Arches have been used extensively in civil and ocean engineering load-bearing structures. The stability of arch structures has attracted considerable attention from both research and engineering communities, and numerous studies on the nonlinear instability behavior of arches can be found in the open literature (Pi and Bradford, 2004a; Pi and Bradford, 2004b; Pi et al., 2011; Liu et al., 2018a; Liu et al., 2018b; He et al., 2020; Liu et al., 2020). In practical engineering, a steel arch may be subjected to thermo-mechanical load that leads to instability of the arch. It is well known that the material property of the steel is dependent on the temperature (Standard Australia, 1998; Heidarpour et al., 2010a; Maruschak et al., 2012; Moghaddasie and Stanciulescu, 2013; Yasniy et al., 2013; Heidarpour et al., 2014; Wang et al., 2020). When the applied temperature is uniformly distributed over the whole arch, the variations of the material property of steels caused by the temperature is uniform, and the steel arch is still isotropic structure. So the classical analysis method can be used to determine the structural behavior of the arch. A considerable amount of literature has been published on the linear or nonlinear behavior of arches subjected to uniform thermal load. Bradford et al. (Bradford, 2006a; Bradford, 2006b; Bradford, 2010) studied nonlinear instability behavior of a pinned arch under uniform thermal loading only. They found that the axial force in the arch produced by temperature is uniform along the arch axis. Pi et al. (Pi and Bradford, 2010a; Pi and Bradford, 2010b; Pi and Bradford, 2014) conducted a series of studies regarding the nonlinear in-plane thermo-elastic behavior of the crown-pinned, pin-ended and fixed steel arch. Heidarpour et al. (2010b) proposed a generic model to investigate the nonlinear elastic behavior of steel-concrete arches subjected to sustained loading at elevated temperatures. Lu et al. (2019) derived the exact solutions for the out-of-plane buckling load of the steel arch in a thermal environment incorporating shear deformations. The dynamic buckling of an arch subjected to transient thermal loading is studied by Keibolahi et al. (2018), from which they verified that the dynamic snap through buckling of the arch may occur when under rapid surface heating. Virgin et al. (2014) considered the influence of temperature and proved that temperature had a great influence on the extreme point of instability of arch. Malekzadeh et al. (2009) investigated the in-plane free vibrations of FGM arches under thermal environment by adopting two-dimensional elasticity theory and first-order shear deformation theory (FSDT). Yang and Huang (Yang et al., 2019a; Yang et al., 2019b; Yang et al., 2020a; Yang et al., 2020b; Yang et al., 2020c). presented an analytical solution of nonlinear in-plane buckling for a novel graphene reinforced composite arch/beam under mechanical/thermal loading, and determined the special parameter switching the buckling mode of the arch.

However, when the applied temperature is gradient distributed along one or two direction of the arch, the variations of the material property of steels caused by the temperature is non-uniform, and so the arch becomes anisotropic structure. The existence of anisotropy and involvement of geometric nonlinearity makes the problem of structural stability of the steel arch more complicated. So far, there has been very limit research investigating the nonlinear instability behavior of arches under gradient thermo-mechanical loadings. Cai et al. (2012) studied the elastic buckling of parabolic arches with rotational constraint under a vertical uniform load and uniform/gradient temperature changes. Pi et al. (Pi and Bradford, 2010c) developed a research on the nonlinear thermo-elastic buckling of pinned arches under temperature gradient field, and they also studied the out-of-plane buckling of fixed and rotationally restrained slender beam under linear gradient thermal action (Pi and Bradford, 2008; Pi and Bradford, 2010a; Pi and Bradford, 2015). Bateni and Eslami. (2014) discussed the effects of temperature gradient on the buckling of FGM arches and concluded that the temperature gradient enhances the buckling resistance of arches under a lateral mechanical load. Similar researches were also observed from Babaei’s report for the Thermo-mechanical nonlinear in-plane analysis of fix-ended FMG shallow arches (Babaei et al., 2018a; Babaei et al., 2018b). Song et al. (2019) carried out a linear analysis on the instability of steel arch under linear temperature gradient field and uniformly distributed radial load according to the effective centroid method and the principle of virtual work. To the best of authors knowledge, no work has been done on the nonlinear behavior and buckling of steel arches under gradient thermal and uniform radial loading.

