Out-of-plane elastic buckling load and strength design of space truss arch with a rectangular section

Abstract

The out-of-plane stability of the two-hinged space truss circular arch with a rectangular section is theoretically and numerically investigated in this paper. Firstly, the flexural stiffness and torsional stiffness of space truss arches are deduced. The calculation formula of out-of-plane elastic buckling loads of the space truss arch is derived based on the classical solution of out-of-plane flexural-torsional buckling loads of the solid web arch. However, since the classical solution cannot be used for the calculation of the arch with a small rise-span ratio, the formula for out-of-plane elastic buckling loads of space truss arches subjected to end bending moments is modified. Numerical research of the out-of-plane stability of space truss arches under different load cases shows that the theoretical formula proposed in this paper has good accuracy. Secondly, the design formulas to predict the out-of-plane elastoplastic stability strength of space truss arches subjected to the end bending moment and radial uniform load are presented through introducing a normalized slenderness ratio. By assuming that all components of space truss circular arches bear only axial force, the design formulas to prevent the local buckling of chord and transverse tubes are deduced. Finally, the bearing capacity design equations of space truss arches are proposed under vertical uniform load.

Graphical abstract

Keywords

torsional stiffness / strength design / elastic buckling / space truss arches / out-of-plane

1 Introduction

Buckling is a key concern in structural research, and can make the structure lose stability and so can reduce the ultimate bearing capacity of the structure. Zhuang et al. [1] constructed a loss function to extract the buckling load based on the Rayleigh principle. Vu-Bac et al. [2,3] proposed a method to accurately restore the target shape of the shell structure, so as to determine the degree of instability caused by buckling and also studied swelling-induced buckling using a new computational framework. Buckling can also affect the ultimate bearing capacity of an arch truss, so research on arch truss stability is of great significance.

The stability of an arch can be divided into out-of-plane stability and in-plane stability. Timoshenko and Gere [4] deduced the in-plane elastic buckling loads of the two-hinge circular arch subjected to uniform radial load, and the deduced result is called the classical solution of in-plane elastic buckling loads. This laid the theoretical foundation for the in-plane stability of arches. Pi et al. [5,6], Guo et al. [7–9], Shi et al. [10] and He et al. [11] studied in-plane elastic buckling performance and elastic-plastic ultimate strength of steel arches, and formulated a complete set of design methods.

Out-of-plane stability is less studied than in-plane stability. Distinct from the in-plane instability mechanism, out-of-plane instability is mainly characterized by bending and torsional deformation, and by in-plane deformations. In the study of out-of-plane first-order elastic buckling loads, Dou et al. [12] derived an analytical solution of biaxial symmetric circular arches based on the relationship between internal force and displacement. The analytical solution considers the effects of sectional asymmetry and shear deformation. Pi et al. [13,14] and Zhao et al. [15] studied the out-of-plane elastic-plastic stability of steel arches and proposed a related design formula.

When the stiffness of chord and transverse tubes of a space truss arch is relatively small, chord and transverse tubes may fail before onset of out-of-plane instability. Guo et al. [16] avoided the elastic buckling of chord tubes by limiting the ratio of global slenderness to chord slenderness of the truss arch. Guo et al. [17] proposed a method considering the effect of chord tubes instability failure to predict out-of-plane elastoplastic instability, using a boundary condition parameter. The buckling of chord and transverse tubes can reduce the ultimate bearing capacity of space truss circular arches. Therefore, it is necessary to study how to prevent the buckling of chord and transverse tubes under various loading modes. However, in the existing literature, there are still deficiencies in the research of chord and transverse tubes.

Bradford and Pi [18] found that the classical solution of out-of-plane buckling loads is not applicable for shallow arches. It is very necessary to discuss whether the classical solution is applicable for the space truss arch with small rise-span ratios. The torsional stiffness of space truss arches needs to be known for the calculation of out-of-plane buckling loads. However, there are relatively large errors in the calculation of torsional stiffness of space truss circular arches in existing research.

This paper mainly studies numerically and theoretically the out-of-plane elastoplastic stability of the two-hinged space truss arch with rectangular section. Torsional stiffness and flexural stiffness are deduced theoretically in order to obtain the calculation formula of the first-order elastic buckling load of such arches. By introducing a reduction coefficient, the design formulas under the radial uniform load and end bending moment are obtained. The theoretical formulas to prevent the buckling of chord and transverse tubes are derived, so as to ensure that the failure mode is global instability failure. Finally, the design formula of space truss arches under the vertical uniform load is obtained. The research results of this paper can be used in the engineering practice of space truss circular arches.

