanics. Many of their studies were based on the general finite element method (FEM). Early studies focused on the construction analysis of concrete structures(Calderón et al., 2011; Choi et al., 1985, 1992; Cruz et al., 1998; Kwak and Kim, 2006; Li, 1994; Mar, 2000); for instance, Choi and Kim (1985) proposed sub-structuring techniques for analyzing multistory frames during construction and presented the concepts of “active”, “inactive” and “deactivated” floors for construction analysis. Choi et al. (1992) presented a simplified building analysis method, named the correction factor method, for multistory frame analysis during construction. Mar (2000) presented a numerical model for the nonlinear and time-dependent analysis of three-dimensional reinforced pre-stressed and composite concrete frames. These studies on the construction analysis of concrete structures mainly focused on time-dependent effects, such as load histories, material nonlinear behavior, and concrete creep and shrinkage. Moreover, the construction mechanical analysis of steel and concrete composite structures has also been considered (Lozano-Galant et al., 2012a, 2012b; Lozano-Galant and Turmo, 2014; Lu et al., 2013). Lozano-Galant et al. (2012) presented the Forward Algorithm, the Forward-Direct Algorithm (Lozano-Galant and Turmo, 2014), and the Backward Algorithm (Lozano-Galant et al., 2012) for simulating the construction of cable-stayed bridges built on temporary supports. Wu et al. (2015) presented a master–slave constraint method to account for changes in the cross-sectional properties of composite members during construction and developed an efficient computational tool to simulate the construction of bridges considering concrete creep and shrinkage. Scholars have also made progress in analyzing the construction mechanics of steel structures (Chen et al., 2015; Tian and Hao, 2015; Tian et al., 2016; Zhao et al., 2016). Tian et al. (2016) proposed a novel asynchronous analysis of the synchronous and asynchronous integral lifting of the steel roof of a China Eastern Airlines hangar. Zhao et al. (2016) proposed a method of incorporating the deformation and residual stress that occur during the erection phase to simulate the influence of the erection process on the buckling behavior of large-span structures. Zhou et al. (2012) proposed a time-dependent analysis based on a matrix mechanics model that considers the temperature variation during construction to optimize the closure scheme and presented a combined brace element to simulate the mechanics of temporary supports. In addition, to develop a general FEM, scholars have presented novel FE methods for time-varying mechanics. For example, Cao et al. (2010) presented a topological FE method for time-varying structures during construction that applies topological theory to FEM analysis. Zhu et al. (2016) applied the vector-form intrinsic FE to construction simulations and proposed a traction cable element for analyzing the lifting process. Unfortunately, these novel FE methods have not been widely used for actual engineering applications.

An overall construction procedure is generally divided into several stages in accordance with the construction scheme, and an FE model can be established and calculated for each stage of construction. Based on the general FEM, there are three main methods for analyzing the mechanical behavior of a structure during construction: the mode variable superimposed (MVS) method (Zhuo, 2001), the element birth and death (EBD) method (ANSYS, Inc., 2004), and the step-by-step modeling (SSM) method (Liu and Guo, 2008). In the MVS method (Zhuo, 2001), which is based on linear elastic theory, an FE model is established for each construction stage using the current load according to the designed configuration. The results for the inner forces and displacements in the current stage are obtained by accumulating the results for the current and all previous stages. In the EBD method (ANSYS, Inc., 2004; Liu and Guo, 2008), the addition and removal of structural members during construction are simulated by “activating” and “killing” elements, respectively, and the structural stiffness matrix is modified during construction. In the SSM method (Liu and Guo, 2008), according to the construction process, a FE model is established for each construction stage through the sequential addition or removal of members, and the structural stiffness matrix is gradually formulated. In this case, the modeling and solution processes are alternated. In the latter two methods, the nonlinear effects of a structure can be considered.

On one hand, because the nonlinear effects on complicated large-span structures during construction cannot be ignored in most cases, the MVS method is not generally used to analyze the mechanical behavior of such structures during construction. On the other hand, in the EBD method, mutation of the structural stiffness and “floating” displacements of the dead elements tend to occur during the calculation. Therefore, the calculation does not converge, and the results for the inner forces and displacements are inaccurate. Several approaches have been proposed to solve these problems in the EBD method; for instance, Tian and Hao (2015) used a positioning principle based on node rectification for installation to improve the convergence of the calculation, although this method could not eliminate the “floating” displacements of the dead elements.

Modern monitoring techniques in construction, which require that the construction mechanical analysis can be interrupted to update the model of the structure, are widely used in the servo control of construction processes (Chacón and Zorrilla, 2015; Luo et al., 2015; Sousa et al., 2013). For this reason, the EBD method cannot be used in the servo control of construction processes because of its one-time continuous calculation. Therefore, the SSM method is more suitable for construction mechanical analysis. However, two problems are encountered with the SSM method, which limit its widespread application: the positioning of newly assembled members and the transformation of the stiffness matrix and the nodal load and displacement vectors between different stages of construction. Unfortunately, limited attention has been given to these problems in the current literature (Chen et al., 2015).

