Buckling capacity of steel circular hollow section considering the restraint of welded hollow spherical joints

Abstract

An integral finite element model of steel circular hollow section and welded hollow spherical joints is established to consider the restraint of welded hollow spherical joints on steel circular hollow section in this paper. The buckling modes are used as the initial geometrical imperfection forms of the steel circular hollow section. Based on the statistics method, a method for calculating the buckling capacity design value of the steel circular hollow section with uniform reliability considering the restraint of welded hollow spherical joints is established, and the formula for calculating error of this value at certain confidence is deduced. The relationships between the buckling capacity design value of steel circular hollow section and the structural parameters of the member and joint are determined quantitatively by linear regression method. A practical formula for the buckling capacity design value of steel circular hollow section considering the restraint of welded hollow spherical joints is derived. Compared to the results calculated by the design code indicates that if the difference of restraint effect which is caused by different welded hollow spherical joint is ignored, the buckling capacity value of steel circular hollow section calculated based on the unified effective length coefficient may be bigger or smaller than that of the actual case. Therefore, refined consideration should be placed on the different restraint of welded hollow spherical joints of different sizes on steel circular hollow section, and the restraint of the end joints should be accurately considered when determining the buckling capacity of steel circular hollow sections.

Keywords

Steel circular hollow section buckling capacity welded hollow sphere joint restraint

Introduction

Grid space structure is widely used in the roof structure of large public buildings such as airport terminal, gymnasium and auditorium due to its large span, light weight, high strength and beautiful shape. The bearing capacity of grid space structure is closely related to the mechanical properties of structural members and joints. As a large number of members in grid space structures bear axial compression loads, the member buckling capacity is always the key point in the structural design. The buckling capacity of slender members has been studied for many years,^1,2 and it is also clearly specified in the grid space grid structure design codes. The stability checking calculation of compressive members is based on the stability coefficient. According to the current structure design codes, such as Chinese design code for steel structures,³ the buckling capacity of the member bearing compression load is calculated based on the stability coefficient and effective length factor. The stability coefficient for certain section is usually determined by the column curves, and the column curves are calculated based on the free members at both ends with the consideration of initial imperfections. The effective length factor of the compression member is used to consider the influence of restraints caused by the joints. By the code method, the effective length coefficient of slender member is determined by the type of the joint. For example, the Chinese Technical Specification for Grid Space Structures⁴ stipulates that for the structure using welded hollow spherical joints, the effective length coefficients are 0.9 for the chord members, 0.9 for web members near the supports and 0.8 for other web members; for the structure using bolted spherical joints, the effective length coefficient is 1.0 for each member. The effective length coefficient for compression members is related to the restraint effect caused by the joints.

In fact, even the same type of joint is used for the grid structure, the restraint effect caused by the joint to the member buckling capacity varies with the joint dimension. The restraint effect on the member is related to the type and dimension of the joint as well as the dimension of the member itself. Therefore, the effective length coefficient of the structural member should differ with the dimensions and type of the joint. If the actual effective length coefficient of members in the grid space structure is larger than that calculated by the code, the buckling capacity of the structural member will be overestimated in current design. Therefore, refined method by which the restraint of joints on the buckling capacity of the structural member can be actually considered needs to be established.

At present, scholars have carried out extensive research on the calculation method of bearing capacity for slender member. Galvão et al.⁵ studied slender frames with semi-rigid connections and the results highlight the importance the stiffness of the semi-rigid connections on the buckling of these structures. But the initial geometrical imperfections of the members were not considered in the research process. There is also a calculation method for the buckling capacity of a single steel hollow section, which does not consider the joint restraint in the calculation process. Gardner and Nethercot^6,7 carried out multiple sets of steel square, rectangular and circular hollow sections tests, which forms the basis for an explicit relationship between cross-sectional slenderness and cross-sectional deformation capacity The buckling capacities of steel square and rectangular hollow section were calculated using the numerical simulation method considering the initial imperfections. Characterisation of local plate imperfection amplitudes is described whereby a model originally devised for hot-rolled carbon steel cross-sections was re-calibrated and applied to stainless steel cross-sections. Of the features investigated, strain hardening characteristics of the material were identified as being primarily responsible for the differences in structural behaviour between hot-rolled and cold-formed sections.

