The rigid connection of beams with columns forms a frame system (e).
When the beams are unlocked from above, the supporting unit of the overlying structure has a transverse rib with a milled end protruding 15-25 mm, through which pressure is transmitted to the column (Fig. a, b, e). Less commonly used is a unit design where the support pressure is transmitted by the internal rib of the beam located above the column flange (c, d). If the transverse support rib of the overlying beam has a protruding end (a, b, d), then the supporting pressure is transmitted first to the support plate of the column head, then to the support rib of the head, and from this rib to the wall of the column (or crossbeam in a through column (e) and then evenly distributed over the cross section of the column. The base plate of the head serves to transfer pressure from the ends of the beam to the supporting ribs of the head, therefore its thickness is determined not by calculation, but by design considerations and is usually taken as 16-25 mm. From the base plate, pressure is transferred to the supporting ribs of the head through horizontal welds, the ends of the ribs are attached to the slab. The leg of these seams is determined by the formula
When installing the base plate on the milled end of the column rod, it ensures complete contact of the plate to the column rib, and the support pressure is transmitted by direct contact of the surfaces, and the welds attaching the base plate are taken structurally.
e) 
In addition, conditions must be met to ensure local stability of the supporting rib.
The bottom of the supporting ribs of the head is reinforced with transverse ribs that prevent them from twisting out of the plane of the column under uneven pressure from the ends of the overlying beams, which arise from inaccurate manufacturing and installation.
From the supporting ribs, pressure is transmitted to the column wall through fillet welds. Based on this, the required length of the ribs.
The estimated length of the seams should not exceed .
The ribs are also checked for shearing:
where 2 is the number of slices;
–thickness of the wall of a column or traverse of a through column.
At high support pressures, shear stresses in the wall exceed the design resistance. In this case, the length of the rib is increased or a thicker wall is adopted. You can increase the wall thickness only at the head of the column (b). This solution reduces metal consumption, but is less technologically advanced to manufacture.
Further distribution of pressure from the column wall over the entire cross-section of the solid column rod is ensured by continuous seams connecting the flanges and the wall.
In through columns (e), pressure from the traverse is transmitted to the branches of the column through fillet welds, the leg of which must be at least:
The column head with supporting ribs of the beams located above the column flanges (c) is designed and calculated similarly to the previous one, only the role of the supporting ribs of the head is performed by the column flanges. If the pressure from the head slab is transmitted to the column through welds (the end of the column is not milled), then the length of the welds attaching one flange of the column to the slab is determined from the condition of their cutting by the reaction of one beam:
,
where is the support reaction of one beam, is the width of the column flange.
If the end of the column is milled, then the welds are made structurally with a minimum leg. To ensure the transfer of support pressure across the entire width of the supporting rib of the beam with a large width of beam chords and narrow column flanges, it is necessary to design a widened cross-beam (Fig. d). It is conventionally assumed that the support pressure from the slab is transferred first completely to the traverse, and then from the traverse to the column flange; in accordance with this, the seams for attaching the traverse to the slab and column are calculated. When the structure is supported on the column from the side (e), the vertical reaction is transmitted through the planed end of the support rib of the beam to the end of the support table and from it to the column flange. The thickness of the support table is taken to be 5-10 mm greater than the thickness of the support rib of the beam. If the support reaction of the beam does not exceed 200 kN, the support table is made from a thick corner with a cut off flange; if the reaction is larger, the table is made from a sheet with a planed upper end. Each of the two seams attaching the table to the column is calculated for 2/3 of the support reaction, which takes into account the possible non-parallelism of the ends of the beam and the table, a consequence of manufacturing inaccuracies and, therefore, uneven pressure transfer between the ends. The required length of one table fastening seam is determined by the formula:
.
Sometimes the table is welded not only along the tanks, but also along the lower end, in this case the total length of the seam is determined by a force equal to
The column head serves as a support for the overlying structures (beams, trusses) and distributes the concentrated load on the column evenly over the cross section of the rod.
The connection between beams and columns can be free or rigid. The hinge joint transmits only vertical loads (a, b, c, d, e).
The rigid connection of beams with columns forms a frame system (e).
When the beams are unlocked from above, the supporting unit of the overlying structure has a transverse rib with a milled end protruding 15-25 mm, through which pressure is transmitted to the column (Fig. a, b, e). Less commonly used is a unit design where the support pressure is transmitted by the internal rib of the beam located above the column flange (c, d). If the transverse support rib of the overlying beam has a protruding end (a, b, d), then the supporting pressure is transmitted first to the support plate of the column head, then to the support rib of the head, and from this rib to the wall of the column (or crossbeam in a through column (e) and then evenly distributed over the cross section of the column. The support plate of the head serves to transfer pressure from the ends of the beam to the supporting ribs of the head, therefore its thickness is determined not by calculation, but by design considerations and is usually taken to be 16-25 mm.
From the base plate, pressure is transferred to the supporting ribs of the head through horizontal welds, and the ends of the ribs are attached to the plate.
The leg of these seams is determined by the formula
.
When installing the base plate on the milled end of the column rod, it ensures complete contact of the plate to the column rib, and the support pressure is transmitted by direct contact of the surfaces, and the welds attaching the base plate are taken structurally.
The width of the supporting rib is determined from the compressive strength condition.
In addition, conditions must be met to ensure local stability of the supporting rib.
.
The bottom of the supporting ribs of the head is reinforced with transverse ribs that prevent them from twisting out of the plane of the column under uneven pressure from the ends of the overlying beams, which arise from inaccurate manufacturing and installation.
From the supporting ribs, pressure is transmitted to the column wall through fillet welds. Based on this, the required length of the ribs.
.
The estimated length of the seams should not exceed .
The ribs are also checked for shear: 
,
where 2 is the number of slices;
– thickness of the wall of the column or traverse of the through column.
At high support pressures, shear stresses in the wall exceed the design resistance. In this case, the length of the rib is increased or a thicker wall is used. You can increase the wall thickness only at the head of the column (b). This solution reduces metal consumption, but is less technologically advanced to manufacture.
Further distribution of pressure from the column wall over the entire cross-section of the solid column rod is ensured by continuous seams connecting the flanges and the wall.
In through columns (e), pressure from the traverse is transmitted to the branches of the column through fillet welds, the leg of which must be at least:
.
The column head with supporting ribs of the beams located above the column flanges (c) is designed and calculated similarly to the previous one, only the role of the supporting ribs of the head is performed by the column flanges. If the pressure from the head slab is transmitted to the column through welds (the end of the column is not milled), then the length of the welds attaching one flange of the column to the slab is determined from the condition of their cutting by the reaction of one beam:
,
where is the support reaction of one beam, is the width of the column flange.
If the end of the column is milled, then the welds are made structurally with a minimum leg. To ensure the transfer of support pressure across the entire width of the supporting rib of the beam with a large width of beam chords and narrow column flanges, it is necessary to design a widened cross-beam (Fig. d). It is conventionally assumed that the support pressure from the slab is transferred first completely to the traverse, and then from the traverse to the column flange; in accordance with this, the seams for attaching the traverse to the slab and column are calculated. When the structure is supported on the column from the side (e), the vertical reaction is transmitted through the planed end of the support rib of the beam to the end of the support table and from it to the column flange. The thickness of the support table is taken to be 5-10 mm greater than the thickness of the support rib of the beam. If the support reaction of the beam does not exceed 200 kN, the support table is made from a thick corner with a cut off flange; if the reaction is larger, the table is made from a sheet with a planed upper end. Each of the two seams attaching the table to the column is calculated for 2/3 of the support reaction, which takes into account the possible non-parallelism of the ends of the beam and the table, a consequence of manufacturing inaccuracies and, therefore, uneven pressure transfer between the ends. The required length of one table fastening seam is determined by the formula:
.
Sometimes the table is welded not only along the tanks, but also along the lower end, in this case the total length of the seam is determined by a force equal to
.
