Steel Structure

Abstract: The load-carrying capacity of double-layer cylindrical reticulated shell is usually determined by the buckling of members, and the imperfection of member has a major influence on member buckling load. The effect of member imperfection, nodal deviation and their couplings on load-carrying capacity of double-layer cylindrical reticulated shells with different rise-to-span ratios and grid numbers are systematically investigated in this study. Furthermore, the method to determine the most unfavorable distribution of member imperfection is developed. Based on the analysis results, the imperfections in member (e. g. , initial curvature of member, residual stress) can be equivalent to member bow imperfection. The shape can be treated as a half-sine curve, and the amplitude can be taken as l/400, which is on the safe side. The effect of member imperfection on load-carrying capacity of reticulated shell is closely related to rise-to-span ratio and grid number of reticulated shell. The reticulated shell with larger rise-to-span ratio and slenderness ratio is more sensitive to member imperfection. In the design of real structure, member imperfection must be considered. Otherwise, the load-carrying capacity will be overestimated. The load-carrying capacity of imperfect reticulated shell is apparently influenced by bending direction of bow imperfection. Form4 is the most unfavorable distribution of bow imperfection for all cases considered. Furthermore, the ultimate loading of reticulated shell with Form1 of bow imperfection, which is obtained by the modified consistent imperfection method other than the modified eigenmode imperfection mode method, is almost equal to the one with Form4 of bow imperfection. The load-carrying capacity of reticulated shell considered in this study is not sensitive to nodal deviation.

Keywords: double-layer cylindrical reticulated shell with welded spherical node; member imperfection; nodal deviation; coupling effect; load-carrying capacity

1 Introduction

Double-layer reticular shell is a kind of typical spatial structures, and has a wide application in public facilities. Different kinds of imperfections inevitably exist in the reticular shell[1], for instance, nodal deviation, initial curvature of member, member eccentricity to joint and error in member length, et al. Double-layer reticular shell with large overall rigidity and so many abundant members is not so sensitive to nodal deviation compared with single one. The load-carrying capacity of double-layer cylindrical reticular shell is usually determined by member buckling, and member imperfection has an evident influence on member buckling load.

In the traditional stability analysis of double-layer reticular shell[2-4], each member is assumed to be straight, without consideration of member imperfection because of the complexity of modeling. In fact, each member in reticular shell is curved in one way or another attributing to machining, transportation and installation. Furthermore, residual stress exists in member in the process of manufacturing. These two types of imperfections belong to member and can decrease the buckling load of member to some extent. Hence, the effect of member imperfection on load-carrying capacity of reticular shell cannot be neglected, and should be taken into consideration in the stability analysis.

Recently, some researchers have paid much attention to the effect of member imperfection on stability of single-layer reticular shell. FAN et al[5] investigated initial member imperfection on stability of single-layer latticed dome, and developed the modified consistent imperfection mode method to obtain the lowest buckling load of reticular dome. YANG et al[6] concluded that the influence of member imperfection on dynamic response of latticed dome cannot be neglected when it enters into plastic state. DING et al[7-8] obtained ultimate bearing capacity of imperfect single-layer latticed shell considering the member buckling effect, and the initial geometrical imperfection of single-layer latticed shell was taken as normal random variable. TIAN et al[9-10] stated that the effect of member imperfection on overall stability should be taken into consideration.

However, there is no related research about overall stability of double-layer cylindrical reticular shell considering member imperfections. Hence, the effect of member imperfection, nodal deviation and their couplings on load-carrying capacity of reticular shell is systematically investigated in the present study. Double-layer cylindrical reticular shells with three rise-to-span ratios (f/S=1/3, f/S=1/4 and f/S=1/5) and three types of grid number (12×11, 10×9 and 8×7) are chosen as the research object. Firstly, the method to determine the shape, amplitude and direction of bow imperfection is presented. Secondly, the influence of member imperfection and its amplitude on load-carrying capacity of reticular shell is extensively investigated. Furthermore, the method to determine the most unfavorable distribution of member imperfection is developed. Finally, effect of nodal deviation and coupling effect of nodal deviation and member imperfection on load-carrying capacity of reticular shell is systematically discussed.

2 Member Imperfection in Reticular Shell

Member buckling has a dominated influence on the load-carrying capacity of double-layer cylindrical reticular shell. Several kinds of imperfections exist in member, i.e., initial curvature of member, residual stress, et al, and they will decrease the buckling load of member to some extent. To figure out the effect of member buckling on the load-carrying capacity of reticular shell, the two types of member imperfections are equivalently treated as bow imperfection in the present study based on Standard for Design of Steel Structures (GB 50017-2017)[11]. Compared with the result obtained by the Code, the shape and amplitude of imperfection member can be determined.