2 Finite element model

All components of space truss circular arches with rectangular section are circular steel tubes. Both Shell181 and Beam188 elements can be used to establish the finite element (FE) model of a space truss arch, but it is simpler to model with beam188 [19]. The beam188 element considering the influence of shear deformation in ANSYS, can be used to solve the problems of linear elasticity, geometric nonlinearity, material nonlinearity and eigenvalue buckling analysis, and is utilized to establish the FE model. Guo et al.’s [20] research shows that the welding residual stress of a space truss arch has little effect on its bearing capacity, so this paper does not take the welding residual stress into consideration.

The overall and cross-sectional dimensions of space truss circular arches are shown in Fig.1, where f is rise-span ratio; S is the arc length of the arch; H is the rise of the arch; R is the curvature radius of the arch; L is the span; Θ, α and β are the included angles (see Fig.1); L_c is the segment length of chord tubes between adjacent transverse tubes; b is the width of the cross-section; a is the height of the cross-section; D_c is the diameter of the chord tube; t_c is the thickness of the chord tube; D_t is the diameter of the transverse tubes; and t_t is the thickness of the transverse tubes. The arch axis is a circular arc. In this paper, the span-span ratio f is in the range of 0.1–0.4, and the span L is in the range of 20–100 m.

The Young’s modulus of steel is E = 206 GPa in elastic analysis and the Poisson’s ratio is 0.3. In elastic-plastic analysis, the constitutive relationship of steel adopts the ideal elastic-plastic model, where the yield stress is f_y = 235 MPa, as shown in Fig.2. Considering the influence of initial geometric imperfections, we assume that the initial geometric imperfections of space truss circular arches are the same as the out-of-plane first-order elastic buckling mode, and the amplitude of the initial imperfections is S/500. The loading cases are shown in Fig.3 and the load is applied to two upper chords. Articulated constraints are applied at both ends of space truss circular arches to restrict the rotation along the tangential direction of the arch axis and to restrict the displacement to three directions.

3 Global elastic buckling load

3.1 Torsional stiffness

The arch axis of space truss circular arches is a circular arc curve, which is inconvenient for calculation of the torsional stiffness. Therefore, the torsional stiffness of space truss circular arches can be calculated by considering the tower truss, since the torsional stiffness of the two is the same [12]. The tower truss can be decomposed into four plane trusses, as shown in Fig.4(a).

As shown in Fig.4(b), according to the relationship between the top section displacement of the tower truss during torsion, the calculation of torsional stiffness is transformed into the bending problem that applies to plane trusses. The calculation formulas are given by Cao [21]:

(1)x={\displaystyle \frac{1}{2+{\displaystyle \frac{{\displaystyle \frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^3\beta }{{\mathrm{s}\mathrm{i}\mathrm{n}}^3\alpha }}-1}{{\displaystyle \frac{4}{3}}{\displaystyle \frac{A_{\mathrm{t}}}{A_{\mathrm{c}}}}{\mathrm{s}\mathrm{i}\mathrm{n}}^3\beta \left(n^2+{\displaystyle \frac{1}{2}}\right)+1}}}}{\displaystyle \frac{M}{b}},

(2)y={\displaystyle \frac{1}{2+{\displaystyle \frac{{\displaystyle \frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^3\alpha }{{\mathrm{s}\mathrm{i}\mathrm{n}}^3\beta }}-1}{{\displaystyle \frac{4}{3}}{\displaystyle \frac{A_{\mathrm{t}}}{A_{\mathrm{c}}}}{\mathrm{s}\mathrm{i}\mathrm{n}}^3\alpha \left(n^2+{\displaystyle \frac{1}{2}}\right)+1}}}}{\displaystyle \frac{M}{a}},

(3)\mathrm{\Delta }{a}_1=\frac{x{L}_{\mathrm{t}}}{E{A}_{\mathrm{t}}{\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha \mathrm{s}\mathrm{i}\mathrm{n}\alpha }+\frac{2L_{\mathrm{t}}^3}{3E{A}_{\mathrm{c}}a}\left(\frac{x}{a}-\frac{y}{b}\right)\left(1+\frac{1}{2n^2}\right),

(4)\mathrm{\Delta }{b}_1=\frac{y{L}_{\mathrm{t}}}{E{A}_{\mathrm{t}}{\mathrm{c}\mathrm{o}\mathrm{s}}^2\beta \mathrm{s}\mathrm{i}\mathrm{n}\beta }+\frac{2L_{\mathrm{t}}^3}{3E{A}_{\mathrm{c}}b}\left(\frac{y}{b}-\frac{x}{a}\right)\left(1+\frac{1}{2n^2}\right),

(5){\phi }_1=\frac{2\mathrm{\Delta }{a}_1}{b}=\frac{2\mathrm{\Delta }{b}_1}{a},

(6)G{I}_{\mathrm{p}1}=\frac{M{L}_{\mathrm{t}}}{{\phi }_1},

where x and y are loads on the plane trusses; ∆a₁ and ∆b₁ are the displacements at the top section of the tower truss; {\phi }_1 is the torsion angle at the top of the tower truss; GI_p1 is torsional stiffness; M is the bending moment on the space truss; a and b are the width of the plane truss; α and β are the included angles; n is the number of chord segments of the plane truss; A_c and A_t are the cross-sectional areas of the chord and transverse tubes, respectively; and L_t is the length of the plane truss.