In this article, the aforementioned problems are investigated to improve the traditional SSM method. First, a new concept of “two moments of a construction stage” is proposed as a basis for defining a construction process. Second, a principle for the positioning of newly assembled members and a method for modifying the structural stiffness matrix and the nodal load and displacement vectors in a construction stage are proposed. Then, an improved step-by-step modeling (ISSM) method for analyzing the mechanical behavior of steel structures during construction is formally presented. To verify the efficiency of the proposed method, a calculation program block implemented in MATLAB based on the proposed method is adopted to analyze the mechanical behavior of a plane frame and a plane arch throughout the entire construction process. Finally, the proposed method is used for analyzing the mechanical behavior of the connective corridor in Shanghai International Financial Centre during construction, and the calculated results are compared with the monitoring results. The ISSM method proposed in this article is an efficient and reliable means of analyzing the mechanical behavior of steel structures during construction.

In this article, the derivation of the ISSM method is demonstrated by planar structure models, that is, each node in an element has two translational degrees of freedom (DOFs) and one rotational DOF. The universal derivation and demonstration of the ISSM method for spatial structural models will be addressed in future reports.

The SSM method considering geometrical nonlinearity

Based on the construction scheme, a construction procedure is generally divided into N construction stages. Accordingly, a time series of the form $0 = t_{0} < t_{1} < t_{2} < \dots < t_{i} < t_{i} < \dots < t_{N} = T$ can be used to represent the various moments of time corresponding to the different stages of construction, where t₀ = 0 represents the initial moment, t_i represents the final moment of the ith construction stage, and t_N = T represents the moment of completion of the entire construction project.

To enable clearer description of the SSM method, this article proposes a novel concept of “two moments of a construction stage”, in which the two moments are the initial and final moments of each construction stage, to define each construction stage. As shown in Figure 1, t_i_–1 represents the initial moment of the ith construction stage and t_i represents the final moment of the ith stage. At the final moment of the i − 1th stage, it is assumed that the i − 1th construction stage has been completed and that the structure tangent stiffness matrix $K'_{i - 1}$ , the nodal load vector P _i–1, and the nodal displacement vector u _i–1 have been obtained. At the initial moment of the ith stage, new members are positioned and assembled; simultaneously, the structural stiffness matrix is modified to the structural initial stiffness matrix $K_{i}$ at this moment, which includes the stiffness contributions of both the existing and the newly assembled members. Therefore, the initial moment of the ith construction stage is different from the final moment of the i − 1th stage.

Figure 1.

Two moments of a construction stage (CS—construction stage).

Because a steel structure is usually in the elastic state during construction, material nonlinearity is generally not considered in the simulation analysis of steel structures during construction, and only geometric nonlinearity is considered. Geometric nonlinear problems are usually solved using incremental theory (Belytschko et al., 2000). In this method, the application of the load Δ F _i that is applied to the structure in the ith construction stage can be divided into several steps. For the jth load step, the load increment $Δ F_{i}^{j}$ and the corresponding displacement increment $Δ u_{i}^{j}$ are obtained. Therefore, the structural incremental equilibrium equations in this stage can be expressed as

{}^{t}{K_{i}^{j}} Δ u_{i}^{j} = Δ F_{i}^{j}

(1)

{}^{t}{K_{i}^{j}} = {}^{0}{K_{i}^{j}} + {}^{L}{K_{i}^{j}} + {}^{σ}{K_{i}^{j}}

(2)

where ${}^{t}{K_{i}^{j}}$ represents the tangent stiffness matrix in the jth load step of the ith construction stage, which consists of the elastic stiffness matrix ${}^{0}{K_{i}^{j}}$ , the displacement stiffness matrix ${}^{L}{K_{i}^{j}}$ , and the stress stiffness matrix ${}^{σ}{K_{i}^{j}}$ .

By adopting the Newton–Raphson iteration algorithm (Belytschko et al., 2000) to solve equation (1), the displacement increment Δ u _i and the stress increment Δ σ _i at the final moment of the ith stage can be obtained. Finally, the displacement vector u _i and the stress vector σ _i at the final moment of the ith stage can be obtained from equations (3) and (4)

u_{i} = u_{i - 1} + Δ u_{i}

(3)

σ_{i} = σ_{i - 1} + Δ σ_{i}

(4)

Furthermore, the structural tangent stiffness matrix is obtained and stored to be modified into the structural initial stiffness matrix in the i + 1th construction stage.

According to the predetermined construction sequence, the mechanical behavior analysis of a steel structure in each construction stage can be conducted following the processes described above. Eventually, the mechanical behavior analysis of the steel structure during construction is completed.

