# In-Plane Instability of Fixed Arches under Linear Temperature Gradient Field and Uniformly Distributed Radial Load

Authors: Xiaochun Song; Hanwen Lu; Airong Liu; Yonghui Huang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper focuses on an in-plane instability analysis of fixed arches under a linear temperature gradient field and a uniformly distributed radial load, which has not been reported in the literature. Combining a linear temperature gradient field and uniformly distributed radial load leads to the changes in axial expansion and curvature of arches, producing the complex in-plane nonuniform bending moment and axial force. Therefore, it is necessary to explore the in-plane thermoelastic mechanism behavior of fixed arches under a linear temperature gradient field and a uniformly distributed radial load in the in-plane instability analysis. Based on the energy method and the exact solutions of internal force before instability, the analytical solutions of the critical uniformly distributed radial load considering the linear temperature gradient field associated with in-plane thermoelastic instability of arches are derived. Comparisons show that agreements of analytical solutions against FE (finite element) results are excellent. Influences of various factors on in-plane instability are also studied. It is found that the change of the linear temperature gradient field has significant influences on the in-plane instability load. The in-plane instability load decreases as the temperature differential of the linear temperature gradient field increases.


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## Body

## 1. Introduction

In practical engineering, a fixed arch under a linear temperature gradient field and a uniformly distributed radial load caused by fire may lose its bearing capacity due to instability which has received concerns recently. Analytical investigations on the buckling of arches subject to a uniformly distributed radial load or a radial point load are extensive [1–13]. Besides the external load, arches may also be under temperature field. The variation of temperature has a great impact on the change of mechanical properties of materials, which may directly affect the safety and stability of arches. Analytical investigations for using the principle of virtual work and energy method on the linear or nonlinear in-plane instability of arches under uniformly temperature field have also been performed. Bradford [14] used the energy method to investigate the nonlinear buckling of shallow arches under fire loading. Pi and Bradford [15] used the principle of virtual work to explore linear in-plane thermoelastic behavior of arches under uniformly temperature field. Heidarpour et al. [16] studied nonlinear thermoelastic behavior of composite steel-concrete arches including partial interaction and thermal loading. Moghaddasiea and Stanciulescu [17] investigated stability boundaries and equilibria of shallow arches under static load and thermal load. Asgari et al. [18] also used the principle of virtual work to conduct nonlinear thermoelastic and in-plane instability analysis of FGM shallow arches. However, the mechanical behavior and the method for solving the instability load for arches under the temperature gradient field are different from those for arches under the uniformly temperature field, and so the temperature gradient field has not been considered in most investigations on in-plane elastic instability of arches. It appears that analytical investigations of in-plane elastic instability of arches under linear gradient temperature gradient field had rarely been reported in the open literature, and only Pi and Bradford [19] explored nonlinear thermoelastic instability of arches subject to linear temperature gradient field, and they define an effective centroid axis of the arch to obtain the effective stiffness of the cross section of the arch and to simplify the instability analysis of arches. Therefore, the instability loads of arches under the temperature gradient field can be obtained by using the method of effective centroid and principle of virtual work.When an arch is on fire, its inside temperature is higher than its outside temperature, resulting in the temperature differential which can form a temperature gradient field state [19]. The temperature gradient field produces both the curvature changes and axial expansion in fixed arches, which in turn induces combined bending and axial actions in the arch [20]. The bending and axial actions produced by the temperature gradient field may lead the arch to buckle in an in-plane symmetric mode or in an in-plane antisymmetric bifurcation mode [19]. When the arch is under linear temperature gradient field and a uniformly distributed radial load, the bending and axial actions of the arch produced by the linear temperature gradient field and the uniformly distributed radial load may also lead the arch to buckle. However, it is not known how to solve the bending moments and the axial forces of fixed arches under linear temperature gradient field and a uniformly distributed radial load, and it is also not known how the temperature gradient fields influence the in-plane instability of fixed arches. To clarify these unknowns, there are strong needs to study the in-plane instability of arches under a linear temperature gradient field and a uniformly distributed radial load.Hence, this paper aims to explore the thermoelastic behavior and in-plane instability of fixed arches under a linear temperature gradient field and a uniformly distributed radial load by using the method of effective centroid and principle of virtual work. The analytical solutions of the in-plane displacements, internal forces, and in-plane instability load of fixed arches are obtained while the influences of the gradient temperature differential, the included angle and the slenderness ratio of arches on the in-plane stability analysis of arches are considered. All analytical solutions are further verified by FE results.


