Advanced Analytical Methods for Fatigue Assessment of Ancillary Systems in Highway Bridges

Abstract

Fatigue failure is a major concern for highway sign structures due to sustained wind-loading events, which have been recognized in many states. To ensure public safety, AASHTO specifies that structures should be designed for infinite life by maintaining wind-induced stress below their constant amplitude fatigue threshold (CAFT). However, existing structures that were not designed for fatigue may contain unnoticed fatigue cracks that are difficult to detect through visual inspection, which is also time-consuming. To address this issue, a simplified analytical inspection tool was developed and implemented into computer software. The tool assesses all critical components according to AASHTO specifications for fatigue and was used to examine a failed structure, which revealed a fatigue damage crack in the vertical weld of the mast-to-arm box connection at the upper chord level. In addition, a spatial interpolation technique was proposed using Isoparametric finite element shape functions to derive wind speed records for unsampled locations from actual data recorded at known locations. This provides a better understanding of the wind events that might be the driving source for fatigue failure of these flexible structures and facilitates fatigue-life prediction by generating a full range of wind loading. Overall, this chapter contributes to improving the safety and efficiency of highway sign structures by providing effective inspection tools and wind-speed interpolation techniques.

1. Introduction

Full-span overhead, cantilever, and butterfly sign support structures are critical ancillary systems on highways that play a crucial role in enhancing accessibility, efficiency, and safety of traffic flow. These systems are responsible for supporting large traffic signs that provide essential information to the public and traffic users. However, these structures have been in service for 30–45 years, and are not designed to withstand fatigue damage resulting from various loading scenarios, such as natural wind gusts, vortex shedding, and truck-induced vibrations [1, 2, 3]. Fatigue damage often occurs in the connection details of these systems, which can lead to catastrophic failures, resulting in injuries, property damage, and traffic closures. As a result, the safety concerns associated with fatigue damage have prompted a comprehensive investigation to accurately simulate and calculate the extent of fatigue damage in these three ancillary systems, following the latest AASHTO standards.

1.1 Objective of chapter

This chapter aims to present an analytical inspection tool that utilizes natural wind-time histories and AASHTO fatigue life calculation procedures to estimate fatigue life consumption in critical spots of cantilevered, overhead, and double-cantilevered (Butterfly) structures. The focus will be on analytical modeling, structural assessment, and software development, with an emphasis on the challenges of wind load event approximation and the importance of accurate lifetime calculations to minimize the unreliability of fatigue life predictions.

2. Source of induced vibrations

Highway sign structures are critical components of the transportation infrastructure, providing safety and guidance to drivers on highways. These structures are slender and tall and are subject to diverse environmental and dynamic loading conditions, which can lead to vibrations and fatigue damage [4]. Identifying and comprehending the sources of induced vibrations that can cause fatigue damage in highway sign structures is crucial for their safety and longevity. Wind loads and traffic loads, are the main sources of induced vibrations that can cause fatigue damage in these structures. Wind-induced vibrations, in particular, are the primary source of fatigue damage in highway sign structures, while traffic-induced vibrations can also contribute to fatigue damage. Furthermore, galloping and vortex shedding are two important phenomena that can cause fatigue damage in these structures as well [5, 6, 7]. Galloping occurs when wind gusts cause the sign structure to oscillate in a twisting motion, potentially leading to high-cycle fatigue damage. Vortex shedding, on the other hand, results from wind flow around a structure, creating vortices that can cause the structure to transversely vibrate, leading to low-cycle fatigue damage. Thus, accounting for the effects of galloping and vortex shedding is crucial when evaluating the potential sources of induced vibrations and fatigue damage in highway sign structures. Proper design, maintenance, and inspection practices that consider these phenomena can aid in the prevention or mitigation of fatigue damage in these structures. However, this chapter will primarily focus on the fatigue damage caused by natural wind.

3. Natural wind modeling

Accurately estimating the wind loading events over the lifetime of a structure is crucial in reducing the uncertainties associated with fatigue life calculations. However, it is challenging to accurately characterize the wind loading events that a structure experiences during its service life, as this requires a stress fluctuation spectrum, which is often difficult to obtain. To overcome this challenge, the developing analytical natural wind-time histories is a reliable method for estimating wind speed fluctuations at specific locations and quantifying the number of stress cycles associated with each stress level that the structure experiences.

