Abstract
In the code for design of steel structures, the effective length factor of the support and other web members of the steel trusses composed of double angle steel were taken as 1.0 and 0.8, respectively. However, the effective length factor of the support and other web members did not take into account factors such as the stiffness provided by non-adjacent members, the stiffness of the joint itself, and the influence of load changes in the code for design of steel structures. To consider the above influencing factors, a finite element model, based on the steel truss atlas, was established in Abaqus, the elastic restraint stiffness of the web member end was obtained through numerical analysis. The equation was established according to the restraints of the web member end, and the effective length factors of the web member were obtained by solving the equation with Matlab. The analysis found that the elastic restraint stiffness of the web member provided by the bottom chord would not increase by increasing the tension of the bottom chord within the range of elastic deformation. The elastic restraint stiffness of the web member provided by the top chord would not weak by increasing the pressure of the top chord within the range of elastic deformation. It was recommended that the effective length factors of the support and other web members of the steel truss should be 0.8 and 0.7, respectively.
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1 Introduction
Steel trusses have been widely used in large-span spatial structures such as factory buildings, hangars, and stadiums. At present, the stability analysis of the structure in the design of truss (Chen, 2008) usually uses a separate stability check of the single compressed diagonals and verticals and chord of the truss, and the restraint effect of adjacent members on the calculated members is calculated by the effective length factors to reflect. In the code for the design of steel structures (2017), the effective length factors of the support and other web members of steel trusses composed of double angle steel were taken as 1.0 and 0.8, respectively. Wang and Chen (2005), Mageirou and Gantes (2006), Bürgermeister and Steup (1974) and Chen and Zhao (2015) studied the effective length of the pressure members in the truss, but did not consider the influence of the stiffness of adjacent members on the effective length of the pressure members. The International Committee for the Development and Research of Tube Structures (CIDECT) carried out a series of studies on the calculated lengths of members in planar hollow steel tube trusses. The monograph (Monty, 1981) and CIDECT (1980, 1984) design guidelines give the effective length factors of the members of rectangular steel tube trusses. Pondal (1988) carried out experiments with two series of 16 test specimens, and obtained an empirical formula for calculating the length factors of the compressed steel tubular truss members by the regression analysis of the test data. Boel (2010) used the finite element method to analyze the stability of the single limb of the lattice beam chord in detail and obtained the length factors under various conditions. Cai and Chen (2008), Hong and Liang (2017) and Kyle and Jeffrey (2018) studied and considered the restraint of the stiffness of adjacent members at the joints, and the effective length of the compressed members in the middle of the truss was reduced. The calculation method and formula of the effective length of the structure are deduced by Jostein (2012), Kishia and Chen (1998) and Saffari and Rahgozar (2008) Due to the thin wall thickness of the members in the steel truss, the plastic deformation capacity of the members is limited. When designing the steel truss, the plastic deformation of the steel truss structure is not considered. During the elastic deformation phase of the steel truss, the effective length factors of the web member should take into account factors such as the stiffness of non-adjacent members, the stiffness of the joint itself, and the influence of the load change of the top chord. Many of these factors are not considered in current design codes for steel structures.
In order to consider the above influencing factors, the finite element model of the steel truss is established by Abaqus finite element software in this paper, and the restraint stiffness of each pressure web member end of the steel truss is obtained through the analysis of the finite element model. The balanced equation of each pressure web member is established according to the restraints of each compressed web member end. The balanced equation is solved by Matlab, and the buckling load is obtained through calculation. Finally, the effective length factors of the web member are obtained by the ratio of the buckling load of the web member to the Euler load.
2 Isolated Member Analysis
The steel truss is composed of double angle steel member and gusset plate. The structural form of the steel truss is shown in Fig. 1a. The steel truss member is composed of the angle steel and filler steel plate. The angle steel and the filler steel plate are connected by welding. The form of the steel truss member is shown in Fig. 1b. The connection between the gusset plate and the steel truss member is connected by welding. The connection form of the gusset plate and the steel truss member is shown in Fig. 1c. The length of the weld is the overlapping length of the gusset plate and the steel truss member, and the thickness of the weld is the same as that of the steel truss member, as shown in Fig. 1b, c.
