Buckling-Controlled Member for Improving the Ductile Behavior of Double-Layer Latticed Space Structures

Abstract

In this paper, a new buckling-controlled member (BCM) is introduced for use in space structures. This member is composed of four components; namely: the encasing, joints, core, and adjustable nuts. The core is intended to act as a structural element to resist the axial loads by its yielding under compression loading. The steel encasing is supposed to confine the steel core. Adjustable steel nuts on the steel core act as lateral bracings and are responsible for lateral load transmission between the encasing and core. The joints at the two ends of the supports of the member. Six experimental tests have been performed under compression load to show the efficiency of the new member. The test results reveal that the proposed member can provide the needed ductility and can delay the brittle buckling of the members. Also, the BCM is capable of considering buckling modes and controlling the plastic range. The experimental and numerical results have also been compared. Additional numerical evaluations have been carried out using finite element models, in which the effects of different parameters of the member have been investigated. The obtained results showed that the arrangement of inner elements is the main factor affecting ductility and postponing the buckling of the members. In the end, the effects of the BCMs on the overall behavior of four double-layer space structures have been studied. The obtained results of analyses indicated that the BCMs can enhance the strength and ductility of space structures, thereby reducing the risk of collapse. Also, the seismic collapse of the space structure was postponed.

Keywords

Space structures compression member force-limiting device (FLD)buckling-controlled member (BCM)ductility brittle buckling

Introduction

Space trusses are used for covering long-span areas with few supports. In the past decades, architects have had an interest in the construction of these structures because of their lightweight, easy fabrication, fast installation, and appealing façade. Research results have shown the tendency of lattice space structures toward brittle and progressive collapse due to the buckling of some compression members. For this purpose, compression members must be effectively prevented from brittle buckling. In recent years, various techniques have been developed to control the behavior of members of space structures. One of these approaches is force-limiting devices.¹ These devices are used to make artificial ductility in compressive members. Several investigations have been carried out in this field such as the introduction of multi-tubular FLD by Parke.² The device was constructed from an outer pipe, an inner pipe, and four middle stripes. Both pipes are subjected to compressive load. The middle stripes, however, are under tensile stresses and are designed so that the four stripes yield first, thereby making the member’s behavior ductile. In 1980, a hydraulic FLD was introduced and employed in three space trusses.³ The results demonstrated that the capacity of the structure with the FLD is 23% more than that of structures without FLD. The device also increases the ductility of the structure significantly. Mukai et al.⁴ studied the effects of multi-tubular FLDs in three space trusses with square-on-square configuration. The results demonstrated that applying an FLD in a member of space structures increases the strength, but the ductility does not substantially improve due to the buckling of the members in the vicinity of the central member. However, using FLDs in all of the members of the upper chord significantly increased the ductility and strength of the space trusses. Abedi and Parke⁵ investigated the effects of a multi-tubular FLD in braced frames. The results showed a significant increase in the ductility and energy absorption of braced frames. Bai and Zhang⁶ evaluated the collapse of a steel-framed roof using FLD. The results showed that FLDs reduce the collapse probability of the roof subjected to wind load and increase the ductility of the roof.

Shen et al.⁷ conducted a parametric study to evaluate the effective parameters on the cyclic behavior of a steel buckling-controlled brace (BCB) with a tube carrying an axial load surrounded by an outer tube. The results demonstrated that the gap between the two tubes is an effective parameter for local and global buckling. Also, the friction between two tubes is a sensitive factor in the cyclic behavior of BCB. In addition, the thickness ratio of the buckling controlled and bearing tube is an important factor in controlling buckling.

Poursharifi et al.⁸ introduced a new type of FLD designed based on buckling restrained braces (BRBs) and accordion metallic dampers. The device was dubbed Accordion Force Limiting Device (AFLD). The experimental studies indicated that the brittle behavior of compressive members can be modified using the proposed FLD. Also, parametric studies were carried out to investigate the effects of gap size on the behavior of the AFLD. The results demonstrated that the gap size is a key factor in designing the AFLD.

Seker and Shen⁹ developed an all-steel tube in tube buckling controlled brace (TinT-BCB). In this study, the behavior of Tint-BCB was evaluated experimentally and numerically. The results demonstrated that the proposed member has a stable and symmetrical cyclic response, with global and local buckling. Also, the proposed member is effective in elongating the cyclic fracture life of braces.

Heidary-Torkamani and Maalek¹⁰ studied a comprehensive parametric investigation on the behavior and mode failure of tube-in-tube buckling restrained brace (TitBRB) members. The results of the study demonstrated that the well-designed TitBRB members can exhibit stable cyclic behavior and high ductility capacity. Therefore, they can be served as effective dampers.