3.1 Spatial wind speed interpolation

The Isoparametric finite element shape functions can be used to estimate wind speed values for unsampled locations by interpolation from surrounding sampled locations. This process involves conventional finite element analysis procedures such as meshing, shape function evaluation, and solving for unknowns. The generated wind speed records, including mean and high speeds, are used to produce a synthetic daily wind time history. Rainflow analysis is then applied to convert the irregular wind speed history into a constant-amplitude cycle-loading. This produces a comprehensive database of wind speed versus a number of cycles for fatigue analysis, which is the main input for the fatigue calculation algorithm. The algorithm calculates the damage associated with each stress level resulting from each wind speed value in the database. Let us consider the state of Kansas as an example to demonstrate the procedure, with more details available in Refs. [4, 8, 9]. After discretizing the domain into the finite number of elements to utilize all the available data, the Kansas map was divided into 12 geometrical interpolation zones to cover the entire domain, combined with quadrilateral and triangular shapes, as shown in Figure 1.

Figure 1.

Kansas interpolation zones.

The coordinates of the center of main cities (element corners) and central coordinates of the county within the element were obtained using ArcGIS [10] in terms of latitude and longitude. The county’s coordinates could be written as a linear combination of the four corner coordinates using the Isoparametric shape functions as in Eq. (1). The shape functions were used to express the coordinates of the center of the county in terms of the coordinates of the cities in each element.

$$\left\{\begin{array}{c}X\\ Y\end{array}\right\}=\left[\begin{array}{ccc}{X}_1& X_2& X_3\\ Y_1& Y_2& Y_3\end{array}\begin{array}{c}{X}_4\\ Y_4\end{array}\right]\left\{\begin{array}{c}\begin{array}{c}{N}_1\\ N_2\end{array}\\ N_3\\ N_4\end{array}\right\}$$E1

where the Ni are the FE shape functions as expressed in Eq. (2) and Eq. (3) for quadrilateral and triangular elements, respectively.

$$\left\{\begin{array}{c}\begin{array}{c}{N}_1=\frac{1}{4}\left(1-\xi \right)\left(1-\eta \right)\\ N_2=\frac{1}{4}\left(1+\xi \right)\left(1-\eta \right)\end{array}\\ N_3=\frac{1}{4}\left(1+\xi \right)\left(1+\eta \right)\\ N_4=\frac{1}{4}\left(1-r\right)\left(1+\eta \right)\end{array}\right\}$$E2

$$\left\{\begin{array}{c}{N}_1=\frac{1}{4}\left(1-\xi \right)\left(1-\eta \right)\\ N_2=\frac{1}{4}\left(1+\xi \right)\left(1-\eta \right)\\ N_3=\frac{1}{2}\left(1+\eta \right)\end{array}\right\}$$E3

Using the unity property of the shape functions, Eq. (1) could be re-written as

$$\left\{\begin{array}{c}\begin{array}{c}X\\ Y\end{array}\\ 1\end{array}\right\}=\left[\begin{array}{ccc}{X}_1& X_2& X_3\\ Y_{\begin{array}{c}1\\ 1\end{array}}& Y_{\begin{array}{c}2\\ 1\end{array}}& Y_{\begin{array}{c}3\\ 1\end{array}}\end{array}\begin{array}{c}{X}_4\\ Y_{\begin{array}{c}4\\ 1\end{array}}\end{array}\right]\left\{\begin{array}{c}\begin{array}{c}{N}_1\\ N_2\end{array}\\ N_3\\ N_4\end{array}\right\}$$E4

This is an underdetermined system of equations; we wish to derive a fourth continuity equation using the physical area of the element. The total area of a quadrilateral element is composed of a summation of the area of four triangles intersected at an arbitrary point inside the element at (X, Y), which represents the coordinates of the county as shown in Figure 2.

Figure 2.

Area decomposition of a quadrilateral element into four triangles.

The area of the quadrilateral could be calculated using the Gauss quadrature formula in two-dimensional regions as

$$A_Q=\underset{-1}{\overset{1}{\int }}\underset{-1}{\overset{1}{\int }}\left|\mathbf{J}\left(\xi ,\eta \right)\right|\mathit{\text{dξdη}}=\sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j\left|\mathbf{J}\left({\xi }_i,{\eta }_j\right)\right|$$E5

where$\left|\mathbf{J}\left(\xi ,\eta \right)\right|$ is the determinant of the Jacobian matrix evaluated at the integration points $\left(\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}\right)$, and $w_i=1$, $w_j=1$, and the Jacobian matrix is given by Eq. (6).