In the SSM method, which considers the geometric nonlinear effects of a structure, two key problems in the initial moment of a construction stage have not been solved. The first problem is the positioning of newly assembled members, which is related to the structural elastic stiffness matrix at the initial moment of a construction stage. When the positioning principle is unreasonable, the structural configuration obtained through simulation does not correspond to the actual configuration, and the results of the mechanical analysis are consequently inaccurate. The second problem is the transformation of the stiffness matrices and the nodal load and displacement vectors between different stages of construction. Because the FE model for a given construction stage includes both the existing members assembled in previous stages and the new members added in the current stage, the number of nodes and elements in the model at this stage differ from those in the previous stage. Therefore, the orders of the structural stiffness matrix and the nodal load and displacement vectors are also different from those in the previous stage. In the traditional SSM method, the structural initial stiffness matrix of the ith construction stage must be completely reconstructed based on the information on the existing and newly assembled members. Consequently, the SSM method is somewhat time-consuming and requires manual intervention. If the structural initial stiffness matrix of the ith construction stage could instead be obtained by modifying the structural stiffness matrix of the i − 1th stage considering the effects of the newly assembled members via matrix update, the efficiency of the SSM could be improved. Thus, it is essential to improve the SSM method in this way to facilitate its widespread use.

Improvement of the SSM method

The two aforementioned problems (the positioning of newly assembled members and the modification of the structural stiffness matrices and the nodal load and displacement vectors at the initial moment of a construction stage) are solved in this section to improve the SSM method.

Modification of the design configuration positioning principle

There are three main principles for the configuration positioning of a structure during construction: the design configuration positioning (DCP) principle, the “floating” configuration positioning (FCP) principle, and the tangent positioning (TP) principle (Zhang et al., 2004).

The DCP principle, in which all nodal coordinates of the structure in each construction stage are determined by the design configuration, as illustrated in Figure 2(a), is adopted in the MVS method. Obviously, when this principle is applied, the influence of the deformations of existing members on both the positions and the stress states of the newly assembled members cannot be considered, which leads to substantial errors due to large deformations.

Figure 2.

Configuration positioning principles: (a) design configuration positioning principle, (b) tangent positioning principle (Zhang et al., 2004): (b1) Case 1, (b2) Case 2, (b3) Case 3, and (c) modified design configuration positioning principle: (c1) Cases 1 and 2, (c2) Case 3.

The FCP principle is a positioning principle that is specific to the EBD method, in which the coordinates of newly added nodes are determined using the unstable “floating” positions of nodes on “dead” elements, which are obtained through solving the equilibrium equation. In the EBD method, the unstable “floating” displacements of the “dead” elements make the newly assembled members deviate from their actual positions. Consequently, the EBD method leads to inevitable deviations between its computed results and the actual structural status. Moreover, due to the “floating” displacements in the EBD method, the structural stiffness matrix may become singular, and the calculation may not converge.

In the TP principle (Zhang et al., 2004), the coordinates of newly added nodes in the structure are determined based on the tangential directions of existing members. Three detailed examples are illustrated in Figure 2(b). At present, the TP principle is mainly applied in bridge engineering.

However, for complicated large-span structures, the structural configuration is generally complex. Moreover, the deformations of existing members influence the positioning of newly assembled members; thus, the coordinates determined by the aforementioned principles are unreasonable. For this reason, the calculated mechanical behavior of the structure may be considerably different from the actual behavior; therefore, it is necessary to explore new positioning principles for complicated large-span structures.

Based on the advantages and disadvantages of the above principles and the considerations related to newly assembled members in practical engineering, this section introduces a new positioning principle, named the modified design configuration positioning (MDCP) principle, which accounts for the deformations of existing members. In this principle, the coordinates of newly added nodes are determined based on the design configuration at the initial moment of a construction stage, whereas the coordinates of the existing nodes are determined based on the configuration of the existing members after deformation. Three cases arise for the positioning of newly assembled members (Figure 2(c), where I′, J′, K′, and L′ denote the deformed positions of nodes I, J, K, and L:

Both nodes of a newly assembled member have been newly added as in the case of member LM in Figure 2(c1).

One node of a newly assembled member has been newly added, whereas the other is already existing, as in the case of member K′L in Figure 2(c1).

Both nodes of a newly assembled member already exist as in the case of member J′K′ in Figure 2(c2).

The MDCP principle simultaneously considers the deformations of existing members assembled in previous construction stages and the DCP principle for newly assembled members and is consistent with common practices regarding newly assembled members in actual engineering. Therefore, this principle is reasonable for the positioning of newly assembled members in construction mechanical analysis.

Modification of the structural stiffness matrix at the initial moment of a construction stage

The initial structural stiffness matrix $K_{i}$ in the ith construction stage is modified at the initial moment of this stage, once the i − 1th (i ≥ 2) construction stage has been completed and the tangent stiffness matrix $K'_{i - 1}$ at the final moment of the i − 1th stage has been obtained. At that moment, because the structural stiffness includes both the structural stiffness at the final moment of the i − 1th stage and the stiffness of the newly assembled members, $K_{i}$ is modified as follows

K_{i} = A_{i}^{T} {K'}_{i - 1} A_{i} + A_{a}^{T} K_{a} A_{a}

(5)

where $K_{a}$ is the elastic stiffness matrix of the newly assembled members and $A_{i}$ and $A_{a}$ are transfer matrices used to expand ${K'}_{i - 1}$ and $K_{a}$ , respectively, so that they can be summed to obtain $K_{i}$ .

