Area-loss effect analysis of pin-jointed structures

Abstract

Corrosion will cause cross-sectional area loss of steel members, which will lead to redistribution of internal force and affect the performance of structures. In this paper, a generalized formula for the internal force variation of pin-jointed structures due to variation of external force, cross-sectional area, and initial length is derived. The area-loss sensitivity coefficient (ASC) and the area-loss evenness coefficient (AEC) based on statistical measures are then defined. By calculating these two coefficients, the influence of single member’s area-loss on the behavior of the structure as well as the evenness of internal force variation are evaluated. Besides, a method to determine the area-loss limit of pin-jointed structures is proposed based on the structural reliability theory and mathematical nonlinear programming. According to the above process, the effect of members’ area loss on structural properties is analyzed without complicated numerical approaches such as FEM, and then the safety of corroded structures is assessed preliminarily. The proposed method is verified by two numerical examples, and provides a convenient tool for pre-manufacturing, maintenance during service life, and structural health monitoring of pin-jointed structures.

Keywords

pin-jointed structure area loss effect analysis mathematical programming safety assessment

Introduction

Pin-jointed structures have the advantages of lightweight, elegant appearance, and high controllability. Therefore, they are widely used in practical engineering such as building structures, bionic engineering and aerospace engineering (Khomami and Najafi, 2021; Chen and Dong, 2013; Puig et al., 2010). The initial stresses of members in pin-jointed structures are always caused by prestress or external loads. Accordingly, pin-jointed structures can be divided into two types (Zhang and OHSAKI, 2011). One is the bar structure. It is composed of bars, which provide material stiffness to the structure. The other is the cable-strut system. It is composed of cables and struts and the initial prestress is essential to stiffen the system which provides geometry stiffness to the entire structure to form an equilibrium and stable configuration (Montuori and Skelton, 2017). The stiffness of this type of structure is collectively provided by the geometry stiffness and material stiffness.

With the increase of the service life, many factors such as long-term wear, accidental loads and especially corrosion will inevitably lead area loss even vitiation of the members; thereby the stiffness distribution of the structures will be changed. At least it will affect the performance degradation of the structure, and at worst it will cause structural damage. In 1982, a two-truck collision in Panama caused the collapse of the Chiriqui River suspension bridge in that several suspender ropes snapped. A subsequent investigation revealed the hanger cables had been significantly weakened by rust. One of the most serious strands had lost two-thirds of its cross section due to corrosion (Hopwood and Havens, 1984). The suspended prestressed cable structure applied in the roof of the ice hockey stadium in Slovakia was built in 1966. The cable roof was exposed to varying environment, resulting in the aging and corrosion of the steel cables. It was found that the corrosion depth had reached 2 mm with approximately 17% of area loss, and all external wires of the strand in specific cables were heavily impaired. (Kmet et al., 2019). In 2005, a roof collapse accident occurred in a swimming pool in the Russian town of Chusovoy due to the severe corrosion of the roof metal frame and the fracture of the members. This was a typical stress corrosion cracking accident (Murphy, 2019). From these engineering accidents, it can be seen that the members’ area losses will lead to the degradation of structural performance (Hamilton, 1995; Shun-Ichi et al., 2004). Therefore, it is necessary to conduct further studies on the cross-sectional area-loss of pin-jointed structures.

Existing researches on the area-loss of members predominantly focused on the corrosion of materials. Daniels (Daniels and Jeffreys, 1945) and Weibull (1949) proposed a fiber bundle model to study the redistribution of internal forces as a result of corrupted steel cables. Furthermore, Coleman (1958) introduced breakdown function to measure the dynamic failure of fiber bundles. Cai et al., (2010) investigated the effects of applied torque on corrosion behavior of 316L stainless steel with crevices. Liu et al. (2020) studied mechanical properties of welded hollow spherical joints under corrosion and de-rusting conditions, and then established a practical corrosion evaluation method with the statistical distribution and corrosion damage kinetics. Jena et al. (2020) proposed a manufacturing process for reduced coating on the surface of carbon steel to study the anti-corrosion and self-cleaning properties of the coating. However, most studies emphasized the performance of the component level under the current state after corrosion, lacking researches on the structure level. In practical engineering, many structures are designed in complex style and composed of plenty of members, of which the contribution to structural stiffness varies greatly. And the degree of corrosion in members usually increase with time. Therefore, it is necessary to determine the member’s area loss of the structure under its service life and establish the relationship between the member’s area loss and the structural performance.

In structural safety studies, the mechanical performance of pin-jointed structures was studied with both geometric and material properties of structural members. Chen et al. (Chen et al., 2014; Zhou et al., 2015) carried out a distributed indeterminacy evaluation considering the effect of member stiffness. And Jiang et al. (2019) studied the influence of members’ material variation on the structural mechanical performance. At present, most studies can only analyze the mechanical properties of the whole structure in the current state (Gholipour et al., 2020; Gao et al., 2021; Kong et al., 2023), and then determine whether the structure is safe. But it provides less help in evaluating the performance degradation of the structure during the remaining service period. So a quick and accurate method is thus essential to determine the limit value of area loss of members which can ensure the structural safety in the present and future service periods.

In this paper, the traditional finite element analysis method is reduced to a generalized formula for the variation of external force, cross-sectional area, and initial length. According to the formula, the part that involves variation of area, the area-loss sensitivity coefficient (ASC) and area-loss evenness coefficient (AEC) are defined in turn to evaluate the impact of the specific member’s area loss on the internal force distribution of overall structures. Furthermore, a systematic method for determining the area-loss limit is also presented based on reliability theory and mathematical nonlinear programming to ensure structural safety. By means of graphical configuration, two numerical examples are designed to illustrate the physical and mathematical implications of the proposed methodology. Finally, a simplified calculation method and its prerequisite for the area-loss limit is discussed.

Generalized formula of internal force variation

For a pin-jointed structure with $n_{n}$ nodes, $n_{e}$ members, and $n_{b}$ constraints, its static equation can be expressed as:

K x = A_{1} q = F

(1)

where

x {ϵ R}^{3 n_{n}}

is the nodal coordinate vector, K is the nonlinear stiffness matrix,

F {ϵ R}^{3 n_{n}}

is the external force vector, and degree of node freedom

n_{a} = 3 n_{n} - n_{b} .

A_{1} ϵ R^{{3 n}_{n} \times n_{e}}

is the equilibrium matrix, corresponding to the force density vector

q

. It can be expressed in the following form:

A_{1} {= (C}^{T} \otimes I_{3}) (C \otimes I_{3} x)^I_{n_{e}} \otimes I_{3,1}

(2)

where

C ϵ R^{n_{e} \times n_{n}}

represents for the member connectivity matrix.

I_{3}

is an identity matrix in 3 order, and

I_{3,1}

is the 3

\times

1 full-one matrix.

q ϵ R^{n_{e}}

is the force density vector of the member.

\hat{*}

(*)^

represents the diagonal matrix of the vector

*

Prestress can rigidify pin-jointed structure (such as cable-strut system) into a structure, thereby having a decisive effect on the stability of the structure. Pellegrino (Pellegrino and Calladine, 1986; Calladine and Pellegrino, 1991) explored that the essence of structural rigidity is that the first-order infinitesimal displacement related to the second-order nonlinear deformation of the pin-jointed structure is activated by prestress. Therefore, the second-order nonlinear deformation as shown in Figure 1 should be considered to calculate the effect of prestress.

V = L - L_{0} = V_{L} + V_{N L}

(3)

where

L

and

L_{0}

represent deformed member length and initial member length;

V_{L}

and

V_{N L}

represent first-order linear deformation and second-order nonlinear deformation.

Figure 1.

Deformation of the single member (Considering second-order nonlinear deformation).