Since the support of the steel truss is fixed on the top of the concrete column in practical engineering, the support of the steel truss is connected with the concrete column, and the stiffness of the concrete column is much greater than that of the steel truss member, so the support of the steel truss can be considered not to rotate. The end of the member of the steel truss connected with the support can be considered as the fixed constraint, and the other end can be considered as elastic constraint. Therefore, the steel truss member connected with the support is classified as member II. The two ends of the member not connected with the support can be considered as the elastic constraints, and the member of the steel truss not connected with the support can be classified as member I.
Because only the pressure member of the steel truss has the problem of stability, the tension element has no problem of stability. Therefore, according to whether the pressure member of the steel truss is connected with the support, the pressure member of the steel truss is divided into two types: the pressure member I and the pressure member II connected with the support. In Fig. 1a, the member of the steel truss connected with the support is blue, and the member of the steel truss not connected with the support is pink.
The pressure member is taken out separately from the steel truss structure system for stability analysis. The pressure member I is taken out separately from the steel truss structure system. The interaction between the end of pressure member I and other members can be reflected by the spring stiffness of the end of the member. The calculation diagram is shown in Fig. 2a. Since the support is fixed restrained, the end of the member II connected to the support can be regarded as a fixed end, and the interaction between the other end of the pressure member II and other members can be reflected by the spring stiffness. The calculation diagram is shown in Fig. 2b.
The classical, elastic critical (buckling or bifurcation) load of a member in concentric pressure is given by the well-known load expression
NE is the Euler load, l is the member length, β the effective (buckling) length factor, and EI the sectional bending stiffness.
If the upper and lower ends of pressure member I are hinged, the effective length coefficient of pressure member I is 1. If the upper and lower ends of pressure member I are rigidly connected, the effective length coefficient of pressure member I is 0.5. The restraint stiffness at both ends of the pressure member has a great influence on the effective length coefficient β of the member. The stiffness of the end of the pressure member is related to the stiffness of other members, the stiffness of the joint plate, and the upper load.
3 Element End Restraint
3.1 Elastic Restraint Stiffness of the Joint
In the finite element analysis, the rotation angle θ is applied to the point K where the axes of the steel truss members intersect, as shown in Fig. 3.
Through the finite element analysis, the reaction force M corresponding to the rotation angle θ is obtained. The relationship curve between the bending moment M and the rotation angle θ obtained through numerical simulation analysis, and the relationship curve between the bending moment M and the rotation angle θ of the joint is shown in Fig. 4. The relationship curve between the bending moment M and the rotation angle θ can be divided into two phases, an elastic phase and an elastoplastic phase. The angle steel in the steel truss has a thin wall thickness and limited plastic deformation capacity. Therefore, the plastic deformation of the steel truss structure is not considered, the elastic restraint stiffness of the joint is taken as the restraint stiffness of the joint.
The elastic restraint stiffness is obtained by the tangent line of the M-θ curve at the origin. The calculation formula (3) of the elastic restraint stiffness is as follows.
In formula (3), K0 is the elastic restraint stiffness of the joint, M is the bending moment in the elastic phase, and θ is the rotation angle corresponding to the bending moment M.
The elastic restraint stiffness obtained by this method includes the following factors, the stiffness provided by non-adjacent members, the stiffness of the joint itself, and the influence of load changes.
3.2 Elastic Restraint Stiffness of the End of the Web Member
The elastic restraint stiffness of the member end is distributed according to the bending stiffness EIi of each member intersecting at the joint of the steel truss. The elastic restraint stiffness of the web member provided by the chord is calculated according to formula (4)–(6).
