The paper presents a method of shaping and describing the geometry of bipolarly prestressed closely spaced built-up member with symmetrical supports and a bisymmetrical cross-section. The following has been defined as a function dependant on the position along the length of the x section of the closely spaced built-up member with determined geometrical parameters: intial elastic _{0} (_{i}_{i}_{i}_{1} and the prestressing zone _{2}, the maximum distance between chords _{max}

bipolarly prestressed closely spaced built-up member

closely spaced built-up member

_{z,ch}

centre of gravity of the cross-section of one chord of the closely spaced built-up member to the z-axis;

maximum deflection;

_{ch,min}

minimum inertia radius of one chord;

_{z,ch}

radius of inertia with respect to the z-axis from one chord of the closely spaced built-up member;

_{i}

distance in the clear between the chords;

_{max}

maximum distance in the clear between the chords in the middle of the member span, equivalent to the spacer thickness;

_{d}

spacer thickness;

_{ch}

cross-sectional area of one chord of the closely spaced built-up member;

Young’s modulus;

_{y,ch}

moment of inertia relative to the y axis of one chord of the closely spaced built-up member;

_{z}

_{1},

_{z}

_{2},

_{z}

_{3}

moment of inertia of a composite section;

_{z,sr}

equivalent moment of inertia to the

total member length;

_{1}

extreme section length with straight member;

_{2}

prestressing zone length; prestressing range;

_{b}

distance between friction grip bolts;

_{s}

distance from the member edge to the first of the bolts joining the chords;

_{cr}

^{Eng}

Engesser critical load capacity;

_{cr}

^{mod}

modified Engesser critical load capacity;

_{e}

Euler critical buckling load;

_{eb}

modified Euler critical buckling load;

_{v}

shear stiffness;

The closely spaced built-up members (CSBUM) are used in engineering structures, such as columns, bracings, chords or diagonal braces of flat and spatial structures, among others: girders, space structures, domes, masts, towers and high-voltage line support structures. They are in the form of at least two component members, called chords, joined together in the welding process or with mechanical fasteners, e.g. rivets, bolts, one-sided bolts: spacerless (Fig.

Among the most commonly used composite CSBUMs sections there are channel sections (Fig.

Examples of composite closely spaced built-up member (CSBUM) sections built of a pair of: (a)–(f) channel sections, (g)–(l) angle sections

Since the early 20^{th} century, CSBUMs made of two angle sections or channel sections have been the standard cross-section of light trusses, welded trusses of medium load, riveted trusses and truss crane beams [

There is an extensive literature on load bearing capacity and stability of the multiple-chord members, including CSBUMs. It should be noted that failure to consider or underestimate shearing force impact on the load bearing capacity of multiple-chord members have caused construction failures and disasters many times in history [

Lue

The interest in cross-sections of cold-formed members, especially thin-walled, has begun to grow since the end of the 20^{th} century. Stone and La Boube [

There are known methods of strengthening compressed members of metal structures by increasing the surface area and/or radius of inertia of the cross-section by joining (welding, gluing, mechanical joining) of additional components, such as sheets or sections to obtain a multiple-chord cross-section. Słowiński and Wuwer [

According to the standard [_{v}_{ch,min}_{ch,min}_{ch,min}

The bipolar displacement prestressing presented in the paper is an innovative method. In the literature on the subject, axially compressed, built-up members, including CSBUMs, shaped in the proposed way, have not been found.

Because in the CSBUMs with compressive axial force it is possible to increase the critical load bearing capacity by introducing bipolar displacement prestressing [

Bipolar prestressing is a controlled, permanent, symmetrical displacement of the CSBUM chord, relative to each other (Fig.

Bipolar prestressing diagram of prestressed CSBUM with symmetrical boundary conditions [

(a) part A, (b) part B, (c) bipolarly prestressed closely spaced built-up member (BPCSBUM)

1 – chord of the CSBUM, 2 – spacer, 3 – spacer connector, 4 – friction grip bolt

Exemplary diagrams of BPCSBUMs

Figure

Figure

As a result of bipolar energy introduced into CSBUM with symmetrical support, a spindle-shaped BPCSBUM is obtained.

There are separated extreme straight lines, located symmetrically to the center, with the length _{1} and _{2} in the middle section, in the BPCSBUM, the chord course of which is non-linear. The division points into sections were associated with cross-sections with friction grip bolts. Thesection _{2}, on which prestresses are introduced in the prestressed member, and the chord course is non-linear, is called the prestressing zone length or the prestressing range. The distance from the edge to the extreme bolt was marked as _{s}_{d}

The transverse dimensions of the CSBUM chord cross-section (flange width – _{f}_{f}_{w}_{w}

It was assumed, in the BPCSBUM shaping, that two following parameters could be controlled: the thickness of the spacer _{d}_{2}.

The spindle shape of BPCSBUM in the prestressing zone determines its geometrical properties. Figure _{i}_{i}_{i}

Geometry of an example BPCSBUM with two-sided pinned support (a) view, (b) cross-sections

In cross-sections, where friction grip bolts are used to join chords, rigid connections were placed due to the lack of free rotation of a single chord (Fig.