For the member i, the internal force is $t_{i} = E_{i} A_{i} V_{i} / L_{0 i}$ , where E, A are Young’s modulus and cross-sectional area of the member. The internal force vector $t ϵ R^{n_{e}}$ can be grouped in the vectorized operation:

t = \hat{E} \hat{A} {\hat{L}}_{0}^{- 1} V = \hat{E} \hat{A} {\hat{L}}_{0}^{- 1} (L - L_{0})

(4)

where

\hat{E} \hat{A} {\hat{L}}_{0}^{- 1}

reflects the axial stiffness of the member. Further, the force density and stress of all members in the structure can be assembled as follows:

q = {\hat{L}}^{- 1} t = \hat{E} \hat{A} (L_{0}^{- 1} - L^{- 1})

(5)

σ = {\hat{A}}^{- 1} t = \hat{E} {\hat{L}}_{0}^{- 1} (L - L_{0})

(6)

Substitute Equation (5) into Equation (1), the following equation can be derived:

K x = A_{1} \hat{E} \hat{A} (L_{0}^{- 1} - L^{- 1}) = F

(7)

According to Equation (7), the mechanical properties of a specific structural system is affected by the variables of $x$ , $L_{0}$ , and $A$ . Hence Equation (7) can be fully differentiated as follows:

K_{T} d x + A_{1} \hat{E} ({\hat{L}}_{0}^{- 1} - {\hat{L}}^{- 1}) d A - A_{1} \hat{E} \hat{A} {\hat{L}}_{0}^{- 2} d L_{0} = d F

(8)

where

K_{T} = \frac{\partial K x}{\partial x}

K_{T}

represents the tangent stiffness matrix of the structure.

As for boundary constraints, $a = {[a_{1}, a_{2,}, \cdot \cdot \cdot, a_{n_{a},}]}^{T} ϵ R^{n_{a}}$ and $b = {[b_{1}, b_{2,}, \cdot \cdot \cdot, b_{n_{b},}]}^{T} ϵ R^{n_{b}}$ are sequence vector of the free and constraint node. Corresponding coordinates of free nodes $x_{a} = G_{a}^{T} x$ and constraint nodes $x_{b} = G_{b}^{T} x$ can be extracted by $G_{a} {ϵ R}^{3 n_{n} \times n_{a}}$ and $G_{b} {ϵ R}^{3 n_{n} \times n_{b}}$ respectively.

{\begin{array}{c} G_{a} (:, i) = I_{3 n_{n}} (:, a_{i}) \\ G_{b} (:, i) = I_{3 n_{n}} (:, b_{i}) \end{array}

(9)

where

G_{a} (:, i)

represents the i-th column of the

G_{a}

matrix,

I_{3 n_{n}} (:, a_{i})

is the

a_{i}

-th column of identity matrix

I_{3 n_{n}}

, and

{[G}_{a} G_{b}]

is unit orthogonal matrix. Equation (9) can be sorted as follows:

x = {[\begin{array}{c} G_{a}^{T} \\ G_{b}^{T} \end{array}]}^{- 1} [\begin{array}{c} x_{a} \\ x_{b} \end{array}] = [G_{a} G_{b}] [\begin{array}{c} x_{a} \\ x_{b} \end{array}]

(10)

Therefore, with the boundary constraints the degrees of freedom is reduced to $n_{a}$ . Now that $x_{a}$ is the generalized coordinate of the structure, the following formula can be obtained:

d x = G_{a} d x_{a}

(11)

Substitute Equation (11) into Equation (8) and multiply it by $G_{a}^{T}$ , equation (12) can be obtained:

d x_{a} = K_{T a}^{- 1} G_{a}^{T} [d F - A_{1} \hat{E} ({\hat{L}}_{0}^{- 1} - {\hat{L}}^{- 1}) d A + A_{1} \hat{E} \hat{A} {\hat{L}}_{0}^{- 2} d L_{0}]

(12)

where

K_{T a}^{- 1} = {({G_{a}^{T} K}_{T} G_{a})}^{- 1}

Similarly, stress matrix $σ (x, L_{0})$ of the members can be fully differentiated in Equation (13):

\hat{E} {\hat{L}}_{0}^{- 1} \frac{\partial L}{\partial x} d x - \hat{E} {\hat{L}}_{0}^{- 2} \hat{L} d L_{0} = d σ

(13)

By virtue of Equation (3), the partial derivative of the length of the deformed member to the nodal displacement is shown below:

\frac{\partial L}{\partial x} = \frac{\partial V_{L}}{\partial x} + \frac{\partial V_{N L}}{\partial x} = B_{L} + B_{N L}

(14)

where

B_{L}

is the linear compatibility matrix of the structure, which is only related to the geometric topology and members’ spatial coordinates.

B_{N L}

is the nonlinear compatibility matrix. Substitute Equation (14) into Equation (13), we conduct the simplified result:

\hat{E} {\hat{L}}_{0}^{- 1} (B_{L} + B_{N L}) d x - \hat{E} {\hat{L}}_{0}^{- 2} \hat{L} d L_{0} = d σ

(15)

B_{L}

and the equilibrium matrix

A_{1}

are related as follows:

B_{L}^{T} = A_{1} {\hat{L}}^{- 1}

(16)

By substituting Equations (11), (12) and (16) into Equation (15), the formula with $d σ$ as the dependent variable can be derived as follows, which could measure influence of $d L_{0}$ and $d A$ on the structure:

d σ = \hat{E} {\hat{L}}_{0}^{- 1} (B_{L} + B_{N L}) {G_{a} K}_{T a}^{- 1} G_{a}^{T} [d F - A_{1} \hat{E} ({\hat{L}}_{0}^{- 1} - {\hat{L}}^{- 1}) d A + A_{1} \hat{E} \hat{A} {\hat{L}}_{0}^{- 2} d L_{0}] - \hat{E} {\hat{L}}_{0}^{- 2} \hat{L} d L_{0}

(17)

Based on Equation (6), the two variables $t (A, σ)$ should be fully differentiated to establish the relationship among $d t$ , $d A$ , and $d σ$ as follows:

d t = \hat{σ} d A + \hat{A} d σ

(18)

By substituting Equations (6), (16), and (17) into Equation (18), we derive the following result:

d t = D_{n} (B_{L} + B_{N L}) {G_{a} K}_{T a}^{- 1} G_{a}^{T} d F + [I_{n_{e}} - {D_{n} (B_{L} + B_{N L}) {G_{a} K}_{T a}^{- 1} G_{a}^{T} B}_{L}^{T}] \hat{σ} d A + D_{n} {\hat{L}}_{0}^{- 1} \hat{L} [(B_{L} + B_{N L}) {G_{a} K}_{T a}^{- 1} G_{a}^{T} B_{L}^{T} D_{n} - I_{n_{e}}] d L_{0}

(19)

where

D_{n} = \hat{A} \hat{E} {\hat{L}}_{0}^{- 1}

denotes the diagonal matrix of members’ stiffness, and the new expression can be rewritten as follows:

d t = d t_{F} + d t_{A} + d t_{L_{0}}

(20)

Equation (20) is the generalized analysis formula for the members’ internal force variation of pin-jointed structures, where $d t_{F} = D_{n} (B_{L} + B_{N L}) {G_{a} K}_{T a}^{- 1} G_{a}^{T} d F$ is the internal force variation caused by the external load, $d t_{L_{0}} = D_{n} {\hat{L}}_{0}^{- 1} \hat{L} [(B_{L} + B_{N L}) {G_{a} K}_{T a}^{- 1} G_{a}^{T} B_{L}^{T} D_{n} - I_{n_{e}}] d L_{0}$ is the internal force variation caused by the initial length error of the member, and $d t_{A} = [I_{n_{e}} - {D_{n} (B_{L} + B_{N L}) {G_{a} K}_{T a}^{- 1} G_{a}^{T} B}_{L}^{T}] \hat{σ} d A$ is the internal force variation caused by area-loss of the member. Jiang et al. (2019) studied the influence of $d L_{0}$ on structure, here we focus on the analysis of $d t_{A}$ .