In the formula, EI is the sum of the bending stiffness of the members that intersect at the joint, EIi is the bending stiffness of the i-th member at the joint, E is the elastic modulus of the steel, Ii is the moment of inertia of the i-th member at the joint, μi is the elastic restraint stiffness distribution coefficient of the i-th member at the joint, Ki is the elastic restraint stiffness assigned to the i-th member at the joint, K0j is the total elastic restraint stiffness at the j (j = A, B … I, c, e … i) joint, j is the j-th joint of the steel truss, i is the member’s number at the j-th joint of the steel truss.
The elastic restraint stiffness Ci provided by the remaining members to the i-th member at the j-th joint is calculated by the formula (7).
4 Finite Element Model of Steel Truss
4.1 Structure of the Steel Truss
Based on the drawings in the steel roof frame atlas (05G511), the finite element model of the steel truss was established in the finite element software Abaqus. When designing the steel truss structure, the space steel truss structure was usually simplified to a plane structure for design calculation, the overall space effect of the steel truss structure was not considered. The finite element model of a single plane steel truss was established in this article, the overall effect of the spatial structure is not considered.
The top chord joint of steel truss is named as a capital letter (A, B, …), the bottom chord joint of steel truss is named as lowercase letter (a, b, …), and the element of steel truss is named as the letter of connection node at both ends of the member, as shown in Fig. 5. The numerical value in Fig. 5 shows the internal force of the steel truss member under unit load. The unit load is applied to the top chord joint where the members intersect. In Fig. 5, the green arrow represents the unit load. A positive value is tension, a negative value is pressure. GWJ indicates the encoding of the steel truss. The number behind GWJ represents the span of the steel truss.
The number of steel truss and the specific dimensions of the members are listed in Table 1.
In Table 1, the letters GWJ represent the steel truss, the numbers 18, 21, and 24 represent the span of the steel truss, and the numbers 1, 3, and 5 represent the vertical load design values of 3.5kN/m2, 4.5 kN/m2, and 5.5 kN/m2, respectively. The symbol ∟ indicates angle steel. The thickness of gusset plate of the steel truss is listed in Table 2.
4.2 Finite Element and Material Models
In the finite element model, the angle steel and the gusset were assigned the solid element (C3D8). As stated in Fukumoto and Morita (2000), the constitutive relation curve of steel is described by the KINH model, as shown in Fig. 6. εy = fy/Es, εb = (fu − fy) /0.1Es, εc = 0.12, where εa, εb and εc are the corresponding strains. where fy is the yield strength, fu is the tensile strength, Es is the elastic modulus of steel, and εy is the yield strain of steel.
4.3 Finite Element Model
Nine finite element steel truss models were established in Abaqus. The roof slope of the steel truss model was 1/10, and the longitudinal spacing between the steel trusses was 6 m. The load of the steel truss includes uniform roof live load and dead weight of prestressed concrete roof slab. The uniform roof live load and the dead weight of prestressed concrete slab are simplified as vertical concentrated load P, and the vertical concentrated load P act on the top chord joints of the steel truss. The simplified calculation diagram of the concentrated load P is shown in Fig. 7. The shaded area in the figure is the calculated area of the uniformly distributed roof live load and the dead weight of the prestressed concrete slab. The shadow area is 6000 mm long and 1500 mm wide. 6000 mm is the spacing between the steel trusses and 1500 mm is the spacing between the top chord joints. The concentrated load P can be obtained by multiplying the shadow area by the uniform roof load and the dead weight of the prestressed concrete slab. The concentrated load P values of the steel truss joints are listed in Table 3.
In the finite element model, the interface between the angle steel and gusset plate adopts the face-to-face contact model, which is divided into normal contact and tangential contact. The normal contact adopts hard contact, and the contact surface can be contacted but not penetrated. Coulomb friction model is used for tangential contact, and the friction coefficient is 0.4. In the finite element model, tie constraint is used between the weld and gusset plate. The mesh size of the model is 5 mm.
Coupling points were created at the centroids of the support base plate of the steel truss to form rigid surfaces. The restraint of the support base plate is applied at the coupling points. The base plate of the support of the steel truss was restrained against all degrees of freedom.