As can be seen from the formulas, if the space truss section is square, i.e., α = β, a = b, the loads x and y of the plane trusses caused by the torsion of the space truss are equal. Therefore, from Eqs. (3) and (4), when the space truss section is square, obtained results for Δa₁ and Δb₁ only take account of the influence of transverse tubes and do not take account of the role of chord tubes, which is obviously incorrect. Therefore, when the plane truss bears the load x or y, the flexural stiffness of chord tubes needs to be considered. The flexural stiffness of the plane trusses considering only chord tubes is:

(7)E{I}_1=2E{I}_{\mathrm{c}}+2E{A}_{\mathrm{c}}{\left(\frac{a}{2}\right)}^2,

(8)E{I}_2=2E{I}_{\mathrm{c}}+2E{A}_{\mathrm{c}}{\left(\frac{b}{2}\right)}^2,

where I_c is the moment of inertia of the chord tubes of the plane trusses. When only chord tubes are considered, the displacement of the plane trusses under the load x and y is:

(9)\mathrm{\Delta }{a}_2=\frac{x{L}_{\mathrm{t}}^3}{3E{I}_1},

(10)\mathrm{\Delta }{b}_2=\frac{y{L}_{\mathrm{t}}^3}{3E{I}_2}.

The total displacement of the plane trusses under the load x and y is:

(11)\begin{array}{rl}\mathrm{\Delta }a=& \mathrm{\Delta }{a}_1+\mathrm{\Delta }{a}_2=\frac{x{L}_{\mathrm{t}}}{E{A}_{\mathrm{t}}{\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha \mathrm{s}\mathrm{i}\mathrm{n}\alpha }\\ & +\frac{2L_{\mathrm{t}}^3}{3E{A}_{\mathrm{c}}a}\left(\frac{x}{a}-\frac{y}{b}\right)\left(1+\frac{1}{2n^2}\right)+\frac{x{L}_{\mathrm{t}}^3}{3E{I}_1},\end{array}

(12)\begin{array}{rl}\mathrm{\Delta }b=& \mathrm{\Delta }{b}_1+\mathrm{\Delta }{b}_2=\frac{y{L}_{\mathrm{t}}}{E{A}_{\mathrm{t}}{\mathrm{c}\mathrm{o}\mathrm{s}}^2\beta \mathrm{s}\mathrm{i}\mathrm{n}\beta }\\ & +\frac{2L_{\mathrm{t}}^3}{3E{A}_{\mathrm{c}}b}\left(\frac{y}{b}-\frac{x}{a}\right)\left(1+\frac{1}{2n^2}\right)+\frac{y{L}_{\mathrm{t}}^3}{3E{I}_2}.\end{array}

The shear stiffness KV of plane truss under the load x and y is:

(13)K{V}_{\mathrm{a}}=\frac{x{L}_{\mathrm{t}}}{\mathrm{\Delta }a},

(14)K{V}_{\mathrm{b}}=\frac{y{L}_{\mathrm{t}}}{\mathrm{\Delta }b}.

The torsional stiffness GI_p of the tower truss is:

(15)G{I}_{\mathrm{P}}=K{V}_{\mathrm{a}}b+K{V}_{\mathrm{b}}a.

Beam188 element is used to establish the FE model of the tower truss, and articulated constraints are applied to each chord tube at the bottom of the tower truss. The bending moment M is due to a small unit load applied to the top of the tower truss. The Young’s modulus of steel is E = 206 GPa. An elastic analysis can verify Eq. (15). It is concluded from Fig.5(a) that the increase of chord tube diameter would lead to the increase of torsional stiffness of the tower truss, while the torsional stiffness of the tower truss obtained by Eq. (6) does not change. It is obvious that Eq. (15) is in better agreement with FE results.

When the number of chord segments n is greater than 20, the disagreement between the torsional stiffness obtained from Eq. (15) and the FE results remains constant as shown in Fig.5(b). This is because the constraint imposed by one end of the tower truss produces an FE calculation result that is the torsional stiffness of a restrained torsion scenario. When the number of chord segments is small, the constraint has a great influence; when the number of chord segments increases, the effect of constraints decreases and the FE calculation results are close to the torsional stiffness of a free torsion scenario.