Equation (5) shows how the structural tangent stiffness matrix $K'_{i - 1}$ at the final moment of the i − 1th construction stage and the elastic stiffness matrix $K_{a}$ of the newly assembled members are expanded and combined to obtain the structural stiffness matrix $K_{i}$ at the initial moment of the ith construction stage.

For the case of a plane structure model, it is assumed that the number of existing nodes at the final moment of the i–1th stage is n and the total number of nodes in the ith stage is m, which includes the nodes on newly assembled members; and the number of coincident nodes between newly assembled members and existing members is l. Thus, the dimensions of the matrices in equation (5) are $(K_{i})_{3 m \times 3 m}$ , $({K'}_{i - 1})_{3 n \times 3 n}$ , $(K_{a})_{3 (m - n + l) \times 3 (m - n + l)}$ , $(A_{i})_{3 n \times 3 m}$ , and $(A_{a})_{3 (m - n + l) \times 3 m}$ . $(K_{i})_{3 m \times 3 m}$ and $(K_{a})_{3 (m - n + l) \times 3 (m - n + l)}$ are expressed as block matrices, and $(K_{i})_{3 m \times 3 m}$ has the form presented in equation (6)

(K_{i})_{3 m \times 3 m} = [\begin{matrix} E_{3 n \times 3 n} & F_{3 n \times 3 (m - n)} \\ F_{3 (m - n) \times 3 n}^{T} & G_{3 (m - n) \times 3 (m - n)} \end{matrix}]

(6)

where $E_{3 n \times 3 n}$ represents the stiffness contribution to $K_{i}$ from members between two of the n existing nodes from the i − 1th stage, $F_{3 n \times 3 (m - n)}$ represents the contribution to $K_{i}$ from members between one of the n existing nodes and one of the m − n newly added nodes, and $G_{3 (m - n) \times 3 (m - n)}$ represents the contribution to $K_{i}$ from members between two of the m − n newly added nodes.

$(K_{a})_{3 (m - n + l) \times 3 (m - n + l)}$ has the form presented in equation (7)

(K_{a})_{3 (m - n + l) \times 3 (m - n + l)} = [\begin{matrix} L_{3 l \times 3 l} & N_{3 l \times 3 (m - n)} \\ N_{3 (m - n) \times 3 l}^{T} & M_{3 (m - n) \times 3 (m - n)} \end{matrix}]

(7)

where for convenience of derivation, $L_{3 l \times 3 l}$ represents the contribution from members among the coincident nodes and is positioned in the upper left corner of $K_{a}$ ; $N_{3 l \times 3 (m - n)}$ represents the contribution to $K_{a}$ from members between a coincident node and a newly added node; and $M_{3 (m - n) \times 3 (m - n)}$ represents the contribution to $K_{a}$ from members between newly added nodes.

$A_{i}$ and $A_{a}$ have the forms as shown in equations (8) and (9)

(A_{i})_{3 n \times 3 m} = [\begin{matrix} I_{3 n \times 3 n} & 0_{3 n \times 3 (m - n)} \end{matrix}]

(8)

(A_{a})_{3 (m - n + l) \times 3 m} = [\begin{matrix} C_{3 l \times 3 l} & D_{3 l \times 3 (n - l)} & 0_{3 l \times 3 (m - n)} \\ 0_{3 (m - n) \times 3 l} & 0_{3 (m - n) \times 3 (n - l)} & I_{3 (m - n) \times 3 (m - n)} \end{matrix}]

(9)

where both $C_{3 l \times 3 l}$ and $D_{3 l \times 3 (n - l)}$ represent the position sub-matrices of the elements related to the coincident nodes in $(K_{i})_{3 m \times 3 m}$ , which can be obtained from the positions of the elements related to the coincident nodes in $({K'}_{i - 1})_{3 n \times 3 n}$ . Substituting equations (8) and (9) into equation (5) yields

A_{i}^{T} {K'}_{i - 1} A_{i} = [\begin{matrix} {({K'}_{i - 1})}_{3 n \times 3 n} & 0_{3 n \times 3 (m - n)} \\ 0_{3 (m - n) \times 3 n} & 0_{3 (m - n) \times 3 (m - n)} \end{matrix}]

(10)

A_{a}^{T} K_{a} A_{a} = [\begin{matrix} C^{T} LC & C^{T} LD & C^{T} NI \\ D^{T} LC & D^{T} LD & D^{T} NI \\ I^{T} N^{T} C & I^{T} N^{T} D & M \end{matrix}]

(11)

Combining the above equations with equation (6) yields

E_{3 n \times 3 n} = {({K'}_{i - 1})}_{3 n \times 3 n} + [\begin{matrix} C^{T} LC & C^{T} LD \\ D^{T} LC & D^{T} LD \end{matrix}]

(12)

F_{3 n \times 3 (m - n)} = [\begin{matrix} C^{T} NI \\ D^{T} NI \end{matrix}]

(13)

G_{3 (m - n) \times 3 (m - n)} = M_{3 (m - n) \times 3 (m - n)}

(14)