In the finite element model of the steel truss, the load is applied in two steps. The first step is to apply the concentrated load P to the top chord joint of the steel truss. The second step is to apply a rotation angle θ to one of the top and bottom chord joints of the steel truss. The relationship between the moment M and the rotation angle θ is obtained by the finite element analysis. The elastic restraint stiffness of the joints and member ends of the steel truss can be obtained by formulas (3)–(7). The finite element model of steel truss is shown in Fig. 8.
5 Finite Element Model Validation
In order to verify the effectiveness of the selected element model, material model and contact model, the corresponding sizes of Ba member and gusset plate in steel truss GWJ18-1 are selected to design the axial compression test. The test specimen size and schematic diagram are shown in Fig. 9. In Fig. 9a, red indicates the fillet weld between the gusset plate and angle steel.
The standard tensile specimen and tensile loading device of the steel are shown in Fig. 10. According to the steel tensile test, the mechanical properties of the steel are shown in Table 4.
The axial compression test is carried out on a 2000kN press machine, and the test loading device is shown in Fig. 11. The strain of the angle steel is measured by the strain gauge in the middle of the test specimen, and the displacement of the test specimen under axial pressure is measured by placing displacement gauge in the bottom plate of the press machine. The arrangement of the strain gauge and displacement gauge is shown in Fig. 11.
According to the axial compression test, the angle steel did not yield in the process of axial compression. The maximum strain of the angle steel was 1074 με, which did not reach the yield strain of the steel 1772 με. The maximum bearing capacity of the test specimen was 395kN. The load–displacement curve of the test specimen is shown in Fig. 12.
The failure of the test specimen is due to the out of plane instability of the upper and lower gusset plates, which makes the gusset plate bend and reach the yield strength of the steel. The failure of the test specimen is due to the out of plane instability of the upper and lower gusset plates, which makes the gusset plate flexural buckling. This is because the length of the gusset plate between the angle steel and the end plate is larger, and the gusset plate between the angle steel and the end plate has no out of plane support. Therefore, the gusset plate of the test specimen is damaged due to out of plane instability under axial pressure. All welds were not damaged during loading. The specific failure mode is shown in Fig. 13.
In ABAQUS software, the above element types, materials and contact models are used to establish the finite element model. The mesh size in the finite element model is 5 mm, and the finite element model is shown in Fig. 14.
In the finite element model, face-to-face contact is adopted between the angle steel and gusset plate, Tie connection is adopted between the weld and angle steel element, and Tie connection is also adopted between the weld and gusset plate. The specific connection method of the angle steel, the gusset plate and the weld in finite element is shown in Fig. 15. In the Fig. 15, the yellow is the weld, the green is the gusset plate, and the orange is the angle steel.
In ABAQUS finite element analysis, reference points are established on the upper and lower plate surfaces of the finite element model, and the degrees of freedom of the end plate surface are coupled to the reference points by coupling command, and boundary conditions are imposed on the reference points. Displacement loading is used in the finite element model. The displacement load is applied at reference point 1 at the top of the finite element, and the fixed constraint is applied at reference point 2 at the bottom of the finite element, as shown in Fig. 14.
The load–displacement curve of the model is obtained by finite element analysis, as shown in Fig. 11. The deformation of the specimen is mainly concentrated in the upper and lower gusset plates, and the deformation mode of gusset plates is shown in Fig. 13b.
The comparative analysis of the test and finite element results shows that the load–displacement curves obtained by the test and finite element analysis are basically consistent, as shown in Fig. 4. The maximum load of the test is 395kN, and the maximum load of the finite element analysis is 415kN. The difference between the maximum load of the test and the finite element analysis is 5.1%. At the same time, the failure modes obtained from the test and finite element analysis are basically the same.
The comparative analysis of the deformation mode, the load–displacement curve and maximum load shows that the results of the test and finite element analysis are basically consistent. The comparison between the experimental and finite element analysis results shows that the selected element type, material model and contact model can be used in the finite element analysis of steel truss.