Wang et al.⁸ studied the influence of initial imperfections on the ultimate bearing capacity of axially compressed steel hollow sections, and gave a local stability calculation method for axial compression members. The test results show that the design of Q690 steel circular hollow section in the existing design standards is mostly conservative. McCann et al.⁹ examined the response of EHS stub columns with parameters variation of imperfection amplitude and imperfection shape. Imperfection sensitivity was found to change with slenderness and aspect ratio. Hassan et al.¹⁰ considered three methods of geometrical imperfections for square and rectangular structural steel hollow sections, base on which a new procedure of determining geometrical imperfection was proposed. Shen and Ahmer¹¹ formed a nonlinear variational approach to model local-global mode interaction. Qi et al.¹² established the member calculating model considering the buckling effect, and calculated the ultimate bearing capacity of the premier geometric faulty structures based on the probability and statistics method. It is shown that the nonlinear design bearing capacity of the lattice shell at a uniform probability level cannot be acquired by the methods suggested by the structure design code. However, the studies above did not examine the buckling capacity of the member in relation to the initial imperfection as well as the joint restraint. Therefore, refined consideration should be carried out.

In this paper, a method for calculating the buckling capacity design value of the steel circular hollow section with uniform reliability considering the restraint of welded hollow spherical joints is established, and the error formula of this design value at a certain confidence level is also derived. This method is verified by comparisons with experimental results recorded in published reference. The relationships between the buckling capacity design value of steel circular hollow section and the geometrical and material parameters of the member and joint are determined quantitatively by linear regression method. A practical formula for the buckling capacity design value of the steel circular hollow section considering the restraint of welded hollow spherical joints is derived.

Simulation of initial geometrical imperfections

Initial geometrical imperfection of the grid space structure members inevitably appears in the process of manufacture, transportation and assembly.^13,14 Initial geometrical imperfections will reduce the buckling capacity of members. Therefore, the effect of initial geometrical imperfections must be taken into account when studying the buckling capacity of steel circular hollow sections with the consideration of joint restrains. The initial geometrical imperfections of the structure member must satisfy the displacement compatibility condition. In this paper, linear buckling mode is taken as the initial geometrical imperfection form of structural members, and l/300 (l is the member length) is taken as the maximum initial geometrical deviation value according to the Technical Specification for Grid Space Structures.⁴ Linear buckling mode is a possible deformation form when a member buckles, and it satisfies displacement continuity conditions.

In reference,⁸ a series of buckling capacity tests were carried out on steel circular hollow sections. Both the two ends of the steel circular hollow section specimen were welded on the steel plates of the upper and lower pedestals in the testing machine, which is shown in Figure 1. The numerical model of the specimen was established by the general finite element software ABAQUS. The initial geometrical imperfections of the specimen were simulated by buckling modes, and the buckling capacity of the specimen was calculated by arc length method with the consideration of geometrical nonlinearity and material nonlinearity. The finite element model is shown in Figure 2. In the finite element model, the mesh size is 1/15 of the longest edge of the steel circular hollow section, and the meshes are refined partially. The mesh element type is linear, finite film strain, complete integral quadrilateral shell element S4 in ABAQUS/Standard. The material model is nonlinear material model, and the yield strength is 690 MPa, which is consistent with the reference.⁸

Figure 1.

Steel circular hollow section specimen.⁸

Figure 2.

Finite element model.

The test result and numerical result of the buckling capacity value of the specimen are listed in Table 1.

Table 1.

Buckling capacity value of steel circular hollow section specimen.

No.	Diameter of the cross section/mm	Wall thickness of the cross section/mm	Length of the specimen/m	Bearing capacity value		Buckling modal order	Relative error/%
No.	Diameter of the cross section/mm	Wall thickness of the cross section/mm	Length of the specimen/m	Test value/kN	Numerical analysis value/kN	Buckling modal order	Relative error/%
1	180	6.0	0.9	2737	2789	5	1.91
2	210	6.0	1.0	3085	3266	3	5.86
3	240	6.0	1.2	3470	3522	6	1.51
4	270	6.0	1.4	3854	3823	2	−0.81
5	360	6.0	1.8	4745	4984	3	5.05
6	420	6.0	2.2	5479	5424	4	1.01

From Table 1, it can be seen that the buckling capacity value of numerical result is very close to that of the test result. The maximum relative error is 5.86%, and the minimum relative error is only −0.81%. The initial geometrical imperfections of the steel circular hollow section specimen could not be accurately known, so the trial calculation needs to be carried out using multi-order buckling modes as the geometrical imperfection forms of the numerical model. For one steel circular hollow section there are different numerical buckling capacity values corresponding to different buckling modes. Generally there would be one value that is most close to that of the test result, and the buckling mode corresponding to this value best fits the actual geometrical imperfection of the very specimen. This indicates that it is feasible to use buckling mode as the initial geometrical imperfection distribution form of steel circular hollow section.