3.2 Flexural stiffness

In space truss circular arches, the transverse tubes mainly connect the upper and lower chord tubes, and do not bear the arch axial force and bending moment. This has little effect on the flexural stiffness of space truss circular arches. Calculation of the flexural stiffness of space truss circular arches only needs to consider the action of chord tubes by parallel axis theorem, and the calculation formula is:

(16)E{I}_{\mathrm{y}}=4E{I}_{\mathrm{c}}+E{A}_{\mathrm{c}}{b}^2,

where EI_c and A_c are the bending stiffness and the cross-sectional areas of the chord, respectively.

3.3 Global elastic buckling load

3.3.1 Radial uniform load

When the load applied to the arch plane reaches a certain value, the arch can leave the original plane and transition to a new equilibrium state due to the combined action of torque around the arch axis and lateral bending moment, which is called out-of-plane buckling of the arch.

The out-of-plane stability of space truss circular arches is studied mainly through the classical solution of out-of-plane elastic buckling loads [22]. Under the radial uniform load, the first-order global buckling loads of space truss circular arches are given by:

(17)q_{\mathrm{c}\mathrm{r}}=\frac{E{I}_{\mathrm{y}}}{R^3}{\left(\frac{\text{π}}{\Theta }\right)}^2\frac{{\left[1-{\left({\displaystyle \frac{\Theta }{\text{π}}}\right)}^2\right]}^2}{1+\lambda {\left({\displaystyle \frac{\Theta }{\text{π}}}\right)}^2},

\lambda =\frac{E{I}_{\mathrm{y}}}{G{I}_{\mathrm{P}}},

where EI_y is calculated by Eq. (16) and GI_p is calculated by Eq. (15), where n = 10. The correctness of Eq. (17) is verified by FE analysis. It is obvious from Fig.6(a) and Fig.6(b) that the first-order elastic buckling load decreases with the increase of the rise-span ratio f and span L. The reduction of transverse tubes diameter D_t leads to the reduction of torsional stiffness and flexural stiffness, which reduces the first-order elastic buckling load, as shown in Fig.6(c). It is found from Fig.6(d) that the elastic buckling load decreases with the reduction of section width b, which is mainly caused by the decrease of flexural stiffness.

3.3.2 End bending moment

Under the action of end bending moment, the first-order elastic global buckling loads of the arch are deduced by the balance method [22], and the formula is:

(18)M_{\mathrm{c}\mathrm{r}}=\frac{E{I}_{\mathrm{y}}+G{I}_{\mathrm{P}}}{2R}-\sqrt{{\left(\frac{E{I}_{\mathrm{y}}-G{I}_{\mathrm{P}}}{2R}\right)}^2+\frac{E{I}_{\mathrm{y}}G{I}_{\mathrm{P}}}{R^2}\frac{{\text{π}}^2}{{\Theta }^2}}.

An FE model is established to verify the correctness of Eq. (18). It is obvious from Fig.7 that as the rise-span ratio increases, the error between the theoretical solution of first-order elastic buckling loads and FE results is increasingly smaller, with the maximum error of about 49.0% and the minimum error of about 1.5%. The theoretical solution is in good agreement with FE results when the rise-span ratio is greater than 0.3, and the maximum error is then about 5.8%.

Since Eq. (18) does not fit well with FE results, the formula is modified as follows:

(19)M_{\mathrm{c}\mathrm{r}}=\mu \left(\frac{E{I}_{\mathrm{y}}+G{I}_{\mathrm{P}}}{2R}-\sqrt{{\left(\frac{E{I}_{\mathrm{y}}-G{I}_{\mathrm{P}}}{2R}\right)}^2+\frac{E{I}_{\mathrm{y}}G{I}_{\mathrm{P}}}{R^2}\frac{{\text{π}}^2}{{\Theta }^2}}\right),

\mu =-0.128+8.26f-20.44f^2+17.35f^3,

f=\frac{H}{L},

where μ is the influence coefficient of rise-span ratio. It can be seen from Fig.8 that the ratio of the calculation result of Eq. (19) to the FE result is close to 1, the average value is 1.0004, the variance is 0.0005, and the coefficient of determination R² is 0.988. If R² ≥ 0.8, then Eq. (19) fits well with the FE results [23].

4 Elastoplastic behavior of space truss circular arches

4.1 Instability deformation mechanism

When space truss circular arches reach the ultimate bearing capacity, the instability deformation mechanism is complex because the arches will produce the out-of-plane and in-plane deformation at the same time. Two models are established for comparison in this paper. The model restricted to the out-of-plane displacement of the space truss arch is called the plane model; the model that considers out-of-plane deformation is called the spatial model.