The connection between beams and columns can be free(hinged) and hard. The free interface transfers only vertical loads. The rigid coupling forms a frame system capable of absorbing horizontal forces and reducing the design moment in the beams. In this case, the beams are adjacent to the column on the side.
With free coupling, the beams are placed on top of the column, which ensures ease of installation.
In this case, the column head consists of a slab and ribs that support the slab and transfer the load to the column rod (Fig.).
If the load is transferred to the column through the milled ends of the supporting ribs of the beams located close to the center of the column, then the cap slab is supported from below by ribs running under the supporting ribs of the beams (Fig. a and b).
Rice. Column heads when supporting beams from above
The ribs of the head are welded to the base plate and to the branches of the column with a through rod or to the wall of the column with a solid rod. The seams attaching the head rib to the slab must withstand the full pressure on the head. Check them using the formula
. (8)
The height of the rib of the head is determined by the required length of the seams that transfer the load to the column core (the length of the seams should not be more than 85∙β w ∙k f:
. (9)
The thickness of the rib of the head is determined from the condition of resistance to crushing under full support pressure
, (10)
where is the length of the crushed surface, equal to the width of the supporting rib of the beam plus two thicknesses of the column head slab.
Having determined the thickness of the rib, you should check it for shearing using the formula:
. (11)
If the wall thicknesses of the channels of a through column and the walls of a continuous column are small, they must also be checked for shear at the point where the ribs are attached to them. You can make the wall thicker within the height of the head.
To give rigidity to the ribs supporting the base plate and to strengthen the walls of the column rod against loss of stability in places where large concentrated loads are transmitted, the vertical ribs that carry the load are framed from below with horizontal ribs.
The head support plate transfers pressure from the overlying structure to the head ribs and serves to fasten the beams to the columns with mounting bolts that fix the design position of the beams.
The thickness of the base plate is assumed to be structurally within 20-25 mm.
When the end of the column is milled, the pressure from the beams is transferred through the base plate directly to the ribs of the head. In this case, the thickness of the seams connecting the slab with the ribs, as well as with the branches of the column, is assigned structurally.
If the beam is attached to the column from the side (Fig.), the vertical reaction is transmitted through the supporting rib of the beam to a table welded to the column flanges. The end of the supporting rib of the beam and the upper edge of the table are attached. The thickness of the table is taken to be 20-40 mm greater than the thickness of the supporting rib of the beam.
Rice. Supporting a beam on a column from the side
It is advisable to weld the table to the column on three sides.
To ensure that the beam does not hang on the bolts and sits tightly on the support table, the supporting ribs of the beam are attached to the column rod with bolts, the diameter of which should be 3 - 4 mm less than the diameter of the holes.
Lecture 13
Farms. general characteristics and classification
A truss is a system of rods connected to each other at nodes and forming a geometrically unchangeable structure. Trusses can be flat (all rods lie in the same plane) and spatial.
Flat trusses (Fig. a) can perceive a load applied only in their plane, and need to be secured from their plane with connections or other elements. Spatial trusses (Fig. b, c) form a rigid spatial beam capable of absorbing loads acting in any direction. Each face of such a beam is a flat truss. An example of a space beam is a tower structure (Fig. d).
Rice. Flat (a) and spatial (b, c, d) trusses
The main elements of the trusses are the belts that form the outline of the truss, and a lattice consisting of braces and posts (Fig.).
1 - upper belt; 2 - lower belt; 3 - braces; 4 - rack
Rice. Truss elements
The distance between the belt nodes is called the panel ( d ) , distance between supports - span ( l ), the distance between the axes (or outer edges) of the chords is the height of the truss ( h f).
Truss chords operate mainly on longitudinal forces and moment (similar to the chords of solid beams); The truss lattice absorbs mainly lateral force.
Connections of elements in nodes are carried out by directly connecting one element to another (Fig. a) or using nodal gussets (Fig. b) . In order for the truss rods to work mainly on axial forces, and the influence of moments can be neglected, the truss elements are centered along axes passing through the centers of gravity.
a – when the lattice elements are directly adjacent to the belt;
b – when connecting elements using a gusset
Rice. Truss nodes
Trusses are classified according to the static diagram, the outline of the chords, the lattice system, the method of connecting elements at nodes, and the amount of force in the elements. According to the static scheme There are trusses (Fig.): beam (split, continuous, cantilever), arched, frame and cable-stayed.
Split beams systems (Fig. a) are used in building coverings and bridges. They are easy to manufacture and install, do not require the installation of complex support units, but are very metal-intensive. For large spans (more than 40 m), split trusses turn out to be oversized and have to be assembled from separate elements during installation. When the number of overlapped spans is two or more, use continuous farms (Fig. b). They are more economical in terms of metal consumption and have greater rigidity, which makes it possible to reduce their height. But when the supports settle, additional forces arise in continuous trusses, so their use on weak subsidence foundations is not recommended. In addition, the installation of such structures is complicated.
a - split beam; 6 - continuous beam; c, e - console;
g - frame; d - arched; g - cable-stayed; z - combined :
Rice. Truss systems
Console trusses (Fig. c, e) are used for canopies, towers, and overhead power line supports. Frame systems (Fig. e) are economical in steel consumption, have smaller dimensions, but are more complex during installation. Their use is rational for long-span buildings. Application arched systems (Fig. e), although they save steel, lead to an increase in the volume of the room and the surface of the enclosing structures. Their use is caused mainly by architectural requirements. IN cable-stayed trusses (Fig. g) all rods work only in tension and can be made of flexible elements, such as steel cables. The tension of all elements of such trusses is achieved by choosing the outline of the chords and lattice, as well as by creating prestress. Working only in tension allows you to fully utilize the high strength properties of steel, since stability issues are eliminated. Cable-stayed trusses are rational for long-span floors and bridges. Combined systems are also used, consisting of a beam reinforced from below with a sprengel or braces, or from above with an arch (Fig. h). These systems are easy to manufacture (due to the smaller number of elements) and are efficient in heavy structures, as well as in structures with moving loads. It is very effective to use combined systems when strengthening structures, for example, reinforcing a beam if its load-bearing capacity is insufficient, with a truss or struts.
Depending on the outlines of belts trusses are divided into segmental, polygonal, trapezoidal, with parallel belts and triangular (Fig.).
The most economical in terms of steel consumption is a truss outlined according to a moment diagram. For a single-span beam system with a uniformly distributed load, this is segmental truss with a parabolic belt (Fig. a ). However, the curvilinear outline of the belt increases the complexity of manufacturing, so such trusses are practically not used at present.
More acceptable is polygonal outline (Fig. b) with a fracture of the belt at each node. It corresponds fairly closely to the parabolic outline of the moment diagram and does not require the manufacture of curvilinear elements. Such trusses are sometimes used to cover large spans and in bridges.
a - segmental; b - polygonal; c - trapezoidal; g - with parallel belts; d, f, g, i - triangular
Rice. Outlines of truss belts:
Farms trapezoidal outlines (Fig. c) have design advantages primarily due to the simplification of the nodes. In addition, the use of such trusses in the coating makes it possible to construct a rigid frame assembly, which increases the rigidity of the frame.
Farms with parallel belts (Fig. d) have equal lengths of lattice elements, the same layout of nodes, the greatest repeatability of elements and parts and the possibility of their unification, which contributes to the industrialization of their production.
Farms triangular the outlines (Fig. e, f, g, i) are rational for cantilever systems, as well as for beam systems with a concentrated load in the middle of the span (rafter trusses). With a distributed load, triangular trusses have increased metal consumption. In addition, they have a number of design flaws. The sharp support unit is complex and allows only hinged coupling with the columns. The middle braces turn out to be extremely long, and their cross-section has to be selected for maximum flexibility, which causes excessive consumption of metal.
According to the method of connecting elements At the nodes, trusses are divided into welded and bolted. In structures manufactured before the 50s, riveted joints were also used. The main types of trusses are welded. Bolted connections, as a rule, with high-strength bolts are used in assembly units.