Momenzadeh et al.¹¹ studied all-steel buckling–controlled braces (BCB) with two various cross sections. Its behavior was compared with that of the conventional brace. Then, the seismic response of a braced frame incorporating BCBs was compared with that of the conventional braces.

Wang et al.¹² presented a new fuse-type energy dissipating bar. This device consists of an inner bamboo-shaped core and an outer restraining tube. Experimental and numerical studies were performed to investigate the behavior of the proposed devices. The results showed stable hysteretic behavior and no local or overall buckling of the proposed device was observed.

Hamid et al.¹³ proposed a novel soft member to reduce the risk of progressive collapse in double-layer space trusses. This member is a combination of three circular tubes made into one component and is used to withstand compressive loads. The results showed that using the proposed member improved the collapse behavior of space trusses. Also, it was found that the combination of the three circular tubes increased the ductility of the member.

Zhang and Deng¹⁴ introduced a buckling-monitoring member to control the buckling of the compression member. For this purpose, finite element analyses were performed to investigate the effects of the important parameters on the behavior of buckling-monitoring members Furthermore, the buckling–monitoring member was applied to monitor the ultimate bearing capacity of a spaces truss.

Mateus et al.¹⁵ presented a buckling-restrained brace (BRB) which was composed of round steel bar cores restrained by inner round steel tubes and an outer square steel tube. The results revealed the applicability, restrained capacity, and end-coupler performance of the proposed brace.

In Table 1, the characteristics of mentioned members have been compared. The shortcomings of these members are as follows:

Excluding the higher buckling mode effects;

Incapability to monitor the buckling of the compression members;

Incapability to improve the bearing capacity and ductility in the members;

Incapability to control the plastic plateau of the members;

Incompatibility with any type of common connection system in structures for example, space structures;

Lack of a design algorithm for the majority of force-limiting devices.

Complexity and difficulty in assembling members with highly skilled labor;

Need for advanced and costly mechanical systems;

Table 1.

The comparison of the characteristics of members.

Type	Type of restraining	Proposed Applications	Effect on structural behavior (studied/not studied)	Design algorithm	Considering higher buckling mode	Buckling mode control	Plastic plateau control
Force Limiting Device(FLD)¹	_	Not specified	Not studied	×	×	Uncontrollable	×
Force Limiting Device(FLD)²	Continuous	Space structure	studied	×	×	Uncontrollable	×
Force Limiting Device(FLD)²
Buckling controlledBrace⁹	Continuous	Common Structures	Studied	×	×	Uncontrollable	×
Buckling controlledBrace⁹
tube-in-tube buckling controlled brace⁹	Continuous	Common Structures	studied	×	×	Uncontrollable	×
tube-in-tube buckling controlled brace⁹
Buckling restrained Brace(BRB)¹⁰	Discontinues	Offshore structure	Not studied	×	×	Uncontrollable	×
Buckling restrained Brace(BRB)¹⁰
Accordion Force Limiting Device(AFLD)⁸	Continuous	Space structure	studied		×	Uncontrollable	×
Accordion Force Limiting Device(AFLD)⁸
Bamboo-shaped energy dissipaters¹²	Discontinues	Common structures	Not studied	×	×	Controllable	×
Bamboo-shaped energy dissipaters¹²
Novel soft members¹³	Continuous	Space structures	Not studied	×	×	Uncontrollable	×
Novel soft members¹³
buckling-monitoring member¹⁴	Continuous	Space structures	Not studied	×	×	Uncontrollable	×
buckling-monitoring member¹⁴
buckling-restrained brace (BRB)¹⁵	Discontinues	Common structures	Not studied	×	×	Controllable	×
buckling-restrained brace (BRB)¹⁵
Buckling Controlled Member (BCM)¹⁶	Discontinues	Space structures	studied	×	×	Controllable	×
Buckling Controlled Member (BCM)¹⁶

To overcome the shortcomings of these members, a new member has been proposed. This member has been named the Buckling-Controlled Member (BCM) and is capable of controlling the capacity and ductility of not only the compressive members but those of the structure itself. For this purpose, an experimental study was conducted on the new proposed member. The experimental results were verified using the ABAQUS finite element analysis package.¹⁷ Subsequently, a parametric study was performed to investigate the effects of important factors on the improvement of the ultimate bearing capacities and ductility of the members. Also, the behaviors of four double-layer space structures were evaluated by employing the proposed device in the critical members.

Buckling-controlled member as a new member for space structures

Similar to BRB elements in the frames, elements can be designed in such a way, in addition to the buckling resistance, it doesn’t interfere with the appearance of the structures and increase the ductility and load-bearing capacity. Also, the buckling can be controlled. Therefore, a new member called the buckling-controlled member (BCM) is introduced.