Area-loss effect of members based on statistical measures

When the assumption of small deformation is introduced, it is considered that $B_{N}$ could be considered as zero, and ${d t}_{A}$ can be written as follow:

{d t}_{A} = [I_{n_{e}} - D_{n} {B_{L} {G_{a} K}_{T a}^{- 1} G_{a}^{T} B}_{L}^{T}] \hat{σ} d \hat{A}

(21)

where

R_{A} = [I_{n_{e}} - D_{n} {B_{L} {G_{a} K}_{T a}^{- 1} G_{a}^{T} B}_{L}^{T}]

is defined as the redistribution matrix of nonlinear internal force considering the area loss. It contains the information of members’ prestress, geometric topology and material parameters. Then the variation of internal force caused by members’ area loss can be further obtained by means of

R_{A}

Under steady load, when the area-loss of member i is $d A_{i}$ and the area-losses of the remaining members are zero, the internal force vector of the structure denotes as:

d t_{A i} = R_{A i} σ_{i} d A_{i} = σ_{i} d A_{i} {[r_{1 i}, r_{2 i}, \cdot \cdot \cdot, r_{j i}, \cdot \cdot \cdot, r_{n_{e} i}]}^{T}

(22)

where

σ_{i} r_{j i} d A_{i}

can be represented by

d t_{A j i}

R_{A i}

is the column vector i in the matrix

R_{A}

σ_{i}

denotes the i-th diagonal element of the member’s stress matrix

\hat{σ}

d {\tilde{A}}_{i}

is the area loss variation of member i.

r_{j i}

is the element on the j-th row and i-th column in the matrix

R_{A}

d t_{A j i}

represents the influence of the area-loss of member i on the internal force variation of member j. Therefore,

d t_{A i}

corresponds to the internal force variation of the structure caused by the area-loss of member i.

When $d A_{i} = 1$ , the area-losses of others are zero, Equation (22) can be rewritten as follows:

d {\tilde{t}}_{A i} = σ_{i} R_{A i} = σ_{i} {[r_{1 i}, r_{2 i}, \cdot \cdot \cdot, r_{j i}, \cdot \cdot \cdot, r_{n_{e} i}]}^{T}

(23)

where

d {\tilde{t}}_{A i}

stands for the variation of the internal force caused by the unit area loss of member i. And the influence of member i on member j increases with the value of

σ_{i} r_{j i}

. The variation of the internal force among single member can be easily calculated by Equation (23), but the performance of the whole structure needs to be assessed by some new coefficients and evaluation methods.

Area-loss sensitivity coefficient

The elements in vector of the internal force variation obtained by Equation (23) belong to multivariate data, which are difficult to evaluate the impact on structural performance directly. Mathematical analysis of multivariate data is usually based on the concept of distance. Here the mathematical concept of Euclidean distance (Balaji and Bapat, 2007) is introduced to measure the internal force variation of structural members caused by unit area loss of member i.

Suppose there are two points in n-dimensional space: $l = {[l_{1}, l_{2,}, \cdot \cdot \cdot, l_{n}]}^{T}$ , and $m = {[m_{1}, m_{2,}, \cdot \cdot \cdot, m_{n}]}^{T}$ , the Euclidean distance between $l$ and $m$ is shown as follows:

D_{E} (l, m) = \sqrt{\sum_{u = 1}^{n} {| (l_{u} - m_{u}) |}^{2}}

(24)

Euclidean distance is one of the most common distance measurement methods. Mathematically, it can be interpreted as the linear distance between two points, which involves all data in vector. By calculating the length of the distance, Euclidean distance can comprehensively measure the size of the vector. The distance between $d {\tilde{t}}_{A i}$ and the origin of n-dimensional space can be calculated by Equation (24) as follows:

D_{E} (d {\tilde{t}}_{A i}, 0) = \sqrt{\sum_{j = 1}^{n_{e}} {(σ_{i} r_{i j} - 0)}^{2}} = | σ_{i} | \sqrt{\sum_{j = 1}^{n_{e}} {(r_{i j})}^{2}}

(25)

where

| σ_{i} |

is the absolute value of stress of member i. Based on Equation (25), the internal force variation of structural members caused by unit area loss of member i can be intuitively compared.

Since the dimension of $D_{E} (d {\tilde{t}}_{A i}, 0)$ is consistent with that of $| σ_{i} |$ , and the stress of some members is relatively large, it is not conducive to intuitive comparison of the values, so dimensionless form of $D_{E} (d {\tilde{t}}_{A i}, 0)$ is defined as the ASC, which is shown as follows:

A S C = \frac{D_{E} (d {\tilde{t}}_{A i}, 0)}{| σ_{a v e} |} = \frac{| σ_{i} |}{| σ_{a v e} |} \sqrt{\sum_{j = 1}^{n_{e}} {(r_{i j})}^{2}}

(26)

where

| σ_{a v e} |

is the average value of

| σ_{i} |

| σ_{a v e} | = \frac{1}{n_{e}} \sum_{i = 1}^{n_{e}} | σ_{i} |

The larger the value of ASC, the greater the internal force changes in structural members. It also shows that the structure is more sensitive to the member’s area-loss. If the number of members is large, the calculation results can be normalized and then sorted in ranking. The ASC corresponding to all members should be arranged in order. The top 10% of members in the ranking (based on actual engineering situation) are defined as area-loss sensitive members. Such members are essential to the entire structure, and many aspects such as pre-manufacturing, anti-corrosion methods, and stress state of the members must be strictly controlled.

Area-loss evenness coefficient

The ASC discussed above can measure the size of the internal force variation. However, this method cannot judge the evenness of multivariate data. For structural safety, the evenness of internal force variation is also one of the important indicators. When the force-transferred path among some members in the structure is unreasonable, internal force variation caused by area loss may have large discreteness. In some special cases, the value of ASC of member i may relatively be small, that is, the area loss will not have much effect on the whole structure. However, due to the large discreteness of internal force variation, it may lead to overstress of some components in the structure. To ensure the safety of the structure, this situation should also be avoided. Therefore, the coefficient of variation (Zahra et al., 2021) is introduced to measure the evenness of internal force variation.

Suppose $l = {[l_{1}, l_{2,}, \cdot \cdot \cdot, l_{n}]}^{T}$ is a point in n-dimensional space, its standard deviation and mean value is $λ$ and $\bar{l}$ respectively. The coefficient of variation for $l$ marked as $ν_{l}$ is shown as follows:

ν_{l} = 100 \times \frac{λ}{\bar{l}} (%) = 100 \times \frac{\sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(l_{i} - \bar{l})}^{2}}}{\bar{l}}

(27)

The coefficient of variation can mathematically describe the evenness of multivariate data, which is a normalized method for degree of evenness of the probability distribution. At the same time, it is a dimensionless quantity, by which the data can be compared directly. However, its disadvantage is that when the mean value is small, the coefficient of variation will be disturbed heavily with low accuracy.

Equation (23) can be rewritten as Equation (28):

d {\tilde{t}}_{A i} = {[d {\tilde{t}}_{A 1 i}, d {\tilde{t}}_{A 2 i}, \cdot \cdot \cdot, d {\tilde{t}}_{A j i}, \cdot \cdot \cdot, d {\tilde{t}}_{A n_{e} i}]}^{T}

(28)

where

d {\tilde{t}}_{A j i}

is a scalar, which also corresponds to the influence of member i on member j.

d {\tilde{t}}_{A j i}

belongs to the entire real number domain, which means negative values may occur. So it may lead to the situation that the mean value as a denominator is small when calculating the coefficient of variation of

d {\tilde{t}}_{A j i}

directly.

To modify the above disadvantage, we take the absolute value of $d {\tilde{t}}_{A j i}$ to calculate its coefficient of variation and define it as AEC:

A E C = \frac{\sqrt{\frac{1}{n_{e} - 1} \sum_{j = 1}^{n_{e}} {(| d {\tilde{t}}_{A j i} | - \bar{| d {\tilde{t}}_{A j i} |})}^{2}}}{\bar{| d {\tilde{t}}_{A j i} |}}

(29)

where

| d {\tilde{t}}_{A j i} |

and

\bar{| d {\tilde{t}}_{A j i} |}

represents for absolute value of

d {\tilde{t}}_{A j i}

and mean value of

| d {\tilde{t}}_{A j i} |

respectively.