Therefore, an analytical study is carried out in this paper to investigate the structural responses and specifically the buckling behavior of arches under gradient thermo-mechanical loadings. The principle of virtual work and mid-plane plane formulations are employed to derive the nonlinear equilibrium paths and critical buckling loads. The temperature-dependent geometric parameters determining the buckling modes of the arch are also obtained. The effects of gradient on the structural response, such as radial and axial displacement, internal force and critical buckling loads are discussed.

As shown in Figure 1, a fixed arch with center angle 2 Θ , arc length S and radius R is subjected to gradient thermal load and uniform radial load, where v and w are the radial and axial displacements in the mid-plane of the arch, respectively. By introducing the linear gradient temperature field, a temperature field T(z) varies continuously along the thickness of the cross section of the arch but uniform over the entire length of the arch.

Before the structural analysis, the following assumptions are adopted as: 1) deformations of the arch satisfy the Euler-Bernoulli hypothesis; 2) expansions of the cross section are ignored; 3) the states of temperature is treated as time-independent; 4) the coefficient of thermal expansion α is independent of the temperature.

According to classic shallow arch theory, the nonlinear strain-displacement relation of the arch is given as ε = w ˜ ′ − v ˜ + v ˜ ′ 2 2 − z v ˜ ″ R (1) where (   ) ′ ≡ d (   ) / d θ , v ˜ = v/R and w ˜ = w/R.

Therefore, the nonlinear stress of the arch under gradient temperature T(z) is σ = E ( z ) { ε − α [ T ( z ) − 20 ] } (2) where the reference temperature is 20°C, the linear gradient temperature T(z) is formulated as T ( z ) = T 1 + T 2 2 + z Δ T h   with   Δ T = T 2 − T 1 (3) where T ₁ and T ₂ are the temperature at the top and bottom of cross section. E(z) is the temperature-dependent elasticity modulus of steel according to the Australian Code AS4100 (Heidarpour et al., 2010a) which is given as E ( z ) = { E 20 + E 20 T ( z ) 2000 ⁡ ln [ T ( z ) / 1100 ]   for   0   o C < Δ T ≤ 600   o C E 20 690 [ 1 − T ( z ) / 1100 ] T ( z ) − 53.5   for   600   o C < Δ T ≤ 1000   o C (4) where E ₂₀ is the elastic modulus of steel at 20°C.

According to the principle of virtual work and mid-plane plane formulations, the nonlinear equilibrium equation of the arch is stated as δ K = ∫ − Θ Θ [ − N R ( δ w ˜ ′ − δ v ˜ + v ˜ ′ δ v ˜ ′ ) − M δ v ˜ ″ − q R 2 δ v ˜ ]   d θ = 0 (5) in which, axial compressive force N and bending moment M are formulated as N = − A 11 [ w ˜ ′ − v ˜ + v ˜ ′ 2 2 − α ( T 1 + T 2 2 − 20 ) ] + B 11 ( v ˜ ″ R + α Δ T h ) (6) M = B 11 [ w ˜ ′ − v ˜ + v ˜ ′ 2 2 − α ( T 1 + T 2 2 − 20 ) ] − D 11 ( v ˜ ″ R + α Δ T h ) (7) where A ₁₁, B ₁₁ and D ₁₁ are the stiffness components defined as { A 11   B 11   D 11 } = b ∫ − h / 2 h / 2 E ( z ) { 1   z   z 2 }   dz (8)

By substituting Eq. 6 into Eq. 7 leads to a new expression of Eq. 7 as M = − ( v ˜ ″ R + α Δ T h ) κ − B 11 A 11 N   with   κ = D 11 − B 11 2 A 11 (9)

Integrating Eq. 5 by parts obtains the equilibrium equation as ( N R ) ′ = 0 (10) N R + ( N R v ˜ ′ ) ′ − M ″ − q R 2 = 0 (11) and also obtains the boundary conditions of the fixed arch as v ˜ = w ˜ = v ˜ ′ = 0 when θ = ± Θ .