The formation of initial geometrical imperfections of space structural members is a stochastic process, so which order of buckling mode can be used to simulate the initial geometrical imperfections of the members is also a stochastic process.^15–17 In order to accurately determine the buckling capacity value of steel circular hollow section in space structures considering the joint restraint, it is necessary to simulate the initial geometrical imperfections by buckling modes of sufficient orders. Therefore, parametric analysis needs to be carried out to derive a practical formula for the buckling capacity value of steel circular hollow section.

The parametric analysis is carried out by finite element program ABAQUS. The finite element numerical model is illustrated in Figure 3. As a commonly used type of joint for grid space structure, the welded hollow spherical joint is contained in the numerical model to consider its restraint on steel circular hollow section. The two hemispheres of welded hollow spherical joint are welded to each other, so an equivalent model can be established with the steel circular hollow section and two half joints which are rigidly fastened to rigid plates at both ends. In the finite element model, shell element model is used for the steel circular hollow section and the joint, and both the geometrical nonlinearity and material nonlinearity are taken into account.^18,19 The complete axial load-displacement curve of steel circular hollow section member with geometrical imperfection considering restraint of welded hollow spherical joints is calculated.

Figure 3.

Finite element model.

Considering the restraint of welded hollow spherical joints at the ends, the factors that may affect the buckling capacity of steel circular hollow section in the space structure are: $l$ , which is the length of steel circular hollow section; $δ$ , which is the wall thickness of the steel circular hollow section; $d$ , which is the outer diameter of the steel circular hollow section; $t$ , which is the wall thickness of the welded hollow spherical joint; $D$ , which is the outer diameter of the welded hollow spherical joint; and the material of steel circular hollow section and welded hollow spherical joint also contributes.²⁰ In order to quantitatively analyze the effect of these parameters on the buckling capacity of steel circular hollow section, parametric analyses were carried out based on finite element models with parameters listed in Table 2. In the process of selecting parameters such as steel circular hollow section length and outer diameter, the requirements of Q235 steel diameter-thickness ratio less than 100.0 and axial compression member slenderness ratio less than 200.0 in reference³ are considered. Based on this, the relationship between the buckling capacity design value and the length of the steel circular hollow sections is taken into account for the parametric analysis. Each type of the parameter of model 1–model 10 has the same value except for the steel circular hollow section length, and the length of model 1–model 10 from 1.2 to 3.9 m. The relationship between the buckling capacity design value and the outer diameter of the steel circular hollow sections is considered for the parametric analysis. Each type of the parameter of model 21–model 30 has the same value except for the cross section outer diameter of the steel circular hollow section, and the outer diameter of model 21–model 30 from 60 to 102 mm.

Table 2.

Parameters of the finite element models.

No.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
d/mm	89	89	89	89	89	89	89	89	89	89	89	89	89	89	89	89	89	89
δ/mm	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	3.5	4.0	4.5	5.0	5.5	6.0	6.5	7.0
l/m	1.2	1.5	1.8	2.1	2.4	2.7	3	3.3	3.6	3.9	1.8	1.8	1.8	1.8	1.8	1.8	1.8	1.8
D/mm	300	300	300	300	300	300	300	300	300	300	300	300	300	300	300	300	300	300
t/mm	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0
Steel	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235
No.	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36
d/mm	89	89	60	63.5	68	70	73	76	83	89	95	102	89	89	89	89	89	89
δ/mm	7.5	8.0	5.0	5.0	5.0	5.0	5.0	5.0	5.0	5.0	5.0	5.0	4.0	4.0	4.0	4.0	4.0	4.0
l/m	1.8	1.8	1.8	1.8	1.8	1.8	1.8	1.8	1.5	1.8	1.8	1.8	1.5	1.5	1.5	1.5	1.5	1.5
D/mm	300	300	300	300	300	300	300	300	300	300	300	300	300	300	300	300	300	300
t/mm	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.0	8.5	9.0	9.5	10.0	10.5
Steel	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235
No.	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54
d/mm	89	89	89	89	89	89	89	89	89	89	89	89	89	89	89	89	89	89
δ/mm	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0	4.0
l/m	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5	1.5
D/mm	300	300	300	300	250	300	320	350	380	400	420	450	480	500	300	300	300	300
t/mm	11.0	11.5	12.0	12.5	14.0	14.0	14.0	14.0	14.0	14.0	14.0	14.0	14.0	14.0	14.0	14.0	14.0	14.0
Steel	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q235	Q345	Q390	Q420

Based on the previous analysis, buckling modes of sufficient orders should be used as the initial geometrical imperfection forms of steel circular hollow section. For each of the 54 numerical models listed in Table 2, the 1st to 100th buckling modes are all used to create imperfect steel circular hollow sections. With the consideration of the joint restraint, the buckling capacity value of the steel circular hollow section with geometrical imperfection of certain buckling mode can be obtained as the peak value of axial load-displacement curve which is calculated by the arc length method. For example, the load-displacement curve of the model no. 4 with geometrical imperfection of its first buckling mode is shown in Figure 4. The peak value of this curve can be used as the buckling capacity value of this imperfect structural member.

