Performance evaluation of a reticulated shell with atmospheric corrosion damage

Abstract

The study on damage evaluation induced by atmospheric corrosion of engineering structures has attracted more and more international concern over the past three decades. However, the effects of atmospheric corrosion on reticulated shells have not been systematically investigated. In this regard, the performance assessment of a reticulated shell subjected to atmospheric corrosion damage is actively conducted in this study. The atmospheric corrosion model of shell elements is first presented, and a refined exponential model for estimating the corrosion depth of steel materials is developed by using the pattern recognition technique. The sensitivity of stiffness matrix to element thickness is established by using Euler–Bernoulli beam element. The sensitivity of mass matrix to element thickness is developed based on lumped mass assumption. Then, the expression of natural frequency sensitivity to element thickness and mass is derived by considering the section loss induced by both the inner and outer surface corrosion. In addition, the explicit expression of frequency sensitivity to mass of spherical joints is also established in detail. The nonlinear static structural analysis is conducted to evaluate effects of atmospheric corrosion on the stress of structural elements. A real reticulated shell constructed in northern China is taken as the example structure to examine the feasibility of the proposed approach and to assess the potential damage caused by atmospheric corrosion to the structure.

Keywords

atmospheric corrosion frequency sensitivity performance evaluation reticulated shell stress change

Introduction

Reticulated shells provide an easy and economical method of roofing large areas and are used frequently by designers who realize the advantages and the impressive beauty of this form of construction. Space shells are of special interest, as they enclose a maximum amount of space with a minimum surface and have proved to be very economical in terms of consumption of constructional materials. The recent development of space structures shows that remarkable progress in the field of reticulated shell has been achieved during the past three decades in many countries. Various new types of reticulated shells have been developed across the world. Steel shells are always subjected to various adverse environmental factors during its long-time service (Cai et al., 1999; Chen et al., 2005; Halama et al., 2011; Ibrahim and Mohammed, 1994). Atmospheric corrosion is the action where engineering metal materials are converted to more stable mineral compounds on exposure to the atmospheric environment for a certain period (Bai and Chen, 2017; Farrow and Graedel, 1996). Atmospheric corrosion can change the physical and chemical properties of the metals as well as weaken the usefulness of construction materials. If the damage accumulation induced by atmospheric corrosion cannot be detected timely, the structural safety will be threatened, and the damage may finally cause partial destruction or even the whole loss of structural function, resulting in huge economic losses and casualty (Chen and Xu, 2007; Cowell and Apsimon, 1996; Purwasih et al., 2018).

The atmospheric corrosion is a normal corrosive phenomenon where the metal materials are exposed to open air, resulting in corrosion due to the chemical or electrochemical action between the metal materials and the humid environment. To date, the research work indicates that there are many factors such as climate condition, atmospheric contamination, and metal surface condition may induce the atmospheric corrosion of steel structures. Climate condition includes humidity, temperature, sunlight, and other climate factors. Temperature can affect metal by atmospheric corrosion in many different manners because change in temperature leads to the change of atmospheric humidity. The influence of relative humidity on atmospheric corrosion is nonlinear, and there exists a certain relative humidity value called critical humidity, beyond which the metal corrosion rate increases remarkably. The influence of atmospheric contamination includes pollutants such as SO₂ and NO₂.

The loss caused by atmospheric corrosion accounts for almost half of all the corrosion loss, which spurs the research on it become a very important field in corrosion since the 1900s. Systemic research in early stages was first performed by Kucera et al. and Shastry et al. (Cai et al., 2018; Feliu and Morcillo, 1993a; Kucera, 1986) and then many scholars have carried on the research on atmospheric corrosion phenomena and prediction model. Considering the reliance of research on measured data, many experiments and field measurements are conducted around the world to explore the mechanism of atmospheric corrosion on metal (Flavio and Stefano, 2002; Herrera and Spencer, 1995). The research includes the 8-year systemic study conducted by Kucera et al. on carbon steel, pure zinc, pure copper, and pure aluminum at 11 experimental spots in Sweden and Czech; the 16-year test executed by Shastry et al. 1988 on four kinds of steel at three spots in America; and the 8-year test conducted by Hou et al. 2004 on four kinds of steel at three spots in Taiwan and so on. Since the 1960s, researches on atmospheric corrosion on metal and field experiments have been financially supported by the Chinese government. In order to accumulate detailed and systemic corrosive data, the Chinese National Scientific Committee and the National Natural Science Funds Committee established the Chinese Atmospheric Corrosive Experimental Network. The data accumulation and research plan have been carried on since 1983, and a great deal of test data is achieved on the site. With application of various monitoring systems and collected data (Juan, 2003; Wang et al., 2009), many prediction models for atmospheric corrosion on metal are put forward in recent years.