Corresponding to $K_{a}$ , the matrix elements related to l coincident nodes are positioned in the upper left corners of $(K_{i})_{3 m \times 3 m}$ and $({K'}_{i - 1})_{3 n \times 3 n}$ ; these matrix elements represent the stiffness contribution only from members between coincident nodes, without any mutual influence between coincident nodes and non-coincident nodes. In addition, $C^{T} LC$ is a $3 l \times 3 l$ submatrix located in the upper left corner of the matrix $A_{a}^{T} K_{a} A_{a}$ in equation (11), that is, the matrix elements related only to newly assembled members between coincident nodes, without any mutual influence between coincident nodes and non-coincident nodes, appear in $C^{T} LC$ . Thus, the other sub-matrices obtained by multiplying $K_{a}$ by the transfer matrix are 0. Therefore, according to the correspondence relationship of the matrix elements in $E_{3 n \times 3 n}$ , it can be found that $C = I_{3 l \times 3 l}$ and $D = 0$ . Therefore, equation (9) can be expressed as follows

{(A_{a})}_{3 (m - n + l) \times 3 m} = [\begin{matrix} I_{3 l \times 3 l} & 0_{3 l \times 3 (n - l)} & 0_{3 l \times 3 (m - n)} \\ 0_{3 (m - n) \times 3 l} & 0_{3 (m - n) \times 3 (n - l)} & I_{3 (m - n) \times 3 (m - n)} \end{matrix}]

(15)

Thus, the transfer matrices A_i and A_a in equation (5) have been obtained as expressed in equations (8) and (15), respectively. The matrix elements that are related only to coincident nodes in both the i − 1th and ith construction stages are all positioned in the upper left corners of the matrices, such that the contributions from members whose end nodes are both coincident nodes appear in $3 l \times 3 l$ sub-matrices in $(K_{i})_{3 m \times 3 m}$ , $({K'}_{i - 1})_{3 n \times 3 n}$ , and $K_{a}$ . Once $A_{i}$ and $A_{a}$ have been obtained, $K_{i}$ can be modified as expressed in equation (5). Consequently, the transformation of the stiffness matrix between different stages of construction can be achieved in the SSM method. For clarity, the expansion of the stiffness matrices is illustrated in Figure 3.

Figure 3.

Expansion of the structural stiffness matrices.

Modification of the nodal load and displacement vector at the initial moment of a construction stage

Similar to the modification of the structural stiffness matrix, the nodal load vector at the initial moment of the ith construction stage $F_{i}$ is obtained as follows

F_{i} = B_{i} F_{i - 1} + B_{a} F_{a}

(16)

where $F_{i - 1}$ is the nodal load vector in the i − 1th construction stage, $F_{a}$ is the load vector for the newly added nodes, and $B_{i}$ and $B_{a}$ are transfer matrices used to expand $F_{i - 1}$ and $F_{a}$ , respectively, so that they can be summed to obtain $F_{i}$ .

The derivation of the transfer matrices for the nodal load vectors is similar to that for the stiffness matrices. It is assumed that the number of nodes in the i − 1th construction stage is n; the number of nodes in the ith stage is m, which includes the newly added nodes; and the number of coincident nodes between the newly assembled members and the existing members is l. Therefore, the dimensions of the matrices and vectors in equation (11) are $(F_{i})_{3 m \times 1}$ , $(F_{i - 1})_{3 n \times 1}$ , $(F_{a})_{3 (m - n + l) \times 1}$ , $(B_{i})_{3 m \times 3 n}$ , and $(B_{a})_{3 m \times 3 (m - n + l)}$ . $(F_{i})_{3 m \times 1}$ and $(F_{a})_{3 (m - n + l) \times 1}$ are expressed as block matrices with the forms shown in equations (17) and (18)

{(F_{i})}_{3 m \times 1} = {\begin{matrix} R_{3 n \times 1} \\ S_{3 (m - n) \times 1} \end{matrix}}

(17)

{(F_{a})}_{3 (m - n + l) \times 1} = {\begin{matrix} P_{3 l \times 1} \\ Q_{3 (m - n) \times 1} \end{matrix}}

(18)

where $R_{3 n \times 1}$ and $S_{3 (m - n) \times 1}$ represent the nodal load subvectors in $F_{i}$ corresponding to the existing and newly added nodes, respectively, and $P_{3 l \times 1}$ and $Q_{3 (m - n) \times 1}$ similarly represent the nodal load subvectors in $F_{a}$ corresponding to the coincident and newly added nodes, respectively.