A smaller value of AEC corresponds to a better evenness of internal force variation, that is, the internal force variation can be uniformly transferred to other members when the area loss occurs in member i. Such member has better adaptability and resistance. Conversely, it means the internal force variation will only delivery to a few members. This may cause excessive stress of such members, thereby resulting in structural instability, partial damage, or even continuous collapse. Similar to ASC, the calculation results of AEC can be normalized and then sorted in ranking. All members in the structure should be arranged from large to small. The top 10% of members in the ranking (based on actual engineering situation) are defined as the area-loss unevenness member and additional attention should also be paid to such members.

Area-loss limit of members based on reliability theory and nonlinear programming

The above coefficients can assess the influence of internal force variation of members, but they are not directly related to structural safety. To ensure the safety of structure, it is necessary to consider the bearing capacity of members to determine the area-loss limit. $d t_{A i j}$ represents for the influence of the area-loss of member j on the internal force variation of member i based on Equation (22). Correspondingly, the elements in the i-th row vector of ${d t}_{A}$ represent all of the internal force variation for member i when the respective area loss of all members occurs. The vector is shown in Equation (30):

{(d t_{A})}_{i}^{T} = {(R_{A})}_{i}^{T} \hat{σ} d \hat{A} = [r_{i 1} σ_{1} d A_{1}, r_{i 2} σ_{2} d A_{2}, \cdot \cdot \cdot, r_{i j} σ_{j} d A_{j}, \cdot \cdot \cdot, r_{{i n}_{e}} σ_{n_{e}} d A_{n_{e}}]

(30)

where

{(R_{A})}_{i}^{T}

is the row vector of

R_{A}

r_{i j}

is the element on the i-th row and j-th column in the matrix

R_{A}

σ_{j}

and

d A_{j}

represents for the stress and area loss of member j.

From Equation (30), it can be seen that ${(d t_{A})}_{i}^{T}$ is a function of $d A_{j}$ $(i = 1,2, \cdot \cdot \cdot, n_{e})$ when $r_{i j}$ and $σ_{j}$ are calculated. Therefore, ${(d t_{A})}_{i}^{T}$ is a multivariate function, by which the area error limit of each member cannot be directly determined.

Therefore, it is necessary to supplement the safety conditions of members and the structure based on reliability theory. Further study as well as area-loss limit of members could be analyzed with above calculations and theories.

Area-loss analysis based on reliability theory

In the reliability theory (Duan et al., 2022), the structural state equation Z can be expressed by the structural resistance R and the external load effect S as Equation (31).

Z = R - S

(31)

Applying Equation (31) to multiple random variables, if the state equation of the structure contains n independent random variables ${Y_{1}, Y_{2}, {\cdot \cdot \cdot, Y}_{i}, \cdot \cdot \cdot, Y_{n}}$ , the mean value and standard deviation of $Y_{i}$ are $μ_{Y_{i}}$ and $λ_{Y_{i}}$ , respectively. The state equation of the structure can be represented by $Z = Z (Y_{1}, Y_{2}, \cdot \cdot \cdot, Y_{n})$ as Equation (32):

Z (Y_{1}, Y_{2}, \cdot \cdot \cdot, Y_{n}) = m_{0} + \sum_{i = 1}^{n} m_{i} Y_{i}

(32)

where

m_{0}

is the constant affecting Z,

m_{i}

is the weight coefficient of random variables.

The safety of the structure is represented by the reliability index β (Alvarez and Hurtado, 2014) as Equation (33):

β = \frac{m_{0} + \sum_{i = 1}^{n} m_{i} μ_{Y_{i}}}{\sqrt{\sum_{i = 1}^{n} (m_{i} λ_{Y_{i}})^{2}}}

(33)

The allowable variation of the internal force is used to calculate the limit of the members’ area-loss, and the critical state equation of the i-th member can be established as Equation (34):

Z_{i} = T_{c i} - T_{i}

(34)

where

T_{c i}

is the allowable maximum of internal force of member i, which can be determined by the type of the structure and safety monitoring requirements. And

T_{i}

is the internal force of member i after considering the area loss.

The expression for $T_{i}$ is shown as follows:

T_{i} = \bar{T_{i}} + \sum_{i = 1}^{n} m_{i j} {d A}_{i}

(35)

where

\bar{T_{i}}

is the internal force of member i regardless of the area loss,

{d A}_{i}

represents the area-loss limit of member i. And

m_{i j}

is the weight coefficient which reflect the influence of member i when the unit area loss of member j occurs.

When the area loss of the member is small relative to the structure, the performance variation of the structure is approximately linear. And the influence of the area losses of different members on the internal force of the same member can be superimposed linearly.

Introducing the area loss of each member as a random variable into the reliability formula, it can be obtained that the reliability index $β_{i}$ when the i-th member reaches the allowable variation of the internal force. $β_{i}$ is expressed in Equation (36).

β_{i} = \frac{T_{c i} - {\bar{T}}_{i} - \sum_{j = 1}^{n_{e}} m_{i j} μ_{j}}{\sqrt{\sum_{j = 1}^{n_{e}} {(m_{i j} λ_{j})}^{2}}}

(36)

where

μ_{j}

and

λ_{j}

are the mean value and standard deviation of the j-th member’s area loss, respectively.

T_{c i} - {\bar{T}}_{i}

is the allowable variation of the internal force of i-th member, it can also be treated as the alarm threshold. When the variation of internal force exceeds the alarm threshold, the relevant monitoring equipment can give warning information. The greater the value is, the higher of the area-loss limit, therefore a reasonable value should be taken according to the type of the structure and safety monitoring requirements (generally, it is recommended to be set to 5%–10% of the initial internal force).

According to reliability theory, if the structure stays safe, its reliability index needs to be greater than a certain value $\bar{β}$ , that is, $β \geq \bar{β}$ . Because $\bar{β}$ is directly related to the failure probability, the larger the $\bar{β}$ , the smaller the failure probability.

For each member, there is a reliability index $β_{i}$ . As long as the minimum value $\min (β_{i})$ is greater than $\bar{β}$ , that is $\min (β_{i}) \geq \bar{β}$ , it can ensure that the failure probability of the structure will be less than a certain value and the structure will stay safe and serviceable. The formula is shown in Equation (37):

\min (β_{i}) = \frac{∆ T_{i}}{\sqrt{\sum_{j = 1}^{n_{e}} {(m_{i j} λ_{j})}^{2}}} \geq \bar{β}

(37)

where

∆ T_{i} = T_{c i} - {\bar{T}}_{i} - \sum_{j = 1}^{n_{e}} m_{i j} μ_{j}

And Equation (37) can be rewritten as follows:

\sum_{j = 1}^{n_{e}} {(m_{i j} λ_{j})}^{2} \leq {(\frac{∆ T_{i}}{\bar{β}})}^{2}

(38)

By the same method, the reliability inequalities of each member can be obtained in turn. And they can be grouped in matrix:

\hat{M} λ \leq T_{β}

(39)

where

\hat{M} = [\begin{array}{c} \begin{array}{c} m_{11}^{2} & m_{12}^{2} \\ m_{12}^{2} & m_{22}^{2} \end{array} & \begin{array}{c} \dots \\ \dots \end{array} & \begin{array}{c} m_{1 n_{e}}^{2} \\ m_{2 n_{e}}^{2} \end{array} \\ \begin{array}{c} ⋮ & ⋮ \end{array} & ⋱ & ⋮ \\ \begin{array}{c} m_{n_{e} 1}^{2} & m_{n_{e} 2}^{2} \end{array} & \dots & m_{n_{e} n_{e}}^{2} \end{array}]

λ = [\begin{array}{c} λ_{1}^{2} \\ \begin{array}{c} λ_{2}^{2} \\ \begin{array}{c} ⋮ \\ λ_{n_{e}}^{2} \end{array} \end{array} \end{array}]

, and

T_{β} = [\begin{array}{c} {(\frac{∆ T_{1}}{\bar{β}})}^{2} \\ \begin{array}{c} {(\frac{∆ T_{2}}{\bar{β}})}^{2} \\ \begin{array}{c} ⋮ \\ {(\frac{∆ T_{n_{e}}}{\bar{β}})}^{2} \end{array} \end{array} \end{array}]

We define $\hat{M}$ as the area loss-internal force inequality coefficient matrix. To obtain $\hat{M}$ , it is necessary to calculate the weight coefficient $m_{i j}$ .