2.1 Shape and amplitude of bow imperfection

Bow imperfection in double-layer reticular shell could be fitted by several straight beam elements, and in the present study the multi-beam method is adopted. The shape of the first Eigen-value buckling mode of a pinned-pinned member satisfies a sinusoidal function with a half wave, shown in Fig.1, which is usually considered as the most unfavorable shape and hence assumed as the shape of bow imperfection. Although each member is semi-rigidly connected in the practical double-layer reticular shell, for simplicity and convenience, the shape of bow imperfection in member is taken as a half-sine wave y=δsin(πx/l), shown in Fig.1, where l is the length of member, and δ is the amplitude of bow imperfection.

Fig.1 Buckling mode and shape of bow imperfection

Imperfect amplitude δ of member can be determined by comparative study between numerical analysis and Standard for Design of Steel Structures (GB 50017-2017)[11], and the hinged member at both ends are discussed here, shown in Fig.1 Two kinds of cross section 127×6.0 and 89×6.0, which will be used in the next section, in double-layer cylindrical reticular shell are investigated here. To consider the effect of member slenderness ratio on imperfect amplitude δ need to be determined, different slenderness ratios are considered here, that is λ=60, λ=80, λ=100, λ=120, λ=140, λ=160 and λ=180. Each member is simulated by 14 elements of Beam188 in ANSYS package. The material of circular steel pipe is Q235 with yield stress 235 MPa and Young’s Modulus 2.1×105MPa. Elastic-plastic strengthening model is applied, and strain hardening rate is taken as 0.02. In the numerical analysis, both geometrical nonlinearity and material nonlinearity are considered, i.e., GMNA.

The limit loads of the two kinds of members with different slenderness ratios can be obtained by following equations [11]:

Where, α1, α2 and α3 are factors, and can be determined by sectional classification. For seamless circular steel pipe, α1=0.41, α2=0.986 and α3=0.152. φ, λ and λn are stability factor, slenderness ratio and normalized slenderness ratio.

Fig.2 Relationship between λ and limit load of member with different imperfect amplitudes

In order to determine imperfection amplitude δ, eight different imperfection amplitudes, i.e., δ=l/1 000, δ=l/500, δ=l/450, δ=l/400, δ=l/350, δ=l/300, δ=l/250 and δ=l/200, are considered in numerical analysis. All numerical results are depicted in solid line in Fig.2, from which it can be concluded that the buckling load of member with δ=l/400 accords well with the one obtained by Code for member with small slenderness ratio, and the buckling load of member with δ=l/500 agrees well with the one obtained by Code for members with moderate slenderness ratios, and the buckling load of member with δ=l/400 conforms to one obtained by Code for members with large slenderness ratios. For the safety, imperfection amplitude δ in all members is taken as l/400 in the following numerical simulation. Axial force-axial/lateral displacement curves of imperfect members with 127×6.0 for λ=80 are plotted in Fig.3, from which it can be concluded that member stiffness and buckling load gradually decreases with the increase of amplitude δ of imperfect member.

Fig.3 Axial force-axial/lateral displacement curves of imperfect members for λ=80 ( 127×6.0)

2.2 Direction of bow imperfection

Direction of bow imperfection can be defined as the angle relative to vertical direction. Direction of bow imperfection in the three dimensional space is random, and can be considered as uniformly distributed random variables in [0°, 360°)[5]. Samples of this random variable can be directly generated by the function RAND in ANSYS package.

Hereto both amplitude and direction of bow imperfection are determined, and double-layer reticular shell with member imperfection can be established. Fig.4 shows one of the randomly generated double-layer cylindrical reticular shells with member imperfection, where δ=l/400, and direction of bow imperfection is a random variable obeying uniform distribution.

Fig.4 Double-layer cylindrical reticular shell with member imperfection (δ magnified by 50 times)

3 Load-Carrying Capacity of Reticular Shell with Member Imperfection

3.1 Model description

Double-layer cylindrical reticular shell with a span S=30m and longitudinal size L=42m is selected as numerical model, shown in Fig.5. To investigate the effect of member imperfection on load-carrying capacity of cylindrical reticular shell with different rise-to-span ratios and member slenderness ratios, three kinds of rise-to-span ratios (f/S=1/3, f/S=1/4 and f/S=1/5, where f and S are rise and span of reticular shell, respectively) and three kinds of gird number 12×11, 10×9 and 8×7 are considered in the present study, shown in Fig.5. Type of different member lengths, number of each type of member and range of slenderness ratio of reticular shell with different rise-to-span ratios and grids numbers are all listed in Tab.1. With the decrease of rise-to-span ratio, slenderness ratio of member in span/lateral direction gradually decrease, while gradually increase with the decrease of grid number.