The load-displacement curves of the two models are compared under the action of full-span vertical uniform loads, as shown in Fig.9. It can be concluded that when the space truss circular arch fails in the out-of-plane stability, it produces in-plane deformation, and the in-plane deformation of spatial model is almost the same as that of plane model.

Therefore, the out-of-plane and in-plane stability must both be considered in the design of space truss circular arches. Guo et al. [24], Shi et al. [10] and He et al. [11] have done the in-plane stability analysis of the truss arch and put forward in-plane design formulas of the truss arch. However, research on the out-of-plane stability and related design theories of the space truss arch is relatively lacking.

When space truss circular arches fail in the out-of-plane stability, there are three failure modes: global instability failure, chord instability failure and transverse instability failure. Among them, chord and transverse instability failures are local instability failures, which greatly reduce the bearing capacity of space truss circular arches. Therefore, the failure mode of all models in this section is the global failure.

4.2 Radial uniform load

Under the action of radial uniform load, the two-hinged space truss circular arches mainly bear the arch axial pressure. Referring to the stability design method of axially compressed columns in the Chinese Code for design of steel structures (GB20017) [25], the reduction coefficient of space truss circular arches is:

(20){\phi }_{\mathrm{a}}=\frac{N_{\mathrm{u}}}{N_{\mathrm{y}}}=\frac{N_{\mathrm{u}}}{4A_{\mathrm{c}}{f}_{\mathrm{y}}},

where N_u is the ultimate compressive strength and N_y is the axial plastic load. The ultimate bearing capacity of space truss circular arches is determined by many factors, such as rise-span ratio, span, chord segment length and so on. Therefore, the normalized slenderness ratio is used to comprehensively consider the influencing factors. The normalized slenderness ratio is:

(21){\lambda }_{\mathrm{N}}=\sqrt{\frac{N_{\mathrm{y}}}{N_{\mathrm{c}\mathrm{r}}}}=\sqrt{\frac{4A_{\mathrm{c}}{f}_{\mathrm{y}}}{q_{\mathrm{c}\mathrm{r}}R}},

where N_cr is the axial force when the space truss arch buckles. It is necessary to establish an FE model for elastic-plastic analysis to obtain the ultimate compressive strength. Fig.10 shows the relationship between the reduction coefficient and the normalized slenderness, and the reduction coefficient decreasing when the rise-span ratio increases. The four curves Fig.10 are the column design curves a, b, c, and d in the Chinese Code for design of steel structures (GB20017). The FE results mainly lie between the curves of class a and class b. This shows that under the action of uniform radial loads, the b curve can be selected to design space truss arches. The design formula is:

(22)\frac{N}{{\phi }_{\mathrm{a}}{N}_{\mathrm{y}}}⩽1,

where N is the maximum axial compressive force of space truss circular arches under ultimate load.

4.3 End bending moment

Under the action of end bending moment, the two-hinged space truss circular arches mainly bear bending moment, and the whole structure can be regarded as in the pure bending state. The upper chord tubes of the space truss circular arches are compressed and the lower chord tubes are tensioned in the pure bending state. When the chord tubes begin to yield, it is considered that the section reaches the plastic hinge moment. In the pure bending state, which is similar to the radial uniform load, the normalized slenderness ratio λ_M and the reduction coefficient {\phi }_{\mathrm{b}} of space truss circular arches are:

(23){\lambda }_{\mathrm{M}}=\sqrt{\frac{M_{\mathrm{y}}}{M_{\mathrm{c}\mathrm{r}}}}=\sqrt{\frac{2A_{\mathrm{c}}{f}_{\mathrm{y}}h}{M_{\mathrm{c}\mathrm{r}}}},

(24){\phi }_{\mathrm{b}}=\frac{M_{\mathrm{u}}}{M_{\mathrm{y}}}=\frac{M_{\mathrm{u}}}{2A_{\mathrm{c}}{f}_{\mathrm{y}}h},

where M_u is the ultimate bending moment of space truss circular arches and M_y is the plastic bending moment of the arch section.

The elastoplastic analysis is carried out through the FE model. Fig.11 shows the relationship between the reduction coefficient and the normalized slenderness ratio. The stability curve of the flexural beam in the European Convention for Constructional Steelwork (ECCS) [26] is quoted in the figure. The ECCS curve for the truss arch subjected to end moment load is defined as:

(25){\phi }_{\mathrm{b}}={\left(\frac{1}{1+{{\lambda }_{\mathrm{M}}}^{2n}}\right)}^{\frac{1}{n}},

where n is the adjustment coefficient and n = 6. It’s obvious from Fig.11 that Eq. (25) is consistent with the FE results and can guide the design of space truss arches in the pure bending state. The design formula is:

(26)\frac{M}{{\phi }_{\mathrm{b}}{M}_{\mathrm{y}}}\le 1,

where M is the maximum bending moment of space truss circular arches under the ultimate load.

