Calculation of deflection of a metal truss. Calculation of a metal truss. Calculation of an arched truss

In various branches of construction, trusses made of profile pipe. Such trusses are structurally metal structures consisting of individual rods and having a lattice shape. Trusses differ from structures made from solid beams by being less expensive and more labor intensive. To connect profile pipes it can be used as welding method, and rivets.

Metal profile trusses suitable for creating any spans, regardless of their length - but for this to be possible, the structure must be calculated with extreme accuracy before assembly. If the calculation metal truss was correct, and all the work on assembling the metal structures was carried out correctly, then the finished truss will only have to be lifted and installed on the prepared frame.

Advantages of using metal rafters

Trusses made from profile pipes have many advantages, including:

• Low weight of the structure;
• Long service life;
• Excellent strength properties;
• Ability to create structures of complex configuration;
• Reasonable cost of metal elements.

Classification of profile pipe trusses

All metal constructions farms have several general parameters, which ensure the division of farms into types.

These options include:

1. Number of belts. Metal trusses can have only one belt, and then the entire structure will lie in one plane, or two belts. In the latter case, the truss will be called a hanging truss. The design of a hanging truss includes two chords - upper and lower.
2. Form. There is an arched truss, straight, single-slope and double-slope.
3. Circuit.
4. Tilt angle.

Depending on the contours there are the following types metal structures:

1. Parallel belt trusses. Such structures are most often used as a support for arranging a roof made of soft roofing materials. A truss with a parallel belt is created from identical parts with identical dimensions.
2. Shed farms. Single slope designs are inexpensive because they require few materials to make. The finished structure is quite durable, which is ensured by the rigidity of the nodes.
3. Polygonal trusses. These structures have very good load-bearing capacity, but you have to pay for it - polygonal metal structures are very inconvenient to install.
4. Triangular trusses. As a rule, trusses with a triangular contour are used to install roofs located at a large slope. Among the disadvantages of such farms it is worth noting a large number of extra costs associated with the mass of waste during production.

How to calculate the angle of inclination

Depending on the angle of inclination, trusses are divided into three categories:

1. 22-30 degrees. In this case, the ratio of the length and height of the finished structure is 5:1. Trusses with such a slope, being light in weight, are excellent for arranging short spans in private construction. As a rule, trusses with such a slope have a triangular outline.
2. 15-22 degrees. In a design with such a slope, the length exceeds the height by seven times. Trusses of this type cannot be more than 20 m in length. If it is necessary to increase the height of the finished structure, the lower chord is given a broken shape.
3. 15 or less. The best option in this case there will be metal rafters from a profile pipe, connected in the shape of a trapezoid - short racks will reduce the impact longitudinal bending on the design.

In the case of spans whose length exceeds 14 m, it is necessary to use braces. The upper belt must be equipped with a panel about 150-250 cm long. When even number panels will result in a structure consisting of two belts. For spans longer than 20 m, the metal structure must be reinforced with additional supporting elements connected by supporting columns.

If you need to reduce the weight of the finished metal structure, you should pay attention to the Polonceau truss. It includes two systems triangular shape, which are connected by tightening. Using this scheme, you can do without large-sized braces in the middle panels.

When creating trusses with a slope of about 6-10 degrees for pitched roofs you need to remember that finished design should not be symmetrical in shape.

Calculation of a metal truss

When making calculations, it is necessary to take into account all the requirements for metal structures state standards. To create the most efficient and reliable design, it is necessary at the design stage to prepare a high-quality drawing, which will display all the elements of the truss, their dimensions and features of connection with the supporting structure.

Before you calculate a farm for a canopy, you should decide on the requirements for the finished farm, and then start from savings, avoiding unnecessary costs. The height of the truss is determined by the type of floor, the total weight of the structure and the possibility of its further displacement. The length of the metal structure depends on the expected slope (for structures longer than 36 m, a construction lift calculation will also be required).

The panels must be selected in such a way that they can withstand the loads that will be placed on the farm. The braces can have different angles, so when choosing panels you also need to take this parameter into account. In the case of triangular grilles, the angle is 45 degrees, and in the case of slanted grilles, it is 35 degrees.

The calculation of a roof made from a profile pipe ends with determining the distance at which the nodes will be created relative to each other. As a rule, this indicator is equal to the width of the selected panels. The optimal pitch for the supports of the entire structure is 1.7 m.

Performing a calculation lean-to farm, you need to understand that as the height of the structure increases, its load bearing capacity. In addition, if necessary, it is worth supplementing the truss diagram with several stiffening ribs that can strengthen the structure.