According to Equation (21), $d \hat{A}$ is a diagonal matrix composed of the area loss of each member. When all area losses of each member take unit area loss, Equation (21) is rewritten to Equation (40):

d t_{A} = (I_{n_{e}} - {{C_{n} B}_{L} K^{+} B}_{L}^{T}) \hat{σ} = R_{A} \hat{σ}

(40)

Written in matrix, Equation (41) can be expressed as follows:

R_{A} \hat{σ} = [\begin{array}{c} \begin{array}{c} σ_{1} r_{11} & σ_{2} r_{12} \\ σ_{1} r_{21} & σ_{2} r_{22} \end{array} & \begin{array}{c} \dots \\ \dots \end{array} & \begin{array}{c} σ_{n_{e}} r_{1 n_{e}} \\ σ_{n_{e}} r_{2 n_{e}} \end{array} \\ \begin{array}{c} ⋮ & ⋮ \end{array} & ⋱ & ⋮ \\ \begin{array}{c} σ_{1} r_{n_{e} 1} & σ_{2} \end{array} r_{n_{e} 2} & \dots & σ_{n_{e}} r_{n_{e} n_{e}} \end{array}]

(41)

The element of the i-th row and the j-th column in above matrix can represent the influence of the unit area loss of member j on the internal force of member i, that is, $σ_{j} r_{i j} = m_{i j}$ . All elements in $\hat{M}$ can be obtained by square each corresponding element in $d t_{A}$ matrix.

There are still two kinds of variables marked as $μ$ and $λ$ in Equation (39), which cannot be solved directly. Here, we introduce the corrosion model of common building steel to determine the mean value of each member’s area loss.

Feliu et al. (Feliu S et al., 1993) conducted statistical processing based on atmospheric corrosion and environmental data of building steels, and established the general corrosion model of them. The model is as follows:

H = U t_{e}^{n}

(42)

where

H

is corrosion depth,

t_{e}

represents for exposed time. U and n are parameters related to the environment and the type of steel.

Liang et al. (Liang and Hou, 2005), considering the domestic climatope, after 16 years of exposure corrosion experiments, fitted a corrosion model of the same form for a variety of steels, and determined the ranges of corresponding parameters. The value of U is mainly related to the environment and increases with the increase of the pollution severity. The steel type has relatively small influence on the value of U, which generally decreases with the increase of the alloy content. The value of U generally ranges from 0.02 mm to 0.10 mm. And the value of n represents the trend of corrosion, which varies greatly with types of steel and environments, which generally ranges from 0.3 to 1.89.

The mathematical model fitted by a large number of experimental data generally conforms to the normal distribution. Therefore, the mean value $μ_{i}^{*}$ ( $i = 1,2, \cdot \cdot \cdot, n_{e}$ ) can be calculated by Equation (42). In statistics, the mean value is the population mean.

According to the law of large numbers (Huang, 2021), the population mean $μ_{i}^{*}$ is approximately equal to the sample mean $μ_{i}$ . Therefore, $μ_{i}^{*}$ can be used to represent the mean value of the area loss when the i-th member is corroded. It is simplified to the inequality group of { $λ_{1}, λ_{2}, \cdot \cdot \cdot, λ_{n_{e}}$ } by substituting $μ_{i}^{*}$ into Equation (39), and the range of $λ_{i}$ ( $i = 1,2, \cdot \cdot \cdot, n_{e}$ ) could be determined.

In statistics, the Pauta criterion (Liu et al., 2018) can be used to describe the distribution of data. That is, 99.73% of the data can be contained within three times the interval of the standard deviation. According to this property, there is the following relationship among the area-loss limit $d A_{i}$ , the mean value $μ_{i}^{*}$ and standard deviation $λ_{i}$ of member i:

d A_{i} = μ_{i}^{*} + 3 λ_{i}

(43)

where mean value

μ_{i}^{*}

can be determined by Equation (42).

It can be seen that to solve the area-loss limit of members, the main problem is to determine standard deviation $λ_{i}$ of area loss.

Area-loss limit based on nonlinear programming

Since ${(d t_{A})}_{i}^{T}$ is a multivariate function, it can be treated as an optimization problem and be solved by mathematical programming (Shi et al., 2000). Based on mathematical programming, the optimal solution of the objective function can be obtained in a given feasible region. The general model of the optimization problem is as follows:

\begin{array}{l} x = {[x_{1}, x_{2}, \cdot \cdot \cdot, x_{p}]}^{T} \\ \min . (o r \max .) f_{i} (x) i = 1,2, \cdot \cdot \cdot, l \\ \begin{array}{l} s . t . g_{j} (x) \geq 0 j = 1,2, \cdot \cdot \cdot, m \\ h_{k} (x) = 0 k = 1,2, \cdot \cdot \cdot, n \end{array} \end{array}}

(44)

where

x_{1}, x_{2}, \cdot \cdot \cdot, x_{p}

are design variables;

f_{i} (x)

is the objective function;

g_{j} (x)

and

h_{k} (x)

are inequality constraint function and equality constraint function, respectively. When at least one of the objective functions or constraint functions is a nonlinear function, this kind of optimization problem belongs to nonlinear programming problem.

Taking Equation (39) as the nonlinear constraint function considering reliability, it can form a group of closed high-dimensional spatial curved surfaces in design space. And each point falling within these spatial curved surfaces represents for a combination of standard deviations of area losses that satisfies the constraints. Obviously, the closer these points are to the origin, the higher the reliability and the smaller the limit of area loss, and vice versa. Therefore, in order to calculate the area-loss limit under the premise of ensuring structural safety, it is necessary to find a point as far as possible away from the origin within the spatial curved surfaces. So this optimization problem is a single-objective nonlinear programming problem. The design variables and the objective function are established as follows:

λ = {[λ_{1}, λ_{2}, \cdot \cdot \cdot, {λ_{i}, \cdot \cdot \cdot, λ}_{n_{e}}]}^{T}

(45)

f_{1} (λ) = \min {- \sum_{i = 1}^{n_{e}} λ_{i}^{2}}

(46)

where

f_{1} (λ)

is a negative sum of variance, the smaller the function is, the larger the member’s area-loss limit is.

Based on the determination of objective function and the nonlinear constraint function considering reliability, the basic framework of nonlinear programming problem has been established. However, other constraints should be abstracted from practical problems to ensure the rationality and feasibility of the solution.

Novák et al. (Novák L and Novák D, 2021) studied coefficient of variation of the functions which contain correlated and uncorrelated random variables in practical problems, they pointed out that the characteristics of materials usually obey normal distribution and the coefficient of variation $ν \leq 0.5$ . The standard deviation is a non-negative number in the mathematical definition. Based on above properties, we can get another group of inequality constraint function named side constraints as follows:

s . t . {0 \leq λ}_{i} \leq λ_{i}^{*} (i = 1,2, \cdot \cdot \cdot, n_{e})

(47)

where

λ_{i}^{*} = ν_{i}^{*} μ_{i}^{*}

, and

ν_{i}^{*}

is the coefficient of variation corresponding to steel and corrosion environment.

Although the points within the spatial curved surfaces are the combination of standard deviation of area error satisfying the constraint conditions, the possibility of each point is not the same. Without additional constraints, the standard deviation combination of small probability may be solved.

We consider that the corrosion model in Equation (42) is obtained by the experiment of multiple groups and multiple specimens, so the mean value calculated conforms to the central limit theorem. Therefore, $μ_{i}^{*}$ and $λ_{i}^{*}$ can be treated as the mean value of maximum probability and the standard deviation of maximum probability, respectively. And the point consists of standard deviation of maximum probability marked as $P_{λ^{*}}$ is defined as the maximum probability point, ${P_{λ^{*}} = (λ}_{1}^{*}, λ_{2}^{*}, \cdot \cdot \cdot, λ_{n_{e}}^{*})$ . Another point marked as $P_{\bar{λ}}$ can be calculated from $P_{λ^{*}}$ under the condition that the distance between these two points is shortest. $P_{\bar{λ}}$ is defined as the reliable constrained confidence point and can be expressed as $P_{\bar{λ}} = ({\bar{λ}}_{1}, {\bar{λ}}_{2}, \cdot \cdot \cdot, {\bar{λ}}_{n_{e}})$ . Then the reliable constrained confidence interval can be designed based on $P_{\bar{λ}}$ to constrain the probability of the target point marked as $P_{λ}$ , $P_{λ} = (λ_{1}, λ_{2}, \cdot \cdot, λ_{n_{e}})$ . The reliable constrained confidence interval is shown in Equation (48):

s . t . {η {\bar{λ}}_{i} \leq λ}_{i} \leq {\frac{1}{η} \bar{λ}}_{i} (i = 1,2, \cdot \cdot \cdot, n_{e})

(48)

where

η

is the reliable constraint confidence coefficient, which is to constrain the discreteness among design variables. The value ranges from 0 to 1.