$$\left[\mathbf{J}\right]=\left[\begin{array}{cc}\frac{\partial x}{\partial \xi }& \frac{\partial y}{\partial \xi }\\ \frac{\partial x}{\partial \eta }& \frac{\partial y}{\partial \eta }\end{array}\right]=\frac{1}{4}\left[\begin{array}{cccc}-\left(1-\eta \right)& \left(1-\eta \right)& \left(1+\eta \right)& -\left(1+\eta \right)\\ -\left(1-\xi \right)& -\left(1+\xi \right)& \left(1+\xi \right)& \left(1-\xi \right)\end{array}\right]\left[\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\\ x_3& y_3\\ x_4& y_4\end{array}\right]$$E6

evaluating Eq. (5) and Eq. (6) yields

$$A_Q=\left(\frac{1}{2}{x}_4\left(y_1-y_3\right)+\frac{1}{2}{y}_2\left(-y_1+y_3\right)+\frac{1}{2}{x}_1\left(y_2-y_4\right)+\frac{1}{2}{x}_3\left(-y_2+y_4\right)\right)$$E7

At the same time, the area of a triangle is given by Eq. (8)

$${\mathbf{A}}_{\mathbf{T}}=0.5\ast \left|\begin{array}{ccc}1& x_i& y_i\\ 1& x_j& y_j\\ 1& x& y\end{array}\right|$$E8

where ($x_i,y_i),\left(x_j,y_j\right),$the coordinates of the base and they change for each triangle while the ($x,y)$ are the coordinates of the apex and they are constant for all the triangles. The area of all triangles could be written as:

$${\mathbf{A}}_{\mathbf{1}}=\frac{1}{2}\left|\begin{array}{ccc}1& x_1& y_1\\ 1& x_2& y_2\\ 1& x& y\end{array}\right|$$

$${\mathbf{A}}_{\mathbf{3}}=\frac{1}{2}\left|\begin{array}{ccc}1& x_3& y_3\\ 1& x_4& y_4\\ 1& x& y\end{array}\right|$$

$${\mathbf{A}}_{\mathbf{2}}=\frac{1}{2}\left|\begin{array}{ccc}1& x_2& y_2\\ 1& x_3& y_3\\ 1& x& y\end{array}\right|$$

$${\mathbf{A}}_{\mathbf{4}}=\frac{1}{2}\left|\begin{array}{ccc}1& x_4& y_4\\ 1& x_1& y_1\\ 1& x& y\end{array}\right|$$E9

Replacing the x in A₁ by

$$x=\sum _{i=1}^4{x}_i{N}_i$$E10

Then we have

$$A_Q-A_2-A_3-A_4=\frac{1}{2}\left|\begin{array}{ccc}1& x_1& y_1\\ 1& x_2& y_2\\ 1& \sum _{i=1}^4{x}_i{N}_i& y\end{array}\right|$$E11

After substituting Eq. (9) and Eq. (10) into Eq. (11) and rearranging terms, we get

$$m=A{N}_1+B{N}_2+C{N}_3+D{N}_4$$E12

where:

$$m=\frac{1}{2}\left(x{y}_1-x{y}_2\right)$$

$$A=\frac{1}{2}{x}_1\left(y_1-y_2\right)$$

$$B=\frac{1}{2}{x}_2\left(y_1-y_2\right)$$

$$C=\frac{1}{2}{x}_3\left(y_1-y_2\right)$$

$$D=\frac{1}{2}{x}_4\left(y_1-y_2\right)$$

Then the Eq. (4) could be re-written as

$$\left\{\begin{array}{c}\begin{array}{c}X\\ Y\end{array}\\ 1\\ \frac{1}{2}\left(x{y}_1-x{y}_2\right)\end{array}\right\}=\stackrel{\mathbf{A}}{\overbrace{\left[\begin{array}{cccc}{x}_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ 1& 1& 1& 1\\ \frac{1}{2}{x}_1\left(y_1-y_2\right)& \frac{1}{2}{x}_2\left(y_1-y_2\right)& \frac{1}{2}{x}_3\left(y_1-y_2\right)& \frac{1}{2}{x}_4\left(y_1-y_2\right)\end{array}\right]}}\left\{\begin{array}{c}\begin{array}{c}{N}_1\\ N_2\end{array}\\ N_3\\ N_4\end{array}\right\}$$E13