By magnitude of maximum effort conventionally distinguish between light trusses with sections of elements made of simple rolled or bent profiles (with forces in the rods N< 3000 kN) and heavy trusses with composite section elements (N> 3000 kN).
The efficiency of trusses can be increased by prestressing them.
Truss lattice systems
The lattice systems used in trusses are shown in Fig.
a - triangular; b - triangular with racks; c, d - diagonal; d - trussed; e - cross; g - cross; and - rhombic; k - half-diagonal
Rice. Truss lattice systems
The choice of lattice type depends on the load application pattern, the outline of the chords and design requirements. To ensure the compactness of the units, it is advisable to have the angle between the braces and the belt in the range of 30...50 0.
Triangular system lattice (Fig. a) has the smallest total length of elements and the smallest number of nodes. There are farms with ascending And downward support braces.
In places where concentrated loads are applied (for example, in places where roof purlins are supported), additional racks or hangers can be installed (Fig. b). These racks also serve to reduce the estimated length of the belt. Racks and suspensions work only on local loads.
The disadvantage of a triangular lattice is the presence of long compressed braces, which requires additional steel consumption to ensure their stability.
IN diagonal in the lattice (Fig. c, d) all the braces have forces of one sign, and the racks have another. A diagonal lattice is more metal-intensive and labor-intensive compared to a triangular lattice, since the total length of the lattice elements is longer and there are more nodes in it. The use of diagonal lattice is advisable for low truss heights and large nodal loads.
Shprengelnaya the grid (Fig. e) is used for off-node application of concentrated loads to the upper chord, as well as when it is necessary to reduce the estimated length of the belt. It is more labor-intensive, but can reduce steel consumption.
Cross the lattice (Fig. e) is used when there is a load on the truss in both one and the other direction (for example, wind load). In farms with belts made of brands, you can use cross a lattice (Fig. g) from single corners with braces attached directly to the wall of the tee.
RhombicAnd semi-diagonal the gratings (Fig. i, j) due to two systems of braces have great rigidity; These systems are used in bridges, towers, masts, and connections to reduce the design length of the rods.
Types of truss rod sections
In terms of steel consumption for compressed truss rods, the most efficient is a thin-walled tubular section (Fig. a). A round pipe has the most favorable distribution of material relative to the center of gravity for compressed elements and, with a cross-sectional area equal to other profiles, has the largest radius of gyration (i ≈ 0.355d), the same in all directions, which makes it possible to obtain a rod with the least flexibility. The use of pipes in trusses allows steel savings of up to 20...25%.
Rice. Types of sections of elements of light shapes
The big advantage of round pipes is good streamlining. Thanks to this, the wind pressure on them is less, which is especially important for high open structures (towers, masts, cranes). The pipes retain little frost and moisture, so they are more resistant to corrosion and are easy to clean and paint. All this increases the durability of tubular structures. To prevent corrosion, the internal cavities of the pipe should be sealed.
Rectangular bent-closed sections (Fig. b) make it possible to simplify the joints of elements. However, trusses made of bent closed profiles with chamferless units require high manufacturing precision and can only be manufactured in specialized factories.
Until recently, light trusses were designed mainly from two corners (Fig. c, d, e, f). Such sections have a wide range of areas and are convenient for constructing joints on gussets and attaching structures adjacent to trusses (purlins, roofing panels, ties). A significant disadvantage of this design form is; a large number of elements with different standard sizes, significant metal consumption for fittings and gaskets, high labor intensity of manufacturing and the presence of gaps between the corners, which promotes corrosion. Rods with a cross-section of two angles formed by a tee are not effective when working in compression.
With a relatively small force, truss rods can be made from single angles (Fig. g). This section is easier to manufacture, especially with unshaped units, since it has fewer assembly parts and does not have gaps closed for cleaning and painting.
The use of t-bars for truss belts (Fig. i) allows one to significantly simplify the knots. In such a truss, the corners of the braces and racks can be welded directly to the wall of the tee without gussets. This halves the number of assembly parts and reduces the labor intensity of manufacturing:
If the truss belt works, in addition to axial force, also in bending (with extra-nodal load transfer), a section of an I-beam or two channels is rational (Fig. j, l).
Quite often, the sections of truss elements are taken from different types of profiles: belts made of I-beams, a lattice made of curved closed profiles, or belts made of T-bars, a lattice made of paired or single corners. This combined solution turns out to be more rational.
Compressed truss elements should be designed to be equally stable in two mutually perpendicular directions. With the same design lengths l x = l y sections made of round pipes and square bent-closed profiles meet this condition.
In trusses made from paired angles, similar radii of inertia (i x ≈ i y) have unequal angles placed together in large shelves (Fig. d). If the estimated length in the plane of the truss is two times less than from the plane (for example, in the presence of a truss), a section of unequal angles put together by small flanges (Fig. e) is rational, since in this case i y ≈ 2i x.
The rods of heavy trusses differ from light ones in having more powerful and developed sections, composed of several elements (Fig.).
Rice. Types of sections of heavy truss elements
Determining the design length of truss bars
The load-bearing capacity of compressed elements depends on their design length:
l ef = μ× l, (1)
Where ts - length reduction coefficient, depending on the method of fastening the ends of the rod;
l- geometric length of the rod (the distance between the centers of nodes or fastening points against displacement).
We do not know in advance in which direction the rod will buckle upon loss of stability: in the plane of the truss or in the perpendicular direction. Therefore, for compressed elements it is necessary to know the design lengths and check the stability in both directions. Flexible stretched rods can sag under their own weight, they are easily damaged during transportation and installation, and under dynamic loads they can vibrate, so their flexibility is limited. To check the flexibility, it is necessary to know the calculated length of the stretched rods.
Using the example of a truss truss of an industrial building with a lantern (Fig.), we will consider methods for determining the estimated lengths. Possible curvature of the truss chords during loss of stability in its plane can occur between the nodes (Fig. a).
Therefore, the calculated length of the chord in the plane of the truss is equal to the distance between the centers of the nodes (μ = 1). The form of buckling from the plane of the truss depends on the points at which the belt is secured against displacement. If rigid metal or reinforced concrete panels are laid along the upper chord, welded or bolted to the belt, then the width of these panels (usually equal to the distance between the nodes) determines the estimated length of the belt. If a profiled decking attached directly to the belt is used as a roofing covering, then the belt is secured against loss of stability along its entire length. When roofing along purlins, the estimated length of the chord from the plane of the truss is equal to the distance between the purlins, secured against displacement in the horizontal plane. If the purlins are not secured with ties, then they cannot prevent the truss chord from moving and the estimated length of the chord will be equal to the entire span of the truss. In order for the purlins to secure the belt, it is necessary to install horizontal connections (Fig. b) and connect the purlins to them. Spacers must be placed in the area of the covering under the lantern.
A - deformation of the upper chord during loss of stability in the plane of the truss; b, c - the same, from the plane of the truss; d - lattice deformation
Rice. To determine the design lengths of truss elements
Thus, the calculated length of the chord from the plane of the truss is generally equal to the distance between the points secured against displacement. The elements that secure the belt can be roofing panels, purlins, connections and struts. During the installation process, when the roof elements have not yet been installed to secure the truss, temporary ties or spacers can be used from their plane.
When determining the design length of lattice elements, the stiffness of the nodes can be taken into account. When stability is lost, the compressed element tends to rotate the node (Fig.d). The rods adjacent to this node resist bending. The greatest resistance to rotation of the node is provided by stretched rods, since their deformation from bending leads to a reduction in the distance between the nodes, while due to the main force this distance should increase. Compressed rods weakly resist bending, since deformations from rotation and axial force are directed in one direction and, in addition, they themselves can lose stability. Thus, the more stretched rods are adjacent to the node and the more powerful they are, i.e. the greater their linear stiffness, the greater the degree of pinching of the rod in question and the shorter its design length. The effect of compressed rods on pinching can be neglected.