Static model of the CSBUM chord in the prestressing zone

Thus, bipolar prestressing of the member was performed in the middle section of the length _{2}, the initial displacement of chords _{0}(

Taking into account the designations from Fig. _{0}_{i}

for _{1};0,5

for _{1}〉

The distance between the chords in the clear is variable on the member length. On the extreme sections with the length _{1} (for _{1}〉 and _{1};_{i}

Functions determining the distance between the chords in the clear were developed based on the curves describing the initial deflection curve (1) and (2) of the member chords in the prestressing zone:

for _{1};0,5

for _{1}〉

The moments of inertia _{i}_{y}

The moment of inertia _{zi}_{i}

After taking into account (3)–(5), the moments of inertia _{zi}_{1}〉 and _{1};

However, for the prestressing zone in the _{1} 0.5

and _{1}〉:

In addition, to maintain the buckling direction, it is necessary to maintain the proportion of moments of inertia of the BPCSBUM:

The moment of inertia of the section _{zi}_{i}_{z,sr}

Given that:

equivalent moment of inertia _{z,sr}

The eccentricity _{zi}

For the extreme sections – for _{1}〉, _{1};

In the prestressing zone, the eccentricity _{zi}

for _{1};0.5

for _{1}〉

At the end of the 19^{th} century, Engesser [_{cr}^{Eng}_{e}

To estimate the critical load capacity of the BPCSBUM, a modification of the Engesser’s formula (_{eb}_{e}

The shear stiffness _{v}

where:

The modified Engesser’s formula (

The issue of stability of the BPCSBUM was solved by the FEM using the commercial ABAQUS/CAE software[

A spatial and shell model was made. The S4R Shell Finite Element, available in the software library, was applied. It is an element with linear shape functions and reduced numerical integration. Simulations for the standard and BPCSBUM were performed with the assumption of the finished element dimension not greater than 10×10 [mm]. An example of finite element grid was shown in Fig.

An example of finite element grid

A model of an ideally elastic-plastic isotropic material was adopted. The material was defined by the Young’s modulus, Poisson’s ratio and density. The standard values specified for steel in [^{3}.

The contact was defined between chords and a spacer and between each of the chords.

The contact between chords and a spacer was defined in the form of general contact with properties of normal behavior as “hard” contact with the possibility of separation after contact. General contact interactions allow to define contact between many regions of the model with a single interaction. The general contact algorithm uses the finite-sliding, surface-to-surface contact formulation and a penalty method to enforce active contact constraints.

The contact between chords was defined in the form of surface-to-surface contact with properties of normal behavior as “hard” contact and tangential behavior using penalty method with friction coefficient 0.1.

The bolt in the middle of the member span joining the chords with the spacer was modelled as a beam-type connector with a diameter corresponding to the diameter of the bolt.

Analysis of the BPCSBUM was divided into three calculation steps:

Initial,

Prestressing,

Buckle.

In the

The calculation step _{max}_{d}

Calculation step

An example of a BPCSBUM – calculation steps: (a) Initial, (b) Prestressing, (c) Buckling analysis

The study covered the standard CSBUM made of the rolled channel sections UPE120 and UPE160 (Tab. _{b}

Calculation example (a) standard CSBUM (b) BPCSBUM

Geometric characteristics of UPE120 and UPE160

Section | _{ch} |
_{y,ch} |
_{z,ch} |
_{z,ch} |
_{z,ch} |

[cm^{2}] |
[cm^{4}] |
[cm^{4}] |
[cm] | [cm] | |

UPE120 | 16.8 | 392 | 60.7 | 1.90 | 2.02 |

UPE160 | 23.7 | 965 | 114 | 2.19 | 2.20 |

A length was assumed for all members _{2} was analysed in two variants: 0.7_{d}_{max}

The critical load capacity of the standard closely spaced built-up member, estimated with the Engesser’s formula, (19) respectively for:

UPE 120: _{cr}^{Eng}

UPE 160: _{cr}^{Eng}

Table

Figures

The result of FEM simulation on BPCSBUM built from a pair of UPE120 channel sections with the length of the prestressing zone _{2} = 2100 mm: (a) 3D view, (b)–(e) 2D view according to the thickness of the spacer _{d}_{d}_{d}_{d}_{d}

The result of FEM simulation on BPCSBUM built from a pair of UPE120 channel sections with the length of the prestressing zone _{2} = 2100 mm: (a) 3D view, (b)–(e) 2D view according to the thickness of the spacer _{d}_{d}_{d}_{d}_{d}

Description of the BPCSBUM geometry

_{2} |
[mm] | 2100 | 2400 | ||||||

_{d} |
[mm] | 4 | 8 | 12 | 16 | 4 | 8 | 12 | 16 |

_{z1} |
[cm^{4}] |
258.50 | |||||||

_{z2}(x=150.0) |
[cm^{4}] |
286.99 | 318.18 | 352.04 | 388.60 | 286.99 | 318.18 | 352.04 | 388.60 |