## 2. Model and Assumptions

The model of the fixed circular arch is shown in Figures1(a) and 1(b). An axis system o∗x∗y∗s∗ is applied to the circular arch. The origin o∗ of the axis system o∗x∗y∗s∗ is the center of cross section in the arch crown, the axis o∗y∗ is the vertical axis of cross section, and o∗s∗ is tangent with the arch axis which is coincident with the arch axis, and pointing to the right side of arch, the axis o∗x∗ is the horizontal axis of cross section of the arch, where R, S, L, f, ϕ, and 2Φ are the radius, length, span, rise, angular coordinates, and including angle of the circular arch, respectively. b and h are the width and height of the cross section, respectively. In addition, Tu and Tb are the cross-sectional temperatures at the top and bottom fiber, respectively; q is the uniformly distributed radial load.Figure 1
Model of the arch. (a) Fixed arch. (b) Cross section.
(a)
(b)The thermoelastic and in-plane instability analyses of this paper use the following assumptions:(1)
There is no correlation between temperatureT and thermal expansion α.(2)
The deformation state and the temperature state are considered to be time-independent.(3)
The deformations of the arch satisfy the theory of Bernoulli–Euler beam.(4)
It is assumed that the section size of the arches is much smaller than the length and radius of the arches.(5)
The expansion of the cross section is assumed to be small and can be ignored in the analysis.(6)
The temperature of environment is assumed to be 20°C, and the temperature gradient is linearly distributed along the spindle axiso∗y∗ of the cross section but evenly distributed along the spindle axis o∗x∗ and the geometric centroid axis o∗s∗. Therefore, the temperature of arbitrary point p of the cross section is not the function of its coordinates x∗ and s∗ but is the function of its coordinate y∗. The temperature of arbitrary point p of the cross section can be given by(1)Ty∗=Tc+y∗ΔThwithTc=Tu+Tb2,ΔT=Tb−Tu.


## 3. Thermoelastic Analysis

### 3.1. Effective Centroid

For a steel arch, the relation between Young’s modulusE and temperature T is defined by [21](2)E=χTE20,where E20 and χT are Young’s modulus of steel at the temperature 20°C and the reduction coefficient, respectively, and χT can be expressed as(3)χT=7T−47806T−4760,for20°C≤T<600°C.The temperature distribution of the cross section is plotted in Figure2(a), and typical changes in Young’s modulus of steel vs. gradient temperature differential ΔT are plotted in Figure 2(b). Figure 2(a) shows that the temperature is distributed in a linear gradient across the cross section of the arch. Figure 2(b) shows that Young’s modulus of low-carbon steel decreases obviously as the gradient temperature differential ΔT increases.Figure 2
Distribution of temperature of cross-section and variations of Young’s modulus. (a) Linear temperature gradient. (b) The variations of Young’s modulus.
(a)
(b)The rectangular solid section shown in Figure3 is used by Pi and Bradford [19] to derive the effective centroid. According to theory of Pi and Bradford [19], it is not precise to conduct thermoelastic analysis of the arch if the axial system o∗x∗y∗s∗ composed of the geometric spindles o∗x∗ and o∗y∗ of the cross section and the geometric centroid axis o∗s∗ is used. Therefore, it is necessary to define an effective centroid axis os to accuracy and simplify the analysis. The position of the geometric centroid o∗ related to the effective axes oxy was given by [19](4)xc=−∫AEx∗,y∗x∗dAEA¯,yc=−∫AEx∗,y∗y∗dAEA¯,where EA¯ is the effective axial stiffness which can be expressed as(5)EA¯=∫AEy∗dA.Figure 3
Effective centroid and geometric centroid.The position of the effective centroido can be expressed as(6)xc∗=−xc,yc∗=−yc.In addition, the axial deformation caused by thermal expansion at the effective centroid can be expressed as(7)αTo=1A∫AαTy∗dA,where α and To are the coefficient of linear expansion and the temperature of the effective centroid, respectively, and To can be expressed as(8)To=Tc+y∗ΔTh,where Tc is the temperature of the geometric centroid.The difference valueyc between geometric centroids and the effective centroids for arches under linear temperature gradient field with the rectangular solid section are shown in Figure 4(a), where the temperatures at the top fiber Tu = 20°C. Figure 4(a) shows that the linear gradient temperature field has a significant influence on the effective centroid of the rectangular section, and yc increases as the gradient temperature differential ΔT increases.Figure 4
(a)yc vs. ΔT and (b) (Tc − To) vs. ΔT for arches under linear temperature gradient field.
(a)
(b)The differences between the effective centroid temperatureTo and the average temperature Tc for arches with a rectangular solid section under linear temperature gradient field are shown in Figure 4(b), where the temperatures at the top fiber Tu = 20°C. Figure 4(b) shows that the linear temperature gradient field has great influence on Tc − To of the rectangular solid section. The value of Tc − To increases as the gradient temperature differential ΔT increases.