The study on damage evaluation induced by atmospheric corrosion of engineering structures has attracted more and more international concern over the past three decades (Chen and Xia, 2017; Weng et al., 2013a). Chen et al. (2005) developed an approach for evaluating corrosion-induced damage of a large spatial frame with special configuration, while the approach cannot be applied to common shell structures because the corrosion effects of spherical joints are not considered. In addition, the effects of atmospheric corrosion on typical reticulated shells have not been systematically investigated. In this regard, the performance assessment of a reticulated shell subjected to atmospheric corrosion damage is actively carried out in this study. The atmospheric corrosion model of shell elements is first presented, and a refined exponential model for estimating the corrosion depth of steel materials is developed. The refined finite element (FE) method for analyzing reticulated shells is developed by considering the contribution of spherical joints. The sensitivity evaluation approach for reticulated shells is proposed, and explicit expression of frequency sensitivity to element thickness and mass of the reticulated shell is derived by considering the section loss induced by both the inner and outer surface corrosion. In addition, the explicit expression of frequency sensitivity to mass of spherical joints is also established in detail. The nonlinear static structural analysis is conducted to evaluate effects of atmospheric corrosion on the stress of structural elements. A real reticulated shell constructed in northern China is taken as the example structure to examine the feasibility of the proposed approach and to assess the potential damage caused by atmospheric corrosion to the structure.

Atmospheric corrosion damage of reticulated shell

One of the widely used empirical models for predicting atmospheric corrosion depth of metal materials is the exponential model (Feliu and Morcillo, 1993a, 1993b)

D = (α_{1} \log N + α_{2}) T^{n}

(1)

where D denotes the corrosion depth in $μ$ m; T denotes the time in year; n denotes the model parameters depending on the type of metal and environmental parameters; the coefficients α₁ and α₂ are the constants depending on the type of metal; N denotes the environmental index which can be expressed based on the atmospheric corrosion data collected from the Chinese atmospheric corrosive experimental network (Hou, 1994; Hou and Liang, 2004)

N = c_{1} + c_{2} + c_{3}

(2)

in which

c_{1} = \frac{H_{80}}{T_{av} \times H_{s}}

(3)

c_{2} = \sqrt{2 [S O_{2}] + 2 [C l^{-}] + [N O_{2}]}

(4)

c_{3} = \frac{1}{2} {(\frac{1}{10} \log \frac{P}{D_{P}})}^{\frac{| 7 - pH |}{23}}

(5)

where c₁ is the sum of the humidity coefficient, c₂ is the air contamination coefficient, and c₃ is rain acidity coefficient; H₈₀ denotes the hours per year for relative humidity larger than 80%; T_av denotes the average temperature per year (°C); P denotes the precipitation per year (mm/a); D_p denotes the days with precipitation per year; H_s denotes the hours with sunshine per year. The sunshine is induced by solar radiation, which can be expressed as the sum of the direct solar radiation, diffuse solar radiation, and reflected solar radiation on a metal surface. In reality, the information of direct, diffuse, and reflected solar radiation on the ground can be obtained through field measurement or simulation.

The Chinese empirical model for predicting the atmospheric corrosion depth of metal structures can be written by

D = [α_{1} \log (c_{1} + c_{2} + c_{3}) + α_{2}] T^{n}

(6)

The regressive technique can be applied to analyzing the test data collected from the national experimental network in China to determine the environmental index N and the corrosion development trend n in the testing cities (see Tables 1 and 2). In addition, the two material constants α₁ and α₂ have also been statistically determined for different types of metal materials as listed in Table 3. However, most of the steel structures are not located in testing cities and the index N, and two material coefficients cannot be determined.

Table 1.

Environmental parameters and index N.

	Item/site	Shijiazhuang	Beijing	Qingdao	Wuhan	Jiangjin	Guangzhou	Wanning	Qionghai
p ₁	Cl^– deposition (mg/100 cm² d)	0.210	0.049	0.250	0.011	0.006	0.024	0.387	0.199
p ₂	SO₂ deposition (mg/100 cm² d)	0.725	0.442	0.704	0.272	0.667	0.107	0.060	0.150
p ₃	NO₂ content (mg/m³)	0.054	0.220	0.038	0.089	0.007	0.035	0.005	0.008
p ₄	Average temperature (°C)	12.9	11.9	12.3	16.8	17.9	22.9	24.2	24.3
p ₅	Hours for relative Humidity >80% (h/a)	3302	2558	4049	4181	5741	4700	6020	6241
p ₆	Precipitation (mm/a)	541.7	586	562	1140	1203	1563	1515	1794
p ₇	Precipitation days (days/a)	74	78	94	116	134	170	124	151
p ₈	pH	6.00	5.5	6.1	6.5	4.4	5.8	5.0	6.9
p ₉	Sunshine hours (h/a)	2640	2559	2161	1621	1317	1607	2026	2072
N	Index	1.934	1.607	2.001	1.438	1.790	1.115	1.483	1.459

Table 2.

Corrosion development trend n in China.

Material type	Beijing	Qingdao	Wuhan	Jiangjin	Guangzhou	Wanning	Qionghai
16MnQ	0.36	0.51	0.43	0.49	0.64	0.59	0.93
12CrMnCu	0.33	0.38	0.36	0.52	0.55	0.76	0.73
09MnNb	0.50	0.49	0.41	0.43	0.53	1.04	0.88
A3	0.35	0.44	0.42	0.45	0.45	1.10	0.66
16Mn	0.36	0.54	0.56	0.42	0.48	1.37	0.94

Table 3.