6 Analysis of Calculation Results
6.1 The Effect of Different Vertical Load on Elastic Restraint Stiffness
Different design values of vertical load are applied to the top chord joints of GWJ18-5. The vertical load design values are 5.5 kN/m2, 4.5 kN /m2, 3.5 kN/m2, 2.5 kN /m2. The design value of the uniform vertical load is simplified to the design value of the concentrated load P. The concentrated load P values of the steel truss joints are listed in Table 5. The concentrated load P was applied to the top chord joint of the steel truss.
The rotation angle θ is applied to one of the top and bottom chord joints of the steel truss. The relationship between the moment M and the rotation angle θ is obtained by the finite element analysis. The elastic restraint stiffness of the joints and member ends of the steel truss can be obtained by formulas (3)-(7). The elastic restraint stiffness of the top and the bottom chord to the web member is calculated and listed in Tables 6 and 7.
It can be seen from Fig. 16 that, in addition to the top chord joint A, the elastic restraint stiffness of the web member provided by the top chord to is higher than the elastic restraint stiffness of the web member provided by the bottom chord within the elastic deformation range of the steel truss. This is because the moment of inertia of the top chord is larger than that of the bottom chord. The elastic restraint stiffness of the web member provided by the bottom chord will not increase by increasing the tension of the bottom chord within the range of elastic deformation, and the elastic restraint stiffness of the web member provided by the top chord will not weak by increasing the pressure of the top chord within the range of elastic deformation.
6.2 Effective Length Factor of the Pressure Web Member
Through the calculation and analysis of the finite element, the elastic restraint stiffness of the web member end of the steel truss is calculated by formula (3)–(7). Since the stability of the members will occur only under pressure, the elastic restraint stiffness of the pressure member end (except the support web member) in the steel truss is listed in Table 8.
The obtained elastic restraint stiffness is applied to the pressure web member end, and a simplified calculation model of the pressure web member is established, as shown in Fig. 17. The buckling load is solved by establishing the balance equation of the pressure web member, and the effective length factors of the pressure web member is solved by the obtained buckling load. The relative displacement of the two ends of the pressure web member is ignored when establishing the balance equation.
The calculation diagram of the pressure web member adopts the small angle θA at one end and the small-angle θB at the other end, as shown in Fig. 17. The second-order differential equation of the pressure web member is established by the bending balance equation of any section, as shown in Fig. 9c. The second-order differential equation of the pressure web member is as follows.
Assuming k2 = P/EI, the general solution of formula (9) and (10) is as follows.
The coefficient B can be obtained by substituting the boundary condition y(0) = 0 into the general solution formula (9).
The coefficients A can be obtained by substituting the boundary conditions y(l) = 0 and x = l into the general solution formula (9).
The boundary condition y'(0) = θA, in which θA = M0/rA, the formula (11) and (12) are substituted into formula (10) to obtain the formula (13) as follows.
Formula (13) can be simplified to obtain formula (14).
The boundary condition y'(l) = θB, in which θB = Ml/rB, the formula (11) and (12) are substituted into formula (10) to obtain the formula (15) as follows.
Formula (15) can be simplified to obtain formula (16).
The Eqs. (14) and (16) are combined, where the values of M0 and Ml are not zero, so the coefficient determinant (17) is zero.
The rA and rB listed in Table 4 are substituted into the above Eq. (17), and kl is obtained by solving the above Eq. (17) with MATLAB.
P is calculated by the formula k2 = P/EI. Assume kl = η, k = η/l, Formula (18) is obtained by combining formula k2 = P/EI and k = η/l.
The calculation formula (19) of P is obtained by formula (18).
where the effective length factor β
The effective length factors of the pressure web members of the steel truss calculated by formula (17)–(20) are listed in Table 9.