This member is composed of four main parts: the steel encasing, the steel core, the steel nuts, and the joints. The components and an actual specimen of the proposed member are shown in Figure 1(a) and (b). In the member, the role of the steel encasing is for confining the steel core. Adjustable steel nuts on the steel core act as lateral bracings. The joints at the two ends of the supports of the member. As a result, a pin-ended member is formed. In the body of the encasing, there are two bean holes at two ends of the elements. The role of these holes is calibrating and keeping the steel core of the element in the encasing.

The components of the BCM (the encasing, joints, core, and adjustable nuts) are made of two different grades of steel. The mechanical properties of steel materials were obtained through tensile tests based on the ASTM A370 standard.¹⁸ The elastic modulus, yield, and ultimate strengths obtained from the tests are given in Table 2.

Table 2.

The mechanical properties of steel encasing and core.

	Specimen number	Young modulus (GPa)	Yield stress (MPa)	Ultimate stress (MPa)	Elongation (%)
Encasing	1	195.9	616	648	6
Encasing	2	198.4	615	665	6
Core	3	197.3	415	695	21
Core	4	197.3	415	692	21

Six test specimens in three groups have been considered. The length of each specimen was 510 mm according to the laboratory facilities. The outer diameter of the encasing is 36 mm and the thickness is 3 mm. The diameter of the core is 14 mm. The outer and inner diameters of the nuts are 28 and 14 mm, respectively. Also, the thickness of the nuts is 12 mm.

Experimental study

In order to determine the behavior of a buckling-controlled member (BCM). Six test specimens in three groups with the following characteristic were performed. Table 3 contains the specifications of the studied groups. It should be noted that the nuts were arranged in two configurations: (1) regular arrangement and (2) irregular arrangement. These arrangements and the details of the proposed BCM are depicted in Figure 2.

Table 3.

Specification of tested groups.

Group	Specimen	Series	Arrangement of nuts	Nut intervals (L) (mm)	Gap size (mm)
1	1	Encasing
2	2	BCM-R	Regular	50	20
	3	BCM-R	Regular	60	20
	4	BCM-IR	Irregular	Matching with four buckling mode shapes	20
3	5	BCM-R	Regular	50	10
3	6	BCM-R	Regular	50	30

The variables in Figure 2 are as follows:

$L$ and $L_{1}$ , $L_{2}$ , $L_{3}$ , $L_{4}$ , $L_{5}$ are nut intervals in regular and irregular arrangements of nuts;

$L_{c o r e}$ is the length of the core;

$L_{E N}$ is the length of the encasing;

$t_{E N}$ is the thickness of the encasing;

$d_{c o r e}$ , $d_{n u t - o u t e r}$ , and $d_{o u t e r}$ are the diameter of the core, the diameter of the nuts, and the outer diameter of the encasing;

Gap is the distance between the core and the encasing.

The first test was carried out on the steel encasing to draw a comparison with the behavior of the BCMs. Figure 3(a) shows the supports and the first specimen for the experimental study. It should be noted that the supports were made of high-strength steel, CK45. The supports have been designed so that the member can rotate up to 10 degrees. The other test specimens were BCMs. In the process of assembling these specimens, the nuts are adjusted on the steel core and fixed in specific intervals by an embedded bolt in the nuts. Then, the core is placed in the steel encasing. In order to transfer loads from the core to the encasing, a gap was considered at the two ends of the specimens. Figure 3(b) and 3(c) show the details and the fabricated test specimens, respectively.

The test setup and loading protocol

A 500 kN capacity universal testing machine was used to extract the behavior of the proposed member. A displacement gage was used to control the buckling of the proposed member. A controller software was used during the tests to record the results. The tests were carried out with a loading rate of 0.01 mm/s and under displacement-control loading type. The values of the applied loads and the relative displacements were recorded in each loading step.

Test results

The applied axial load, the displacement of the core, and the lateral displacement of the encasing were the recorded parameters in the test. The axial load-axial displacement responses of the buckling-controlled members were derived from the experimental tests. The results for each group are as follows:

Group 1

The axial load-axial displacement response of the steel encasing has been illustrated in Figure 4(a). The results showed the linear behavior of the member before buckling and a sudden reduction in the maximum capacity of the member upon reaching the critical load. Also, Figure 4(b) depicts the buckling of the encasing in the UTM system.

Group 2

In this section, the experimental results of the BCMs are presented. The nuts are arranged in regular intervals (specimens 2, 3) and based on the four buckling mode shapes (specimen 4) of the steel core. Figure 5 shows the test results. It can be concluded that the brittle behavior of the encasing changes to a ductile behavior when the buckling of the member is delayed. It is also observed that the plastic region of the member is 40 mm due to the 20 mm gap size between the core and the encasing. After the core buckles, the load transfers from the core to the encasing. Therefore, a sudden spike occurs in the axial load-axial displacement curve.