Eq. (13) cannot be solved directly since it produces a singular matrix. However, the Moore-Penrose [11] inverse A⁺ could be calculated easily for this square matrix. Eq. (13) is written in compact form as

$$\mathit{b}=\mathbf{A}x$$E14

The Pseudo-inverse of the singular matrix A is:

$${\mathbf{A}}^{+}={\left({\mathbf{A}}^{\mathbf{T}}\mathbf{A}\right)}^{-1}{\mathbf{A}}^{\mathbf{T}}$$E15

And the solution of Eq. (13) will therefore be approximately obtained as.

$$\mathit{x}={\mathbf{A}}^{+}\mathit{b}$$E16

The solution x, in this case, is not exact. Instead, it minimizes the quantity$‖\overrightarrow{b}-A\overrightarrow{x}‖.$By knowing the county center coordinates X and Y and the nodal cities’ coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄), the shape functions (N₁, N₂, N₃, N₄) could be calculated by solving Eq. (16). After calculating the shape functions {$N_i$}, medium and high wind speeds were interpolated using the nodal values for the element surrounding the county as follows:

$$\mathrm{HWS}=N_1\ast {\mathit{HWS}}_1+N_2\ast {\mathit{HWS}}_2+N_3\ast {\mathit{HWS}}_3+N_4\ast {\mathit{HWS}}_4$$E17

$$\mathrm{MWS}=N_1\ast {\mathit{MWS}}_1+N_2\ast {\mathit{MWS}}_2+N_3\ast {\mathit{MWS}}_3+N_4\ast {\mathit{MWS}}_4$$E18

where HWS and MWS are the interpolated high and mean wind speeds for each given day in a county, HWS_i and MWS_i are the actual high and mean wind speeds for the same day for the corner city of the element considered. It is worth mentioning that this work is not intended to produce wind-speed surfaces. Instead, it produces discrete wind speed records at the center of each county. For the triangular elements, the county coordinates could be obtained easily by inverting the coordinates matrix and multiplying it by the county coordinates vector.

3.2 Synthetic wind-time histories

The spatial and temporal variation of wind velocity has two components: a daily mean component U(z) and daily fluctuating component u (z, t), expressed through U (z, t) = U(z) + u (z, t), where U (z, t) is the varying wind speed profile during the day [12]. Because wind is a random process with dynamic behavior that cannot be entirely predicted, several researchers examined different techniques like the autoregressive (AR) model, or the real wind accord to simulate the wind fluctuation with good accuracy [13, 14]. However, for the purpose of this algorithm, it is more suitable to use the well-established Kaimal spectrum [15] to simulate the power spectral density. Eq. (19) and the weighted amplitude wave superposition represented by Eq. (20) [16] was used to generate daily time history for the entire time span.

$$S_K\left(f\right)=\frac{200U_{\ast }^{2}z}{U_z{\left(1+50\frac{\mathit{fz}}{U_z}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$E19

where $S_K$ is the Kaimal spectrum, z is the height above the ground (10 m (33 ft.)), $U_{\ast }$is shear velocity, $U_z$ is the mean wind velocity at z, and f is the specified frequency.

$$u\left(t\right)=\sum _{i=1}^N\sqrt{2S_i{f}_i∆f}.cos\left(2\pi f_{i}t+{\varnothing }_i\right)$$E20

In Eq. (20), a sensitivity calculation was performed in which 798, 80, and 40 cosine waves were used to build synthetic wind speed histories for the city of Wichita over a 45-year period and extract the number of wind cycles corresponding to each speed. The Rainflow method was used to establish the distribution of speed versus the number of cycles for the three cosine-wave sets (Figure 3). The figure shows that the overall distribution was almost identical, and the cycle variation followed a Gaussian distribution. Discretization using 80 waves was an excellent trade-off between computational speed and accuracy of results to generate the 45-year wind database. More details are in Ref. [1]. Table 1 summarizes the main parameters used in the final wind-speed simulation. Figure 4 shows a sample of generated wind-time histories for various mean wind speeds.

Figure 3.

Speed vs. number of cycles for 45 years in Wichita for different cosine-wave sets.

Parameter	Value
Surface roughness class	Open terrain (k = 0.005)
Height above ground	33 ft.
$U_z$	Vary
$U_{\mathit{max}}$	Vary
Fluctuation wind speed spectrum	Kaimal
Length of time history	One day
Time step	1 s
Frequency range	3–300 Hz
Number of cosine waves in superposition	80

Table 1.

Main parameters in wind-speed simulation.

Figure 4.

Wind-time histories for various mean wind speeds.