Material coefficients f₁ and f₂ in China.

Material type	α ₁	α ₂
16MnQ	33.88	51.88
12CrMnCu	36.24	51.40
09MnNb	35.11	68.95
A3	39.36	61.34
16Mn	39.55	65.94

To this end, a pattern recognition technique is presented to estimate its environmental index N and the corrosion development trend n. The nine important environmental parameters in equations (4) to (6) are normalized for each city using the following equation

P_{j} = \frac{(p_{j} - p_{j}^{min})}{(p_{j}^{max} - p_{j}^{min})}

(7)

where $p_{j}$ denotes the value of the jth environmental parameter in a city (j = 1, 2,…, 9 in this study); $x_{j}^{min}$ and $x_{j}^{max}$ denote the minimum and maximum values of the jth environmental parameter among all the cities concerned, including one with unknown N and n.

The matching index, M_d, of the city with unknown N and n to a city with known N and n can be computed as

M_{d} (a, b) = 1 - \sum_{j = 1}^{m} \frac{| P_{j} (a) - P_{j} (b) |}{m}

(8)

where a denotes the city with unknown model parameters N and n; b refers to the city with known model parameters; and m is the number of environmental parameters, which is equal to 9 in this study. The smallest value of M_d then indicates that the environmental index N and the corrosion development trend n of the relevant city b can be used for the city a. This approach can be used to estimate the environmental index and the corrosion development trend for the city with unknown N and n.

Structural model of reticulated shell

As typical spatial structures, reticulated shells can be analyzed based on FE method by adopting truss or beam elements. For reticulated shells, three-dimensional (3D) beam elements with 6° degrees of freedoms at each node can be adopted to simulate static and dynamic responses under external excitations. The mass matrix of the ith element of a reticulated shell in the local coordinate system (LCS), $M_{i}^{e}$ , can be constructed based on lumped mass assumption (Xu and Qu, 2003)

M_{i}^{e} = \frac{ρ_{i} A_{i} l_{i}}{2} [\begin{matrix} I_{m} & 0 \\ 0 & I_{m} \end{matrix}] = \frac{ρ_{i} A_{i} l_{i}}{2} M_{B}^{lump}

(9)

I_{m} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(10)

where ρ_i is the material density of the ith element; A_i and l_i are the cross-sectional area and length of the ith element. The element mass matrix in the global coordinate system (GCS) M_i can be expressed as the multiple of the element stiffness matrix in the LCS with the coordinate transformation matrix $T_{i}^{e}$

M_{i} = T_{i}^{eT} M_{i}^{e} T_{i}^{e}

(11)

The effect of a spherical joint on the performance of a reticulated shell can be considered as a lumped mass acting on the structure. Commonly, a spherical joint is a hollow joint as shown in Figure 1, and the mass matrix of the ith welded spherical joint in the GCS $M_{i}^{s}$ can be deduced as

M_{i}^{s} = \frac{4 π ρ_{i}^{s}}{3} (R_{i, out}^{3} - R_{i, in}^{3}) I_{m}

(12)

where $ρ_{i}^{s}$ is the material density of the hollow spherical joint; R_i,out and R_i,in denote the outer and inner radii of the hollow spherical joint.

Figure 1.

Configuration of typical spherical joints in the reticulated shell.

In reality, structural elements are combined through hollow spherical joints to form a reticulated shell. Thus, the volume and mass of a hollow spherical joint should eliminate the effects of the connected structural elements. The actual volume can be expressed as

V_{i}^{s} = \frac{4 π}{3} (R_{i, out}^{3} - R_{i, in}^{3}) - \sum_{j = 1}^{ns} V_{i, j}^{s}

(13)

where $V_{i, j}^{s}$ is the eliminated spherical crown area induced by the jth structural element connected to the ith hollow spherical joint; ns is the number of elements of the ith hollow spherical joint

M_{i}^{s} = ρ_{i}^{s} (\frac{4 π}{3} (R_{i, out}^{3} - R_{i, in}^{3}) - \sum_{j = 1}^{ns} V_{i, j}^{s}) I_{m}

(14)

The global mass matrix M and stiffness matrix K of the reticulated shell can be expressed as follows

M = \sum_{i = 1}^{ne} T_{i}^{T} M_{i}^{e} T_{i} + \sum_{i = 1}^{nj} {(T_{i}^{s})}^{T} M_{i}^{s} T_{i}^{s}

(15)

K = \sum_{i = 1}^{ne} T_{i}^{T} K_{i}^{e} T_{i}

(16)

where ne is the number of elements; $K_{i}^{e}$ is the element stiffness matrix in the LCS; T_i is the freedom transform matrix from the LCS to the GCS for structural element; nj is the number of hollow spherical joints; $T_{i}^{s}$ is the freedom transform matrix from the LCS to the GCS for hollow spherical joints.