Figure 4.

Axial load-displacement curve.

Buckling capacity of steel circular hollow section

For each of the 54 models listed in Table 2, the 1st to 100th buckling modes are all used to create imperfect steel circular hollow sections. There would be one buckling capacity value corresponding to one imperfect model. So for one steel circular hollow section of the given size, it has 100 possible buckling capacity values. Theoretically there are countless imperfect forms for one steel circular hollow section, so these 100 values need to be analysed statistically to establish a reasonable method for the buckling capacity value of steel circular hollow sections with initial imperfection and joints restraint.

The initial geometrical imperfection of steel circular hollow section is unpredictable, so the buckling capacity value is a stochastic variable essentially. The 100 buckling capacity values calculated with the 100 buckling modes are just samples of this stochastic variable. The population of this stochastic variable can be analysed based on the sample using statistical method. Then the method for calculating the buckling capacity design value of the steel circular hollow section with uniform reliability considering the restraint of welded hollow spherical joints can be established accordingly.

Basic assumptions

Multiple independent factors cause initial geometrical flaw of steel circular hollow section in space structures. As each one of the factors plays a less prominent part, initial geometrical imperfection of steel circular hollow section is normally distributed as the probability theory describes.

Initial geometrical imperfection of steel circular hollow section can be represented by deviation vectors of finite element joint coordinate. According to Design Specification for Grid Space Structures,⁴ some assumptions are made: The mean value of the finite element joint coordinate deviation of the member is 0; the tolerance of the joint coordinate deviation is l/300 (where l is the length of the steel circular hollow section), and the standard deviation of the joint coordinate deviation is a half of its tolerance. Accordingly the element joint coordinate deviation of steel circular hollow section distributes in normal fashion of N(0, (l/600)²), and range from −l/300 to l/300. The geometrical initial imperfections of the steel circular hollow section are independent multiple random variables, because deviation of the element joint coordinate is thought to be independent.

Distribution of buckling capacities of steel circular hollow sections

Take the fourth model of steel circular hollow section in Table 2 as an example. One hundred buckling modes are used to create initial geometrical imperfect steel circular hollow section. A 100 sized sample of the buckling capacity of steel circular hollow section can be obtained by FEM analysis. Based on this sample, the population distribution of the buckling capacity of this steel circular hollow section considering the joint restraint can be estimated.

Based on the analysis in Section ‘Basic assumptions’, it can be assumed that the bearing capacity values of imperfect steel circular hollow section considering the joint restraint are normally distributed, and a further nonparametric K-S (Kolmogorov-Smirnov) test is carried out for the verification.

Let the cumulative frequency distribution function of the sample be $g_{n} (x)$ , and $g_{n} (x)$ is distributed as the theoretic distribution function $g (x)$ . The statistic for K-S test is given as:

T_{n} = \underset{1 \leq i \leq n}{m a x} | g (x_{i}) - g_{n} (x) |

(1)

Where $x_{i}$ is the $i$ th value in the sample; n is the sample size.

The inspection level of 0.05 is used to test the population distribution. T_n = 0.050, p = 0.675 and p > 0.050, it indicates that the buckling capacity values of the imperfect steel circular hollow section considering the joint restraint are normally distributed.

The mean µ and variance $σ^{2}$ of the population are calculated based on the sample by the maximum likelihood method. The probability density function of the imperfect steel circular hollow section is given by:

f (x; μ, σ^{2}) = \frac{1}{σ \sqrt{2 π}} \exp [- \frac{1}{2 σ^{2}} {(x - μ)}^{2}]

(2)

Where $x$ is random variable. It stands for the buckling capacity value of the steel circular hollow section here.

Create a likelihood function as following:

\begin{matrix} M = \prod_{i = 1}^{n} \frac{1}{σ \sqrt{2 π}} \exp [- \frac{1}{2 σ^{2}} {(x_{i} - μ)}^{2}] \\ = {(\frac{1}{2 π σ^{2}})}^{\frac{n}{2}} \exp [- \frac{1}{2 σ^{2}} {(x_{i} - μ)}^{2}] \end{matrix}

(3)

Constitute equations set as follows:

{\begin{matrix} \frac{\partial}{\partial μ} \ln M = \frac{1}{σ^{2}} \sum_{i = 1}^{n} (x_{i} - μ) = 0 \\ \frac{\partial}{\partial (σ^{2})} \ln M = - \frac{n}{2} \cdot \frac{1}{σ^{2}} + \frac{1}{2 σ^{4}} \sum_{i = 1}^{n} {(x_{i} - μ)}^{2} = 0 \end{matrix}

(4)

Use $\hat{µ}$ as the estimator of µ, and use ${\hat{σ}}^{2}$ as the estimator of $σ^{2}$ . Expressions of $\hat{µ}$ and ${\hat{σ}}^{2}$ can be acquired by solving equation (4).