The complete procedures for generating single daily time history are shown in Figure 5 and were implemented in C# code to produce a 45-year database of wind-time histories and daily synthetic wind profiles for all counties in Kansas. After generating the database, the Rainflow counting technique [17, 18] described in detail in Section 5 was implemented to convert the irregular wind-time histories into a usable number of constant amplitude cycles.

Figure 5.

Flowchart for wind-time histories generation.

4. Structural modeling

The finite element software Staad Pro V8i SS6 [19] was chosen to model the flexible sign structures and execute the static analysis. The structures were modeled using a combination of beam elements, truss elements, and plate elements with appropriate cross-sectional dimensions. The upper and lower main chords were modeled using a 2-node frame element, and they ran continuously. At the same time, the secondary members were connected at the intersection nodes and released to act as a pin connection. Moreover, the main truss chords are connected to the plates that make up both types of connections, old and new. The base support is simulated as a fixed support. In addition to that, the material used is steel with an elastic modulus of 200 GPa (29,000 ksi) for the cantilevered structures.

4.1 Mast-to-arm connection

The gusseted box connection was used in some structural models, while the ring-stiffened connection was used for other models. A 4-noded plate elements were used to model all the plates in the connection and attached directly to the main upper and lower chords at the center node. The steel plates have the same thickness as the actual structures and are connected to the pole through another plate with the same fillet weld thickness and properties. The pole was divided into sub-elements, and the plate nodes merged with pole nodes, as shown in Figure 6. A convergence study was conducted to verify that the model converged and to find out the mesh size that provides a mesh-independent solution for the critical stress.

Figure 6.

(a) post-to-chord connection (b) The model.

4.2 Sign(s) modeling

The highway sign structures carry the sign(s) on the truss, and the sign placement varies with location, length, and height of the sign and the number of the total signs. The sign is modeled in the structure by creating a frame attached to the truss using four elements, and the sign ribs are distributed within the frame depending on the number and spacing of the ribs. The wind load was calculated based on the size of the sign then distributed equally over the ribs. Since the sign location is sensitive to the cantilever structure, the frame and the ribs are created on the structure truss at their respective location in the actual structure, Figure 7 shows the geometry and details of the overhead structures [20].

Figure 7.

(a and b) Geometry and details of the overhead truss model (c) Model of four-chord box truss.

4.3 Dynamic amplification factor (DAF)

This study utilized the static solution for certain analytical models. Due to the dynamic nature of the wind, the calculated stresses were amplified using an overall blanket average (DAF). The analytical wind modeling was carried out over a range of frequencies [3–300 HZ], assuming harmonic excitations the DAF was calculated by averaging the frequency-response curve, shown in Figure 8 for this particular range of frequencies as in Eq. (21).

Figure 8.

Frequency-response curve and average DAF $\left(\mathrm{\xi }=0.02\right)$.

$$\mathit{DAF}=\frac{\underset{0}{\overset{1.4}{\int }}\frac{\mathit{dR}}{\sqrt{{\left(1-R^2\right)}^2+{\left(2\mathit{\xi R}\right)}^2}}}{1.4}$$E21

where $\xi$ is the damping ratio and R = $\frac{\omega }{{\omega }_n},\omega :\text{the excitation frequency},{\omega }_n:\text{natural frequency of the strucutre}.$

4.4 Wind loading calculations

The wind loading resulting from certain wind speed was evaluated by calculating the wind pressure using AASHTO 2015 [19], Eq. (22).

$$\mathit{Pz}=0.00256K_{z}G{V}^2{I}_r{C}_d\left(\mathit{psf}\right)$$E22

where K_z is the height and exposure factor calculated based on the height of the member and conservatively taken not to be less than 1.0, if the structure is located on a bridge, this value is taken to be 1.3. G is the gust factor = 1.14, V is the applied wind velocity (mph), I_r is the importance factor = 1.0. The drag coefficient (C_d) was considered based on the object size and shape. For the truss members, the value of C_d was taken to be 1.2, while for the signs, the value of C_d was determined based on the aspect ratio. After generating the pressure resulting from each wind speed, the pressure is multiplied by the surface area where it is applied to generate wind force.

5. The concept of fatigue and cycles counting

Metal fatigue is a phenomenon where a structure or its component undergoes damage or failure prematurely due to repetitive loading. To assess and predict the extent of fatigue damage, various analysis methods have been developed. The AASHTO Specification, for instance, uses the stress-life method to estimate fatigue lives of full-span highway sign support structures. A critical tool in this method is the S-N diagram, which plots alternating stress range versus cycles to failure. These diagrams are developed using experimental fatigue test data for different materials and structural components [21].