Group 3

In order to evaluate the effects of gap size between the steel core and the encasing on the behavior of the proposed member, two specimens with gap sizes of 10 and 30 mm (specimens 5 and 6) were examined. The results are illustrated in Figure 6. The same results are derived in this section, with the difference that the lengths of the plastic region are 20 and 60 mm. Therefore, it can be concluded that the plastic region is a function of the gap size and is an important parameter in postponing the member’s buckling. In other words, the length of the plastic region can be controlled by the gap size.

Comparison between experimental and numerical results

In this section, the experimental results are used to verify the accuracy of the finite element model of the member using ABAQUS.¹⁷ All the components of the proposed member are modeled using solid elements. A friction interaction was considered between the core and the encasing for the longitudinal behavior and a hard contact-type interaction was used for the normal behavior. The friction coefficient of steel on a steel surface is considered equal to 0.3.¹⁹ Also, tie constraint is used to connect the nuts, steel bars, and supports.²⁰ The axial stress-axial strain behaviors of the core, encasing, and nuts, obtained from a series of coupon tests (as mentioned in section 2), have been used in defining the material behaviors in the nonlinear finite element analyses. The nonlinear static analysis with the modified Arc-Length-Riks method was carried out. An initial member length imperfection of 0.1% is considered in the middle of the steel core and encasing.²¹ Based on the mesh sensitivity analysis, finer mesh grids were used to discretize the core and coarse mesh grids were used for the joints in order to increase the speed of the analysis. The C3D8R element (in ABAQUS), an 8-node solid element, was used for all the components of the member, with reduced integration and hourglass control. Nonlinear static analyses were performed on the BCM specimens and the numerical results were compared with the experimental results. Figure 7(a1) shows the experimental and numerical results for the encasing specimen. The figure shows that the buckling loads obtained from the numerical modeling and the test are equal to 160 kN (at a displacement of 1.387 mm) and 153 kN (at a displacement of 5.32 mm), respectively. Also, the results demonstrate that the error percentage of the model and the test in predicting the stiffness of the specimen is sufficiently small. In addition, the results of the other specimens are compared and shown in Figure 7(a1) to (a6). The comparison of numerical and experimental results is presented in Table 4. The table indicates that the error percentage of the numerical modeling and the tests in predicting the initial stiffness is between 8% and 18%. Also, the error percentages in estimating the length and slope of the plastic region are 3%–24% and 10%–25%, respectively. In addition, the comparison of results indicates an error percentage of 2%–13% in determining the ultimate strength of the specimens.

Table 4.

Comparison of numerical and experimental results.

		Initial stiffness (kN/mm)	Plastic length (mm)	The slope of the plastic region	Ultimate strength (kN)
Specimen 1	Test	61.7	-	-	153
	Numerical modeling	70.1	-	-	165
	Error (%)	13.6	-	-	7.8
Specimen 2	Test	23.17	35.38	1.27	213
	Numerical modeling	25.66	37.06	1.4	209
	Error (%)	11	5	10	−2
Specimen 3	Test	17.95	34.85	0.68	210.7
	Numerical modeling	22.05	35.92	0.8	214
	Error (%)	17	3	18	2
Specimen 4	Test	20.02	34.77	0.96	217.42
	Numerical modeling	25.61	40.23	1.2	217.049
	Error (%)	18	16	25	0
Specimen 5	Test	15.7	24.89	1.53	228.91
	Numerical modeling	16.9	30.75	1.9	237.56
	Error (%)	8	24	24	4
Specimen 6	Test	12.95	30.77	0.57	210.35
	Numerical modeling	17.44	37.06	0.66	236.69
	Error (%)	12	20	16	13

Therefore, it can be seen that a good agreement exists between the experimental and numerical axial load-axial displacement responses.

Figure 7(b1) to (b6) illustrate the good agreement between the deformed states of the experiments and the numerical models. Meanwhile, the test results exhibit that nut arrangement has a significant effect on the deformed shape of the core; so much so that the entire steel core buckles when the nuts are arranged regularly, whereas the two ends of the steel core buckle when the nuts are arranged irregularly. This is due to the concentration of nuts in the middle of the core. Based upon the suitable agreement between the test data and the numerical results, the introduced finite element model can be confidently employed in the parametric study.

The experimental results demonstrate that the behavior of the proposed BCM has five parts as shown in Figure 8. These parts are as follows:

(1) Part OA (Region-I: Linear Elastic Range). During the initial stage of loading, stress alters linearly proportional to strain up to the proportional limit stress (F_el). The relationship between stress and strain in this region is F_e = E $. ε_{e}$ in the range F<F_el and $ε$ < $ε_{e l}$ , where E is the initial elastic modulus of the steel core.