### 3.2. Differential Equations of Equilibrium

The in-plane thermoelastic analysis of fixed arches under a linear temperature gradient field and a uniformly distributed radial load is presented in this section. It is known that the axial compressive force and the bending moment of arches under a linear temperature gradient field and a uniformly distributed radial load are variable. In addition, before carrying out the in-plane instability of an arch, it is essential to accurately determine the axial compressive force of the arch. Therefore, the preinstability analysis of the arch is needed. Based on the energy method, the energy differential equation for thermoelastic analysis of the arch under a linear temperature gradient field and a uniformly distributed radial load can be expressed as(9)δ∏=∫−ΦΦ∫ARσss0δεss0dAdϕ−∫−ΦΦqR2δv˜dϕ=0,where εss0 is the linear preinstability longitudinal normal strain, which can be expressed as(10)εss0=w˜′−v˜−yRv˜″+w˜′.σ
ss0 is the preinstability longitudinal normal stress, which can be expressed as(11)σss0=Eεss0−αΔTowithΔTo=To−20°C,where w˜=w/R and v˜=v/R are defined as the dimensionless axial displacement and radial displacement, respectively.Substituting equations (10) and (11) into equation (9) can be obtained as(12)δ∏=∫−ΦΦNRδw˜′−δv˜+Mδv˜″+δw˜′dϕ−∫−ΦΦqR2δv˜dϕ=0.The initial expression of axial compressive forceN and bending moment M are given by(13)N=−EA¯w˜′−v˜−αΔTo,(14)M=−EIx¯Rv˜″+w˜′+αRΔTh,where EIx¯ is the effective stiffness, which can be expressed as(15)EIx¯=∫Ay2Ey∗dA.Integrating by parts of equation (12), the differential equations can be obtained as(16)M″−N−qR2=0,(17)−M′−N′R=0.


### 3.3. Displacements

The static boundary conditions for fixed arches(18)v˜=0,w˜=0atϕ=±Φ,(19)v˜′+w˜=0atϕ=±Φ,are required. The dimensionless displacement v˜ and w˜ for arches under linear temperature gradient field and a uniformly distributed radial load can be solved by substituting equations (13), (14), (18), and (19) into equations (16) and (17):(20)v˜=EA¯αΔTo−qREA¯ΨR2+rxt2ϕsinϕ+cosϕ−cosΦΦsinΦ+Φ2cosϕcosΦ−1,(21)w˜=EA¯αΔTo−qREA¯ΨR2+rxt2ΦΦsinϕcosΦ−ϕcosϕsinΦ+2R2sinΦEA¯αΔTo−qREA¯ΨΦsinϕ−ϕsinΦ,where rxt=EI¯/EA¯.(22)Ψ=ΦR2+rxt2cosΦsinΦ+Φ−2R2sin2Φ.The distributions of the dimensionless displacementv˜ and w˜ along the arch axis given by equations (20) and (21) are plotted in Figures 5(a) and 5(b) for arches with bottom fiber temperature of cross section Tb = 50°C, 100°C, and 200°C, respectively, where the top fiber temperature of cross section Tu = 20°C, the included angle 2Φ = 60°, and S/rx=60° with rx=I/A.Figure 5
Distributions of displacement along the arch axis for arches with differentTb. (a) Radial displacement. (b) Axial displacement.
(a)
(b)Figures5(a) and 5(b) show that the dimensionless radial displacement v˜ of arches under a linear temperature gradient field and a uniformly distributed radial load are symmetrically distributed along the arch axis, while the dimensionless axial displacement w˜ of arches are antisymmetrically distributed along the arch axis. In addition, Figure 5(a) shows that the dimensionless radial displacement v˜ increases as the bottom fiber temperature of cross section Tb increases. Figure 5(b) shows that the axial displacements of the left half of the arch increases as the bottom fiber temperature of cross section Tb increases, while the dimensionless axial displacements w˜ of the right half of the arch decreases as the bottom fiber temperature of cross section Tb increases. Figure 5(b) also shows that the axial displacements at the crown and two ends are equal to zero.The distributions of the dimensionless displacementv˜ and w˜ along the arch axis given by equations (20) and (21) are plotted in Figures 6(a) and 6(b) for arches with the included angle 2Φ = 30°, 60°, and 90°, respectively, where the top fiber temperature of cross section Tu = 20°C, the bottom fiber temperature of cross section Tb = 100°C, and S/rx=100.Figure 6
Distributions of displacement along the arch axis for arches with various included angles. (a) Radial displacement. (b) Axial displacement.
(a)
(b)Similarly, Figures6(a) and 6(b) show that the dimensionless radial displacement v˜ of arches under a linear temperature gradient field and a uniformly distributed radial load are symmetrically distributed along the arch axis, while the dimensionless axial displacement w˜ of arches are antisymmetrically distributed along the arch axis. In addition, Figure 6(a) shows that the radial displacements increase as the included angle 2Φ increases. Figure 6(b) shows that the axial displacements of the left half of the arch increase as the included angle 2Φ increases, while the axial displacements of the right half of the arch decrease as the included angle 2Φ increases.