The compressed belt is weakly pinched at the nodes, since the linear stiffness of the tensile lattice elements adjacent to the node is low. Therefore, when determining the estimated length of the belts, we did not take into account the rigidity of the nodes. The same applies to support braces and racks. For them, the design lengths, as for the belts, are equal to the geometric length, i.e. the distance between the centers of nodes.
For other lattice elements, the following scheme is adopted. In the nodes of the upper chord, most of the elements are compressed and the degree of pinching is small. These nodes can be considered hinged. In the nodes of the lower chord, most of the elements converging in the node are stretched. These nodes are elastically clamped.
The degree of pinching depends not only on the sign of the forces of the rods adjacent to the compressed element, but also on the design of the unit. If there is a gusset that tightens the knot, the pinching is greater, therefore, according to the standards, in trusses with knot gussets (for example, from paired angles), the estimated length in the plane of the truss is 0.8× l, and in trusses with elements abutting end-to-end, without nodal gussets - 0.9× l .
In the event of loss of stability from the plane of the truss, the degree of pinching depends on the torsional rigidity of the chords. The gussets are flexible from their plane and can be considered as sheet hinges. Therefore, in trusses with nodes on gussets, the estimated length of the lattice elements is equal to the distance between the nodes l 1 . In trusses with chords made of closed profiles (round or rectangular pipes) with high torsional rigidity, the coefficient of reduction of the design length can be taken equal to 0.9.
The table shows the calculated lengths of elements for the most common cases of flat trusses.
Table - Design lengths of truss elements
Note. l-geometric length of the element (distance between the centers of nodes); l 1 - the distance between the centers of nodes secured against displacement from the plane of the truss (truss chords, braces, covering slabs, etc.).
Selection of cross-sections for compressed and tensile elements
Selection of cross-section of compressed elements
The selection of sections of compressed truss elements begins with determining the required area from the stability condition
, (2)
.
1) It can be tentatively assumed that for the belts of light trusses l = 60 - 90 and for the lattice l = 100 - 120. Greater flexibility values are obtained with less effort.
2) Based on the required area, a suitable profile is selected from the assortment, its actual geometric characteristics A, i x, i y are determined.
3) Find l x = l x /i x and l y = l y /i y , For greater flexibility, the coefficient j is specified.
4) Do a stability check using formula (2).
If the flexibility of the rod was previously set incorrectly and the test showed overstress or significant (more than 5-10%) understress, then the section is adjusted, taking an intermediate value between the preset and actual flexibility value. Usually the second approach achieves its goal.
Note. Local stability of compressed elements made from rolled sections can be considered ensured, since the rolling conditions determine the thickness of the flanges and walls of the profiles to be greater than required from the stability conditions.
When choosing the type of profiles, you need to remember that a rational section is one that has the same flexibility both in the plane and from the plane of the truss (the principle of equal stability), therefore, when assigning profiles, you need to pay attention to the ratio of the effective lengths. For example, if we are designing a truss from angles and the calculated lengths of the element in the plane and from the plane are the same, then it is rational to choose unequal angles and place them together in large shelves, since in this case i x ≈ i y, and when l x = l y λ x ≈ λ y . If the estimated length is out of plane l y is twice the design length in the plane l x (for example, the upper chord in the area under the lantern), then a more rational section would be a section of two unequal angles placed together with small shelves, since in this case i x ≈ 0.5×i y and at l x =0.5× l y λ x ≈ λ y . For lattice elements at l x =0.8× l y the most rational would be a section of equal angles. For truss chords, it is better to design a section of unequal angles placed together with smaller flanges in order to provide greater rigidity from the plane when lifting the truss.
Selection of the section of tensile elements
The required cross-sectional area of the stretched truss rod is determined by the formula
. (3)
Then, according to the assortment, the profile with the nearest larger area is selected. In this case, checking the accepted cross-section is not required.
Selection of rod cross-sections for maximum flexibility
Truss elements should generally be designed from rigid bars. Rigidity is especially important for compressed elements, the limit state of which is determined by loss of stability. Therefore, for compressed truss elements, SNiP establishes requirements for maximum flexibility that are more stringent than in foreign regulatory documents. The maximum flexibility for compressed elements of trusses and connections depends on the purpose of the rod and the degree of its loading: , where N - design force, j×R y ×g c - load-bearing capacity.
Tension bars should also not be too flexible, especially when subjected to dynamic loads. Under static loads, the flexibility of tensile elements is limited only in the vertical plane. If tension members are prestressed, their flexibility is not limited.
A number of light truss rods have low forces and, therefore, low stresses. The cross-sections of these rods are selected for maximum flexibility. Such rods usually include additional posts in a triangular lattice, braces in the middle panels of trusses, bracing elements, etc.
Knowing the estimated length of the rod l ef and the value of the ultimate flexibility l pr, we determine the required radius of gyration i tr = l ef/l tr. Based on it, in the assortment we select the section that has the smallest area.
Columns serve to transfer the load from the structures above through the foundation to the ground. Depending on how the load is applied to the column, centrally compressed, eccentrically compressed and compressed-flexural columns are distinguished. Centrally compressed columns operate on a longitudinal force applied along the axis of the column and causing uniform compression of its cross section. Eccentrically compressed columns and compressed-bending columns, in addition to axial compression from longitudinal force, also work on bending from moment.
The columns consist of three main parts: rod , which is the main load-bearing element of the column; head , serving as a support for overlying structures and securing them to the column; bases , distributing the concentrated load from the column over the surface of the foundation, providing attachment using anchor bolts.
Columns differ: by type - constant and variable in height sections; according to the design, the sections of the rod are solid (solid-walled) and through (lattice).
When choosing the type of column section, it is necessary to strive to obtain the most economical solution, taking into account the magnitude of the load, the convenience of connecting supporting structures, operating conditions, and manufacturing capabilities.
The main type of solid columns, along with rolled ones, is a welded I-beam, composed of three sheets of rolled steel, which is most convenient to manufacture using automatic welding and allows for simple joining of supporting structures. The core of a through column consists of two branches (rolled channels or I-beams), interconnected by connecting elements in the form of strips or braces, which ensure the joint operation of the branches and significantly affect the stability of the column as a whole and its branches.
A triangular lattice of braces is more rigid than slats, since it forms a truss in the plane of the column face, all elements of which work under axial forces. It is recommended for use in columns loaded with a longitudinal force of more than 2500 kN or with a significant distance between branches (more than 0.8 m). The planks create a non-bracing system in the plane of the column face with rigid nodes and bending elements.
For inspection and possible painting of internal surfaces in through columns of two branches, a gap of at least 100 mm is established between the flanges of the branches.
Column design diagram
Rice. 4.1. Column design diagram
Calculated column length lef taking into account the methods of fixing the column in the foundation and pairing it with the beam adjacent in the upper part, it is assumed to be equal to:
lef = μ l,
Where l – geometric column length;
μ – coefficient of effective length, taken depending on the conditions for fastening its ends and the type of loading (under the action of a longitudinal force on the column from above: μ = 1 – with hinged fastening of both ends of the column; μ = 0.7 – when one end of the column is rigidly fastened and the other is hinged).
When beams are supported on a column from above, the column is treated as hinged at the upper end. Fastening the column to the foundation can be hinged or rigid. If the foundation is sufficiently massive, and the base of the column is developed and has reliable anchorage, the column can be considered pinched in the foundation.
Calculation of the strength of elements subject to central compression by force N should be performed according to the formula
Where An– net cross-sectional area.
Calculation of column stability under central compression is performed according to the formula
Where φ – stability coefficient for central compression, taken according to conditional flexibility for various types of stability curves according to Table. 3.11.
4.1. Calculation of rolling column
Example 4.1. Select a solid column made of rolled wide-flange column I-beams with a height l= 6 m. The column is hinged at the bottom and top. Design longitudinal force N= 1000 kN. Construction material – steel class C245 with design resistance Ry γ With= 1.