(2) Part AB (Region-II: Nonlinear Elastic Range). This range is the region between the proportional limit and yield point. The stress-strain relationship in this region is F_e = F_el+ E_t. ( $ε$ - $ε_{e l}$ ), where E_t is the tangent modulus which can be obtained from the following equation:

E_{t} = \frac{Δ F}{Δ ε}

(1)

(3) Part BC (Region-III: Plastic Range). This range is the plastic region which is a function of gap size and encasing. It should be noted that these regions are related to the behavior of the steel core;

L = f (gap size, encasing)

(2)

(4) Part CD (Region-IV: strength Hardening Range). At the end of the plastic region, strength hardening starts due to the simultaneous performance of the steel encasing and core and the gap size. This range is the region between the end of region-III and ultimate strength.

(5) Part DE (Region-V: Post-Buckling Range). This range shows the post-buckling region of the member. When the strength of the member reaches the ultimate strength, the continued loading causes the buckling of the member.

Parametric study

In this section, the effects of nut arrangement and gap size between the core and the encasing are evaluated. Therefore, different specimens in two groups have been considered. Table 5 gives the specification of the studied groups. The naming of each specimen is according to the geometrical parameters such as the arrangement of nuts (regular or irregular), nut interval, and gap size between the core and the encasing. For instance, the first specimen has been named BCM-R-D40-G30, where BCM is the abbreviation for the Buckling-Controlled Member, R stands for Regular, D stands for nut distance (in mm), and G is the size of the gap (in mm). It is worth noting that IR and N are used instead of R and D in the naming and numbering of specimens with irregular arrangements of nuts.

Table 5.

The specification of the studied groups.

Group	Parameter	Arrangement	Numbering	Nut intervals (mm) (D)	Gap size (mm) (G)	Designation
1	Nut intervals	Regular	N1	10	30	BCM-R-D. . .-G. . .
			N2	20
			N3	30
			N4	40
			N5	50
			N6	60
			N7	70
			N8	80
			N9	90
			N10	100
		Irregular	N11	40-50-70-70	30	BCM-IR-N. . .-G. . .
			N12	40-70-60-70
			N13	Matching with four buckling mode shapes
			N14	20-30-40-40-50-60
			N15	100-62.5-37.5-25-12.5-12.5-12.5
			N16	30-40-50-60-70
			N17	30-30-40-50-70
2	Gap size	Regular	N17	50	10	BCM-R-D. . .-G. . .
			N18		20
			N19		30
			N20		40
			N21		41
			N22		42
			N23		43
			N24		44
			N25		45
			N26		46
			N27		47
			N28		50

The numerical results are presented in the following parts:

Group 1 (the effects of nut arrangement)

In this part, the effect of nut arrangement on the behavior of the proposed member is investigated. Regular and irregular arrangements of nuts were considered.

Regular nut arrangement

Regular nut arrangement (group 1) on the steel core causes the buckling of the member to be delayed and results in the creation of the plastic region before the member buckles. Also, the results show that increasing the inter-nut distance causes the slope of the plastic region to decrease, such that the BCM-R-D10-G30 specimen has the maximum slope and the BCM-R-D100-G30 specimen has the minimum slope. In other words, by increasing the number of nuts on the core, the behavior of the proposed member becomes similar to the tensile behavior of steel before the overall buckling (see Figure 9).

Irregular nut arrangement

The results of the analysis performed on the proposed member with irregular nut arrangement demonstrated that similar to the previous specimens, the buckling of the member is postponed. Compared to the other specimens, the BCM-IR-N13-G30 specimen has a suitable behavior in terms of its plastic region and ultimate strength (see Figure 10(a)) due to the arrangement of nuts based on the higher buckling modes. Also, the effect of nut arrangement on the behavior of the proposed member has been studied. For this purpose, the behavior of the member with a regular nut arrangement has been compared to that of the member with an irregular nut arrangement. Both arrangements have the same number of nuts. The results indicate that the concentration of nuts in the central part of the core increases the slope of the plastic region (see Figure 10(b)).

Group 2 (the effects of gap size)

Another parameter to be considered in this study is the effect of gap size on the behavior of the member. Therefore, many gap sizes, from 10 to 47 mm, were considered. Analysis results indicate that the plastic region is a function of the gap size between the core and the encasing (see Figure 11(a)). The plastic length increases linearly for gap sizes that are 44 mm and smaller, and undergo a sudden decrease for larger gap sizes.

For instance, the length of the plastic region in specimens with a gap size of 10 and 44 mm are 20 and 88 mm, respectively. Whereas the length of the plastic region in the specimen with a gap size of 45 mm is 86 mm. In addition, the results show a sudden spike in the axial load-axial displacement curves of the members, and the ultimate strength decrease as the gap size increases. Therefore, it can be concluded that the gap size is an important parameter affecting the ductility, strength, and delay in the overall buckling of members (see Figure 11(b)). Meanwhile, the results indicate that for gap sizes larger than 40 mm, the behavior and the ultimate load of the member remain almost unchanged.