Performance evaluation of atmospheric corrosion damages

Frequency sensitivity of reticulated shell

A reticulated shell is inevitably affected by atmospheric corrosion, which results in the corrosion-induced section reduction of its structural elements. The reduction of element section will eventually induce the variation of structural performance. A reticulated shell consists of a large number of steel elements with complicated configuration. Performance analysis of a complex reticulated shell is tedious and commonly hampered by computational cost of dynamic analysis. It is beneficial to put forward effective approaches to evaluate the structural performance with atmospheric corrosion damages. Because most measurable dynamic characteristics of a steel space structure are its natural frequencies, the effects of atmospheric corrosion on natural frequencies of a reticulated shell are investigated in this study. The sensitivity analysis of dynamic properties of structural parameters is very important and widely accepted in the structural performance assessment (Weng et al., 2011, 2013b). Sensitivity coefficients are defined as the rate of change of the natural frequency f_j with respect to a change in the element thickness t_i due to atmospheric corrosion (Li et al., 2015). The sensitivity of the frequency to the ith element thickness t_i can be expressed as

\frac{\partial f_{r}}{\partial t_{i}} = \frac{1}{8 π^{2} f_{r}} \cdot \frac{ϕ_{r}^{T} (\frac{\partial K}{\partial t_{i}} - 4 π^{2} f_{r}^{2} \frac{\partial M}{\partial t_{i}}) ϕ_{r}}{ϕ_{r}^{T} M ϕ_{r}}

(17)

where M and K are the mass matrix and stiffness matrix of the reticulated shell, respectively. f_r and φ_r are the rth natural frequency and modal vector of the reticulated shell, respectively.

To facilitate the sensitivity analysis on a reticulated shell with complicated configuration, the frequency to the ith element thickness t_i can be expressed by using element stiffness and mass matrices in the LCS with the aid of the coordinate transformation matrices of the ith element

\frac{\partial f_{r}}{\partial t_{i}} = \frac{ϕ_{r}^{T} (\sum_{i = 1}^{ne} T_{i}^{eT} \frac{\partial K_{i}^{e}}{\partial t_{i}} T_{i}^{e}) ϕ_{r} - 4 π^{2} f_{r}^{2} ϕ_{r}^{T} (\sum_{i = 1}^{ne} T_{i}^{eT} \frac{\partial M_{i}^{e}}{\partial t_{i}} T_{i}^{e} + \sum_{i = 1}^{nj} {(T_{i}^{s})}^{T} \frac{\partial M_{i}^{s}}{\partial t_{i}} T_{i}^{s}) ϕ_{r}}{8 π^{2} f_{r} ϕ_{r}^{T} M ϕ_{r}}

(18)

where ne is the number of all the elements.

Sensitivity coefficients of stiffness matrices

The sensitivity coefficient depends on the determination of $\partial K_{i}^{e} / \partial t_{i}$ and $\partial M_{i}^{e} / \partial t_{i}$ . This can be computed based on the type of cross-section of a structural element. The stiffness matrix of the ith beam element in the LCS $K_{i}^{e}$ can be expressed as the combination of the cross-sectional area A_i and the second moment of inertia $I_{x}^{i}$ , $I_{y}^{i},$ and $I_{z}^{i}$ , respectively, by using the Euler–Bernoulli beam element

K_{i}^{e} = K_{i, 1}^{e} A_{i} + K_{i, 2}^{e} I_{x}^{i} + K_{i, 3}^{e} I_{y}^{i} + K_{i, 4}^{e} I_{z}^{i}

(19)

where E_i and G_i are the Young’s modulus and shear modulus of the element material; $I_{x}^{i}$ , $I_{y}^{i},$ and $I_{z}^{i}$ are the second moment of inertia with respect to the local coordinates x, y, and z, respectively, of the ith structural element

K_{i, 1}^{e} = \frac{E_{i}}{l_{i}} [\begin{matrix} 1 \\ 0 & 0 \\ 0 & 0 & 0 & sym \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(20)

K_{i, 2}^{e} = \frac{G_{i}}{l_{i}} [\begin{matrix} 0 \\ 0 & 0 \\ 0 & 0 & 0 & sym \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(21)

K_{i, 3}^{e} = \frac{2 E_{i}}{l_{i}^{3}} [\begin{matrix} 0 \\ 0 & 0 \\ 0 & 0 & 6 & sym \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - 3 l_{i} & 0 & 2 l_{i}^{2} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 6 & 0 & 3 l_{i} & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 3 l_{i} & 0 & l_{i}^{2} & 0 & 0 & 0 & 3 l_{i} & 0 & 2 l_{i}^{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(22)

K_{i, 4}^{e} = \frac{2 E_{i}}{l_{i}^{3}} [\begin{matrix} 0 \\ 0 & 6 \\ 0 & 0 & 0 & sym \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 3 l & 0 & 0 & 0 & 2 l_{i}^{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - 6 & 0 & 0 & 0 & - 3 l_{i} & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 3 l & 0 & 0 & 0 & l_{i}^{2} & 0 & - 3 l & 0 & 0 & 0 & 2 l_{i}^{2} \end{matrix}]

(23)

Therefore, the sensitivity of the stiffness matrix of the ith element to the thickness t_i is