To generate the wind-time histories for the 45 years of data, an irregular variation of speed with time was observed. The Rainflow counting technique, developed by Matsuishi and Endo [18], was adapted to extract each wind speed cycle. This technique identifies closed hysteresis loops in a non-periodic stress response and converts irregular time histories to cycles. The algorithm borrowed from ASTM E1049 [17] was implemented into a computer code to extract the cycle database for 45 years. By implementing the Rainflow counting technique for each daily wind-time history, a speed-cycle matrix was generated that represented the number of cycles for each wind speed in a day, grouped in 0.5 mph range scale. This approach helps in predicting the extent of fatigue damage caused by wind loading in a structure, Figure 9 shows an example of the Rainflow analysis.

Figure 9.

Rainflow counting example.

6. Stress analysis and damage calculation

6.1 AASHTO S-N curves

AASHTO manual [22] provides S-N curves for different connection types based on a wide range of laboratory fatigue tests of full-scale structures. Eq. (23) could express the number of cycles to failure, where N_i is the number of cycles to failure at i-th stress range,$∆{\sigma }_i$is the member stress value corresponding to a wind speed value. A’ is a constant associated with the component provided in AASHTO manual [22].

$$N_i=\frac{A^{\prime }}{∆{\sigma }_i^3}$$E23

Table 2 describes the S-N equation for different components that were used in this study. Each steel S-N curve has a flat plateau described as the threshold in the table. Below this value, stress is assumed to have no contribution to the cumulative damage and the components have infinite life [23].

Table 2.

S-N equations for different structure components used by AASHTO.

6.2 Fatigue damage and miner rule

Once the structural analysis is completed for all the wind speeds in the time period where the structure is investigated, the member end forces resulting from the FE solution were collected for each structural component associated with a wind speed. The axial stress was calculated in each critical component, as indicated in Table 3. The resulting stress was amplified using the DAF. Then, the calculated axial stress in each critical member is associated with an appropriate S-N curve from the AASHTO to calculate the needed number of cycles to failure. Moreover, the fractional damage was calculated by using Miner rule Eq. (24) to estimate the damage consumption by finding the ratio of the number. of stress cycles experienced by the structure to the number of cycles required for failure. As indicated by the S-N curve, only the stresses greater than the threshold were assumed to contribute to the damage.

Table 3.

Stress calculation in different structure spots.

$$D_i=\frac{n_i}{N_i}$$E24

where D_i is the damage in a specific member at a particular stress range, n_i is the number of cycles at i-th stress range, obtained from Rainflow analysis, N_i is the number of cycles to failure at the same stress range obtained from the S-N curve. In the scope of this work, wind time histories were generated for the 45 years of data [1], these histories represent highly irregular variations of speed with time. Rainflow counting technique [18] was used to convert the irregular time histories to cycles.

$$D=\sum _i{D}_i$$E25

The cumulative damage was determined by adding all the fractional damages associated with each wind speed using Eq. (25).

6.3 Case study: Cantilever structure

A cantilever highway sign structure that has been in service for 32 years was selected to test the validity of the developed approach. The structure is located in Sedgwick County, Kansas, over northbound I-235 at ramp to West Street. The structure consisted of three panels spaced at 7 ft. (2133.6 mm) and supported over a single tapered pole that has a total height of 27 ft. (8229.6 mm) and base outer diameter OD of 16 in. (406.4 mm). The main truss has design model 1 properties and consisted of multiple angle sections $3^{\prime \prime }\times 3^{\prime \prime }\times {\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}^{\prime \prime }$connected by welded angle-to-gusset connections. The full geometric details are shown Figure 10 and the general structural properties are presented in Table 4.

Figure 10.

Sedgwick structure geometry (all dimensions are in feet/inches).

Structural data		Original project data		Sign and attachment
Structure type	Cantilever	Date let	1987	height (ft.)	length (ft.)
material	Steel	Inspection Date	2019	6.5	12.5
Arm truss span	30 ft.			2.5	7.5
Vertical clearance	18 ft.

Table 4.

Sedgwick structure information.

On October 30, 2019, Kansas Department of Transportation (KDOT) had performed a comprehensive field inspection to assess the condition of all the structure components as part of their regular inspection plans. The visual inspection revealed different corrosion levels for the entire structure, 50% corrosion staining was observed on the anchor bolts hardware, and full corrosion staining on the full height of the pole. In addition to that, the connection plates have corrosion staining reflected through corrosion bleed-out emanating from the weld copings. A complete fatigue crack in the vertical weld of mast-to-arm box connection at one side of the upper chord level was also observed. Close-up view of the vertical column-to-mast arm connection is shown in Figure 11. Obviously, the crack occurred in the entire weld toe resulting in a complete separation of the vertical splice plate from the column.