{\begin{matrix} \overset{}{\hat{μ}} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} = \overline{X} \\ {\hat{σ}}^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \overline{X})}^{2} \end{matrix}

(5)

The estimators of the mean µ and standard variance σ of the population of the example imperfect steel circular hollow section considering the joint restraint are 533 and 36 kN respectively.

Buckling capacity design value with uniform reliability

Use $P_{c r}$ to stand for one of the buckling capacity values of the 100 imperfect steel circular hollow sections. The reliability of $P_{c r}$ as the buckling capacity design value of steel circular hollow section is expressed by equation (6). That is to say, the probability of the event that $P_{c r}$ happens to be the minimum value is

γ = \frac{1}{\sqrt{2 π}} \int^{} \begin{matrix} + \propto \\ b \end{matrix} \exp (- \frac{x^{2}}{2}) d x

(6)

Where

b = (P_{c r} - \hat{μ}) / \hat{σ} .

The buckling capacity design value of steel circular hollow section is defined by:

P_{c r d} = \hat{μ} - 2 \hat{σ}

(7)

The buckling capacity design value $P_{c r d}$ has a uniform probability level of 0.977 according to equation (6).

Three statistical samples of buckling capacity values of the No. 4 steel circular hollow section are calculated using 1^st~100^th, 101^th ~200^th and 201^th ~300^th buckling modes as the initial geometrical imperfection forms. The minimum values of the three samples are listed in Table 3 respectively, and the corresponding reliabilities calculated based on equation (6) are also listed in the fourth column.

Table 3.

Minimum value and its reliability of statistical samples of buckling capacity values.

No.	Buckling mode	Minimum value of the sample	Reliability
1	1–100	494	0.861
2	101–200	601	0.030
3	201–300	579	0.102

It can be seen from Table 3 that different initial geometrical imperfections of steel circular hollow section will lead to different buckling capacity samples. If the minimum value of the sample is used as the buckling capacity design value, it depends on the selected initial geometrical imperfection of the steel circular hollow section, and a stable buckling capacity design value of steel circular hollow section with uniform reliability cannot be obtained. It is also not economical to just take the minimum value of the sample as the buckling capacity design value without consideration of reliability.

Error of the buckling capacity design value

Theoretically, if buckling capacity values of infinite imperfect steel circular hollow sections are calculated, the accurate value of the bearing capacity design value $P_{c r a}$ at 0.977 probability level can be acquired. Actually, the buckling capacity sample is not infinite and the sample size is limited by the calculation cost. Therefore, it is necessary to determine the difference between the buckling capacity design value $P_{c r d}$ calculated ground on finite sample and the accurate value $P_{c r a}$ .

The buckling capacity values of steel circular hollow sections distribute in the normal fashion of $N (μ, σ^{2})$ , the mean of the sample $\overset{}{\bar{X}} = \frac{1}{n} \cdot \sum_{i = 1}^{n} x_{i}$ distributes in the normal fashion of $N (μ, \frac{σ^{2}}{n})$ . Let $u = \frac{(\overset{}{X} - μ)}{\sqrt{\frac{σ^{2}}{n}}}$ , so $u$ distributes in the normal fashion of $N (0, 1)$ . The variance of the buckling capacity values of steel circular hollow sections is supposed to have the same value with that of the variance estimator. So the error of buckling capacity design value inferred in this paper with a confidence level of $1 - α$ is given by:

\begin{matrix} e = | P_{c r d} - P_{c r a} | = | (\hat{μ} - 2 \overset{}{\hat{σ}}) - (μ - 2 σ) | \\ = | \overset{}{\bar{X}} - μ | = \frac{μ_{1 - 0.5 α} σ}{\sqrt{n}} = \frac{μ_{1 - 0.5 α} \overset{}{\hat{σ}}}{\sqrt{n}} \end{matrix}

(8)

Where $u_{1 - 0.5 α}$ is the quantile that makes the probability of $| u | < u_{1 - 0.5 α}$ is $1 - α$ .

According to equation (8), the error of the buckling capacity design value is relative to the reciprocal of the arithmetic square root of the sample size.