\frac{\partial K_{i}^{e}}{\partial t_{i}} = K_{i, 1}^{e} \frac{\partial A_{i}}{\partial t_{i}} + K_{i, 2}^{e} \frac{\partial I_{x}^{i}}{\partial t_{i}} + K_{i, 3}^{e} \frac{\partial I_{y}^{i}}{\partial t_{i}} + K_{i, 4}^{e} \frac{\partial I_{z}^{i}}{\partial t_{i}}

(24)

The first derivatives to the thickness change caused by the reduction of the inner surface of the element can be derived as

\frac{\partial A_{i}}{\partial t_{i}^{in}} = 2 π r_{i, in}; \frac{\partial I_{y}^{i}}{\partial t_{i}^{in}} = \frac{\partial I_{z}^{i}}{\partial t_{i}^{in}} = π r_{i, in}^{3}; \frac{\partial I_{x}^{i}}{\partial t_{i}^{in}} = 2 π r_{i, in}^{3}

(25)

in which r_i,in is the inner radius of circular cross-section of the element as shown in Figure 2. The first derivatives to the thickness change caused by the reduction of the outer surface of the shell element can be given as

\frac{\partial A_{i}}{\partial t_{i}^{out}} = 2 π r_{i, out}; \frac{\partial I_{y}^{i}}{\partial t_{i}^{out}} = \frac{\partial I_{z}^{i}}{\partial t_{i}^{out}} = π r_{i, out}^{3}; \frac{\partial I_{x}^{i}}{\partial t_{i}^{out}} = 2 π r_{i, out}^{3}

(26)

in which r_i,out is the outer radius of circular cross-section of the structural element.

Figure 2.

Configuration of typical structural elements in the reticulated shell.

Thus, the sensitivity of the stiffness matrix of the ith element thickness $t_{i}^{in}$ and $t_{i}^{out}$ can be expressed as

\frac{\partial K_{i}^{e}}{\partial t_{i}^{in}} = π r_{i, in}^{3} (\frac{2}{r_{i, in}^{2}} K_{i, 1}^{e} + 2 K_{i, 2}^{e} + K_{i, 3}^{e} + K_{i, 4}^{e})

(27)

\frac{\partial K_{i}^{e}}{\partial t_{i}^{out}} = π r_{i, out}^{3} (\frac{2}{r_{i, out}^{2}} K_{i, 1}^{e} + 2 K_{i, 2}^{e} + K_{i, 3}^{e} + K_{i, 4}^{e})

(28)

Sensitivity coefficients of mass matrices

The sensitivity of the mass matrix of the ith element to the thickness t_i is

\frac{\partial M_{i}^{e}}{\partial t_{i}} = \frac{ρ_{i} l_{i}}{2} M_{B}^{lump} \frac{\partial A_{i}}{\partial t_{i}}

(29)

Thus, the sensitivity coefficients of the mass matrix of the ith element thickness to the $t_{i}^{in}$ and $t_{i}^{out}$ can be expressed as

\frac{\partial M_{i}^{e}}{\partial t_{i}^{in}} = π r_{i, in} ρ_{i} l_{i} M_{B}^{lump}

(30)

\frac{\partial M_{i}^{e}}{\partial t_{i}^{out}} = π r_{i, out} ρ_{i} l_{i} M_{B}^{lump}

(31)

The volume of the jth spherical crown $V_{i, j}^{s}$ , induced by the jth structural element connected to the ith hollow spherical joint, can be expressed as

\begin{matrix} V_{i, j}^{s} = π h_{i, j} t_{i}^{s} (R_{i, out} + R_{i, in}) \\ = π h_{i, j} (R_{i, out}^{2} - R_{i, in}^{2}) (j = 1, 2, \dots, ns) \end{matrix}

(32)

where $t_{i}^{s}$ is the thickness of the ith hollow spherical joint; h_i,j is the height of the jth spherical crown of the ith hollow spherical joint. Therefore, the mass matrix of the ith hollow spherical joint is written by

M_{i}^{s} = π ρ_{i}^{s} (\frac{4}{3} (R_{i, out}^{3} - R_{i, in}^{3}) - \sum_{j = 1}^{ns} π h_{i, j} (R_{i, out}^{2} - R_{i, in}^{2})) I_{m}

(33)

The sensitivity of the mass matrix of the ith spherical joint to the thickness $t_{i}^{s}$ is

\frac{\partial M_{i}^{s}}{\partial t_{i}^{s}} = π ρ_{i}^{s} (\frac{4}{3} \frac{\partial (R_{i, out}^{3} - R_{i, in}^{3})}{\partial t_{i}^{s}} - \sum_{j = 1}^{ns} h_{i, j} \frac{\partial (R_{i, out}^{2} - R_{i, in}^{2})}{\partial t_{i}^{s}}) I_{m}

(34)

Thus, the sensitivity coefficients of the mass matrix of the ith spherical joint to the thickness $t_{i}^{s, in}$ and $t_{i}^{s, out}$ can be expressed as

\frac{\partial M_{i}^{s}}{\partial t_{i}^{s, in}} = π ρ_{i}^{s} (- 4 R_{i, in}^{2} + 2 \sum_{j = 1}^{ns} h_{i, j} R_{i, in}) I_{m}

(35)

\frac{\partial M_{i}^{s}}{\partial t_{i}^{s, out}} = π ρ_{i}^{s} (4 R_{i, out}^{2} - 2 \sum_{j = 1}^{ns} h_{i, j} R_{i, out}) I_{m}

(36)

After determining all the sensitivity coefficients of the element mass and stiffness matrices with respect to the thickness change, the change in the rth natural frequency, $Δ f_{r}$ , due to the reduction in the thickness t_i of the ith structural element, $Δ t_{i}$ , can be determined according to equation (27).