The size of the reliable constrained confidence interval is affected by $η$ and decreases with the increase of it. The closer the target point $P_{λ}$ to the reliable constrained confidence point $P_{\bar{λ}}$ , the higher the confidence level of the standard deviation combination. Therefore, an appropriate $η$ should be taken, and the target point constrained by above all constraint functions will satisfy the reliability and the possibility.

The standard deviation combination in the target point is the optimal solution we required, and the area-loss limit can be conveniently calculated by substituting the combination into Equation (43).

Numerical examples

In Section 3 and 4, the ASC, the AEC, and the area-loss limit based on reliability theory and nonlinear programming are proposed. To verify the rationality and accuracy of them, an algorithm is programmed by Matlab, and a triangular truss (without prestressed members) and a cable dome (with prestressed members) are taken as two examples to analyze the effect of the members area-loss on the structural performance. In addition, the analytical solution of the internal force caused by member’s area loss is also deduced in the first example. Compared with the solution, the result calculated by proposed method is verified to be valid and accurate.

Example 1. Triangular truss

The triangular truss with non-prestressed member is shown in Figure 2. It consists of 3 steel bars, which are connected by a hinged joint. The top nodes are constrained by displacement in the x-, y-, and z-directions. The elastic modulus $E$ , the yield strength $σ_{b}$ and the initial cross-sectional area $A_{0}$ of three members are the same. The bars are numbered as bar ①, bar ②, and bar ③, of which the bar ③ is vertically downward. And a vertical downward external load $F$ is applied to the hinged joint, the geometric parameters is shown in Figure 2.

Figure 2.

Triangular truss.

Relationship between internal force variation and member’s area loss

According to Equation (21), the relationship between internal force variation and member’s area loss can be quickly calculated. As a simple structure, the theoretical solution of the internal force caused by member’s area loss can be solved by mechanical analysis easily. The solution could provide a theoretical basis for verifying the accuracy of the proposed method.

The theoretical solution of the internal force of member i with initial area is as follows:

F_{i} = {\begin{array}{c} \frac{F {[L R]}_{i} {[L^{2} R]}_{3}}{\sum_{k = 1}^{3} {[L^{3} R^{2}]}_{k}} i = 1, 2 \\ \frac{F \sum_{k = 1}^{2} {[L^{3} R^{2}]}_{k}}{\sum_{k = 1}^{3} {[L^{3} R^{2}]}_{k}} i = 3 \end{array}

(49)

where

F_{i}

represents for the internal force of member i,

L

and

R

are geometric parameters shown in Figure 2, and

{[a b]}_{i} = a_{i} b_{i}

Under the premise of not exceeding the yield strength of members, it is assumed that the areas of three members are weakened to $A_{1}$ , $A_{2}$ , and $A_{3}$ in sequence, respectively.

When the above-mentioned area loss occurs in a single member j, the theoretical solution of the internal force of member i can be written as follows:

F_{i j} = {\begin{array}{c} \frac{F {[L R]}_{i} {[L^{2} R]}_{3} \cdot A_{j ʘ 3}}{\sum_{k = 1}^{3} {[L^{3} R^{2}]}_{k} \cdot A_{j ʘ k}} i = 1, 2; j = 1, 2, 3 \\ \frac{F \sum_{k = 1}^{2} {[L^{3} R^{2}]}_{k} \cdot A_{j ʘ k}}{\sum_{k = 1}^{3} {[L^{3} R^{2}]}_{k} \cdot A_{j ʘ k}} i = 3; j = 1, 2, 3 \end{array}

(50)

where

F_{i j}

represents for the internal force of member i with the area loss in member j.

A_{j ʘ k}

is a tensor representation method of member’s area and can be expressed as

A_{j ʘ k} = {\begin{array}{c} 0 j = k \\ j j \neq k \end{array}

According to Equation (49) and Equation (50), the internal force variation of member i caused by the area loss of member j can be calculated in Equation (51).

{{∆ F}_{i j} = F}_{i j} - F_{i}

(51)

Based on relationship between internal force variation and area loss of members, the ASC, the AEC, and the area-loss limit can be analyzed.

Reliability and nonlinear programming

In this example, the structure consists of three bars. According to Equations (45) and (46), the design variable can be written as $λ = {[λ_{1}, λ_{2} {, λ}_{3}]}^{T}$ , and the objective function is $f_{1} (λ) = \min {- \sum_{i = 1}^{3} λ_{i}^{2}}$ . Based on equation (41), the reliability index of the structure has the following relation:

\min {\frac{∆ T_{1}}{\sqrt{\sum_{j = 1}^{3} {(m_{1 j} λ_{j})}^{2}}}, \frac{∆ T_{2}}{\sqrt{\sum_{j = 1}^{3} {(m_{2 j} λ_{j})}^{2}}}, \frac{∆ T_{3}}{\sqrt{\sum_{j = 1}^{3} {(m_{3 j} λ_{j})}^{2}}}} \geq \bar{β}

(52)

Equation (52) is equal to ${β_{1} \geq \bar{β}} \cap {β_{2} \geq \bar{β}} \cap {β_{3} \geq \bar{β}}$ , and can be rewritten into a group of inequality equations in Equation (53).

{\begin{array}{c} \frac{m_{11}^{2} λ_{1}^{2}}{{(∆ T_{1} / \bar{β})}^{2}} + \frac{m_{12}^{2} λ_{2}^{2}}{{(∆ T_{1} / \bar{β})}^{2}} + \frac{m_{13}^{2} λ_{3}^{2}}{{(∆ T_{1} / \bar{β})}^{2}} \leq 1 (53 A) \\ \frac{m_{21}^{2} λ_{1}^{2}}{{(∆ T_{2} / \bar{β})}^{2}} + \frac{m_{22}^{2} λ_{2}^{2}}{{(∆ T_{2} / \bar{β})}^{2}} + \frac{m_{23}^{2} λ_{3}^{2}}{{(∆ T_{2} / \bar{β})}^{2}} \leq 1 (53 B) \\ \frac{m_{31}^{2} λ_{1}^{2}}{{(∆ T_{3} / \bar{β})}^{2}} + \frac{m_{32}^{2} λ_{2}^{2}}{{(∆ T_{3} / \bar{β})}^{2}} + \frac{m_{33}^{2} λ_{3}^{2}}{{(∆ T_{3} / \bar{β})}^{2}} \leq 1 (53 C) \end{array}

The solutions to the group of inequality equations are three ellipsoid regions in space, as shown in Figure 3. The overlapped ellipsoid region satisfies Equation (52), and the points within it can ensure the reliability of the structure.

Figure 3.

Feasible region of critical state equation.

Taking appropriate $μ_{i}^{*}$ and $ν_{i}^{*}$ , the side constraints of design variables can be determined by Equation (47) as follows:

{\begin{array}{c} {0 \leq λ}_{1} \leq μ_{1}^{*} ν_{1}^{*} \\ {0 \leq λ}_{2} \leq μ_{2}^{*} ν_{2}^{*} \\ {0 \leq λ}_{3} \leq μ_{3}^{*} ν_{3}^{*} \end{array}

(54)

Similarly, according to Equation (48), the reliable constrained confidence interval of the truss can be calculated and be expressed in Equation (55) when appropriate $η$ is taken.