Rice. 4.2. Rolling column section
We determine the estimated lengths of the column in planes perpendicular to the axes x-x And ooh:
The pre-flexibility of medium-length columns with a force of up to 2500 kN is set within λ = 100...60. We accept λ = 100.
The conditional flexibility of the column is determined by the formula
V′′ (see Table 3.12) we determine the stability coefficient under central compression j= 0,560.
We calculate the required cross-sectional area:
Find the required radii of gyration:
From the assortment we accept wide-flange I-beam Ι 23 K2/GOST 26020-83, having a cross-sectional area A= 75.77 cm 2; radii of gyration і X= 10.02 cm and і y= 6.04 cm.
Defining flexibility:
λ X = lX/і X= 600 / 10,02 = 59,88; λ y = ly/і y= 600 / 6,04 = 99,34.
Conditional maximum flexibility of the column
According to conditional flexibility y define j= 0,564.
We check the stability of the column in the plane of least rigidity (relative to the axis y-y):
The section has been accepted.
If the stability condition of the column is not met, the section dimensions are adjusted (the adjacent number of rolled products is accepted according to the assortment) and re-checked.
4.2. Calculation and design of a continuous welded column
Example 4.2. Select a solid welded column of symmetrical I-section, made of three rolled sheets, according to example 3.4. At the bottom, the column is rigidly clamped in the foundation, at the top it is hinged to the beams. Markings: top of the working platform deck 13 m. Material of construction according to table. 2.1 – steel class C245 with design resistance Ry= 24 kN/cm2. Working conditions factor γ With= 1.
Design diagram of the column in Fig. 4.1. Longitudinal force N, compressing the column, is equal to two reactions (transverse forces) from the main beams resting on the column:
N = 2Q max = 2 1033.59 = 2067.18 kN.
The geometric length of the column (from the foundation to the bottom of the main beam) is equal to the level of the working platform floor minus the actual construction height of the floor, consisting of the height of the main beam on the support h o , height of the deck beam hbn and flooring thickness tn, plus the depth of the column base below the finished floor level (a depth of 0.6 - 0.8 m is accepted):
If there is an auxiliary beam in the beam cage (when the beams are coupled by floor), the height of the beam is added to the height of the floor hbv.
Calculated column lengths in planes perpendicular to the axes x-x And ooh:
Rice. 4.3. Section of a solid welded column
Set by the flexibility of an average length column within λ = 100 – 60 for columns with a force of up to 2500 kN; λ = 60 – 40 – for columns with a force of 2500 –4000 kN; for more powerful columns, flexibility is accepted λ = 40 – 30.
We accept λ = 80.
Conditional flexibility of the column
According to the conditional flexibility for an I-section with a stability curve type ′′ V′′ we determine the stability coefficient under central compression j= 0.697 (see Table 3.11).
Required cross-sectional area of the column
Required radii of gyration of the section:
ix = iy = lx/l= 813 / 80 = 10.16 cm.
Using from the table. 4.1 dependencies of the radius of gyration on the type of section and its dimensions (height h and width b), we define for an I-beam:
h =ix/k 1 = 10.16 / 0.43 = 23.63 cm;
b =iy/k 2 = 10.16 / 0.24 = 42.33 cm;
For technological reasons (from the condition of automatic welding of waist seams), the wall height hw should not be less than the width of the belt bf. We assign section dimensions, linking them with the standard width of the sheets: 
Further calculations are carried out only relative to the axis ooh, since the flexibility of the rod relative to this axis will be almost twice as great as relative to the axis x-x.
The wall thickness is set to the minimum based on the condition of its local stability and is taken within the range of 6 - 16 mm.
Limiting conditional flexibility
Wall flexibility (ratio of design wall height to thickness hw/tw) in centrally compressed I-beam columns, according to the condition of local wall stability, should not exceed 
where the values are determined from the table. 4.2.
Determine the wall thickness at 
We accept a wall from a sheet with a cross-section of 400´8 mm with a cross-sectional area
If, for design reasons, the wall thickness tw accepted less tw, min from the condition of local stability, then the wall should be strengthened with a paired or one-sided longitudinal stiffener rib dividing the design compartment of the wall in half (Fig. 4.4). Longitudinal ribs should be included in the design cross-section of the rod:
Acalc =A+å Ap.
Legend:`
l– conditional flexibility of the element, taken into account for stability under central compression;
`l 1 – conditional flexibility of the element, taken into account for stability in the plane of the moment.
Notes: 1. Box-shaped profiles include closed rectangular profiles (composite, bent rectangular and square).
2. In a box section with m> 0 value ` luw should be determined for a wall parallel to the bending moment plane.
3. For values 0 < m < 1.0 value ` luw should be determined by linear interpolation between the values calculated using m= 0 and m= 1,0.
Shelf overhang width ratio bef = (bf – tw)/2 = (40 – 8) / 2 = 19.6 cm
to shelf thickness tf in centrally compressed elements with conditional flexibility
l= 0.8 – 4 according to the condition of local stability of the shelf should not exceed
from where we determine the minimum thickness of the shelf:
Required area of one shelf
Rice. 4.4.
Required shelf thickness
We accept
Section height
h = hw + 2tf= 400 + 2 ∙ 1.2 = 42.4 cm.
Shelf area
We calculate the geometric characteristics of the section:
- square
– moment of inertia about the axis ooh(we neglect the moment of inertia of the wall)
– radius of inertia
– actual flexibility
– conditional flexibility
– stability coefficient under central compression
General stability of the column relative to the y-y axis
Checking the overall stability of the column relative to the axis y-y:
Where gWith= 1 – coefficient of working conditions according to table. 1.3.
Undervoltage in the column
The section has been accepted.
If the column stability condition is not met, the section dimensions are adjusted and re-checked. Adjustment, as a rule, is made by changing the size of the shelves, subject to the obligatory observance of the condition of their local stability.
To strengthen the contour of the section and the wall of the column when 
install transverse stiffeners located at a distance a= (2,5...3)hw one from the other; Each sending element must have at least two ribs (see Fig. 4.4). Minimum dimensions of protruding part br and thickness tr transverse stiffeners are taken in the same way as in the main beam.
We check:
installation of transverse stiffeners is not required.
In places where ties, beams, struts and other elements adjoin the column, stiffeners are installed in the zone of concentrated force transmission, regardless of the wall thickness.
The connection between the chord and the wall is calculated for shear according to the formula
Where T = QficSf/I– shearing force per unit length of the belt caused by
conventional shear force
Qfic = 7,15 ∙ 10 –6 (2330 – E/Ry)N/φ ,
Here φ – stability coefficient for central compression, taken when calculating based on the conditional flexibility of the column relative to the axis x- x;
Sf– static moment of the column belt relative to the axis x- x;
Ix– moment of inertia of the column section.
In centrally compressed columns, the shear force is insignificant, since the transverse force arising from random influences is small. The connection between the wall and the shelves is made by automatic welding. The minimum leg of the weld is adopted structurally depending on the maximum thickness of the elements being welded ( t max = tf= 12 mm) kf= 5 mm.
4.3. Calculation and design of a through column
Example 4.3. Select a through column from two channels connected by strips (Fig. 4.5), according to example 4.2.
Rice. 4.5.
Calculation of through columns relative to the material axis x- x determine the profile number, and by calculation relative to the free axis y- y, produced in the same way as solid columns, but with the flexibility of the rod replaced by reduced flexibility, the distance between the branches is assigned, which ensures the equal stability of the rod in two mutually perpendicular planes.
4.3.1. Calculation of a column for stability relative to the material axis x-x
It is recommended to pre-specify flexibility: for medium-length columns 5 - 7 m with a design load of up to 2500 kN, flexibility is accepted l= 90 – 50; with load 2500 – 3000 kN – l= 50 – 30, for taller columns it is necessary to set the flexibility to be slightly greater.
Ultimate column flexibility 
Where 
– coefficient taking into account the incomplete use of the bearing capacity of the column, taken to be at least 0.5. When the column's load-bearing capacity is fully utilized lu= 120.