Stress variation induced by damages

The thickness reduction of structural elements and spherical joints in a reticulated shell due to atmospheric corrosion will cause stress variations and affect the force-bearing capacity and performance of the shell. The nonlinear static analysis is carried out to evaluate the stress changes of the shell under gravity forces and service loads. The static equivalent equation of the original shell and damaged shell can be expressed as

Kx = G + F

(37)

K_{c} x_{c} = G_{c} + F

(38)

where K_c is the structural stiffness matrix with corrosion damages; x and x_c are the structural deformation vectors without/with corrosion damages; G and G_c are the structural gravity forces without/with corrosion damages; F is service load acting on the shell. The Newton–Raphson iteration method is thus used to ensure that the solution converge at each load increment and the iteration process is finished when the error is smaller than the tolerance Tol

x_{c}^{{1}} = x_{c}^{{0}} + Δ x_{c}^{{1}}

(39)

‖ x_{c}^{{n + 1}} - x_{c}^{{n}} ‖ \leq Tol

(40)

The flowchart for the evaluation procedure of reticulated shell subjected to atmospheric corrosion is displayed in Figure 3.

Figure 3.

Evaluation procedure of reticulated shell subjected to atmospheric corrosion.

Case study

Description of a reticulated shell

Owing to the wide application of reticulated shells in recent years, a Kiewitt-type reticulated shell constructed in northern China is selected as an engineering case in this study to evaluate the effects of atmospheric corrosion damages on structural performance. The span and height of the example reticulated shell are 65 and 9.9 m, respectively. The FE model of the shell is constructed using 3D beam elements. Figure 4(a) and (b) displays the configuration of the reticulated shell, which has 169 nodes and 456 steel elements, respectively. As displayed in Figure 5, the shell consists of 48 radial elements, 168 circular elements, and 240 skew elements, respectively. For the sake of convenience in the subsequent discussion, the elements in each component of the reticulated shell are numbered differently. The radial elements are numbered from 1 to 48 (denote rm). The circular elements are numbered from 49 to 216 (denote cm), and the skew elements are counted from 217 to 456 (denote sm). The 48 radial elements radiate from the shell vertex (node 1) in eight directions respectively. The 168 circular elements are located on six concentric circles, counted from the smallest circle 1 to the largest circle 6 (denote cm1 to cm 6), with node 1 being the center. All the nodes on the outer most circle (circle 6) are rigidly constrained based on design configuration. Each element is constructed using thin-wall steel tube having a wall thickness of 5 mm. The outer diameter and cross-sectional area for all the radial and circular elements are 180 mm and 0.002749 m², respectively. The outer diameter and cross-sectional area for all the skew elements are 159 mm and 0.002419 m², respectively. The outer and inner radii of a spherical joints are 170 and 150 mm, respectively. The Young’s modulus for the steel material is 2.07 × 10⁵ MPa. The density and Poisson’s ratio of the material are 7800 kg and 0.28, respectively. The reticulated shell is covered with roof slabs having an equivalent mass of 20 kg/m². The FE model of the shell is constructed by using 3D beam elements. The mass matrix includes the contribution of structural elements and other components such as roof slabs. The structural elements used in the radial elements are made of Q235 steel (material type A3) with a yielding stress of 235 MPa.

Figure 4.

Configuration of a reticulated shell.

Figure 5.

Components of the reticulated shell: (a) radial elements, (b) circular elements, and (c) skew elements.

Dynamic characteristics of reticulated shell

The dynamic characteristics analysis is conducted based on the established FE model of the shell. The first 50 natural frequencies of the shell are computed, and the first eight vibration modes are depicted in Table 4 and Figure 6. It is seen that the natural frequencies of the shell are very close, and there exists many duplicate natural frequencies which relate to the symmetric mode shapes. The ratio of height to span (f/L) is 1:6.54 which is relatively small. The mode shapes demonstrate the coupled vibration in both horizontal and vertical directions. Owing to the symmetric axes of the shell, there exist many symmetric mode shapes such as the 1st and 2nd, 3rd and 4th, and 5th and 6th mode shapes. The parametric study is carried out to investigate the effects of various factors on the dynamic features, which includes the ratio of height to span (f/L) and boundary constraint. The values of various factors are as follows: (1) ratio of height to span (f/L): 1/8, 1/6.54, 1/5, 1/4 and (2) boundary constraint: rigid and joint constraints.

Table 4.

The first 50 frequencies of the reticulated shell (Hz).