### 3.4. Internal Forces

Substituting equations (24)–(26) into equations (14) and (15) leads to the solution of axial compressive force N and bending moment M for arches under a linear temperature gradient field and a uniformly distributed radial load, which can be expressed as(23)N=qR−2rxt2ΦcosϕsinΦΨqR−EA¯αΔTo,(24)M=2rxt2sinΦΦcosϕ−sinΦRΨqR−EA¯αΔTo−EI¯αΔTh.To demonstrate the influences of the linear temperature gradient field onN and M given by equations (23) and (24). The variations of Nc with the included angle 2Φ for arches with various bottom fiber temperatures of cross section Tb (Tb = 20°C, 200°C, and 400°C) are plotted in Figure 7(a), where Nc is the central axial compressive force, and the top fiber temperature of cross section Tu = 20°C, q = 2000 kN/m. In addition, the variations of Mc with the included angle 2Φ for arches having different bottom fiber temperatures of cross section Tb (20°C, 200°C, and 400°C) are plotted in Figure 7(b), where Mc is the central bending moment and the top fiber temperature of cross section Tu = 20°C, q = 2000 kN/m.Figure 7
Influences of the temperature gradient on internal forces. (a) Axial forces. (b) Bending moments.
(a)
(b)Figure7(a) shows that Nc increases as the bottom fiber temperature of cross section Tb increases. Figure 7(a) also shows that Nc increases as the included angle 2Φ increases initially and, after that, it decreases as the included angle 2Φ increases in case the included angle of arches reaches to a certain value. Figure 7(b) shows that Mc decreases as the bottom fiber temperature of cross section Tb increases. Figure 7(b) also shows that Mc decreases as the included angle 2Φ increases initially and, after that, it increases as the included angle 2Φ increases once included angle of arches attach a certain value.To demonstrate the influences of the slenderness ratioS/rx on N and M given by equations (23) and (24), the variations of Nc with 2Φ for arches with different slenderness ratio S/rx (S/rx=50,100,150) are plotted in Figure 8(a), where the top fiber temperature of cross section Tu = 20°C and the bottom fiber temperature of cross section Tb = 100°C, q = 2000 kN/m. In addition, the variations of Mc with 2Φ for arches with different slenderness ratio S/rx (S/rx=50,100,150) are plotted in Figure 8(b), where the top fiber temperature of cross section Tu = 20°C and the bottom fiber temperature of cross section Tb = 100°C, q = 2000 kN/m.Figure 8
Influences of the slenderness ratio on the internal forces. (a) Axial forces. (b) Bending moments.
(a)
(b)Figure8 shows that both the central axial force Nc and the central bending moment Mc increase as the slenderness ratio S/rx increases.


## 3.1. Effective Centroid

For a steel arch, the relation between Young’s modulusE and temperature T is defined by [21](2)E=χTE20,where E20 and χT are Young’s modulus of steel at the temperature 20°C and the reduction coefficient, respectively, and χT can be expressed as(3)χT=7T−47806T−4760,for20°C≤T<600°C.The temperature distribution of the cross section is plotted in Figure2(a), and typical changes in Young’s modulus of steel vs. gradient temperature differential ΔT are plotted in Figure 2(b). Figure 2(a) shows that the temperature is distributed in a linear gradient across the cross section of the arch. Figure 2(b) shows that Young’s modulus of low-carbon steel decreases obviously as the gradient temperature differential ΔT increases.Figure 2
Distribution of temperature of cross-section and variations of Young’s modulus. (a) Linear temperature gradient. (b) The variations of Young’s modulus.
(a)
(b)The rectangular solid section shown in Figure3 is used by Pi and Bradford [19] to derive the effective centroid. According to theory of Pi and Bradford [19], it is not precise to conduct thermoelastic analysis of the arch if the axial system o∗x∗y∗s∗ composed of the geometric spindles o∗x∗ and o∗y∗ of the cross section and the geometric centroid axis o∗s∗ is used. Therefore, it is necessary to define an effective centroid axis os to accuracy and simplify the analysis. The position of the geometric centroid o∗ related to the effective axes oxy was given by [19](4)xc=−∫AEx∗,y∗x∗dAEA¯,yc=−∫AEx∗,y∗y∗dAEA¯,where EA¯ is the effective axial stiffness which can be expressed as(5)EA¯=∫AEy∗dA.Figure 3
Effective centroid and geometric centroid.The position of the effective centroido can be expressed as(6)xc∗=−xc,yc∗=−yc.In addition, the axial deformation caused by thermal expansion at the effective centroid can be expressed as(7)αTo=1A∫AαTy∗dA,where α and To are the coefficient of linear expansion and the temperature of the effective centroid, respectively, and To can be expressed as(8)To=Tc+y∗ΔTh,where Tc is the temperature of the geometric centroid.The difference valueyc between geometric centroids and the effective centroids for arches under linear temperature gradient field with the rectangular solid section are shown in Figure 4(a), where the temperatures at the top fiber Tu = 20°C. Figure 4(a) shows that the linear gradient temperature field has a significant influence on the effective centroid of the rectangular section, and yc increases as the gradient temperature differential ΔT increases.Figure 4
(a)yc vs. ΔT and (b) (Tc − To) vs. ΔT for arches under linear temperature gradient field.
(a)
(b)The differences between the effective centroid temperatureTo and the average temperature Tc for arches with a rectangular solid section under linear temperature gradient field are shown in Figure 4(b), where the temperatures at the top fiber Tu = 20°C. Figure 4(b) shows that the linear temperature gradient field has great influence on Tc − To of the rectangular solid section. The value of Tc − To increases as the gradient temperature differential ΔT increases.