According to the experimental and parametric studies, it can be concluded that the behavior of the proposed member is dependent upon the arrangement of nuts (regular or irregular), nut interval, and gap size between the core and the encasing. This means that unlike material properties, any change in these parameters affects the ductility and the ultimate strength of the member.

Design procedure of the BCM

In this section, a design procedure for the proposed BCM is presented. This procedure applies to space structures. The design procedure is as follows:

Determine the size of the gap (denoted by G and a recommended value of 30 mm); the gap size is used to calculate the length of the steel core (L_core) (see Figure 3(c)). The insertion length of the end nuts into the encasing is considered 2G.

L_{c o r e} = L_{0}

(3)

L_{E N} = L_{c o r e} - 2G

(4)

where L₀ and L_EN are the length of the member and the steel encasing, respectively.

2. Design of steel core;

In this step, the cross-sectional area of the steel core (A_b) is obtained. For this purpose, the critical compression members are first identified. Then, the diameter of the steel core was determined based on equation (5).

\begin{array}{l} N_{c r} = 1.5 A_{b} . f_{y} \\ A_{b} = \frac{π D_{c o r e}^{2}}{4} \end{array}

(5)

where f_y is the yield strength of the steel core, N_cr is the force of the critical compression member, and D_core is the diameter of the steel core.

3. Selection of a space (c) between the adjustable nuts and the encasing (c = 1 mm is recommended);

4. Design of the steel encasing;

4.1. Consider the diameter-to-thickness ratio of the steel encasing based on equation (6) to avoid local buckling.

\frac{D_{E N}}{t_{E N}} 0.11 \frac{E_{E N}}{f_{y}}

(6)

At first, this ratio is assumed to be equal to $0.11 \frac{E_{E N}}{f_{y}}$ . Therefore, the diameter of the encasing is obtained based on the thickness $t_{E N}$ :

D_{E N} = 0.11 \frac{E_{E N}}{f_{y}} t_{E N}

(7)

The thickness $t_{E N}$ will be obtained as follows:

4.2. The encasing of the proposed BCM should restrain the buckling of the steel core based on equation (8).²².

M_{Y, E N} \geq M_{E N}

(8)

M_{Y, E N} = S_{E N} . f_{y}

(9)

M_{E N} = \frac{N_{c r} . c}{1 - \frac{N_{c r}}{N_{E N}}}

(10)

N_{E N} = \frac{π^{2} . E_{E N} . I_{E N}}{L_{E N}^{2}}

(11)

S_{E N} = \frac{I_{E N}}{y}

(12)

where $M_{Y, E N}$ is the yield flexural strength of the encasing, $M_{E N}$ is the flexural strength of the encasing $S_{E N}$ , is the elastic section modulus, I_EN is the moment of inertia of the steel encasing, and y is the distance from the neutral axis to the farthest fiber. At first, substituting equations (9) and (10) into equation (8) gives the thickness of the encasing. Then the diameter of the section is determined using equation (7).

5. Determining the nut interval (L) and the arrangement of nuts:

For regular arrangement, the intervals are recommended to be between 5 and 7 cm, based on the results of the parametric studies in Section 7-1-1 and for irregular arrangement, the nut intervals are matching with four buckling mode shapes;

6. Determining the diameter of the adjustable nuts based on the following equation:

d_{nut, outer} = D_{EN} - 2 t_{EN} - 2 c

d_{nut, inner} = D_{core}

(13)

where D_EN is the outer diameter, t_EN is the thickness of the steel encasing, and d_nut,outer and d_nut,inner are the outer and inner diameters of the nuts, respectively.

Effects of buckling-controlled member (BCM) on the behavior of double-layer space structures

In this section, the effects of the proposed BCMs on the behavior of space structures have been assessed. For this purpose, three double-layer space structures, a flat roof, a barrel vault roof, and a braced dome have been considered (see Figure 12). The behavior evaluation of double-layer space structures has been conducted in two states:

(a) State 1: the collapse behavior of the first three space structures has been evaluated under the snow loading;

(b) State 2: the seismic collapse behavior of the last one has been assessed under the ground motion records (it is noted that the seismic collapse behavior of space structure has not been presented in this paper).

It is worth noting that the behavior evaluation of the space structures should be carried out according to the following proposed procedure:

Configuration processing of the space structure models using the FORMIAN software;

Determining the buckling behavior of the members;

In this step, the axial load-axial displacement buckling responses of the member are extracted with an initial member imperfection of 0.1% at the middle of the member;

3. Carrying out the stability analyses under loading.

3.1. a static stability analysis has been carried out under incremental symmetric snow loading, considering the material and geometric nonlinearities. In the present study, to determine the equilibrium paths through limit points into the post-critical range, the “Arc-Length-Type Method” was used;

3.2. an incremental dynamic analysis has been conducted under the ground motion records to assess the seismic stability of the studied space structure.