Let's be flexible l = 50.
Conditional flexibility
According to the table 3.12 we determine the type of curve in accordance with the type of the accepted section (type ′′ b′′). According to table. 3.11 conditional flexibility = 1.7 corresponds to the stability coefficient under central compression j = 0,868.
Find the required cross-sectional area using the formula
Required area of one branch
Required radius of gyration relative to the axis x-x
According to the required area Ab and radius of gyration ix We select from the assortment (GOST 8240-93) two channels No. 36, having the following section characteristics:
Ab= 53.4 cm 2; A= 2Ab= 53.4 × 2 = 106.8 cm 2; Ix= 10820 cm 4; I 1 = 513 cm 4;
ix= 14.2 cm; i 1 = 3.1 cm; wall thickness d= 7.5 mm; shelf width bb= 110 mm; reference to the center of gravity z o = 2.68 cm; linear density (weight of 1 linear meter) 41.9 kg/m.
If the maximum channel profile = 2 = 22926.7 cm 4.
Radius of inertia
Column Bar Flexibility
λ y = ly/iy = 813 / 14,65 = 55,49.
Given flexibility
Conditional reduced flexibility
According to the table 3.11 depending on the type of stability curve ″ b″ we determine the stability coefficient under central compression φ = 0,830.
We check:
Column stability relative to the axis y- y secured.
Undervoltage in the column
which is permissible in a composite section according to SNiP.
In columns with lattice, the stability of an individual branch in the area between adjacent lattice nodes must also be checked.
Design force
Nb = N/2 = 2067.18 / 2 =1033.59 kN.
Estimated length of the branch (see Fig. 34)
l 1 = 2b o tgα= 2 · 28.64 · 0.7 = 40.1 cm.
Sectional area of the branch Ab= 53.4 cm 2.
Section radius of gyration [ 36 relative to the axis 1-1 i 1 = 3.1 cm.
Branch flexibility
Conditional branch flexibility
Central compression stability coefficient for stability curve type ″ b″ φ = 0,984.
We check the stability of a separate branch:
The column branch in the area between adjacent lattice nodes is stable.
Triangular lattice calculation
The calculation of a triangular lattice of a through column is performed as a calculation of a truss lattice, the elements of which are calculated for the axial force from the conventional transverse force Qfic(see Fig. 4.8). When calculating the cross braces of a cross lattice with struts, one should take into account the additional force that arises in each brace from the compression of the column branches. The force in the brace is determined by the formula
Section of a brace from an equal angle ∟ 50 × 50 × 5 , previously accepted when calculating the through column rod ( Ad= 4.8 cm 2), we check for stability, for this we calculate:
– estimated length of the brace
ld = bo/cos α = 28.64 / 0.819 = 34.97 cm;
– maximum flexibility of the brace
Where iyo= 0.98 cm – minimum radius of gyration of the angle section relative to the axis yO- yO(by assortment);
– conditional flexibility of the brace
– φ min = 0.925 – minimum stability coefficient for the type of stability curve ″ b″;
– γ With= 0.75 – coefficient of working conditions, taking into account the one-sided attachment of a brace from a single corner (see Table 1.3).
We check the compressed brace for stability using the formula
The stability of the brace is ensured.
Spacers serve to reduce the design length of a column branch and are calculated for a force equal to the conventional shear force in the main compressed element ( Qfic/2). Usually they are taken with the same cross-section as the braces. We calculate the attachment point of the brace to the column branch using mechanized welding for the force in the brace Nd= 16.37 kN. We calculate the weld based on the metal of the fusion boundary.
The forces perceived by the seams are calculated using the following formulas
- at the butt
Nabout = (1 – α )Nd= (1 – 0.3) 16.37 = 11.46 kN;
NP = α Nd= 0.3 · 16.37 = 4.91 kN.
Specifying the minimum leg of the seam at the feather kf= tyy– 1 = 5 – 1 = 4 mm, find the estimated seam lengths:
- at the butt
lw,about = Nabout/(β zR wz γwzγ c) = 11.46 / (1.05 · 0.4 · 16.65 · 1 · 1) = 1.64 cm;
lw,P= NP/(β zRwzγ wzγ c) = 4.91 / (1.05 · 0.4 · 16.65 · 1 · 1) = 0.7 cm.
We accept the minimum structural length of the weld at the butt and feather lw,about = lw,P= 40 + 1 = 50 mm.
If it is not possible to place the welds within the width of the branch, then to increase the length of the seams it is possible to center the braces on the face of the column.
When dividing a column into dispatch marks due to transportation conditions, the dispatch elements of through columns with gratings in two planes should be strengthened with diaphragms located at the ends of the dispatch element. In through columns with a connecting grid in the same plane, diaphragms should be placed along the entire length of the column at least every 4 m. The thickness of the diaphragm is taken to be 8 - 14 mm (Fig. 4.9).
Rice. 4.9.
4.4. Design and calculation of column heads
The main beam rests on the column from above, and the interface is assumed to be hinged. Longitudinal compressive force N from the main beams is transmitted through a support slab planed on both sides with a thickness ton= 16 – 25 mm directly on the ribs of the head of a solid column and on the diaphragm in a through column.
The ends of the column, ribs and diaphragm are milled. The transfer of force from the ribs to the wall of the column and from the diaphragm to the walls of the branches of the column is carried out by vertical welds. The plate is used to fasten the beams to the column with mounting bolts that fix the design position of the beams. The welds attaching the slab to the column are designed structurally with a leg of the minimum size, taken according to the greatest thickness of the joined elements (see Table 3.6). The dimensions of the slab in plan are taken to be larger than the contour of the column by 15 - 20 mm in each direction to accommodate welds.
To impart rigidity to the vertical ribs and diaphragm, as well as to strengthen the walls of the column rod or branches of the through column from loss of stability in places where large concentrated loads are transmitted, the vertical ribs from below are framed by a horizontal stiffener.
4.4.1. Solid column head
The head consists of a plate and ribs (Fig. 4.10).
Rice. 4.10.
The required area of the vertical paired rib is determined from the collapse condition:
Fin thickness
where is the conditional length of the distribution on-
load equal to the width of the supporting rib of the main beam bh plus two thicknesses of the column head slab ( ton accepted 25 mm).
Rib width (protruding part)
We take two vertical ribs with a cross-section of 140´22 mm.
We check the vertical rib for local stability.
The height of the support rib is determined based on the placement of welds that ensure force transmission N from the ribs to the wall of the column.
We specify the leg of the weld seam kf= 7 mm (within design requirements kf , min = 7 mm for mechanized sheet welding t max = 25 mm and – the smallest thickness of the elements to be connected).
Required seam length
Taking into account 1 cm for compensation of defects in the end sections of the seam along its length, we finally accept the height of the rib hr= 45 cm.
The estimated length of the seam should be no more than 85 β fkf.
We check it using the formula
For thin walls of a solid column, the wall thickness tw check for shear along the edges of the fastening of the supporting vertical ribs. Required wall thickness
which is greater than the accepted wall thickness tw= 8 mm. We locally strengthen the column wall by replacing a section of the wall within the height of the head with a thicker insert. We accept the thickness of the insert t ′ w= 18 mm.
To reduce stress concentration when butt welding elements of different thicknesses, we perform bevels with a slope of 1:5 on an element of greater thickness. The width of the horizontal stiffening ribs is taken equal to the width of the vertical support ribs bs= br= 140 mm. The thickness of the rib is determined from the condition of its stability:
it must be at least We accept a paired rib from a sheet with a section of 140×10 mm.
4.4.2. Head of a through column
The head consists of a plate and a diaphragm, supported by a horizontal stiffener (Fig. 4.11).
Rice. 4.11.
The calculation is carried out similarly to the calculation of the head of a solid column.
Diaphragm thickness td determined by calculation of crushing due to axial force N:
where is the conditional length of the concentrated load distribution (see clause 4.4.1).
We accept td= 22 mm.