## 3.2. Differential Equations of Equilibrium

The in-plane thermoelastic analysis of fixed arches under a linear temperature gradient field and a uniformly distributed radial load is presented in this section. It is known that the axial compressive force and the bending moment of arches under a linear temperature gradient field and a uniformly distributed radial load are variable. In addition, before carrying out the in-plane instability of an arch, it is essential to accurately determine the axial compressive force of the arch. Therefore, the preinstability analysis of the arch is needed. Based on the energy method, the energy differential equation for thermoelastic analysis of the arch under a linear temperature gradient field and a uniformly distributed radial load can be expressed as(9)δ∏=∫−ΦΦ∫ARσss0δεss0dAdϕ−∫−ΦΦqR2δv˜dϕ=0,where εss0 is the linear preinstability longitudinal normal strain, which can be expressed as(10)εss0=w˜′−v˜−yRv˜″+w˜′.σ
ss0 is the preinstability longitudinal normal stress, which can be expressed as(11)σss0=Eεss0−αΔTowithΔTo=To−20°C,where w˜=w/R and v˜=v/R are defined as the dimensionless axial displacement and radial displacement, respectively.Substituting equations (10) and (11) into equation (9) can be obtained as(12)δ∏=∫−ΦΦNRδw˜′−δv˜+Mδv˜″+δw˜′dϕ−∫−ΦΦqR2δv˜dϕ=0.The initial expression of axial compressive forceN and bending moment M are given by(13)N=−EA¯w˜′−v˜−αΔTo,(14)M=−EIx¯Rv˜″+w˜′+αRΔTh,where EIx¯ is the effective stiffness, which can be expressed as(15)EIx¯=∫Ay2Ey∗dA.Integrating by parts of equation (12), the differential equations can be obtained as(16)M″−N−qR2=0,(17)−M′−N′R=0.


## 3.3. Displacements

The static boundary conditions for fixed arches(18)v˜=0,w˜=0atϕ=±Φ,(19)v˜′+w˜=0atϕ=±Φ,are required. The dimensionless displacement v˜ and w˜ for arches under linear temperature gradient field and a uniformly distributed radial load can be solved by substituting equations (13), (14), (18), and (19) into equations (16) and (17):(20)v˜=EA¯αΔTo−qREA¯ΨR2+rxt2ϕsinϕ+cosϕ−cosΦΦsinΦ+Φ2cosϕcosΦ−1,(21)w˜=EA¯αΔTo−qREA¯ΨR2+rxt2ΦΦsinϕcosΦ−ϕcosϕsinΦ+2R2sinΦEA¯αΔTo−qREA¯ΨΦsinϕ−ϕsinΦ,where rxt=EI¯/EA¯.(22)Ψ=ΦR2+rxt2cosΦsinΦ+Φ−2R2sin2Φ.The distributions of the dimensionless displacementv˜ and w˜ along the arch axis given by equations (20) and (21) are plotted in Figures 5(a) and 5(b) for arches with bottom fiber temperature of cross section Tb = 50°C, 100°C, and 200°C, respectively, where the top fiber temperature of cross section Tu = 20°C, the included angle 2Φ = 60°, and S/rx=60° with rx=I/A.Figure 5
Distributions of displacement along the arch axis for arches with differentTb. (a) Radial displacement. (b) Axial displacement.
(a)
(b)Figures5(a) and 5(b) show that the dimensionless radial displacement v˜ of arches under a linear temperature gradient field and a uniformly distributed radial load are symmetrically distributed along the arch axis, while the dimensionless axial displacement w˜ of arches are antisymmetrically distributed along the arch axis. In addition, Figure 5(a) shows that the dimensionless radial displacement v˜ increases as the bottom fiber temperature of cross section Tb increases. Figure 5(b) shows that the axial displacements of the left half of the arch increases as the bottom fiber temperature of cross section Tb increases, while the dimensionless axial displacements w˜ of the right half of the arch decreases as the bottom fiber temperature of cross section Tb increases. Figure 5(b) also shows that the axial displacements at the crown and two ends are equal to zero.The distributions of the dimensionless displacementv˜ and w˜ along the arch axis given by equations (20) and (21) are plotted in Figures 6(a) and 6(b) for arches with the included angle 2Φ = 30°, 60°, and 90°, respectively, where the top fiber temperature of cross section Tu = 20°C, the bottom fiber temperature of cross section Tb = 100°C, and S/rx=100.Figure 6
Distributions of displacement along the arch axis for arches with various included angles. (a) Radial displacement. (b) Axial displacement.
(a)
(b)Similarly, Figures6(a) and 6(b) show that the dimensionless radial displacement v˜ of arches under a linear temperature gradient field and a uniformly distributed radial load are symmetrically distributed along the arch axis, while the dimensionless axial displacement w˜ of arches are antisymmetrically distributed along the arch axis. In addition, Figure 6(a) shows that the radial displacements increase as the included angle 2Φ increases. Figure 6(b) shows that the axial displacements of the left half of the arch increase as the included angle 2Φ increases, while the axial displacements of the right half of the arch decrease as the included angle 2Φ increases.