Figure 11.

Crack in the weld toe in Sedgwick structure, courtesy of KDOT. (Bureau of structural and geotechnical services).

AASHTO 2015 Structural Supports for Highway Sign LTS specifies an infinite life for both mast-arm-to-pole connections, namely, the gusseted box connection, and the ring-stiffened box if they were detailed as per the AASHTO 2015 recommendations. These connections were tested experimentally in full size and they did not develop any fatigue cracking under both in-plane and out-of-plane loading scenarios. However, in all the tested specimens, the fatigue cracking occurred in other critical locations such as the tube-to-transverse-plate welds in the mast arm, the pole, and/or hand holes [22, 24]. The connection between the side plate and the pole falls under the category of E` details in AASHTO specification having a CAFT of 2.6 ksi (18 MPa). The wind loading event for the whole structure’s service life revealed a range of wind speeds (1–33 mph, 1.6–53 km/h) with the corresponding number of cycles that the structure might experience. After providing the software with the necessary information, including an approximated average corrosion reduction factor of about 17%, which reflects the existing conditions of this structure, the software starts to build and run the analysis. Upon performing the required successive analyses by STAAD pro, the software read back and classified the end forces for each component. The fatigue engine evaluated the fatigue life consumption for each component in the model and displayed the results in the results screen. Based on Miner’s rule, the stressed component approaches the end of its life when the cumulative damage index exceeds unity. Thus, the end of fatigue life was detected only for the Mast-to-arm connection colored in red with a damage index of 1.116. The stress variation with wind speed for the connection along with the wind speed cycles were plotted and shown in Figure 12. The stress life method assumed that the stresses greater than CAFT (18 MPa or 2.6 ksi for this connection) contribute to the fatigue damage accumulation. From the plot, only the stresses resulting from the wind speeds (21–33 mph, 34–53 km/h) will cause fatigue damage. The number of cycles to failure associated with each stress is calculated using the S-N equation with A’ = 3.29 × 10⁸ ksi. Table 5 shows a sample fatigue damage calculation as per Miner’s rule. The possibility of this complete fatigue crack is likely due to the poor quality of the weld, some defects are very likely to exist resulting in local stress concentration yielding rapid fatigue damage accumulation. Moreover, the harsh corrosion environment introduced discontinuities along the weld length which resulted in significantly lower fatigue resistance. It is important to note here that the ancillary structure under consideration was designed 32 years prior to the inspection year when no fatigue design provisions were available. Another important reason to note here is the fact that the geographical area plays a vital rule in the analysis results since the variation in fatigue life is extremely correlated to the difference in wind environment in various sites, Sedgwick County is known for the strong wind records and many ancillary structures have shown a different level of wind-related distress as per KDOT. In addition, the wind-induced fatigue damage was evaluated for all different critical spots in the structure, namely, pole-to-base plate weld connection, chord-to-transverse plate weld connection, anchor bolts, and all the truss members. The fatigue lives of all the previous details were found to be infinite for this particular structure. This is attributed to the fact that at lower wind speeds, all the stresses experienced by these details are below the threshold stress thus no fatigue damage was developed. At the same time, under higher wind speeds, the stresses are higher than the threshold causing a tendency to develop fatigue damage, but the number of cycles is low to an extent not causing this damage to be significant resulting in extended fatigue life.

Figure 12.

Stress variation with wind speed in the connection.

Speed (mph)	Stress (ksi)	Ni (Cycle)	ni (Cycle)	Di
21	2.82	17421692.22	3,647,732	0.2094
22	3.17	12235811.27	2,582,522	0.2111
23	3.42	9730332.478	2,102,996	0.2161
24	3.67	7864766.583	1,105,565	0.1406
25	4.03	5975640.433	590647.5	0.0988
26	4.28	4981930.229	260,106	0.0522
27	4.53	4196883.267	149,004	0.0355
28	5.08	2969547.639	133981.5	0.0451
29	5.33	2568838.623	138694.5	0.0540
30	5.74	2065094.187	102,642	0.0497
31	6.19	1644141.092	40,464	0.0246
32	6.49	1425231.089	4365	0.0031
33	6.84	1216291.56	405	0.0003
Sum = 1.116

Table 5.