{\begin{array}{c} {η {\bar{λ}}_{1} \leq λ}_{1} \leq {\frac{1}{η} \bar{λ}}_{1} \\ {η {\bar{λ}}_{2} \leq λ}_{2} \leq {\frac{1}{η} \bar{λ}}_{2} \\ {η {\bar{λ}}_{3} \leq λ}_{3} \leq {\frac{1}{η} \bar{λ}}_{3} \end{array}

(55)

The target point could be calculated based on the above formula. The feasible region, reliable constrained confidence interval as well as three typical points are shown in Figures 4 and 5:

Figure 4.

Feasible regions and two typical points.

Figure 5.

(a) feasible region and reliable constrained confidence interval. (b)Target feasible region and target point.

In Figure 4, the part of the overlapped ellipsoid region satisfies the requirement that the variable is positive and reliability by Equation (52). The cube is the feasible region satisfied by the side constraints in Equation (54), of which the vertex farthest from the origin point is the maximum probability point $P_{λ^{*}}$ . Taking this point as the center to make a ball which is tangent to the feasible region considering reliability, that is, the part of the overlapped ellipsoid region. The tangent point is the reliable constrained confidence point $P_{\bar{λ}}$ , and the line between the two points, as shown in Figure 4, represents for the shortest distance. The reliable constrained confidence interval calculated by $P_{\bar{λ}}$ and Equation (55) is shown by the small cube in Figure 5(a) and (b). The intersection region of the reliable constrained confidence interval and the feasible region considering reliability is the target feasible region to be solved. And the target point $P_{λ}$ farthest from the origin point within the target feasible region could be calculated as shown in Figure 5(b).

Here, a specific example of a triangular truss is given. For the triangular truss shown in Figure 2, the elastic modulus $E$ of the members is $2.06 \times 10^{5}$ MPa, the yield strength $σ_{b} = 345 M P a$ , and the initial cross-sectional area $A_{0} = 1 \times 10^{- 3} m^{2}$ . Geometric parameters of the truss are as follows: $R_{1} = 4 m$ , $R_{2} = 3 m$ , $R_{3} = 7 m$ . And a vertical downward load F = 1000N is applied to the vertex of the truss. The failure probability of the truss is taken less than 0.001‰ and the alarm threshold of the internal force is taken 10% of the initial internal force.

The internal force variation caused by the specific members’ area loss can be calculated by Equation (21). The result can be compared with the theoretical solution by Equation (51). $γ_{i j}$ is used to represent the ratio of the difference between the result calculated by the two methods to the initial internal force of members. The smaller the value of $γ_{i j}$ is, the closer the results calculated by two methods. $γ_{i j}$ is shown in the histogram, and the maximum of $γ_{i j}$ marked as $γ_{\max}$ is also shown in broken line graph in Figure 6. It can be seen that when the ratio of area loss to initial area is 30%, $γ_{\max}$ close to 3%, and meanwhile $γ_{i j}$ has a nonlinear increasing relationship with the variation of area loss. When the ration is greater than 35%, $γ_{\max}$ increases sharply and it has been greater than 4% which means a relatively large error between the two results. It can be considered that the method shown in Equation (21) is applicable to the case that the single member’s area loss does not exceed 30% of its initial area, which provides a basis for the subsequent calculation of area-loss limit.

Figure 6.

The relationship between $γ_{\max}$ and the variation in area-loss.

The ASC and the AEC are calculated to analyze the influence of member’s area loss by Equation (26) and Equation (29). According to $Φ (- β) \leq 10^{- 6}$ , $β \geq \bar{β} = 4.753$ can be calculated.

In the corrosion model, U, n, and

t_{e}

is taken as 0.08 mm, 1.2 and 12 years, respectively. The mean value

μ^{*}

of each member could be calculated by Equation (42). And the coefficient of variation

ν_{i}^{*}

is taken as 0.5, the reliable constraint confidence coefficient

η

is taken as 0.95. The standard deviation

λ_{i}

of each member could be calculated. And the area-loss limit can be finally determined by Equation (43) as shown in Table 1.

Table 1.

Area-loss effect and limit of members in triangular truss.

Member index	Stress (Pa)	Length (m)	ASC	AEC	$\frac{d A}{A_{0}}$
①	$6.622 \times 10^{5}$	5	0.490	0.263	15.523%
②	$5.619 \times 10^{5}$	5.657	0.400	0.263	18.620%
③	$1.072 \times 10^{6}$	4	0.890	0.263	11.209%

It can be seen from the results in Table 1 that the ASC of bar ③ is the largest, which means it is the most sensitive bar among all members. And bar ③ has the greatest impact on the structure in case of area loss. The sensitivity of the rest two bars is related to the distance of bar end to the end of bar ③, and the closer the distance is, the more sensitive the bar is. The AEC of three bars is almost the same, which may be due to the simple form of the structure, and the three bars are forced through the same vertex. It makes the internal force variation could be transferred uniformly when the members’ area loss occurs.

Based on reliability theory and nonlinear programming, the mean value $μ^{*}$ and the standard deviation $λ_{i}$ could be calculated. The mean value of three bars is same, $μ^{*}$ = 86.491 ${m m}^{2}$ . And the standard deviations of them are $λ_{1}$ = 22.913 ${m m}^{2}$ , $λ_{2}$ = 33.238 ${m m}^{2}$ , $λ_{3}$ = 8.533 ${m m}^{2}$ , respectively. By Equation (43), the area-loss limits of members are calculated as $d A_{1}$ = 155.23 ${m m}^{2}$ , $d A_{2}$ = 186.205 ${m m}^{2}$ , and $d A_{3}$ = 112.089 ${m m}^{2}$ , respectively. Where bar ② has the largest limit of area loss, the ratio of it to initial area is 18.62%. The $γ_{\max}$ corresponding to the ratio in Figure 6 less than 1.5, it can be seen that the results are accurate and can provide a theoretical basis for alarm threshold of members and structural healthy monitoring.

Example 2. Cable dome

Cable dome is a simple kind of pin-jointed structure with prestressed members. The cable dome in this study is circumferentially divided into 12 sections, with a span of 60 m. The layout and geometric dimensions of the dome are shown in Figure 7(a). Steel cables are used as tension members with an ultimate tensile strength of 1670 MPa, and the elastic modulus of steel cables marked as $E_{s}$ is $1.95 \times 10^{5}$ MPa. The steel bars are used as members in compression with the yield strength of 345 MPa, and the elastic modulus of steel bars marked as $E_{b}$ is $2.06 \times 10^{5}$ MPa. The geometric parameters of the structure are shown as R₁ = 10.8 m, R₂ = 9.6 m and the outermost nodes are constrained by the displacement in the x-, y-, and z-directions. In addition, a vertical downward load F₁ = 75 kN is applied to the central vertex of the structure; vertical downward loads F₂ = 45 kN are applied to the top of the middle ring of the bars; vertical downward loads F₃ = 85 kN are applied to the top of the outermost ring of the bars. According to the symmetry of dome, all members can be classified into 11 groups. The failure probability of the truss is taken less than 0.001‰ and the alarm threshold of the internal force is taken 10% of the initial internal force. The material information of members and geometric parameters are shown in Table 2 and Figure 7(b).

Figure 7.

Cable dome: (a) axonometric drawing (b) geometric parameters and external force.

Table 2.

Information of the members in cable dome.

Member group	B1	B2	B3	RC1	RC2	RC3
Number	1	12	12	12	12	12
Section area( $m^{2}$ )	$1.15 \times 10^{- 3}$	$3.92 \times 10^{- 4}$	$1.15 \times 10^{- 3}$	$5.25 \times 10^{- 4}$	$5.25 \times 10^{- 4}$	$5.25 \times 10^{- 4}$
Initial pre-force( $k N$ )	−461.53	−70.40	−159.33	368.85	480.46	688.85
Member group	DC1	DC2	DC3	HC1	HC2
Number	12	12	12	12	12
Section area( $m^{2}$ )	$5.25 \times 10^{- 4}$	$5.25 \times 10^{- 4}$	$8.39 \times 10^{- 4}$	$5.25 \times 10^{- 4}$	$1.45 \times 10^{- 3}$
Initial pre-force( $k N$ )	115.51	207.39	472.03	376.57	858.22

Relationship between internal force variation and member’s area loss

Analysis of the internal force variation of the dome caused by members’ area loss can be carried out. According to the symmetry of the structure, members in the same group have the same properties. Therefore, only one representative member in each group needs to be selected for calculation. According to Equation (21), the internal force variation of all kinds of members in dome caused by specific member’s unit area loss is shown in Figure 8.