No.	Freq.	No.	Freq.	No.	Freq.	No.	Freq.	No.	Freq.
1	4.066	11	4.541	21	4.818	31	5.111	41	5.285
2	4.066	12	4.541	22	4.875	32	5.189	42	5.303
3	4.406	13	4.561	23	4.953	33	5.189	43	5.303
4	4.406	14	4.594	24	4.953	34	5.226	44	5.332
5	4.421	15	4.617	25	4.987	35	5.227	45	5.332
6	4.421	16	4.695	26	4.987	36	5.269	46	5.378
7	4.451	17	4.695	27	4.998	37	5.270	47	5.4372
8	4.452	18	4.708	28	5.069	38	5.271	48	5.440
9	4.483	19	4.726	29	5.069	39	5.271	49	5.440
10	4.483	20	4.818	30	5.109	40	5.276	50	5.465

Figure 6.

The first 8 mode shapes of the reticulated shell.

The variation of the first 50 natural frequencies with the f/L ratio plotted in Figure 7(a) reveals that the obvious effects of the f/L ratio on shell dynamic properties. By keeping the shell span and decreasing the height, the shell configuration will be transferred from a 3D spatial structure to a 2D plane one. Simultaneously, the magnitude of natural frequencies gradually reduces. The effects of boundary conditions are analyzed and provided in Figure 7(b). It is clear that the difference of natural frequencies and mode shapes between rigid and joint constraints is quite small.

Figure 7.

Variation of natural frequencies with structural parameters. (a) Frequency variation with f/L; (b) Frequency variation with constraint.

Atmospheric corrosion damage of shell

The corrosion model is applied to predict the atmospheric corrosion depth of the steel elements of the reticulated shell. The national experimental network has obtained the environmental index N and the atmospheric corrosion trend n for the seven cities but not for Shijiazhuang. Therefore, the environmental parameters of Shijiazhuang are collected and listed in Table 1. The environmental index N of Shijiazhuang is calculated as 1.9336. To determine the atmospheric corrosion trend n for Shijiazhuang, the pattern recognition technique expressed by equations (7) and (8) is used. The seven matching index are depicted in Figure 8 for the seven cities, and the best match to Shijiazhuang is Qingdao. Thus, it is reasonable to use the coefficient n of Qingdao for Shijiazhuang, that is, n = 0.44 for Q235 steel according to Table 2. The constants f₁ and f₂ are 39.36 and 61.34, respectively, for Q235 steel according to Table 3.

Figure 8.

Matching degree of Shijiazhuang to other seven cities.

With all the model parameters determined, the corrosion depth of steel elements in the space structure can be predicted using equation (6). Figure 9(a) and (b) shows the variation of corrosion depth and corrosion rate of Q235 steel with time in year. It is seen that the corrosion depth of the material increases with time, and the corrosion rate sharply decreases with the increasing time. In addition, the corrosion rate of material depth is much faster in the first 5 years than later.

Figure 9.

Variation of corrosion damage of materials with time: (a) corrosion depth and (b) corrosion rate.

Shell performance with atmospheric corrosion damage

Effects of atmospheric corrosion on natural frequencies

The reticulated shell concerned is made of steel elements of circular hollow section. The outer surfaces of all the structural elements are painted to prevent atmospheric corrosion, but the inner surfaces of all the structural elements are not painted because of operation inconvenience. In this regard, the atmospheric corrosion on the inner surfaces of the structural elements should be considered. Nevertheless, the coating on outer surface may be damaged in the open air. Therefore, the atmospheric corrosion on both the inner and outer surfaces is also considered as an extreme case and compared with the case of inner surface corrosion only. For either case, the sensitivity coefficients of natural frequencies to the change in element thickness are first computed. The change in the natural frequency, Δf, due to the reduction in thickness Δt, is then computed. Finally, the sum of the changes in natural frequencies due to the thickness reductions of all the structural elements gives the final change in natural frequencies, from which the effect of atmospheric corrosion damage on natural frequencies of the reticulated shell can be evaluated.

Figures 10 and 11 display the sensitivities of the first eight natural frequencies to the thickness change of each element due to inner and outer surface corrosion, respectively. Figures 12 and 13 display the spatial distribution of sensitivity coefficients of the first eight natural frequencies to the thickness change of each element. It is clear that the element distributions of frequency sensitivity for inner and outer corrosion are similar for the first eight natural frequencies while the sensitivity magnitudes of inner corrosion are slightly larger than those of outer corrosion. Further examination on sensitivity distribution reveals that the structural elements with large sensitivity coefficients are radial elements and circular elements. The first two natural frequencies are more sensitive to the thickness change of radial elements and some skew elements within the first two circles. The third and fourth natural frequencies are more sensitive to the thickness change of circular elements in the three and four circles. While for the other higher natural frequencies, the sensitive elements are the radial elements and skew elements parallel to radial elements. To conclude the element distribution with large frequency sensitivity to element thickness, one can find that the radial elements are most sensitive to section loss caused by atmospheric corrosion. Simultaneously, some circular elements and skew elements close to the radial elements are also sensitive to section loss.

Figure 10.

Sensitivity of first eight natural frequencies to thickness change of all the structural elements (inner surface corrosion).

Figure 11.