$(B_{i})_{3 m \times 3 n}$ and $(B_{a})_{3 m \times 3 (m - n + l)}$ can be expressed as block matrix with the forms shown in equations (19) and (20)

{(B_{i})}_{3 n \times 3 m} = [\begin{matrix} I_{3 n \times 3 n} & 0_{3 n \times 3 (m - n)} \end{matrix}]

(19)

{(B_{a})}_{3 m \times 3 (m - n + l)} = [\begin{matrix} {C'}_{3 l \times 3 l} & {D'}_{3 l \times 3 (m - n)} \\ 0_{3 (n - l) \times 3 l} & 0_{3 (n - l) \times 3 (m - n)} \\ 0_{3 (m - n) \times 3 l} & I_{3 (m - n) \times 3 (m - n)} \end{matrix}]

(20)

where $C'_{3 l \times 3 l}$ and $D'_{3 l \times 3 (m - n)}$ are position sub-matrices that represent the positions of the elements related to the coincident nodes in $(F_{i})_{3 m \times 1}$ . The above formulas yield

R_{3 n \times 1} = C' P + D' Q + F_{i - 1}

(21)

S_{3 (m - n) \times 1} = Q

(22)

Similar to the stiffness matrix modification method, the load vector elements related to the coincident nodes are positioned in the upper parts of $F_{i}$ , $F_{i - 1}$ and $F_{a}$ . Therefore, according to the principle of correspondence for vector elements, it can be found that $D'_{3 l \times 3 (m - n)} = 0$ and $C'_{3 l \times 3 l} = I_{3 l \times 3 l}$ , and thus, equation (20) can be expressed as follows

{(B_{a})}_{3 m \times 3 (m - n + l)} = [\begin{matrix} I_{3 l \times 3 l} & 0_{3 l \times 3 (m - n)} \\ 0_{3 (n - l) \times 3 l} & 0_{3 (n - l) \times 3 (m - n)} \\ 0_{3 (m - n) \times 3 l} & I_{3 (m - n) \times 3 (m - n)} \end{matrix}]

(23)

Because there is a one-to-one correspondence between the elements in the nodal displacement vector and those in the nodal load vector, the nodal displacement vector at the initial moment of the ith construction stage u _i can be similarly be obtained as expressed in equation (24)

u_{i} = B_{i} u_{i - 1} + B_{a} u_{a}

(24)

where $u_{i - 1}$ is the nodal displacement vector in the i − 1th construction stage and $u_{a}$ is the displacement vector for the newly added nodes. It is obvious that there is no displacement of the newly added nodes at the initial moment of the ith stage; therefore, $u_{a} = 0$ and equation (24) can be expressed as follows

u_{i} = B_{i} u_{i - 1}

(25)

where $B_{i}$ is given in equation (19).

Thus, the transfer matrices $B_{i}$ and $B_{a}$ have been obtained, and the nodal load vector $F_{i}$ and the nodal displacement vector u _i can be calculated by modifying the corresponding vectors for the previous construction stage as expressed in equations (16) and (25), respectively. For clarity, the expansion of the nodal load vectors is illustrated in Figure 4.

Figure 4.

Expansion of the nodal load vectors.

Analysis process using the ISSM method

As described above, the problems of the positioning of newly assembled members and the modification of the structural initial stiffness matrices and the nodal load and displacement vectors between different stages of construction have been solved. In other words, the improvement of the SSM method has been completed. The flowchart presented in Figure 5 illustrates the ISSM method.

Figure 5.

Flowchart of the ISSM method.

In this article, the application scope of the ISSM method includes all kinds of planar steel structures and any other steel structures that can be simplified as planar ones.

Illustrative examples

A calculation program block based on the ISSM method proposed in this article was compiled in MATLAB. To verify the validity and accuracy of the ISSM method, a plane frame and a plane shallow arch were used as numerical examples. The mechanical behavior of each of these structures throughout the entire construction process was analyzed, and in this section, the results obtained using the proposed method are compared with those obtained using other methods.

Plane frame

Description of the frame structure

The first structure considered is a single-bay plane frame with fixed column bases and three floors, where the first floor is 4.5 m in height and the other two floors are each 3 m in height, as shown in Figure 8. The cross section of every column and beam is H400 × 200 × 6 × 8. Q345 grade steel is adopted to account for an ideal elastoplastic model with elasticity modulus of $E = 2.06 \times 10^{11} N / m^{2}$ . A uniformly distributed load of $q = 120 kN / m$ is applied along the beam on each floor during the assembly of the corresponding members, and the structural weight is not considered.

A total of three construction stages are considered. The erection of each story of the frame is treated as one construction stage, as shown in Figure 6(b).

Figure 6.

(a) Frame model, (b) construction stages of the frame, (c) positions of newly assembled members: (c1) the first story, and (c2) the second story.

The FE model of the structure consists of 51 beam elements, and each node of each element has two translational DOFs and one rotational DOF. The numbers of the nodes and crucial members are presented in Figure 6(a).

Results and discussion of the construction mechanical analysis

The DCP, FCP, TP, and MDCP principles were each used to determine the positions of the newly assembled members, with the results shown in Figure 6(c). The MVS, EBD, and ISSM methods with both the TP and MDCP principles were used to analyze the mechanical behavior of the frame throughout the entire construction process. Figure 7(a) shows the variation in the displacements of Nodes 29 and 44 throughout the construction steps, and Figure 7(b) shows the variation in the inner forces of several crucial members with construction steps. Pictures of displacement results obtained using these methods are shown in Figure 12(a) of Appendix 2.

Figure 7.

(a) Variation in the vertical displacements with construction steps and (b) variation in the inner forces of crucial elements with construction steps.