By substituting Eqs 9, 10 into Eq. 11, the equilibrium equation of arches is rewritten as v ˜ i v μ 2 + v ˜ ″ = Q   with   Q = q R − N N (12) where μ is defined as μ 2 = N R 2 κ (13)

By solving Eq. 12 and considering the fixed-end boundary conditions, obtains v ˜ = Q μ 2 [ μ 2 θ 2 − β 2 2 + β ( cos ⁡ μ θ − cos ⁡ β ) sin ⁡ β ]   with   β = μ Θ (14)

By substituting Eq. 14 into Eq. 10, the axial displacement w ˜ is solved as w ˜ = Q θ ( 1 − Q ) ( θ 2 − Θ 2 ) 6 + Q B 11 A 11 R ( θ − Θ ⁡ sin ⁡ μ θ sin ⁡ β ) + Q ( 1 + Q ) μ 2 ( Θ ⁡ sin ⁡ μ θ sin ⁡ β − θ ) + Q 2 Θ 2 ⁡ sin ⁡ μ θ ⁡ cos ⁡ μ θ 4 μ ⁡ sin 2 ⁡ β + Q 2 Θ ( 3 ⁡ cos ⁡ β − 4 ⁡ cos ⁡ μ θ ) θ 4 μ ⁡ sin ⁡ β (15)

Substituting Eqs 14, 15 into Eqss 6, 7, the exact expression of the axial compressive force N and bending moment M of the arch are determined as N = − A 11 Q 2 Θ 2 4 ⁡ sin 2 ⁡ β − A 11 Q ( 2 + Q ) Θ 2 6 + A 11 Q ( 1 + Q ) μ 2 − A 11 Q ( 3 P + 4 ) Θ 4 μ ⁡ tan ⁡ β + α A 11 ( T 1 + T 2 2 − 20 ) + B 11 α Δ T h (16) M = B 11 Q 2 Θ 2 4 ⁡ sin 2 ⁡ β + B 11 Q ( 2 + Q ) Θ 2 6 + B 11 2 Q A 11 R ( 1 − β ⁡ cos ⁡ μ θ sin ⁡ β ) + B 11 Q ( 3 P + 4 ) Θ 4 μ ⁡ tan ⁡ β − B 11 Q ( 1 + Q ) μ 2 + D 11 Q R ( β ⁡ cos ⁡ μ θ sin ⁡ β − 1 ) − α B 11 ( T 1 + T 2 2 − 20 ) − D 11 α Δ T h (17)

Eqs 10, 16 shows that the axial compressive force of the arch is constant through the arch when the gradient temperature is given.

By substituting Eq. 14 into Eq. 6 and integrating it over the entire length of the arch, the nonlinear equilibrium equation is obtained as A 1 Q 2 + B 1 Q + C 1 = 0 (18) where A 1 = 5 12 ⁡ sin 2 ⁡ β − cos 2 ⁡ β 6 ⁡ sin 2 ⁡ β + 3 ⁡ cos ⁡ β 4 β ⁡ sin ⁡ β − 1 β 2 (19) B 1 = 1 3 + 1 β ⁡ tan ⁡ β − 1 β 2 (20) C 1 = r 11 2 β 2 λ 2 − α S 2 4 λ 2 h 2 ( T 1 + T 2 2 − 20 + B 11 Δ T A 11 h ) (21) where r 11 = κ / A 11 / h , λ is a geometrical parameter defined as λ = S h Θ 2 (22)

By combining Eqs 14, 18, the solutions of the nonlinear equilibrium conditions of the fixed arch under gradient thermo-mechanical loadings.

Typical symmetric limit point buckling may occur when the arch is subjected to a gradient thermal and uniform radial load. When this happens, the arch reaches a limit point load and buckle in a limit point instability mode. The limit point loads are located at the local extrema of the nonlinear equilibrium path defined in Eq. 18, and the equation of equilibrium between limit point loads and axial force parameter β can be determined by using partial derivative method as A 2 Q 2 + B 2 Q + C 2 = 0 (23) where A 2 = β ∂ A 1 ∂ β − 4 A 1 ,   B 2 = − 4 A 1 + β ∂ B 1 ∂ β − 2 B 1 ,   C 2 = − 2 B 1 + β ∂ C 1 ∂ β (24)

Solving Eqs 18, 23 simultaneously, the critical axial force parameter β and the limit point buckling load Q can be obtained, respectively.