Beam188 is utilized to simulate members of double-layer cylindrical reticular shell in ANSYS package. All joints at the bottom of reticular shell in longitudinal edge lines are hinged (supported in x, y and z directions) and all joints in edge line along span direction are only supported in vertical direction (y direction), shown in Fig.5. External load is uniformly exerted in vertical direction. The material of steel pipe is Q235 with yield stress 235MPa and Young’s Modulus 2.1×105MPa. Elastic-plastic strengthening model is applied, and strain hardening rate is taken as 0.02. Both geometrical nonlinearity and material nonlinearity analysis are considered, and arc-length method is utilized to obtain equilibrium paths and limit loads of double-layer cylindrical reticular shells.

Fig.5 Geometric sizes and boundary conditions of double-layer cylindrical latticed shells

Tab.1 Geometric sizes and cross-sectional properties of members in the double-layer reticular shells

3.2 Load-carrying capacity of perfect reticular shell

Load-carrying capacity of perfect double-layer cylindrical reticular shells with different grid numbers (14×13, 12×11 and 10×9) and rise-to-span ratios (f/S=1/3, f/S=1/4 and f/S=1/5) is firstly determined. Through numerical GMNA, limit loads of all perfect double-layer cylindrical reticular shells are obtained and listed in Tab.2.

Tab.2 Limit loads of all perfect reticular shells (kN·m－2)

3.3 Effect of member imperfection on load-carrying capacity of reticular shells

To figure out the effect of member imperfection on load-carrying capacity of double-layer cylindrical reticular shells, lots of geometrically and materially nonlinear analyses of cylindrical reticular shells with member imperfection are carried out in this section.

The multi-beam method is utilized to simulate member imperfections. The shape of member imperfection is a half-sine wave y=δsin(πx/l), where l is the length of member, and δ=l/400 determined in Section 2. Direction of bow imperfection uniformly distributed in the three dimensional space.

Each member with bow imperfection can be fitted by a certain number of straight elements. The computational accuracy and efficiency of limit load are apparently influenced by the number of equivalent straight elements. The shape of member with bow imperfection, that is a half-sine wave, cannot be fitted well if the number of equivalent straight elements is too small, while cost much computational effort if the number is too large. Through trial calculations, authors of the present paper found that more than 10 equivalent straight elements is suitable to fit the shape of bow imperfection with a half-sine wave, and which can reach a balance between computational accuracy and computational effort.

Limit loads of imperfect double-layer cylindrical reticular shells with different grid numbers (12×11, 10×9 and 8×7) and rise-to-span ratios (f/S=1/3, f/S=1/4 and f/S=1/5) are obtained considering both geometrical and material nonlinearity. There are totally 3(grid numbers)× 3(f/S)×50(samples of bending direction of bow imperfection)=450 randomly generated reticular shells. Scatter plot of limit load of imperfect reticular shell and its reduction compared with ideal one is shown in Figs. 6~8.

Fig.6 Limit loads and their reduction of reticular shells with member imperfection (12×11)

Fig.7 Limit loads and their reduction of reticular shells with member imperfection (10×9)

Fig.8 Limit loads and their reduction of reticular shells with member imperfection (8×7)

From Fig.6a), Fig.7a) and Fig.8a) we can clearly see the limit load of imperfect shell gradually decreases with the decrease of rise-to-span ratio f/S. From Fig.6b), Fig.7b) and Fig.8b) we can conclude that the limit load reduction of imperfect shell also gradually decreases with the decrease of rise-to-span ratio f/S compared with ideal reticular shell.

Probabilistic statistical parameters (for instance, mean value, standard deviation, coefficient of variation, the minimum, the maximum, et al. ) for all limit loads of imperfect reticular shells with different grid numbers and rise-to-span ratios are listed in Tab.3, where SD, CV and MR stand for standard deviation, coefficient of variation and mean reduction of limit loads, respectively. From Tab.3 we can see that for the reticular shell with same f/S, SD and CV gradually increase from dense grid to sparse grid, i.e., from 12×11 to 8×7, which indicates that the dispersion degree of limit load of reticular shell with sparse grid is higher than the one with dense grid. The mean reduction (MR) of limit load of imperfect shell compared with perfect one is shown in Figs.6b), 7b) and 8b) and Tab.3, from which we can conclude that for the reticular shell with same f/S, mean reduction of limit load apparently increase from dense grid to sparse grid. This can be attributed to the fact that the member slenderness ratio of reticular shell with sparse grid (for instance, 8×7) is larger than the one of reticular shell with dense grid (for instance, 12×11). Hence, we can draw the conclusion that the reticular shell with larger slenderness ratio is more sensitive to member imperfection. In the design and numerical analysis of double-layer cylindrical reticular shell, member imperfection must be considered to incorporate the effect of member buckling on load-carrying capacity of whole structure. Otherwise, the load-carrying capacity will be overestimated, especially for the reticular shell with larger member slenderness ratio. For instance, for grid number 8×7, the limit load of perfect reticular shell will be higher than the actual limit load of imperfect one about 20%.

Tab.3 Probabilistic statistical parameters for limit loads of reticular shells with member imperfection