4. Identification of the critical compression members:

In this step, the critical compression members are identified. Using the results of the nonlinear static and dynamic analyses in step 3, the locations of the buckled members are determined. Then, they are categorized into different sets of buckled members (e.g. first set, second set, etc). The effects of member buckling on the response of the double-layer space structures can be classified as follows²³:

(i) Overall collapse;

(ii) Local collapse with snap-through;

(iii) Local collapse without snap-through.

5. Designing of the BCM;

After identifying the critical compressive members (buckled members), the BCMs are designed based on the load-carrying capacity of the critical members. In this step, the arrangements of nuts (regular or irregular), nut intervals (L), core diameter (d_core), nut diameter (D_outer,nuts), the thickness of encasing (t_EN), and the space between the adjustable nuts and encasing (c) must be determined based on the proposed BCM design procedure.

6. Replacement of the critical members by BCMs;

It should be noted that BCMs are used in a small number of critical compressive members. The strategy of replacement of BCMs is that after designing the BCMs using the proposed procedure, the axial displacement-axial load response of the BCMs is extracted using numerical modeling in ABAQUS. Then the compressive behavior of the critical compressive members is replaced by this behavior.

7. Re-evaluation of structural stability by re-carrying out the static and dynamic stability analyses of the space structure model equipped with BCMs.

Studied space structures

State 1- the collapse behavior evaluation of space structures under symmetric snow loading

In this section, the collapse behavior of double-layer space structures has been evaluated under snow loading using nonlinear static analysis. The specifications of the studied space structures are given in Table 6.

Table 6.

The mechanical properties of materials and the section properties of members in the studied space structures.

Modulus of elasticity(E) $M P a$	Poisson’s ratio( $ν$ )	Yield stress( $F_{y}$ ) $M P a$	Coefficient of thermal expansion( $α$ ) 1/ϒ c		Mass Density $k g / m^{3}$
2.1×10⁵	0.3	400	12×10⁻⁶		7850
The specifications of studied space structures	Double-layer flat roof		Double-layer barrel vault roof		Double-layer dome
	size	Rise-to-span ratio	size	Rise-to-span ratio	size	Rise-to-span ratio
	30×30×2.1 m	0	30×42×1.5 m	0.3	30×2 m	0.3
Range of section properties	Chord	Web	Chord	Web	Chord	Web
Section Area A(cm²)	13.95–28.27	13.95	5.37–17.72	17.72	5.37–8.06	7.73
Diameter D(mm)	80–100	80	60–100	100	77–85	85
Thickness t_w (mm)	6–10	6	3–6	6	3–6	3

These space structures were designed according to a total factored load of 395 kg/m² (1.4×100 kg/m² dead load+1.7×150 kg/m² snow load), applied as concentrated nodal loads, while snow load was applied in two symmetric and asymmetric patterns based on the sixth volume of the national design code of buildings of Iran.²⁴ The mechanical properties of the materials and the section properties of the members in the studied space structures are given in Table 6, respectively. After designing the studied space structures, the behavior of members in tension and compression was extracted. Then the nonlinear analysis was carried out using ABAQUS. Both geometric and material nonlinearities were considered in the analyses. It should be noted that the modified Arc-Length-Riks method was used to trace the equilibrium paths after reaching the limit points into the post-critical range.

Numerical results

In this section, the behavior of the studied space structures was evaluated based on the proposed procedure. At first, the location of critical members (see Figure 13(a)–(c)) in each case was determined using nonlinear static analysis. It is noted that the location of the critical members in all models is different. The extracted member behavior in tension and compression with slenderness ratios of $λ$ = 85 and $λ$ = 78 and with an initial member imperfection of 0.1% at the middle of the member is depicted in Figure 13(d). Based on the capacity of the critical members, the BCMs were designed according to the proposed design procedure in section 8. Then, the behavior of the designed BCMs was extracted via numerical modeling in ABAQUS 6.12. As an example, the behavior of the designed the BCM for a critical member of the double-layer flat roof is shown in Figure 13(e).

After obtaining the behavior of the BCMs, the critical members were replaced by the proposed members, and the behavior of the space structures was re-evaluated. The percentage of the BCMs used in each model was 3%–4%. Also, the displacement-load proportional factor (LPF) responses of the models have been depicted in Figure 14. LPF is the collapse load ratio, calculated by dividing the collapse load obtained from the nonlinear static analysis by the design load. The results demonstrate that the BCMs used in the critical members improve the behavior of the structures. In the models without BCMs, the structures collapse under the applied loads after reaching the critical loads. Whereas, by replacing the critical members with BCMs, the structures can maintain their resistance without strength degradation. In other words, the behavior of space structures could be altered from sudden loss of strength to ductile behavior using BCMs. The collapse of the structure is therefore postponed.