Beyond symmetric limit point buckling, the anti-symmetric bifurcation buckling may also occur to the arch under gradient thermal and uniform radial load. The differential equation for bifurcation equilibrium of the arch is determined by the author previous research as (Heidarpour et al., 2010b) v ˜ b i v μ 2 + v ˜ ″ b = 0 (25) where v ˜ b is the bifurcation buckling displacement. Then, a characteristic equation can be determined by solving Eq. 25 incorporating the fixed-ended boundary conditions of the arch as ( sin ⁡ β − β ⁡ cos ⁡ β ) sin ⁡ β = 0 (26)

When sin ⁡ β − β ⁡ cos ⁡ β = 0 in Eq. 26, leads to a critical axial force β b parameter for bifurcation buckling as β b = μ b Θ = 1.4303 π (27)

Substituting β b into Eq. 13 leads to the axial compressive force N _b of the arch as N b = ( 1.4303 π ) 2 κ ( S / 2 ) 2 (28) and substituting β b into Eq. 18 leads to the nonlinear equilibrium equation for bifurcation buckling as 0.416664 Q 2 + 0.333331 Q + ( 1.4303 π ) 2 r 11 2 λ 2 − α S 2 4 λ 2 h 2 ( T 1 + T 2 2 − 20 + B 11 Δ T A 11 h ) = 0 (29) which is a mono basic quadratic equation, and the existence of real solutions for Eq. 29 requires that 0.333331 2 − 4 × 0.416664 × [ ( 1.4303 π ) 2 r 11 2 λ 2 − α S 2 4 λ 2 h 2 ( T 1 + T 2 2 − 20 + B 11 Δ T A 11 h ) ] ≥ 0 (30) from which a critical parameter λ b 1 defined as the minimum geometric parameter triggering bifurcation buckling of the arch is solved as λ b 1 = 302.8644722 r 11 2 − 3 . 750026 α S 2 h 2 ( T 1 + T 2 2 − 20 + B 11 Δ T A 11 h ) (31)

When sin ⁡ β = 0 in Eq. 26, leads to a critical axial force parameter β s for lowest buckling of the arch as β s = μ s Θ = π (32)

Substituting β s into Eq. 18 and solving the lowest buckling load as q R = π 2 κ ( S / 2 ) 2 (33)

When the axial force parameter β approaches to critical axial force parameter β s and leads to a real solutions for the lowest buckling load given by Eq. 33, the condition need to be satisfied as lim β → β s ( B 1 2 − 4 A 1 C 1 ) = 4 π 4 r 11 2 λ 2 − α π 2 S 2 λ 2 h 2 ( T 1 + T 2 2 − 20 + B 11 Δ T A 11 h ) − 4 ≥ 0 (34)

Similarly, a critical parameter λ s triggering lowest buckling of the arch can be solved from Eq. 34 as λ s = r 11 2 π 4 − α S 2 π 2 4 h 2 ( T 1 + T 2 2 − 20 + B 11 Δ T A 11 h ) (35)

When the arch having λ < λ s under a given gradient temperature, there is no buckling for the arch instead of nonlinear bending.

The nonlinear elastic behavior and buckling of shallow arch under gradient thermal and uniform radial loading has been studied in this paper. Exact solutions for the structural responses, internal force and critical buckling loads of the arch have been derived by adopting the principle of virtual work and mid-plane plane formulations. The effects of gradient temperature on the radial displacement, axial displacement, internal force and critical buckling loads have been discussed comprehensively. From the numerical discussion, the effects of the gradient temperature are quite significantly on the structural behavior of the arch. The arch deflects in its concave direction and buckles when a critical temperature gradient and external load are reached. It is also found that the arch under gradient thermal and uniform radial loadings can buckle in a limit point instability mode or a bifurcation mode. The critical gradient temperature change Δ T can be determined by setting the limit point load obtained from Eq. 23 to be equal to the bifurcation buckling load obtained from Eq. 29 at β = 1.4303 π , from which a critical parameter λ b 2 that is switching the buckling mode can also be solved when a gradient temperature is given. The value of three critical parameters λ b 1 , λ b 2 , and λ s decreases with an increase of the gradient temperature change Δ T or slenderness S/h.