5 Local buckling failure

5.1 Chord buckling failure

Chord tubes mainly bear pressure and tension in space truss circular arches. When the segment length of the chord is too long or the stiffness is too small, the chord also suffers from instability failure. The normalized slenderness ratio λ_c of a single segment chord is:

(27){\lambda }_{\mathrm{c}}=\frac{L_{\mathrm{c}}}{\text{π}{i}_{\mathrm{y}}}\sqrt{\frac{f_{\mathrm{y}}}{E}},

where i_y is the radius of gyration of the chord. Under the radial uniform load, space truss circular arches mainly bear the arch axial pressure, and the ultimate compressive strength can be calculated by Eq. (20). When considering the buckling failure of chord tubes, the reduction coefficient of ultimate bearing capacity is:

(28)\omega =\frac{N_{\mathrm{u}}^{\prime }}{N_{\mathrm{u}}},

where {N^{\prime }}_{\mathrm{u}} is the ultimate bearing capacity considering chord buckling failure, and N_u is the ultimate bearing capacity not considering chord buckling failure, which is calculated by Eq. (20). Fig.12 shows the relationship between the normalized slenderness ratio λ_c of a single segment chord tube and the reduction coefficient ω. It can be concluded from the figure that when the normalized slenderness ratio λ_c is small, the ultimate bearing capacity will decrease significantly. The ultimate bearing capacity will first increase and then remain constant with the increase of the normalized slenderness ratio λ_c; that is, the structure presents global instability failure. Therefore, before the space truss circular arches reach the ultimate bearing capacity, the buckling failure of chord tubes must be prevented.

Since the segment length of the upper chord is slightly longer than that of the lower chord, the upper chord is more prone to buckling failure than the lower chord. Therefore, taking the upper chord as an example, this paper expounds how to prevent the buckling failure of the chord. The arch axial pressure of space truss circular arches is mainly borne by chord. It is assumed that all chords and transverse tubes bear only axial force under the radial uniform load, and the four chords are uniformly compressed. Therefore, when these arches reach the ultimate load, the axial pressure on a single chord is:

(29)N_1=\frac{N_{\mathrm{u}}}{4}.

Consideration of a segment chord from the upper chord can establish a simplified model, as shown in Fig.13. The constraint condition of the chord tube in the simplified model is considered to be hinged at both ends, and this simplification is biased towards safety. The arc length of the segment chord tube is very small compared with that of complete arches, so the segment chord tube can be considered as a straight tube with initial defects. The included angle θ of the segment chord to the circle is:

(30)\theta =\frac{L_{\mathrm{c}}}{R}.

The span of the segment chord is:

(31)l=2\left(R+\frac{a}{2}\right)\mathrm{s}\mathrm{i}\mathrm{n}\frac{\theta }{2}.

The rise height of the segment chord is:

(32)h=\left(R+\frac{a}{2}\right)\left(1-\mathrm{c}\mathrm{o}\mathrm{s}\frac{\theta }{2}\right).

When the segmental chord is equivalent to a straight tube with initial defects, the initial defects are:

(33)\frac{h}{l}=\frac{1-\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{\theta }{2}}}{2\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{\theta }{2}}}.

In the simplified model, when the chord tube is mainly subject to axial compression, due to the existence of initial defects, the bending moment of the mid-span section is the largest and the axial compressions of each section are equal, so the stress of the mid-span section is the largest. The maximum stress of chord tubes should be less than the yield strength in order to prevent the buckling failure of chord tubes. The maximum stress is:

(34)\sigma =\frac{M{D}_{\mathrm{c}}}{2I_{\mathrm{c}}}+\frac{N_1}{A_{\mathrm{c}}}\le f_{\mathrm{y}},

(35)M=N_{1}h,

(36)I_{\mathrm{c}}=\frac{\text{π}{D_{\mathrm{c}}}^4}{64}\left(1-\frac{{\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^4}{{D_{\mathrm{c}}}^4}\right),

(37)A_{\mathrm{c}}=\text{π}{\left(\frac{D_{\mathrm{c}}}{2}\right)}^2-\text{π}{\left(\frac{D_{\mathrm{c}}-2t_{\mathrm{c}}}{2}\right)}^2,

where σ is maximum stress of chord mid span section; M is the mid-span bending moment; I_c is the moment of inertia of the chord section; and A_c is the cross-sectional area of the chord.

Substituting Eqs. (35)–(37) into Eq. (34) lead to:

(38)N_1\le \frac{\text{π}\left[{D_{\mathrm{c}}}^4-{\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^4\right]f_{\mathrm{y}}}{32h{D}_{\mathrm{c}}+4{D_{\mathrm{c}}}^2+4{\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^2}.