It can be seen from Table 9 that the maximum value of effective length factors of the web member of the steel truss is 0.667, and the maximum value of the average value is 0.651. Therefore, the effective length factors of the web member in the steel truss can be taken as 0.7. The determination of the factors already includes the influence of the tension of the top chord, the pressure of the bottom chord, the joint stiffness, and other members of the steel truss on the effective length factors of the web member. Compared with the effective length factors of 0.8 specified in the code for design of the steel structures (GB50017-2017), the value in the code for design of the steel structures (GB50017-2017) is more conservative.
6.3 Effective Length Factor of the Support Web Member
Through the calculation and analysis of the finite element, the elastic restraint stiffness of the support web member end of the steel truss is calculated by formula (3)–(7), and the elastic restraint stiffness of the top end of the support web members is listed in Table 10.
Because the support of the steel truss was applied fixed-end restraint, the stiffness of the support web member connected with the support is large. The end of the support web member connected with the support can be assumed as fixed-end restraint, the other end restraint of the support web member connected with the support can be reflected by the spring stiffness of the end of the support web member. The obtained elastic restraint stiffness rB is applied to the end of the support web member, and a simplified calculation model of the support web member is established, as shown in Fig. 18. The relative displacement of the two ends of the support web member is ignored when establishing the balance equation.
The calculation diagram of the support web member adopts the small-angle θB at one end, as shown in Fig. 18. The second-order differential equation of the support web member is established by the bending balance equation of any section, as shown in Fig. 10c. The second-order differential equation of the support web member is as follows.
Assuming k2 = P/EI, the general solution of formula (22)–(23) is as follows.
The coefficient B can be obtained by substituting the boundary condition y(0) = 0 into the general solution formula (22).
The coefficients A can be obtained by substituting the boundary conditions y'(0) = 0 into the general solution formula (23).
By substituting coefficients A and B into formula (22), the general solution formula is as follows.
The boundary condition y(l) = 0 and x = l is substituted into the general solution formula (24), and the formula (25) is obtained
Formula (27) can be simplified to obtain formula (28).
The boundary condition y'(l) = θB, in which θB = Ml/rB, formula (24) and (25) are substituted into formula (23) to obtain the formula (29) as follows.
Formula (29) can be simplified to obtain formula (30).
The Eqs. (28) and (30) are combined, where the values of M0 and Ml are not zero, so the coefficient determinant (31) is zero.
The rB listed in Table 10 are substituted into the above Eq. (31), and kl is obtained by solving the above Eq. (31) with MATLAB.
The effective length factors of the pressure web members of the steel truss calculated by formula (17)–(19) are listed in Table 11.
It can be seen from Table 11 that the maximum value of effective length factors of the support web member of the steel truss is 0.682, and the maximum value of the average value is 0.666. It is considered that the lower end of the support web member is not fixed completely and the axial force of the support web member is large. Therefore, the effective length factors of the support web member in the steel truss can be taken as 0.8. The determination of the factors already includes the influence of the tension of the top chord, the pressure of the bottom chord, the joint stiffness, and other members of the steel truss on the effective length factors of the support web member. Compared with the effective length factors of 1.0 specified in the code for design of the steel structures (GB50017-2017), the value in the code for design of the steel structures (GB50017-2017) is more conservative.
7 Conclusion
-
1.
According to the finite element analysis, the elastic restraint stiffness of the pressure web member provided by the bottom chord would not increase by increasing the tension of the bottom chord within the range of elastic deformation, and the elastic restraint stiffness of the pressure web member provided by the top chord would not weak by increasing the pressure of the top chord within the range of elastic deformation.
-
2.
Through calculation and analysis, it can be seen that the effective length factor of the pressure web member given in the code is conservative. It is recommended that the effective length factors of the support web member and the pressure web member of the steel truss be 0.8 and 0.7, respectively.
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The research is supported by Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2019L0547).
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Zheng, L., Qu, X., Gao, Y. et al. The In-Plane Effective Length Factor of Web Members of the Steel Truss. Int J Steel Struct 21, 800–819 (2021). https://doi.org/10.1007/s13296-021-00474-1
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DOI: https://doi.org/10.1007/s13296-021-00474-1