Stress and damage in connection associated to each wind speed.

7. Process generalization

The synthetic wind time histories generated can be extended to any location with updated wind speed records using the same spatial interpolation technique. In cases where the wind speed records are insufficient, temporal interpolation can also be utilized to provide a more comprehensive understanding of wind fluctuations throughout the life of the structure.

While bridge failures can occur due to various reasons, the study of vibrations in a bridge’s structural integrity provides insight into how it will perform under severe stress from vibration loading. Similar to highway sign structures, bridge structures are also susceptible to sustained wind loading events that can lead to fatigue failure over time.

To improve the safety and efficiency of bridge structures, effective inspection tools and wind-speed interpolation techniques compliant with AASHTO specifications for fatigue can be developed. This process of calculating fatigue damage in highway sign structures can be generalized to bridge structures. By implementing these techniques, the public’s safety and economic stability can be ensured.

8. Conclusion and recommendation

The aim of this chapter is to establish an analytical framework for evaluating fatigue damage in highway sign structures and to develop computer software that can serve as a cost-effective inspection tool. Analytical models were created for a cantilever structure by utilizing past wind event histories to simulate damage in different critical truss components. The key finding from this study is that the pole-to-mast arm connection at the weld is the most critical fatigue detail due to its lower CAFT, which makes it susceptible to fatigue damage accumulation. The prevalence of fatigue failure in flexible highway structures underscores the importance of timely identification of faulty connections. As such, the developed software is expected to have a significant impact on state highway decision-making and inspection inventories.

Acknowledgments

This work was supported by Kansas Department of Transportation, Bureau of structural and geotechnical services through the grant KTRAN: KSU-20-3 (KDOT).

Nomenclature

$\xi$	Damping ratio
${\omega }_n$	Natural frequency of the structure
$\omega$	Excitation frequency
$R$	Frequency ratio
$U_z$	the mean wind velocity at height z
f	Specified frequency
$U_{\ast }$	Shear velocity
${\sigma }_u^2$	variance of the turbulent wind component
$\gamma$	Scale factor
$t$	Time in seconds
$K_z$	Height and exposure factor
$G$	Gust factor
$I_r$	Importance factor
$C_d$	Drag coefficient
$V$	Wind velocity
$N_i$	the number of cycles to failure at i-th stress range
$A^{\prime }$	Stress constant associated with structural component
$∆{\sigma }_i$	Member stress
$D_i$	Incremental damage in member
$D$	Total damage in member
$n_i$	Wind speed cycles
${\sigma }_{x1}$	The axial stress in first quadrant in the main members
${\sigma }_{x2}$	The axial stress in second quadrant in the main members
${\sigma }_{x3}$	The axial stress in third quadrant in the main members
${\sigma }_{x4}$	The axial stress in fourth quadrant in the main members
$\mathit{My}$	The moment about y-axis
$\mathit{Mz}$	The moment about z-axis
$2C_3$	The moment arm
$r_o$	Outer radius
$r_i$	Inner radius
${\sigma }_x$	The axial stress in the pole element
$\mathit{Fx}$	Axial force in the pole
$A$	Cross sectional area of the pole
$I$	Second moment of area
$t_m$	Thickness of the main hollow member
$t_{w3}$	Thickness of the main members weld
$d_b$	Diameter of the bolts in the mast-arm connection
$G$	Half of the width of the connection plate
$J$	Half of the hight of the connection plate
$T$	Tension force in the main members
$I_{\mathit{yy}}$	the moment of inertia along the vertical axis passing through the centroid of the secondary element.
$I_{\mathit{yz}}$	The product of inertia
$I_{\mathit{zz}}$	the moment of inertia along the axis perpendicular to both horizontal and vertical axis through centroid of the secondary element
$y$	Vertical distance from the centroid to the point where the stress is calculated
$z$	Horizontal distance from the centroid to the point where the stress is calculated
$t_p$	Thickness of the plate
$t_{w1}$	Inner weld thickness in the base plate
$t_{w2}$	outer weld thickness in the base plate
${\sigma }_w$	Total axial stress in the weld
$S_x$	Membrane stress in the plate
$t_w$	Weld thickness in the ring stiffened connection
$w_i$	Weight function
$\mathbf{J}$	Jacobian matrix
${\mathbf{A}}^{+}$	Moore-Penrose matrix
$S_K\left(f\right)$	Kaimal spectrum
$f$	specified frequency
${\varnothing }_i$	Phase angle
k	Surface roughness class