According to equations (7) and (8), buckling capacity design values and relative errors of 54 steel circular hollow sections considering welded hollow spherical joint restraints listed in Table 2 can be calculated. The results are shown in Table 4.

Table 4.

Buckling capacity design values and relative errors of steel circular hollow sections.

No.	1	2	3	4	5	6	7	8	9
Buckling capacity design value/kN	715	623	588	455	416	355	283	151	91.08
Relative error/%	5.13	6.77	3.11	4.11	3.51	3.07	5.16	6.41	3.92
No.	10	11	12	13	14	15	16	17	18
Buckling capacity design value/kN	42.3	317	481	643	842	989	1078	1278	1325
Relative error/%	4.96	9.59	9.11	4.32	3.47	2.95	7.1	1.76	1.43
No.	19	20	21	22	23	24	25	26	27
Buckling capacity design value/kN	1510	1638	111	218	309	387	593	471	621
Relative error/%	2.45	1.75	2.97	4.95	11.5	8.06	5.04	5.49	6.09
No.	28	29	30	31	32	33	34	35	36
Buckling capacity design value/kN	839	1023	1491	623	623	623	623	623	623
Relative error/%	3.48	2.68	1.95	6.78	6.78	6.78	6.78	6.78	6.78
No.	37	38	39	40	41	42	43	44	45
Buckling capacity design value/kN	623	623	623	623	935	623	827	1190	1396
Relative error/%	6.78	6.78	6.78	6.78	1.05	6.8	4.21	6.17	1.49
No.	46	47	48	49	50	51	52	53	54
Buckling capacity design value/kN	1485	1378	1083	864	738	623	976	1028	1069
Relative error/%	1.29	1.29	0.73	3.04	1.65	5.33	4.46	6.26	5.44

Buckling capacity of steel circular hollow section considering the restraint of welded hollow spherical joint

It can be seen from the analysis in Section ‘Buckling capacity of steel circular hollow section’ that it is necessary to calculate the sample of plenty of data by statistical method when determining the buckling capacity design value of steel circular hollow sections considering the joints restraint. Although this method is fully justified by the stochastic characteristic, it is not practical due to the massive calculation work. A formula for the buckling capacity design value of steel circular hollow sections considering the joints restraint needs to be derived. Therefore, the influence of the steel circular hollow section length $l$ , the cross section wall thickness $δ$ of the steel circular hollow section, the cross section outer diameter $d$ of the steel circular hollow section, the wall thickness $t$ of welded hollow spherical joint, the outer diameter $D$ of welded hollow spherical joint and the material design strength on the buckling capacity design value of steel circular hollow section should be analysed respectively. The relationship between the buckling capacity design value of steel circular hollow section and each parameter should be quantitatively determined by linear regression method.

The buckling capacity design value $P_{c r d}$ of the steel circular hollow section models in Table 2 are all listed in Table 4. Each type of the parameter of model 1–model 10 has the same value except for the steel circular hollow section length $l$ . The curve of $l$ – $P_{c r d}$ of model 1–model 10 is shown in Figure 5. It can be seen from Figure 5 that the buckling capacity design value decreases with the increment of the steel circular hollow section length.

Figure 5.

The curve of $l$ – $P_{c r d}$ of model 1–model 10.

Each type of the parameter of model 11–model 20 has the same value except for the cross section wall thickness $δ$ of the steel circular hollow section. The curve of $δ$ – $P_{c r d}$ of model 11–model 20 is shown in Figure 6. It can be seen from Figure 6 that the buckling capacity design value increases with the increment of the cross section wall thickness of the steel circular hollow section.

Figure 6.

The curve of $δ$ – $P_{c r d}$ of model 11–model 20.

Each type of the parameter of model 21–model 30 has the same value except for the cross section outer diameter $d$ of the steel circular hollow section. The curve of $d$ – $P_{c r d}$ of model 21–model 30 is shown in Figure 7. It can be seen from Figure 7 that the buckling capacity design value increases with the increment of the cross section outer diameter of steel circular hollow section.

Fig. 7.

The curve of $d$ – $P_{c r d}$ of model 21–model 30.

Each type of the parameter of model 31–model 40 has the same value except for the wall thickness $t$ of welded hollow spherical joint. The curve of $t$ – $P_{c r d}$ of model 31–model 40 is shown in Fig.ure 8. It can be seen from Figure 8 that the wall thickness of welded hollow spherical joint has no influence on the buckling capacity design value of steel hollow section.

Figure 8.

The curve of $t$ – $P_{c r d}$ of model 31–model 40.