The simplified model is established as an FE model to verify Eq. (38). Beam188 is used to establish the FE model of the simplified model. The constitutive relationship adopts the ideal elastic-plastic model as shown in Fig.2. The FE model is hinged as shown in Fig.13, and the out-of-plane displacement is limited. As shown in Fig.14(a), the bearing capacity of the chord clearly increases with the increase of chord diameter D_c and the maximum error is 5.6%. However, changing the span l of segmental chord has little effect on the bearing capacity and the maximum error is 4.8%, as shown in Fig.14(b).

Substituting Eq. (29) into Eq. (38) leads to:

(39)N_{\mathrm{u}}\le \frac{\text{π}\left[{D_{\mathrm{c}}}^4-{\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^4\right]f_{\mathrm{y}}}{8h{D}_{\mathrm{c}}+{D_{\mathrm{c}}}^2+{\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^2}.

Under the action of end bending moment, space truss circular arches are in a pure bending state, which would make the lower chord tensioned and the upper chord compressed, so the upper chord is more prone to buckling failure. The ultimate moment of such arches can be calculated by Eq. (24):

(40)M_{\mathrm{u}}={\phi }_{\mathrm{b}}{M}_{\mathrm{y}}.

Therefore, the axial pressure on a single upper chord is:

(41)N_2=\frac{M_{\mathrm{u}}}{2a}.

Similarly, it can be concluded that the ultimate bending moment of these arches should meet the following formula to avoid chord buckling failure:

(42)M_{\mathrm{u}}\le \frac{\text{π}a\left[{D_{\mathrm{c}}}^4-{\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^4\right]f_{\mathrm{y}}}{16h{D}_{\mathrm{c}}+{{2D}_{\mathrm{c}}}^2+{2\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^2}.

Under the vertical uniform load, space truss circular arches would bear the arch axial pressure and bending moment at the same time. Therefore, in order to avoid the chord buckling failure, the following formula must be satisfied:

(43)\frac{N}{4}+\frac{M}{2a}\le \frac{\text{π}\left[{D_{\mathrm{c}}}^4-{\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^4\right]f_{\mathrm{y}}}{32h{D}_{\mathrm{c}}+4{D_{\mathrm{c}}}^2+4{\left(D_{\mathrm{c}}-2t_{\mathrm{c}}\right)}^2},

where M and N are the maximum bending moment and axial compressive force respectively under the ultimate load obtained by FE elastoplastic analysis.

5.2 Transverse buckling failure

In space truss circular arches, transverse tubes make the upper and lower chords deform cooperatively. In a similar way to that of the chord, the bearing capacity of space truss circular arches will be reduced. Therefore, in the design of space truss circular arches, it is necessary to prevent the instability failure of transverse tubes.

Under the action of the radial uniform load, space truss circular arches mainly bear the arch axial pressure. It is assumed that the chord tubes are uniformly compressed and the transverse tubes only bear the axial force. The calculation of the transverse tubes can be simplified. The simplified model is shown in Fig.15.

The load F is the axial pressure of a single chord, which can be calculated by:

(44)F=\frac{N_{\mathrm{u}}}{4}.

The axial pressure N_b of transverse tubes is:

(45)N_{\mathrm{b}}=F\sqrt{1+\frac{a^2}{{L_{\mathrm{c}}}^2}}.

In order to prevent the transverse tubes from being damaged, the axial pressure of the transverse tubes must meet the following formula:

(46)\{\begin{array}{rl}& N_{\mathrm{b}}\le {\displaystyle \frac{{\text{π}}^{2}E{I}_{\mathrm{b}}}{{L_{\mathrm{c}}}^2+a^2}},\\ & N_{\mathrm{b}}\le f_{\mathrm{y}}{A}_{\mathrm{b}},\end{array}

where I_b is the moment of inertia of the transverse tubes. Substituting Eqs. (44)–(45) into Eq. (46) leads to:

(47)\{\begin{array}{rl}& N_{\mathrm{u}}\le {\displaystyle \frac{4{\text{π}}^{2}E{I}_{\mathrm{b}}{L}_{\mathrm{c}}}{{\left({L_{\mathrm{c}}}^2+a^2\right)}^{\frac{3}{2}}}},\\ & N_{\mathrm{u}}\le {\displaystyle \frac{{4f}_{\mathrm{y}}{A}_{\mathrm{b}}{L}_{\mathrm{c}}}{\sqrt{{L_{\mathrm{c}}}^2+a^2}}}.\end{array}

Under the action of end bending moment, the upper chord of space truss circular arches is compressed and the lower chord is tensioned, so the transverse tubes at the top of the arch truss are more prone to instability and failure. The calculation of transverse tubes can be simplified. The simplified model is shown in Fig.15. The load F is the axial pressure of a single transverse tube, which can be calculated by:

(48)F=\frac{M_{\mathrm{u}}}{2a}.