## 3.4. Internal Forces

Substituting equations (24)–(26) into equations (14) and (15) leads to the solution of axial compressive force N and bending moment M for arches under a linear temperature gradient field and a uniformly distributed radial load, which can be expressed as(23)N=qR−2rxt2ΦcosϕsinΦΨqR−EA¯αΔTo,(24)M=2rxt2sinΦΦcosϕ−sinΦRΨqR−EA¯αΔTo−EI¯αΔTh.To demonstrate the influences of the linear temperature gradient field onN and M given by equations (23) and (24). The variations of Nc with the included angle 2Φ for arches with various bottom fiber temperatures of cross section Tb (Tb = 20°C, 200°C, and 400°C) are plotted in Figure 7(a), where Nc is the central axial compressive force, and the top fiber temperature of cross section Tu = 20°C, q = 2000 kN/m. In addition, the variations of Mc with the included angle 2Φ for arches having different bottom fiber temperatures of cross section Tb (20°C, 200°C, and 400°C) are plotted in Figure 7(b), where Mc is the central bending moment and the top fiber temperature of cross section Tu = 20°C, q = 2000 kN/m.Figure 7
Influences of the temperature gradient on internal forces. (a) Axial forces. (b) Bending moments.
(a)
(b)Figure7(a) shows that Nc increases as the bottom fiber temperature of cross section Tb increases. Figure 7(a) also shows that Nc increases as the included angle 2Φ increases initially and, after that, it decreases as the included angle 2Φ increases in case the included angle of arches reaches to a certain value. Figure 7(b) shows that Mc decreases as the bottom fiber temperature of cross section Tb increases. Figure 7(b) also shows that Mc decreases as the included angle 2Φ increases initially and, after that, it increases as the included angle 2Φ increases once included angle of arches attach a certain value.To demonstrate the influences of the slenderness ratioS/rx on N and M given by equations (23) and (24), the variations of Nc with 2Φ for arches with different slenderness ratio S/rx (S/rx=50,100,150) are plotted in Figure 8(a), where the top fiber temperature of cross section Tu = 20°C and the bottom fiber temperature of cross section Tb = 100°C, q = 2000 kN/m. In addition, the variations of Mc with 2Φ for arches with different slenderness ratio S/rx (S/rx=50,100,150) are plotted in Figure 8(b), where the top fiber temperature of cross section Tu = 20°C and the bottom fiber temperature of cross section Tb = 100°C, q = 2000 kN/m.Figure 8
Influences of the slenderness ratio on the internal forces. (a) Axial forces. (b) Bending moments.
(a)
(b)Figure8 shows that both the central axial force Nc and the central bending moment Mc increase as the slenderness ratio S/rx increases.