The height of the diaphragm is determined from the condition of cutting the walls of the column branches ( d= 7.5 mm – wall thickness for the adopted channel):
hd = N/(4dRsγ c) = 2067.18 / (4 · 0.75 · 13.92 · 1) = 49.5 cm.
We accept hd= 50 cm.
We check the diaphragm for shear as a short beam:
Where Q = N/2 = 2067.18 / 2 = 1033.59 kN .
The strength condition is not met. We accept the thickness of the diaphragm td= 25 mm and check again:
We determine the leg of the weld, made by mechanized welding and ensuring the attachment of the diaphragm to the wall of the column branches (calculation for metal fusion boundary):
Where lw = hd– 1 = 50 – 1 = 49 cm – estimated seam length equal to the height of the diaphragm minus 1 cm, taking into account defects in the end sections of the seam.
We accept the seam leg kf= 7 mm, which corresponds to its minimum value for mechanized welding of elements t= 25 mm.
The estimated length of the flank seam should be no more than 85 β fkf. We check: lw = 49 < 85 × 0,9 × 0,7 = 53,5 см. Условие выполняется.
The thickness of the horizontal stiffener is taken ts= 10 mm, whichever is greater 
Width bs we assign from the stability condition of the edge:
We accept bs= 30 cm.
4.5. Design and calculation of the column base
The base is the supporting part of the column and serves to transfer forces from the column to the foundation. For relatively small design forces in columns (up to 4000 - 5000 kN), bases with traverses are used. The force from the column rod is transmitted through welds to the slab resting directly on the foundation. For a more uniform transfer of pressure from the slab to the foundation, the rigidity of the slab, if necessary, can be increased by installing additional ribs and diaphragms.
The base is secured by fixing its design position on the foundation with anchor bolts. Depending on the fastening, the column is hinged or rigidly connected to the foundation. In a hinged base, anchor bolts with a diameter of 20–30 mm are attached directly to the base plate, which has a certain flexibility that ensures compliance under the action of random moments (Fig. 4.12).
Rice. 4.12. Column base at Rice. 4.13.
To allow some movement (straightening) of the column during its installation in the design position, the diameter of the holes in the slab for anchor bolts is taken to be 1.5 - 2 times larger than the diameter of the anchors. Washers with a hole that is 3 mm larger than the diameter of the bolt are put on the anchor bolts, and after tensioning the bolt with a nut, the washer is welded to the plate. With rigid coupling, anchor bolts are attached to the column core through traverse outriggers, which have significant vertical rigidity, which eliminates the possibility of column rotation on the foundation. In this case, bolts with a diameter of 24–36 mm are tightened with a tension close to the design resistance of the bolt material. The thickness of the anchor plate is tap= 20 – 40 mm and width bap equal to four diameters of the bolt holes (Fig. 4.13).
The design of the base must correspond to the method of coupling it with the foundation adopted in the design diagram of the column. A column base with rigid fastening to the foundation was accepted for calculation and design.
4.5.1. Determining the dimensions of the base plate in plan
We determine the design force in the column at the base level, taking into account the column’s own weight:
Where k= 1.2 – design factor that takes into account the weight of the lattice, base elements and column head. The pressure under the slab is assumed to be uniformly distributed. In a centrally compressed column, the dimensions of the slab in plan are determined from the strength condition of the foundation material:
Where y– coefficient depending on the nature of the distribution of local load over the crushing area (with uniform stress distribution y =1);
Rb , loc– design resistance of concrete to crushing under the slab, determined by the formula
Rb , loc= αφ bRb= 1 ∙ 1.2 ∙ 7.5 = 9 MPa = 0.9 kN/cm 2,
Where a= 1 – for concrete class below B25;
Rb= 7.5 MPa for concrete class B12.5 – the calculated compressive strength of concrete corresponding to its class and taken according to table. 4.3;
jb– coefficient that takes into account the increase in the compressive strength of concrete in cramped conditions under the base plate and is determined by the formula
Here Af 1 – area of the upper edge of the foundation, slightly larger than the area of the base plate Af.
Table 4.3
Design resistance of concreteR b
|
Strength class |
|||||||
|
Rb, MPa |
Coefficient jb no more than 2.5 is accepted for concrete of classes higher than B7.5 and no more than 1.5 for concrete of class B7.5 and lower.
Let's ask in advance jb= 1,2.
Base plate calculation
Slab dimensions (width B and length L) are assigned according to the required area Af, are linked to the contour of the column (the overhangs of the base plate must be at least 40 mm) and are consistent with the assortment (Fig. 4.14).
Rice. 4.14.
Set the width of the slab:
B = h + 2tt + 2c= 36 + 2 1 + 2 4 = 46 cm,
Where h= 36 cm – height of the section of the column rod;
tt= 10 mm – traverse thickness (take 8 – 16 mm);
With= 40 mm – minimum overhang of the cantilever part of the slab (preliminarily assumed to be 40 – 120 mm and, if necessary, specified in the process of calculating the thickness of the slab).
Required slab length
For a centrally compressed column, the base plate should be close to square (recommended aspect ratio L/IN≤ 1.2). We accept a square slab with dimensions IN= L= 480 mm.
Slab area Af= LB = 48 · 48 =2304 cm 2.
The area of the foundation edge (we set the dimensions of the upper edge of the foundation 20 cm larger than the dimensions of the base plate)
Actual ratio
Design resistance of concrete to crushing under the slab
Rb , loc = 1 ∙ 1.26 ∙ 7.5 = 9.45 MPa = 0.95 kN/cm2.
Checking the strength of concrete under the slab:
Reducing the size of the slab is not required, since it was adopted with minimal dimensions in plan.
4.5.2. Determining the thickness of the base plate
The thickness of the base slab, supported on the ends of the column, traverses and ribs, is determined from the condition of its bending strength from the resistance of the foundation, equal to the average stress under the slab:
In each section, the maximum bending moments acting on a strip 1 cm wide are determined from the design uniformly distributed load
Location on 1 , supported on four sides:
Where a 1 = 0.053 – coefficient that takes into account the reduction in the span moment due to the support of the slab on four sides and is determined from table. 4.4 depending on the ratio of the larger side of the plots b to less a.
Table 4.4
Oddsa 1 for calculating the bending of a slab supportedon four sides
|
b/a |
||||||||||
Values b And a determined by dimensions in the light:
b = 400 – 2d= 400 – 2 × 7.5 = 385 mm; A= 360 mm; b/A = 385 / 360 = 1,07.
Location on 2 , supported on three sides:
Where b– the coefficient is taken according to the table. 4.5 depending on the ratio of the fixed side of the plate b 1 = 40 mm to free A 1 = 360 mm.
Table 4.5
Oddsb to calculate the bending of a slab supported on three edges
|
b 1 /a 1 |
||||||||||
Relationship between the parties b 1 /a 1 = 40 / 360 = 0.11; in relation to the parties b 1 /a 1 < 0,5 плита рассчитывается как консоль длиной b 1 = 40 mm (Fig. 4.15).
Bending moment
On the cantilever section 3
Rice. 4.15.
When a slab is supported on two edges converging at an angle, the bending moment for the safety factor is calculated as for a slab supported on three sides, taking the size a 1 diagonally between edges, size b 1 equal to the distance from the top of the corner to the diagonal (Fig. 4.16, A).
If there is a sharp difference in the magnitude of the moments in different sections of the slab, it is necessary to make changes to the slab support scheme in order to, if possible, equalize the values of the moments. This is done by setting diaphragms and ribs. We divide the slab on the site 1 half diaphragm thickness td= 10 mm (see Fig. 4.15).
Aspect Ratio
b/a= 38,5 / 17,5 = 2,2 > 2,
When the slab is supported on four edges with the aspect ratio b/a> 2 bending moment is determined as for a single-span beam slab with a span A, freely lying on two supports:
By highest value From the bending moments found for various sections of the slab, we determine the required moment of resistance of a 1 cm wide slab:
where is the thickness of the slab?