_{z,sr} |
[cm^{4}] |
268.48 | 279.39 | 291.24 | 304.04 | 269.90 | 282.37 | 295.92 | 310.54 |

_{2} |
[mm] | 2100 | 2400 | ||||||

_{d} |
[mm] | 4 | 8 | 12 | 16 | 4 | 8 | 12 | 16 |

_{z}1 |
[cm^{4}] |
457.42 | |||||||

_{z}2(x=150.0) |
[cm^{4}] |
501.02 | 548.42 | 599.62 | 654.60 | 501.02 | 548.42 | 599.62 | 654.60 |

_{z,sr} |
[cm^{4}] |
472.68 | 489.27 | 507.19 | 526.43 | 474.86 | 493.82 | 514.30 | 536.29 |

Critical load capacity of BPCSBUM estimated by modified Engesser’s (28) and FEM formula is presented in Table

Critical load capacity of BPCSBUM

_{2} |
_{d}=S_{max} |
_{cr}^{mod} |
_{cr,PBSB}^{MES} |
ζ_{1} |
ζ_{2} |
---|---|---|---|---|---|

[mm] | [mm] | [kN] | [kN] | [%] | [%] |

2100 | 4 | 601.2 | 594.4 | 1.14 | 18.72 |

8 | 624.9 | 631.8 | -1.09 | 23.40 | |

12 | 650.6 | 671.1 | -3.06 | 28.48 | |

16 | 678.3 | 712.0 | -4.73 | 33.95 | |

2400 | 4 | 615.2 | 570.9 | 7.76 | 21.49 |

8 | 643.4 | 608.1 | 5.81 | 27.05 | |

12 | 673.9 | 647.7 | 4.05 | 33.08 | |

16 | 706.8 | 689.5 | 2.51 | 39.57 | |

2100 | 4 | 1060.3 | 1043.0 | 1.66 | 17.32 |

8 | 1096.5 | 1097.0 | -0.05 | 21.32 | |

12 | 1135.5 | 1154.2 | -1.62 | 25.64 | |

16 | 1177.3 | 1180.2 | -0.25 | 30.26 | |

2400 | 4 | 1083.1 | 1005.8 | 7.69 | 19.84 |

8 | 1125.9 | 1061.0 | 6.12 | 24.57 | |

12 | 1172.1 | 1119.1 | 4.74 | 29.69 | |

16 | 1221.7 | 1187.8 | 2.85 | 35.17 |

analytically obtained percentage comparison of the critical load capacities of BPCSBUM (_{cr}^{S}_{cr,} PBSBMES

increased critical load capacity of BPCSBUM (_{cr}^{mod}_{cr}^{Eng}

Differences between the obtained analytically critical load bearing capacity of BPCSBUM and FEM were within the following ranges:

-4.73% ÷ 7.76% for 2x UPE 120;

-1.62% ÷ 7.69% for 2x UPE160.

The results for the BPCSBUM analyzed in the example are shown in Figures

Comparison of numerical (FEM) and analytical (mod) results for BPCSBUM: (a) 2xUPE120, _{2} = 2100 mm, (b) 2xUPE120, _{2} = 2400 mm

Comparison of numerical (FEM) and analytical (mod) results for BPCSBUM: (a) 2xUPE160, _{2} = 2100 mm, (b) 2xUPE160, _{2} = 2100 mm

Figure

Critical load bearing capacity of BPCSBUM built of: (a) 2xUPE120, (b) 2xUPE160

horizontal axis – thickness of spacer _{d}

vertical axis – a dimensionless coefficient, i.e. the proportion of the critical load bearing capacity of BPCSBUM _{cr}^{mod}_{cr}^{Eng}

Graphs for the prestressing zone length were drawn up _{2} = 0.7_{2} = 0.8

(1) The studies presented in this paper relate to the BPCSBUM. The literature on CSBUMs is extensive, but there are no studies on BPCSBUM for which a correct description of geometry is indispensable to start static and strength analyzes.

(2) A high convergence of critical load bearing capacity of BPCSBUM estimated from the modified Engesser’s (23) and FEM formula was obtained. For considered prestressing zone length _{2} = 0.7_{2} = 0.8

(3) In connection with the possibility of applying bipolar prestressing by displacement to reinforce the structure of CSBUMs:

an equivalent moment of inertia _{z,sr}

using the relationship (25), it is possible to predict an increase in the load bearing capacity of the CSBUM under bipolar prestressing.

(4) For the BPCSBUM considered in the example, the predicted load bearing capacity gain with a 4 mm spacer is nearly 20%. However, when using a 16 mm spacer, it is 30–40%. Therefore it is possible to

Further analytical, numerical and experimental tests are planned for the load bearing capacity and stability of the BPCSBUM, in particular with other chord sections, different spacer thickness and the prestressing zone lengths.