Several conclusions can be drawn:

The inner forces obtained using the ISSM method in combination with the MDCP principle are consistent with those obtained using the EBD method, which indicates that the ISSM method proposed in this article is effective. Moreover, the inner forces and displacements obtained using the ISSM method in combination with the MDCP principle are nearly identical to those obtained using the MVS method, which indicates that the nonlinear effect on the structure is not obvious.

At the final moment of Stage 1, the cambering “floating” displacement of Node 29 given by the EBD method is 2.09 mm. Because of this “floating” displacement, the final vertical displacement of Node 29 obtained from the EBD method is 2.11 mm smaller than those given by other methods. This result demonstrates that the “floating” displacements of dead elements substantially influence the final configuration of the structure, and this effect can be avoided using the ISSM method proposed in this article.

The final vertical displacements of Nodes 29 and 44 obtained using the TP principle are 0.69 and 1.52 mm smaller, respectively, than those obtained using the MDCP principle. Furthermore, there are differences between the axial forces on the beams obtained using the TP principle and those obtained using the other principles because the lengths of the second- and third-floor beams as determined using the TP principle are smaller than the designed lengths. Consequently, the structural stiffness determined using this principle is higher than the stiffness determined using the other principles. Therefore, it is not reasonable to apply the TP principle to this structure.

The computational time of the ISSM method is 52.03 s, which is 3.26 s less than that of the SSM method. This finding indicates that the proposed modification method for the structural stiffness matrix in the ISSM method can improve the computation efficiency.

Overall, the ISSM method in combination with the MDCP principle proposed in this article is the most reasonable analysis approach for this structure.

Shallow arch

Description of the structure

The second structure considered is a shallow arch with a span of 24 m, a rise of 1.2 m, and fixed skewbacks, as shown in Figure 8(a). The cross section of each member is H800 × 250 × 8 × 12. Q345 grade steel is used to account for an ideal elastoplastic model with an elasticity modulus of $E = 2.06 \times 10^{11} N m$ . During construction, a nodal load of $P_{1} + P_{2} = 378 kN$ is applied on Node 25 and Node 26, a nodal load of $P_{1} = 18 kN$ is applied on the remaining nodes, and the structural weight is not considered.

Figure 8.

(a) Shallow arch model, (b) construction stages of the plane shallow arch, and (c) positions of the newly assembled members.

The arch is constructed using the cantilever erection technique without a bracing structure, in which the members are assembled from each skewback toward the mid-span. Five construction stages are considered. In each of the first four stages, three members are assembled at each end of the arch; in the fifth stage, the last member (Element 25) is assembled between Nodes 25 and 26. The construction process is illustrated in Figure 8(b).

The FE model of the structure consists of 25 identical beam elements of 1.113 m in length, each with three DOFs per node. The numbers of nodes and crucial members are presented in Figure 8(a).

Results and discussion of the construction mechanical analysis

The DCP, FCP, and MDCP principles were each used to determine the positions of the newly assembled members, as shown in Figure 8(c). The MVS, EBD, and ISSM methods were used to analyze the mechanical behavior of the shallow arch throughout the entire construction process. Figure 9(a) shows the variation in the displacements of Nodes 7, 13, 19, and 25 with the construction steps, and Figure 9(b) shows the variation in the inner forces of Elements 5 and 7 with the construction steps. The pictures of displacement results obtained using these methods are shown in Figure 12(b) of Appendix 2.

Figure 9.

(a) Variation in the vertical displacements with construction steps and (b) variation in the inner forces of crucial elements with construction steps.

Several conclusions can be drawn:

As shown in Figure 9(a), when the EBD method is used for calculation, the nodes on the dead elements show different levels of “floating” displacements, with the exception of Node 7. In particular, the maximum vertical floating displacement of 72.9 mm occurs on Node 25 at the final moment of Stage 3. Therefore, upon completion of the construction of the shallow arch, the vertical displacements of these nodes as obtained using the EBD method are markedly different from those obtained using the ISSM method, indicating that the “floating” displacements of dead elements severely influence the configuration of the structure during construction. This influence can be prevented using the ISSM method proposed in this article.

As shown in Figure 9(b), from Stages 1 to 4, the inner forces obtained using all three methods are nearly equal, which indicates that there is no obvious nonlinear effect on the arch before closure. In Stage 5, the axial forces on Elements 5 and 7 as obtained using the three methods are slightly different. The bending moments of Elements 5 and 7 obtained using the ISSM method are larger than those obtained using the MVS method, with a deviation of 10.13% for Element 5 and 9.34% for Element 7 at the final moment of Stage 5. The reason for this difference is that there is an obvious nonlinear effect of the shallow arch after closure. By contrast, the moments of Elements 5 and 7 obtained using the ISSM method and the EBD method are only slightly different, which indicates that the ISSM method proposed in this article adequately accounts for nonlinear effects.

The computational time of the ISSM method is 103.08 s, which is 7.08 s less than that of the SSM method. This finding indicates that the proposed modification method for the structural stiffness matrix in the ISSM method can improve the computation efficiency.

Overall, the ISSM method proposed in this article is reasonable.