Sensitivity of first eight natural frequencies to thickness change of all the structural elements (outer surface corrosion).

Figure 12.

Spatial distribution of the sensitivity coefficients of first eight natural frequencies (inner surface corrosion).

Figure 13.

Spatial distribution of the sensitivity coefficients of first eight natural frequencies (outer surface corrosion).

The changes in natural frequencies of the reticulated shell due to inner and double surface corrosion are displayed in Figure 14 for the corrosion period of 1, 4, 10, and 20 years, respectively. The changes in the first five natural frequencies with corrosion time are shown in Figure 15. Clearly, the changes in natural frequencies increase as corrosion year increases. The changes in the all the first 20 natural frequencies are negative, indicating that the natural frequencies are reduced. This is because, while the reduction of element thickness due to corrosion causes the stiffness reduction, it also causes the mass reduction. The change in a natural frequency is negative mainly because the relative extent of the stiffness reduction effects is larger than those of mass reduction effects. It is seen that the 16th natural frequency of the structure has the maximum change due to both inner and double surface corrosion. The maximum change in the 16th natural frequency due to inner surface corrosion is about 0.1% in 20 years. Even though both the inner and outer surface corrosions are considered, the maximum change in the second natural frequency is about 1.4% in 20 years. Therefore, it can be concluded that the natural frequencies of the reticulated shell considered in this study are only slightly affected by the atmospheric corrosion of materials. However, this conclusion may not be applicable to other steel space structures in other places.

Figure 14.

Frequency changes for different corrosion periods.

Figure 15.

Variation of the first five natural frequencies with time.

Effects of atmospheric corrosion on structural stresses

The effects of atmospheric corrosion on element stresses of the reticulated shell under gravity forces are examined in this study. The atmospheric corrosion may induce the reduction of cross-sectional areas of the shell elements, which may cause the reduction of both dead loads and element stiffness. Therefore, the variations of dead loads and stiffness may induce the redistribution of the structural stresses. In addition, the gravity forces of other structural components, such as joints and cladding, are regarded as constant forces acting on the structural nodes without changes. Nonlinear static analyses of the shell are carried out for 10-, 20-, and 50-year atmospheric corrosion, and the variation of maximum stress of elements are displayed in Figure 16 under inner surface corrosion.

Figure 16.

(a) Variation of maximum stress of elements (MPa): 10-year corrosion, (b) 20-year corrosion, and (c) 50-year corrosion.

It is seen from Figure 16 that the maximum stress changes under for 10-, 20-, and 50-year atmospheric corrosion are 0.11, 0.14, and 0.21 MPa, respectively. It is found that the numbers of structural elements with large stress change increase with the increasing corrosion year. There exist both negative and positive stress changes under corrosion damage, which denotes the stress redistribution in the shell. However, the structural elements of increasing stress are more than those of decreasing stress. It can also be found that with the increase of the atmospheric corrosion year, the variation of maximum element stress also increases while this change is quite small compared to element’s ultimate stress. Similar observations can be made for the structural elements with double surface corrosion. A careful investigation on stress changes in all the structural elements reveals that for the concerned reticulated shell, the structural elements of higher stress level have small stress change only while large stress changes occur in the structural elements of lower stress level.

Concluding remarks

The performance assessment of a reticulated shell subjected to atmospheric corrosion damage is actively carried out in this study. The damage model of structural elements of a reticulated shell is first presented based on atmospheric corrosion theory. The refined exponential model for estimating corrosion depth of steel materials at a city is developed by using the pattern recognition technique to determine the model parameters. The sensitivity coefficients of stiffness and matrices to element thickness are deduced based on beam elements and lumped mass assumption. In addition, the expression of frequency sensitivity to mass of spherical joints is established in detail. Then, the expression of natural frequency sensitivity to element thickness is derived by considering the section loss induced by both the inner and outer surface corrosion. The nonlinear static structural analysis is conducted to evaluate effects of atmospheric corrosion on the stress of structural elements.

The made observations demonstrate that the corrosion depth of the material increases with time, and the corrosion rate sharply decreases with the increasing time. In addition, the corrosion rate of material depth is much faster in the first 5 years than later. The element distributions of frequency sensitivity for inner and outer corrosion are similar to the first eight natural frequencies while the sensitivity magnitudes of inner corrosion are slightly larger than those of outer corrosion. The changes in natural frequencies of the reticulated shell increase as corrosion year increases and the frequency changes are negative, indicating that the natural frequencies are reduced. The stress changes increase with increasing atmospheric corrosion year, and they are larger in the case of double surface corrosion than in the case of inner surface corrosion only. For the concerned reticulated shell, the structural elements of higher stress level have smaller stress change. It is worthwhile to point out that the results obtained from the reticulated shell cannot be applied to other spatial shells in other places. The framework proposed here can be used to perform evaluation case by case.

Footnotes

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: The authors are grateful for the financial support from the National Natural Science Foundation of China (Grant No.: 51678463), the Research Project of Ministry of Housing and Urban-Rural Development of China (Grant No.: 2017K5-003), and the Chenguang Science and Technology Plan of Wuhan (Grant No.: 2016070204010107).

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