## 4. Instability Analysis

From the previous analysis, the bending moment, the axial displacements, and radial displacements increase as the gradient temperature differential increases for fixed arches under the variational linear gradient thermal load and the sustained uniformly distributed radial load. This indicates that the equilibrium configuration of fixed arches under a linear temperature gradient field and a uniformly distributed radial load changes with the gradient temperature differential. This change of the equilibrium configuration may result in an instability configuration being attained in the high gradient temperature differential. When the gradient temperature differential is sufficiently high, the fixed arch may buckle under the sustained uniformly distributed radial load which is smaller than its instability load in a smaller gradient temperature differential. It has been shown that [19] antisymmetric bifurcation is the dominant instability mode for most shallow arches under a linear gradient thermal load. It has also been shown that [2] the antisymmetric bifurcation is the dominant instability mode for most nonshallow arches under a uniformly distributed radial load. Therefore, the antisymmetric bifurcation instability of the arch is studied in this section. During the antisymmetric bifurcation instability of the arch, the axial inextensibility condition in the following equation should be satisfied:(25)w˜b′−v˜b=0,where v˜b and w˜b are the dimensionless radial and axial instability displacements, respectively. In order to perform an antisymmetric bifurcation instability analysis, second-order terms of the strain v˜′+w˜2/2 need to be included in the strain of equation (10), which can be expressed as(26)ε=w˜′−v˜+12v˜′+w˜2−yRv˜″+w˜′.According to the principle of virtual work, using the condition of equation (25) and substituting the nonlinear strain given by equation (26) and the bending moment M given by equation (14) lead to the virtual work statement for instability equilibrium as(27)δΠb=∫−ΦΦEI¯Rv˜b″+v˜b′δv˜b″+v˜b′+NRv˜b′+w˜bδv˜b′dϕ=0.Integrating equation (27) into parts results in the differential equation for the dimensionless radial instability displacement as(28)v˜biv+1+ω2v˜b″+ω2v˜b=0,where(29)ω=1−NR2EI¯.The general solution of equation (28) is obtained as(30)v˜b=C1sinϕ+C2sinωϕ+C3cosϕ+C4cosωϕ.Substituting the boundary conditions of fixed archesv˜b=v˜b′=0 at ϕ=±Φ into equation (30) leads to a group of four homogeneous equations for C1, C2, C3, and C4 as(31)ΩC1C2C3C4T=0,where the coefficient matrix Ω is given by(32)Ω=sinΦsinωΦcosΦcosωΦ−sinΦ−sinωΦcosΦcosωΦcosΦωcosωΦ−sinΦ−ωsinωΦcosΦωcosωΦsinΦωsinωΦ.The existence of nontrivial solutions forC1–4 requires the determinant of the matrix Ω of equation (32) to vanish, resulting in(33)cotωΦ−ωΦcotΦΦtanωΦ−ωΦtanΦΦ=0.The antisymmetric solutions obtained from the second factor of equation (33) satisfy the axial inextensibility condition for bifurcation instability of an arch given by equation (25), which can represent bifurcation instability modes of the arch as pointed out in [15]. The lowest antisymmetric solutions of fixed one can be obtained from the second factor of equation (33) as(34)ωΦ=ηπ,where the coefficient η is related to the included angle 2Φ as shown in Figure 9.Figure 9
Coefficientη for antisymmetric bifurcation instability of fixed arches.In addition, substituting the solution (34) into equation (29) leads to lowest antisymmetric bifurcation instability load for fixed arches as(35)Ncr=EI¯ηπ2S/221−Φ2ηπ2.Substituting the axial forceN obtained from equation (23) into equation (35) leads to an equation as(36)qR−2rxt2ΦcosϕsinΦΨqR−EA¯αΔTo=EI¯ηπ2S/221−Φ2ηπ2.The lower bound and upper bound of the antisymmetric bifurcation instability loads can be given by equation (36) by setting ϕ=±Φ and ϕ=0. In addition, the average antisymmetric bifurcation instability loads can be used and given by averaging equation (23) over the arch length as(37)qcr=EI¯ΨR3Φ2ηπ2−Φ2Φ−2rxt2Φsin2Φ−2rxt2Φsin2ΦEA¯αΔToRΨ−2rxt2Φsin2Φ.Typical changes of the lower and upper bounds of the dimensionless in-plane antisymmetric bifurcation instability loads and the average dimensionless in-plane antisymmetric bifurcation instability loadsqcrR/NE2 vs. the included angle 2Φ are plotted in Figure 10. Figure 10 shows that the lower and upper bounds of the critical instability loads are very close to the critical average instability load given by equation (37) for fixed arches.Figure 10
Upper and lower bounds of the instability loads and the average dimensionless in-plane antisymmetric bifurcation instability loads. (a)10°≤2Φ<40°. (b)40°≤2Φ<180°.
(a)
(b)Typical changes of the dimensionless in-plane antisymmetric bifurcation instability loadqcrR/NE2 vs. 2Φ obtained from equation (37) for fixed arches having different bottom fiber temperatures Tb are plotted in Figure 11, where(38)NE2=E20I1.4303π2S/22.Figure 11
Antisymmetric bifurcation instability loads for arches having differentTb.In Figure11, arches with rectangular solid section have its dimensions as follows: the width b = 2000 mm and the height h = 500 mm. Besides, Poisson’s ratio ν and Young’s modulus E at T = 20°C of the material are assumed as ν = 0.32 and E20 = 200 GPa, respectively.Figure11 shows that the gradient temperature differential influences the in-plane antisymmetric bifurcation instability loads of fixed arches significantly. As expected, the dimensionless instability load qcrR/NE2 decreases as the gradient temperature differential increases, and it is also shown from Figure 11 that as the included angle 2Φ increases, the critical instability load decreases rapidly for fixed shallow arches with 2Φ < 40° and the critical instability load becomes nearly a constant when fixed nonshallow arches 2Φ > 40°.Typical changes of the dimensionless in-plane antisymmetric bifurcation instability loadqcrR/NE2 vs. 2Φ obtained from equation (37) for fixed arches with different slenderness ratios S/rx are plotted in Figure 12, where the dimensions of the rectangular solid section and the properties of the materials are the same as those in Figure 11.Figure 12
Antisymmetric bifurcation instability loads for arches having different slenderness ratios.Figure12 shows that the slenderness ratio S/rx influences the in-plane antisymmetric bifurcation instability loads of fixed arches significantly. As expected, the dimensionless instability load qcrR/NE2 decreases obviously as the slenderness ratio S/rx increases for shallow arches about 2Φ < 60° and decreases slightly as the slenderness ratio S/rx increases for nonshallow arches about 2Φ > 60°.