Engineering application

Engineering situation

The connective corridor in the Shanghai International Financial Centre is a steel braced-frame system with three floors. The corridor is supported on two core tubes, and the ends of main beams are resting on sliding supports. The construction site of the corridor is presented in Figure 10(a) and (b), and the plan-view of the corridor is shown in Figure 10(c) and (d).

Figure 10.

Continued

A total of four construction stages are considered, and temporary supports are used during installation of members, as shown in Figure 10(e). In the first three construction stages, each story of the frame is installed; and in the fourth stage, the temporary supports are removed.

One of the main frames of the connective corridor is analyzed in this section (Figure 10(c)). The FE model of the frame consists of 198 nodes and 223 beam elements. Only the in-plane load-deformation of the frame is considered in the numerical simulation and the structural monitoring; accordingly, for the FE model, each node of an element has two translational DOFs and one rotational DOF. The typical nodes and members are numbered in Figure 10(e). The self-weight of all the members and components, including both the primary ones and the secondary ones, is considered in the FE model of the frame.

Monitoring strategy

The stresses on crucial members and the deformations of main beams on the first floor are monitored to ensure the safety of the corridor during construction. Vibrating Wire Strain Gages were used to measure the strain of members, while Inclinometers were used to measure the displacements of the corridor; automatic dataloggers were used for datalogging. The schematic diagram of the monitoring system is shown in Figure 10(f), and the pictures of the sensors and the datalogger on the construction site are shown in Figure 10(g).

Comparison between calculation and measurement

The ISSM method in combination with the MDCP principle proposed in this article was used to analyze the mechanical behavior of the connective corridor throughout the entire construction process. The calculated and measured results of the vertical displacements on Nodes 46 and 54 with the construction steps are shown in Figure 11(a), and the results of the stresses on Elements 147 and 152 with the construction steps are shown in Figure 11(b).

Figure 11.

(a) Variation in the vertical displacements of crucial nodes with construction steps and (b) variation in the stresses of crucial elements with construction steps.

As shown in Figure 11, several conclusions can be drawn:

The variation tendency of the calculated results obtained using the ISSM method proposed in this paper matched well with that of the measured results. During the installation process (Stages 1, 2, and 3), the calculated results are slightly different from the measured results.

In Stage 4, because the temporary supports were being removed, the vertical displacements of Nodes 47 and 54 markedly increased. Moreover, the axial compression of Element 152 turned into axial tension; the moment of Element 147 significantly increased after removing the temporary supports. The calculated results of stresses on the tension flange of the column obtained using the proposed method in this article are smaller than the measured results, with a deviation of 53.3% at the final moment of Stage 4. The reasons for the differences between the calculated and measured results are the distinction between the actual construction load and the theoretical load, temperature action, the installation errors of members, and the asynchronism of removing temporary supports.

Overall, the ISSM method proposed in this article can accurately simulate the construction process of planar steel structures and any other steel structures that can be simplified as planar ones.

Conclusion

To solve the two key problems encountered in the traditional SSM method, the ISSM method was proposed in this article. Two illustrative examples and an engineering application were presented to verify the validity and accuracy of the proposed method for analyzing the mechanical behavior of steel structures during construction. Several conclusions were drawn:

A new concept of “two moments of a construction stage” was proposed to more clearly define the construction process in the SSM method, and it was noted that the problems of the positioning of newly assembled members and the transformation of the structural stiffness matrices and the nodal load and displacement vectors arise at the initial moment of each construction stage.

The MDCP principle was proposed for the positioning of newly assembled members at the initial moment of a construction stage. Because this principle is consistent with common member assembly practices of assembled members in practical engineering, it introduces little deviation in the modeled lengths of the newly assembled members compared with their true lengths, and it is also easy to realize. The results of the illustrative examples demonstrated that the principle proposed in this article is more suitable than other positioning principles for the newly assembled members of large-span steel structures.

The transfer matrices necessary for expanding the structural stiffness matrices and the nodal load and displacement vectors were obtained, thereby resolving the problem of the transformation of the stiffness matrix and the nodal load and displacement vectors between different stages of construction.

Based on the MDCP principle and the modification method for the structural stiffness matrix and the nodal load and displacement vectors proposed in this article, the ISSM method considering nonlinear effects on structures was proposed and a calculation program block based on this method was compiled. This method permits interruption partway through the analysis and is therefore suitable for adoption in the servo control of construction processes, and it also avoids the “floating” displacements that arise in the EBD method. Comparisons of the results obtained for two illustrative examples using the ISSM method and other methods demonstrated the validity of the ISSM method.

The engineering application demonstrates that the ISSM method in combination with the MDCP principle proposed in this article is effective and suitable for construction mechanics analysis of planar steel structures and steel structures that can be simplified as planar ones.

Footnotes

Appendix 1

Appendix 2 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was conducted with financial support from the National Natural Science Foundation of China for Critical Techniques of Construction Process Analysis of Large and Complex Spatial Steel Structures (Project designation: 51078289) and for Numerical Model and Loading Behavior Analysis Method of the Steel Structure in Service (Project designation: 51678431).

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