We accept a sheet with a thickness of 30 mm.
When determining the bending moment M 1 ׳ in a strip 1 cm wide for the section of the slab in question 1 it is allowed to take into account the unloading influence of adjacent cantilever sections along the long sides (as in a continuous beam) according to the formula
M 1 ׳ = M 1 – M 3 =q(α 1 a 2 – 0,5c 2) = 0.9 (0.053 ∙ 36 2 – 0.5 ∙ 5 2) = 50.57 kN∙cm.
4.5.3. Traverse calculation
The thickness of the traverse is accepted tt= 10 mm.
The height of the traverse is determined from the condition of placing the vertical seams for attaching the traverse to the column rod. For the safety factor, it is assumed that all the force is transmitted to the traverses through four fillet welds (welds connecting the column rod directly to the base slab are not taken into account).
We accept the weld leg kf= 9 mm (usually set within 8 – 16 mm, but not more than 1.2 t min). Required length of one seam made
mechanized welding, based on the fusion boundary
lw = N/(4β zkf Rwzγ wzγ c) = 2184 / (4 ∙ 1.05 ∙ 0.9 ∙ 16.65 ∙ 1 ∙ 1) = 34.7 cm<
< 85 β f kf= 85 · 0.9 · 0.9 = 68.85 cm.
We accept the height of the traverse taking into account the addition of 1 cm for defects at the beginning and end of the seam ht= 38 cm.
We check the strength of the traverse as a single-span, double-cantilever beam resting on the branches (flanges) of the column and receiving back pressure from the foundation (Fig. 4.16, b).
Rice. 4.16.
Where d= B/2 = 48 / 2 = 24 cm – width of the cargo area of the traverse.
Where σ = Mop/Wt= 178.8 / 240.7 = 0.74 kN/cm2;
τ = Qetc/(ttht) = 432 / (1 38) = 11.37 kN/cm2.
The traverse cross-section is accepted.
Required leg of horizontal seams for force transmission ( Nt= qtL) from one traverse per slab
where å lw = (L– 1) + 2(b 1 – 1) = (48 – 1) + 2 (4 – 1) = 53 cm – total length of horizontal seams.
We accept the weld leg kf= 12 mm, which is equal to the maximum permissible leg kf, max = 1.2 tt= 1.2 · 1 = 12 mm.
4.5.4. Calculation of slab reinforcement ribs
For the designed base, it is necessary to install stiffeners
there is no support plate on the cantilever section, so the calculation is given as an example for other options for designing the column base (see Fig. 4.16, A).M r And Qr according to the formula
Where σ = Mr/Wr = 6Mr/(trhr 2) = 6 270 / (1 10 2) = 16.2 kN/cm 2;
τ = Qr/(trhr) = 108 / (1 10) = 10.8 kN/cm2.
Rib accepted.
We check the welds attaching the rib to the traverse (rod) of the column for the resultant tangential stresses from bending and shearing.
We assign a suture leg kf= 10 mm.
We check the shear strength of the metal of a seam made by mechanized welding (estimated length of the seam lw = hr– 1 = 10 – 1 = 9 cm:
We check the strength of the seams along the fusion boundary:
Required leg of welds for attaching ribs to the base plate
kf = Qr/ = 108 / = 0.77 cm.
We accept the seam leg kf= 8 mm.
The column rod is fastened to the base plate using a structural weld with a 7 mm leg (when welding sheets t max = tp= 30 mm).
STEEL COLUMN
BUILDINGS AND STRUCTURES
Centrally compressed columns are used to support interfloor floors and coverings of buildings, work platforms and overpasses. The column structure consists of the rod itself and supporting devices - the head and base. The overlying building structures that directly load the column rest on the head, the column rod transmits the load from the head to the base and is the main structural element, and the base transfers the entire received load from the rod to the foundation.
Column types
There are three types of columns used in building frames:
— columns of constant cross-section;
— columns of variable cross-section (stepped);
— columns of separate type.
Columns of constant section used in craneless buildings and in buildings with the possibility of using suspended and bridge electric lifting mechanisms with a lifting capacity of up to 20 tons, as a rule, with a useful height from the floor level to the bottom of the trusses of no more than 12 m.
When using cranes with a lifting capacity of more than 15 tons, stepped columns consisting of two parts, the upper part is usually a welded or rolled I-beam, the lower part consists of a tent and crane branch that are connected to each other either by ties in the form of a solid sheet or by a through lattice of hot-rolled angles.
Separate type columns are used in buildings with cranes with a lifting capacity of more than 150 tons and a height of 15-20 m. The tent and crane struts in this design are connected to each other by a series of horizontal slats that are flexible in the vertical plane, due to which the load perception is separated, the crane strut receives only the vertical force from the overhead crane, and the tent branch collects all the loads from the frame and covering of the building.
Column sections
Column rods are made from single wide-flange I-beams or made up of several rolled profiles; composite rods are divided into through and solid. Through ones, in turn, are divided into unbraced, lattice and perforated.
Solid columns most often they are a welded or rolled wide-flange I-beam, where the welded option has an advantage due to the ability to select the optimal cross-section to ensure the required rigidity in the column while simultaneously saving material. Quite easy to manufacture are cross-section columns that are equally stable in two directions. With the same dimensions, the cross section outperforms the I-beam due to greater rigidity. Solid columns also include closed-section columns, which can be composed of paired rolled channels, bent electric-welded profiles or round pipes. A significant disadvantage of this option is the inaccessibility of the inner surface for maintenance, which can lead to rapid corrosive wear.
Through columns – A typical structural design consists of two branches (made of channels, I-beams or pipes) interconnected by lattices ensuring the joint operation of the branches of the column rod. Grating systems are used from braces, braces and struts, and the non-bracing type in the form of planks. The column lattice is usually placed in two planes and is made from single corners, giving preference to a shapeless connection, with fastening directly to the shelves of the rod branches. To prevent twisting of such columns and maintain their contour, diaphragms are installed at the ends.
Column parts and assemblies
Column heads. There are two design solutions for supporting trusses and crossbars on columns, with a hinged free connection - the beams are usually installed on top, with hinged and rigid connections they are attached to the side.
With top connection, the column head consists of a base plate and stiffeners that transfer the load to the column body. The ribs of the head are welded to the slab and branches of the column with a through rod or to the walls of the column with a solid rod. The height and thickness of the ribs are determined based on the required length of the welds, which must withstand the full pressure on the head and the resistance to collapse under the influence of support pressure. To compensate for the skew of the connecting flanges, giving additional stability and rigidity to the vertical ribs, they are, if necessary, framed with transverse ribs. The base slab is usually a planed plate with a thickness of 20...30mm, for light columns 12...30mm, the size of the slab contour in plan is assigned to be larger than the column contour by 15...20mm.
With lateral attachment, the support reaction is transmitted through the supporting rib of the adjacent beam to a table welded to the column floors. The end of the supporting rib of the beam and the table are milled, the thickness of the table is taken to be 20...40 mm greater than the thickness of the supporting rib.
Column base are the supporting part of the column and serve to transfer force from the column to the foundation. The structural solution of the base depends on the type and height of the cross-section of the rod, the method of mating with the foundation and the method of installation of the columns. They are divided into common and separate bases, which can be without traverses, with common or separate traverses, single-walled or double-walled. The main dimensions of the base plate are determined depending on the type of bases and bending calculations. The holes for the anchor bolts are laid 20...30 mm larger than their diameter, the tension is carried out through washers, which are then welded to the slab. To ensure the rigidity of the base and reduce the thickness of the support, traverses, ribs and diaphragms are installed, but due to this, the base with traverses is larger in size compared to one without traverses. The bases of through columns are usually designed of a separate type, each branch has its own loaded base. However, if the height of the column section is less than 1 m, it is permissible to use a common base, as with the solid columns discussed above.
Consoles They are used to support crane beams on columns of constant cross-section; single-walled ones are predominantly used; if it is necessary to transmit large forces, double-walled ones are used.