Each type of the parameter of model 41–model 50 has the same value except for the outer diameter $D$ of welded hollow spherical joint. The curve of $D$ – $P_{c r d}$ of model 41–model 50 is shown in Figure 9. It can be seen from Figure 9 that with the increment of the outer diameter of welded hollow spherical joint the buckling capacity design value of steel hollow section increases first and then decreases.

Figure 9.

The curve of $D$ – $P_{c r d}$ of model 41–model 50.

Each type of the parameter of model 51–model 60 has the same value except for the material design strength $f$ . The curve of $f$ – $P_{c r d}$ of model 51–model 60 is shown in Figure 10. It can be seen from Figure 10 that the buckling capacity design value of steel hollow section increases with the increment of material design strength $f$ .

Figure 10.

The curve of $f$ – $P_{c r d}$ of model 51–model 60.

Construct dimensionless variables $D^{2} / δ l$ and P_crd/πfd². The relationship curve of $D^{2} / δ l$ and P_crd/πfd² is shown as Figure 11. Based on Figure 11 it indicates that these two dimensionless variables are linearly related with a linear correlation coefficient of 0.972. By least square method the relationship of these dimensionless variables can be expressed as

\frac{P_{c r d}}{π f d^{2}} = - 0.0413 + 0.0103 \frac{D^{2}}{δ l}

(9)

Therefore, the expression of $P_{c r d}$ is given as follows:

P_{c r d} = (- 0.0413 + 0.0103 \frac{D^{2}}{δ l}) π f d^{2}

(10)

By equation (10) the restraint of welded hollow spherical joint on steel circular hollow section is finely considered. It corresponds with the actual case. Based on the current design code,³ the restraint of welded hollow spherical joint on buckling capacity of steel circular hollow section is considered by a unified effective length coefficient. The different restraint of welded hollow spherical joints of different sizes on steel circular hollow section is not finely taken into account. The buckling capacity values of model 2, model 13, model 26, model 36, model 43 and model 53 in Table 2 are respectively calculated based on the code and equation (10). The calculation results are shown in Table 5.

Table 5.

Buckling capacity values of steel circular hollow section based on the code^[3] and Eq. (10).

Steel circular hollow section model	Buckling capacity value/kN		Relative difference/%
Steel circular hollow section model	By the code	By equation (10)	Relative difference/%
Model 2	508	605	−16.03
Model 13	465	391	18.93
Model 26	286	241	18.67
Model 36	508	605	−16.03
Model 43	508	719	−29.35
Model 53	827	985	−16.04

Figure 11.

Relationship curve of $D^{2} / δ l$ and P_crd/πfd2.

It can be seen from Table 5 that the buckling capacity values of model 2, model 36, model 43 and model 53 calculated by the code are smaller than those calculated by equation (10). The buckling capacity values of model 43 calculated by the code is 29.35% smaller than that calculated by equation (10). The buckling capacity values of model 13 and model 26 calculated by the code are bigger than those calculated by equation (10). The buckling capacity values of model 13 calculated by the code is 18.93% bigger than that calculated by equation (10).

Therefore, when the restraint of welded hollow spherical joints is finely considered, the buckling capacity value of steel circular hollow section could be quite different from that calculated based on the current code. If the difference of restraint effect which is caused by different welded hollow spherical joint is ignored, the buckling capacity value of steel circular hollow section calculated based on the unified effective length coefficient may be bigger or smaller than that of the actual case. If the buckling capacity value calculated by the code is smaller than that calculated by equation (10), steel circular hollow section of bigger size would be used in the design, which makes the design uneconomical. If the buckling capacity value calculated by the code is bigger than that calculated by equation (10), steel circular hollow section of smaller size would be used in the design, which makes the design unsafe. Therefore, refined consideration should be placed on the different restraint of welded hollow spherical joints of different sizes on steel circular hollow section in the design, and the restraint of the end joints should be accurately considered when determining the buckling capacity of steel circular hollow sections.

Conclusions

In this paper, buckling mode is used as the initial geometrical imperfection form of steel circular hollow section. There are different numerical buckling capacity values corresponding to different buckling modes for one steel circular hollow section. Generally, there would be one value that is most close to that of the test result. This indicates that the buckling mode corresponding to this value best fits the actual geometrical imperfection, and it is feasible to use buckling mode as the initial geometrical imperfection distribution form of steel circular hollow section.

By nonparametric test, the buckling capacity value of geometrical imperfect steel circular hollow section considering the restraint of welded hollow spherical joint is verified to distribute normally. A new way of calculating the buckling capacity design value of steel circular hollow section at a uniform probability with the consideration of the restraint of welded hollow spherical joint is inferred, and its error expression at certain confidence level is also proposed.