From the analysis results, it can be drawn that the BCMs improve structural parameters such as strength, stiffness, and ductility. In order to evaluate the effects of the proposed member, the ratios of the ultimate strength and stiffness of the structures were calculated. The calculations, as shown in Table 7, demonstrate the increase of these parameters in the models. Also, the ductility ratios of the models have been determined. The ductility ( $μ$ ) is calculated as the ratio of ultimate displacement $δ_{u}$ to the yield displacement $δ_{y}$ .

Table 7.

Effects of BCMs on the structural important parameters.

	Double-layer flat roof				Double-layer barrel vault				Double-layer dome roof
	Ultimate strength (kN)	Stiffness (MPa)	Ductility ( $μ$ )	Energy absorption (kN.m)	Ultimate strength kN)	Stiffness (Mpa)	Ductility ( $μ$ )	Energy absorption (kN.m)	Ultimate strength (kN)	Stiffness (Mpa)	Ductility ( $μ$ )	Energy absorption (kN.m)
Without BCM	8489.6	180	4.19	1335.06	6528.78	367	1.12	671.26	67.58	3.80	1.41	7.14
With BCM	13,018.5	210	6.88	2109.77	9863	724	8.69	1873.14	90.10	5.60	5.07	15.98
Ratio	1.53	1.17	1.64	1.58	1.51	1.97	7.79	2.79	1.33	1.47	3.59	2.24

μ = \frac{δ_{u}}{δ_{y}}

(8)

$δ_{u}$ is the displacement at the descending branch of the axial load-axial displacement response, where the applied load decreases to 80% of the peak value. The yield displacement is computed corresponding to the first yielding of the members. Furthermore, the ratio of energy absorption for the structures was determined. This ratio relates the energy absorbed by the models with BCM to the energy absorbed by models without BCM. According to the obtained results, the ratio of energy absorption in double-layer flat roof, double-layer barrel vault roof, and double-layer dome roof are respectively 1.58, 2.79, and 2.24. Also, the ductility ratios in the studied models are 1.64, 7.79, and 3.59, respectively. Therefore, it can be concluded that the used BCMs in the critical members have a significant effect on the absorbed energy and the increase in the ductility of structures.

According to the results of the aforementioned analyses, the benefits of the using BCMs in space structures are as follows:

Preventing the brittle collapse of the structure;

Increasing the stiffness, ductility, and Energy absorption capacity of the space structure;

Increasing the load-carrying capacity of the space structure.

Conclusions

Single and double-layer space structures are extensively used for covering long-span areas. However, in these structures, the buckling of compressive members can lead to progressive collapse and loss of life and property. Therefore, in the present study, a new member called the Buckling-Controlled Member (BCM) has been introduced. This member has low steel consumption and can be easily implemented without the need for an expert workforce. Also, this member significantly improves the ductility of structures and can consider the higher buckling modes and control the plastic range.

In order to evaluate the behavior of the proposed member, six experimental specimens with the same length were tested. Also, a design procedure was presented to design the BCMs and the effects of the proposed member on the behavior of the studied space structures. For this purpose, three double-layer space structures were selected. Based on the analysis results, the locations of the buckled members were determined and the BCMs were designed using the presented design procedure and based on the capacity of the critical buckled members. The behavior of the BCMs was then assigned to the critical members and the effect of the proposed member on the behavior of the space structures was investigated. It is noted that the effects of BCMs on the behavior of space structures have been investigated in the three space structures using nonlinear static analysis. The obtained results can be summarized as follows:

The buckling-controlled member (BCM) has a high potential to postpone the post-buckling of the members.

The proposed member has high ductility and capacity, which can be controlled by the gap and the encasing.

The arrangement of nuts on the core (regular or irregular arrangements) affects the member’s behavior, such that the buckling of the member is delayed by creating a plastic region in the member. Also, the results showed that the slope of the plastic region increases as the inter-nut distances decrease, leading to the behavior of the member to approach the tensile behavior of steel.

The parametric studies showed that the arrangement of nuts corresponding to the four buckling mode shapes has a suitable behavior in terms of plastic region, ultimate strength, and energy absorption.

The gap size between the steel core and the encasing is the main parameter in providing ductility and delaying the overall buckling of the member. Also, the results demonstrated that the gap size controls the length of the plastic region and the ultimate strength. Meanwhile, it is an effective parameter in eliminating the sudden spike seen in the behavior diagram of the member.

The analysis results demonstrated that using BCMs in the highly stressed members of the studied space structures improved their strength and ductility, to the degree that the behavior of the space structure is altered from a fast rate of strength reduction to ductile behavior. As a result, the collapse of the space structures is delayed.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mohammad Reza Chenaghlou

Data availability statement

All data, models, experimental results, or code generated or used during the study are available from the corresponding author by request

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