Figure 8.

Internal force variation caused by member’s area loss in cable dome.

The plane coordinate of each point in Figure 8 represents the influence relationship between members, and the vertical coordinate of each point is the value of the internal force variation. Taking the point at row j and column i as an example, it represents that when the unit area loss occurs in member i, the internal force in member j will change by $d t_{A j i}$ .

According to Figure 8, the points with larger values in vertical coordinate are distributed among RC1, RC2, and RC3, which indicates that these three groups of cables have a greater influence on themselves.

Area loss analysis and safety assessment

The ASC and the AEC can be obtained by Equation (26) and Equation (29) as shown in Table 4.According to

Φ (- β) \leq 10^{- 6}

β \geq \bar{β} = 4.753

can be calculated. Taking the actual structure into consideration, the cables (RC1, RC2, RC3) at the top of the dome are more prone to be corroded and the exposed time of them are longer than the indoor members. And the parameters in corrosion model of bars have a little different from that of cables. The specific parameters of cable dome are shown in Table 3. And the mean value

μ^{*}

of each member could be calculated by Equation (42).

Table 3.

Parameters of corrosion model in cable dome.

Member group	U	N	$t_{e}$
B1 B2 B3	0.07	1.1	11
RC1 RC2 RC3	0.08	1.3	15
Other members	0.08	1.1	10

Table 4.

Area-loss effect and limit of members in cable dome.

Member group	ASC (norma-lized)	Ranking	AEC (norma-lized)	Ranking	$λ_{i}$ ( ${m m}^{2}$ )	$d A$ ( ${m m}^{2}$ )	$\frac{d A}{A_{0}}$
B1	0.018	10	0.275	11	21.078	121.305	10.55%
B2	0.029	9	1.000	1	11.775	68.917	17.58%
B3	0.010	11	0.588	6	21.078	121.305	10.55%
RC1	0.456	3	0.801	2	3.993	116.052	22.11%
RC2	0.520	2	0.772	3	2.325	111.049	21.15%
RC3	1.000	1	0.662	5	1.059	107.249	20.43%
DC1	0.050	8	0.682	4	13.047	79.247	15.09%
DC2	0.101	7	0.370	8	13.047	79.247	15.09%
DC3	0.208	4	0.444	7	16.829	101.397	12.09%
HC1	0.142	6	0.368	9	13.047	79.247	15.09%
HC2	0.199	5	0.357	10	22.523	134.747	9.29%

The coefficient of variation $ν_{i}^{*}$ of cables and bars is taken as 0.35 and 0.38, respectively. And reliable constraint confidence coefficient $η$ is taken as 0.95. According to above parameters and nonlinear programming, the standard deviation $λ_{i}$ of each member could be calculated. Finally, the area-loss limit can be calculated by Equation (43) as shown in Table 4.

It can be seen from Table 4, the cables of RC1, RC2, and RC3 have a relatively higher ranking of the ASC which reflects that these cables have a greater impact on the whole structure than other members. And the ASC of RC3 is the largest, so this group of cables should be defined as the area-loss sensitive members. B1, B2, and B3 have a lowest ranking of the ASC, it can be considered that the steel cables are more sensitive than bars, that is these members are more essential to the dome. In the ranking of the AEC, B2 is higher than other members. And this group of bars should be defined as the area-loss unevenness members. The cables of RC1, RC2, and RC3 also have a relatively higher ranking of the AEC, which means these members cannot uniformly transfer the internal force variation to other members. When area loss occurs in such members, specific members are prone to large variation of internal force and their safety will be reduced. Based on the ASC and the AEC, the members belong to B2, RC1, RC2, and RC3should be paid additional attention to. The pre-manufacturing error and the anti-corrosion measures for such members should be strictly controlled.

Based on reliability theory and nonlinear programming, the area-loss limits of each member can be obtained. Among them, RC1, RC2, RC3 in cables and B1, B3 in bars have a larger area-loss limit than other members. They are $d A_{B 1} = d A_{B 2}$ = 121.305 ${m m}^{2}$ , $d A_{R C 1}$ = 116.052 ${m m}^{2}$ , $d A_{R C 2}$ = 111.049 ${m m}^{2}$ , and $d A_{R C 3}$ = 107.249 ${m m}^{2}$ , respectively. The maximum ratio of the area-loss limit to initial area is 22.11%. The area-loss limit of B2 is the smallest and it is $d A_{B 2}$ = 68.917 ${m m}^{2}$ . The results could determine the area-loss limit of members which could ensure the structure works safely and internal force variation of members will less than the alarm threshold. It could provide theoretical basis for the setting alarm threshold for member’s internal force, structural healthy monitoring and anti-corrosion measures of members.

Discussion

The method proposed in Section 4 to solve the target point $P_{λ}$ is accurate but complicated. In the case of some conditions, the method can be simplified to quickly solve the target point $P_{λ}$ .

The set of critical state equations of the reliability index can be determined by Equation (38), by which the unique set of positive solutions to equations can be calculated. The set of solutions is just the coordinates of point $P_{λ}$ , $P_{λ} = (λ_{1}, λ_{2}, \cdot \cdot, λ_{n_{e}})$ . It can be easily seen that $P_{λ}$ is the farthest point from the origin point within the reliability feasible region. The maximum probability point $P_{λ^{*}}$ can be calculated by Equation (47), ${P_{λ^{*}} = (λ}_{1}^{*}, λ_{2}^{*}, \cdot \cdot \cdot, λ_{n_{e}}^{*})$ . When $\frac{λ_{1}}{λ_{1}^{*}} = \frac{λ_{2}}{λ_{2}^{*}} = \dots = \frac{λ_{n_{e}}}{λ_{n_{e}}^{*}} = c$ (c is a constant), origin point, $P_{λ}$ , and $P_{λ^{*}}$ are collinear. In this case, $P_{λ}$ is also the nearest point to $P_{λ^{*}}$ within the reliability feasible region. Therefore $P_{λ}$ is the target point as shown in Figure 9.

Figure 9.

Simplified solution of target point.

However, in practical engineering, it is difficult to meet the situation that the coordinates of $P_{λ}$ and $P_{λ^{*}}$ are completely proportional. In mathematics, the correlation coefficient is a parameter used to describe the degree of correlation between variables. The greater the correlation coefficient is, the more similar the variables are. When the correlation coefficient is 1, there is a linear correlation between variables. When the correlation coefficient of the two points is large enough (recommended to be greater than 0.95), $P_{λ}$ is within the reliable constrained confidence interval and approximately equal to the target point.

Conclusion

(1) A generalized formula is proposed for members’ internal force variation of pin-jointed structures. According to this formula, internal force variation of members can be obtained conveniently when the external force, cross-sectional area, and initial length changed respectively. The formula is validated with analytical solutions and provides a fast calculation method for area loss analysis without the complicated numerical approaches such as FEM.

(2) The area-loss sensitivity coefficient (ASC) of the member is defined for the pin-jointed structures, which can evaluate the degree of influence on the structure. Besides, the area-loss evenness coefficient (AEC) is proposed to assess the evenness of the internal force variation. Two types of essential members can be selected accordingly, which provides a theoretical basis for maintenance during the service life and selection of members in structural health monitoring.

(3) The critical state equations and the reliability inequalities of the structure are established based on the structural reliability theory. Based on the mathematical nonlinear programming, the complex multivariate data can be calculated by nonlinear optimization, and then a complete method for determining the area-loss limit of pin-jointed structure is proposed. Example analysis shows that the method is accurate and feasible which provides a scientific guidance for setting alarm threshold for member’s internal force and the determination of area-loss limit in aged structures.

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: This work was supported by the National Natural Science Foundation of China (Grant No. 51878600 and 51978605) and National Key R&D Program of China (2017YFC0806100).

ORCID iDs

Man-yu Deng

Xing-fei Yuan

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