The relationships between the buckling capacity design value of steel circular hollow section considering the joint restraint and the geometrical and material parameters of the member and joint are determined quantitatively by linear regression method. A practical formula for the buckling capacity design value of the steel circular hollow section considering the restraint of welded hollow spherical joints is derived.

When the restraint of welded hollow spherical joints is finely considered, the buckling capacity value of steel circular hollow section could be quite different from that calculated based on the current code. If the difference of restraint effect which is caused by different welded hollow spherical joint is ignored, the buckling capacity value of steel circular hollow section calculated based on the unified effective length coefficient may be bigger or smaller than that of the actual case. According to the calculation example of this paper, the relative difference of the stability bearing capacity of steel circular hollow section calculated by these two methods range between −29.35% and 18.93%. Refined consideration should be placed on the different restraint of welded hollow spherical joints of different sizes on steel circular hollow section, and the restraint of the end joints should be accurately considered when determining the buckling capacity of steel circular hollow sections.

One concern about the findings is that there are many types of joints applied in spatial latticed structures, and they offer different restraint. This paper focuses on welded hollow spherical joints, and future research should be undertaken to study the design stability bearing capacity of steel circular hollow section considering the restraint effect of other kinds of joints, such as bolt-ball joint, intersecting joint and cast steel joint.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful for the financial support from National Key Research and Development Plan of China (2021YFB2600500).

ORCID iD

Lin Qi

References

Bardi

Kyriakides

. Plastic buckling of circular tubes under axial compression—Part I: Experiments. Int J Mech Sci 2006; 48(8): 830–841.

Lee

Chang

Park

, et al. Compressive strength of girth-welded stainless steel circular hollow section members: stub columns. J Constr Steel Res 2014; 92: 15–24.

GB50017-2017. Code for design of steel structures. Beijing,China: China Construction Industry Press, 2017.

JGJ 7-2010. Technical specification of spatial grid structure. Beijing,China: Construction Industry Press, 2010.

Galvão

Silva

ARD

Silveira

RAM

, et al. Nonlinear dynamic behavior and instability of slender frames with semi-rigid connections. Int J Mech Sci 2010; 52(12): 1547–1562.

Gardner

Saari

Wang

. Comparative experimental study of hot-rolled and cold-formed rectangular hollow sections. Thin-Walled Struct 2010; 48(7): 495–507.

Gardner

Nethercot

. Numerical modeling of stainless steel structural components: a consistent approach. J Struct Eng 2004; 130: 1586–1601.

Wang

Guo

Bai

, et al. Experimental and numerical study on the stability capacity of Q690 high-strength circular steel tubes under axial compression. Int J Steel Struct 2017; 17(3): 843–861.

Mccann

Fang

Gardner

, et al. Local buckling and ultimate strength of slender elliptical hollow sections in compression. Eng Struct 2016; 111: 104–118.

10.

Hassan

Salawdeh

Goggins

. Determination of geometrical imperfection models in finite element analysis of structural steel hollow sections under cyclic axial loading. J Constr Steel Res 2018; 141: 189–203.

11.

Shen

Wadee

. Length effects on interactive buckling in thin-walled rectangular hollow section struts. Thin-Walled Struct 2018; 128: 152–170.

12.

Zhang

Huo

. Design bearing capacity of the initial imperfect lattice shell. Adv Struct Eng 2016; 19: 14–22.

13.

Shayan

Rasmussen

KJR

Zhang

. On the modelling of initial geometric imperfections of steel frames in advanced analysis. J Constr Steel Res 2014; 98: 167–177.

14.

Zhang

Jin

Shao

, et al. Practical advanced design considering random distribution of initial geometric imperfections. Adv Struct Eng 2011; 14(3): 379–389.

15.

Fraser

Budiansky

. The buckling of a column with random initial deflections. J Appl Mech 1969; 36(2): 233–240.

16.

Papadopoulos

Soimiris

Papadrakakis

. Buckling analysis of I-section portal frames with stochastic imperfections. Eng Struct 2013; 47: 54–66.

17.

Stefanou

Papadopoulos

Manolis

. Buckling load variability of cylindrical shells with stochastic imperfections. Int J Reliab Saf 2011; 5: 191–208.

18.

Graham

Deodatis

. Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties. Prob Eng Mech 2001; 16(1): 11–29.

19.

Stefanou

Papadrakakis

. Stochastic finite element analysis of shells with combined random material and geometric properties. Comput Methods Appl Mech Eng 2004; 193(1): 139–160.

20.

. Load-carrying capacity and practical design method of welded hollow spherical joints in space latticed structures. Adv Steel Constr 2010; 6: 976–1000.