Using the same method as above, under the action of end bending moment, the maximum bending moment of space truss circular arches should meet the following formulas:

(49)\{\begin{array}{rl}& M_{\mathrm{u}}\le \frac{2a{\text{π}}^{2}E{I}_{\mathrm{b}}{L}_{\mathrm{c}}}{{\left({L_{\mathrm{c}}}^2+a^2\right)}^{\frac{3}{2}}},\\ & M_{\mathrm{u}}\le \frac{{2af}_{\mathrm{y}}{A}_{\mathrm{b}}{L}_{\mathrm{c}}}{\sqrt{{L_{\mathrm{c}}}^2+a^2}}.\end{array}

Under the vertical uniform load, space truss circular arches bear the arch axial pressure and bending moment at the same time. In order to prevent the buckling failure of the transverse tubes before the overall instability failure, the following formulas must be met in the design:

(50)\{\begin{array}{rl}& \frac{N}{4}+\frac{M}{2a}\le \frac{{\text{π}}^{2}E{I}_{\mathrm{b}}{L}_{\mathrm{c}}}{{\left({L_{\mathrm{c}}}^2+a^2\right)}^{\frac{3}{2}}}\\ & \frac{N}{4}+\frac{M}{2a}\le \frac{f_{\mathrm{y}}{A}_{\mathrm{b}}{L}_{\mathrm{c}}}{\sqrt{{L_{\mathrm{c}}}^2+a^2}}\end{array},

where M and N are the maximum bending moment and axial compressive force respectively under the ultimate load obtained by FE elastoplastic analysis.

6 Vertical uniform load

Design formulas of space truss circular arches under radial uniform load and end bending moment have been given. However, in practical application, space truss circular arches are rarely in a pure pressure or pure bending state, and most of them bear arch axial pressure and bending moment at the same time. The stress of space truss circular arches is complex under the action of compression and bending because there are many influencing factors. It is difficult to deduce a design formula to predict the ultimate bearing capacity of the arch. The interaction equation is given in this paper, which can provide a conservative lower boundary for ultimate bearing capacity. The interaction equation is:

(51)\frac{N}{{\phi }_{\mathrm{a}}{N}_{\mathrm{y}}}+\frac{M}{{\phi }_{\mathrm{b}}{M}_{\mathrm{y}}}⩽1,

where M and N are the maximum bending moment and axial compressive force respectively under the ultimate load obtained by FE elastoplastic analysis.

Instability failure of chord and transverse tubes also reduces load-carrying capacity. Therefore, the design of space truss arches must meet Eq. (43) and Eq. (50), so that the failure mode of the space truss circular arch is global instability failure. It is important to establish an FE model to verify Eq. (51), as shown in Fig.16. It is obvious from the figure that Eq. (51) provides a conservative lower boundary for the ultimate bearing capacity of space truss circular arches. It can be seen from Fig.16(a) that space truss circular arches mainly bear the arch axial pressure under the action of full-span vertical uniform loads, while the bending moment has little effect. Under the half-span vertical uniform load, it can be concluded from Fig.16(b) that space truss circular arches bear both pressure and bending moment, and the bending moment is relatively large. The effect of bending moment decreases as the rise-span ratio increases.

7 Conclusions

The out-of-plane stability of the two-hinged space truss circular arch with rectangular section is studied in this paper. The flexural stiffness and torsional stiffness of space truss circular arches are deduced theoretically in order to obtain the out-of-plane global elastic buckling load. The classical solution of flexural torsional buckling load does not work for the arch with small rise-span ratio under the end bending moment, and it is modified based on the FE results by the influence coefficient.

In the elastic-plastic analysis, class b curve in GB2017 and ECCS curve can simulate the relationship between the reduction coefficient and normalized slenderness ratio, and can guide the design of space truss arch under the radial uniform load and end bending moment.

The buckling failure of chord and transverse tubes would reduce the ultimate bearing capacity of a space truss arch. In the local buckling failure analysis, it is assumed that all components in the arch truss are only subjected to axial force, and the chord and transverse tubes are simplified. Under the radial uniform load, end bending moment, full-span and half-span vertical uniform load, the theoretical formulas for prevention of local instability failure of chord and transverse tubes, before the space truss circular arch reach the ultimate bearing capacity, are deduced theoretically.

Under the action of full-span and half-span vertical uniform load, space truss arches are subject to the combination of bending moment and axial compressive force. The interaction equation is proposed based on the design method of space arch truss in pure compression state and pure bending state, and provides a conservative lower boundary for the out-of-plane elastic-plastic buckling strength of the arches. It can be used to guide the design of space truss circular arches.

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