## 5. Comparisons with FE Results

To verify the validity and accuracy of analytical solutions of axial force and bending moment and antisymmetric bifurcation instability loads for fixed arches under a linear temperature gradient field and a uniformly distributed radial load, the scheme of analytical results comparison is adopted with the FE results simulated by ANSYS. In the FE analysis, a rectangular solid section was used and its dimensions are the widthb = 2000 mm and the height h = 500 mm. Besides, Poisson’s ratio ν and Young’s modulus E with T = 20°C of the steel are set as ν = 0.3 and E20 = 200 GPa, respectively. Beam 3 of ANSYS is a linear 2-node beam element in 2-D, which is used to formulate the FE model. Convergence studies were demonstrated to show that 300 beams and 3 elements can get accurate converged results. Hence, the arch is divided into 300 elements. In addition, according to equation (2), Young’s modulus of the steel decreases as temperature increases, and so the relation of Young’s modulus and temperature needs to be set in ANSYS based on equation (2).The analytical solutions of the axial force and the bending moment obtained from equations (23) and (24) for arches under linear temperature gradient field and a uniformly distributed radial load are compared with the FE results in Figure 13(a) as the variations of Nc with the included angle 2Φ for arches with the slenderness ratio S/rx=100,150, respectively, and in Figure 13(b) as the variations of Mc with 2Φ for arches having the slenderness ratio S/rx=100and150, respectively, where Tu = 20°C and Tb = 200°C. Figure 13 shows that the analytical solutions of the axial force and the bending moment agree with the FE results very well.Figure 13
Comparisons of analytical solutions with FE results for internal forces. (a) Bending moments. (b) Axial forces.
(a)
(b)The analytical solutions of the in-plane antisymmetric bifurcation instability load obtained from equation (37) for arches under a linear temperature gradient field and uniformly distributed radial load are compared with the FE results as the variations of the dimensionless instability load qcrR/NE2 with 2Φ in Figure 14(a) for arches with the slenderness ratio S/rx=100 and in Figure 14(b) for arches with the slenderness ratio S/rx=150, where Tu = 20°C and Tb = 200°C.Figure 14
Comparisons of analytical solutions with FE results for critical instability load. (a)S/rx=100. (b)S/rx=150.
(a)
(b)Above all, the analytical solutions of the critical instability load agree with the FE results very well for fixed shallow arches (2Φ < 90°), and the analytical solutions of the instability load agree with the FE results not excellent for fixed nonshallow arches (2Φ > 90°) because the assumption of the axial inextensibility condition is used in instability analysis. For the instability analysis of shallow arches (2Φ < 90°), the assumption is perfectly satisfied, but for the instability analysis of nonshallow arches (2Φ > 90°), the assumption is not perfectly satisfied, and the partial extensibility of the section of nonshallow arches exists. Hence, the analytical solutions can be used to predict the in-plane antisymmetric bifurcation instability load accurately for shallow fixed arches and approximately for nonshallow fixed arches under a linear temperature gradient field and a uniformly distributed radial load.


## 6. Conclusion

This paper presented the in-plane antisymmetric bifurcation instability of fixed circular arches under a linear temperature gradient field and a uniformly distributed radial load. To derive the critical instability load, accurate preinstability thermoelastic analyses for solutions of the displacements and the internal forces were conducted. The following was found:(1)
The radial displacements of arches under a linear temperature gradient field and a uniformly distributed radial load are symmetrically distributed along the arch axis, while the axial displacements are antisymmetrically distributed along the arch axis. In addition, the radial displacement increases as the bottom fiber temperature of cross section increases. The axial displacements of the left half of the arch increases as the bottom fiber temperature of cross section increases, while the axial displacements of the right half of the arch decreases as the bottom fiber temperature of cross section increases.(2)
The central axial force increases as the bottom fiber temperature of cross sectionTb increases, while the central bending moment decreases as the bottom fiber temperature of cross section Tb increases.(3)
The central axial force increases as the slenderness ratioS/rx increases, while the central bending moment increases as the slenderness ratio S/rx increases.Based on classical instability theory and accurate preinstability internal forces, the analytical solutions of in-plane antisymmetric bifurcation instability load for arches under a linear temperature gradient field and a uniformly distributed radial load were obtained. The following was found:(1)
The lower and upper bounds of the critical instability loads are very close to the critical average instability load(2)
The critical instability load of an arch decreases as the gradient temperature differential increases(3)
The critical instability load of an arch decreases obviously as the slenderness ratioS/rx increases for shallow arches about 2Φ < 60° and decreases slightly as the slenderness ratio S/rx increases for nonshallow arches about 2Φ > 60°In addition, the analytical solutions of the axial force and the critical instability load agreed with the FE results very well. It was indicated that the analytical solutions for the in-plane antisymmetric bifurcation instability load of fixed arches derived in this paper can be used to predict the in-plane antisymmetric bifurcation instability load of both shallow and nonshallow arches under a linear temperature gradient field and a uniformly distributed radial load.This paper mainly deepens the academic understanding of the in-plane thermoelastic instability of fixed arches under linear temperature gradient field and a uniformly distributed radial load. In engineering practice, arches are usually influenced by the gradient temperature field, so it is worth investigating the in-plane instability of arches considering the effect of the gradient temperature field. Of course, in addition to the linear temperature gradient field, there are also nonlinear temperature gradient fields on hollow sections such as box section and circular pipe section. The in-plane thermal instability of such arches considering the influences of shear deformation is worth to be researched in the future.



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*Source: 5938030-2019-08-18.xml*