Calculation examples

When selecting pipes for metal trusses, you should consider the following recommendations:

• For arranging structures less than 4.5 m wide, pipes with a cross section of 40x20 mm and a wall thickness of 2 mm are suitable;
• For a structure width of 4.5 to 5.5 m, 40 mm square profile pipes with a 2 mm wall are suitable;
• For metal structures bigger size the same pipes as in the previous case, but with a 3 mm wall, or pipes with a cross section of 60x30 mm with a 2 mm wall are suitable.

The last parameter that should also be paid attention to when calculating is the cost of materials. First, you need to consider the cost of the pipes (remembering that the price of pipes is determined by their weight, not their length). Secondly, it’s worth asking about the cost complex works for the production of metal structures.

Recommendations for choosing pipes and manufacturing metal structures

Before cooking farms and picking optimal materials For future design, it is worth familiarizing yourself with the following recommendations:

• When studying the range of pipes available on the market, you should give preference to rectangular or square products - the presence of stiffeners significantly increases their strength;
• Selecting pipes for rafter system, it would be best to opt for stainless steel products made of high quality steel (pipe sizes are determined by the project);
• When installing the main elements of the truss, tacks and double corners are used;
• In the upper chords, I-angles with different sides, the smaller of which is necessary for docking;
• For mounting the lower belt, corners with equal sides are quite suitable;
• The main elements of large-sized structures are attached to each other with overhead plates;
• The braces are mounted at an angle of 45 degrees, and the racks are mounted at a 90-degree angle.
• When a metal truss for a canopy is welded, it is worth making sure that each weld is sufficiently reliable (read also: " ");
• After welding work metal elements the structure remains to be covered protective compounds and paint.

Conclusion

Trusses made from profile pipes are quite versatile and suitable for solving a wide range of problems. Making trusses cannot be called simple, but if you approach all stages of work with full responsibility, the result will be a reliable and high-quality structure.

2.6.1. General concepts.

A flat rod system, which, after including hinges in all nodes, remains geometrically unchanged is called a truss.

Examples of farms are shown in Fig. 2.37..

In real rod structures that fit the definition of “truss,” the rods in the nodes are connected not by hinges, but by beams, rivets, welding, or embedded (in reinforced concrete structures). However, in the design diagrams of such structures, hinges can be introduced into the nodes, but on the condition that

· the rods are perfectly straight;

· the axes of the rods intersect in the center of the node;

· concentrated forces are applied only to nodes;

dimensions cross sections the rods are significantly less than their length.

Fig.2.37.. Statically definable flat trusses.

Under these conditions, the truss rods work only in tension or compression, only longitudinal forces arise in them.

This circumstance greatly simplifies the calculation rod system and allows you to obtain results with a sufficient degree of accuracy.

To determine the forces in the truss rods using the section method, you must:

1) Conduct the section in such a way that it

· crossed the axis of the rod in which the force is determined;

· crossed, if possible, no more than three rods;

· divided the farm into two parts.

2) Direct the longitudinal forces in the rods in the positive direction, i.e. from the node.

3) Select equilibrium equations for part of the truss that would include only one required force. Such equations are, for example,

· the sum of moments relative to the point at which the lines of action of forces in the truss rods cut by the section intersect; Such points are usually called moment points;

the sum of the projections of forces onto vertical axis for bracing trusses with parallel chords.

4) To determine the forces in the racks, cut out the nodes if no more than three rods meet in them.

5) To simplify the determination of the arms of internal forces relative to the moment point when drawing up moment equations, if necessary, replace the required forces with their projections onto mutually perpendicular axes.

2.6.2. Determination of forces in truss rods.

To determine the forces in the truss rods it is necessary:

· determine the reactions of the supports;

· using the section method to determine the required forces;

· check the results obtained.

The reactions of supports in simple beam trusses shown in Fig. 2.37 are determined in the same way as in single-span beams using equations of the form

To check the support reactions we use the equation

Let's consider the calculation algorithm using a specific example.

A design diagram of the farm is given (Fig. 2.38).

It is required to determine the forces in rods 4-6, 3-6, 3-5, 3-4, 7-8.

The solution of the problem.

1) Determining the reactions of the supports.

To do this, we use the equilibrium equation:

We write the equations using the accepted sign rule:

Solving the equations, we find

We check the reactions of the supports using the equation.

2) Determining the forces in the truss rods.

a) Efforts in rods 4-6, 3-6, 3-5.

To determine the forces in the indicated rods, we cut the truss with a section ah-ah into two parts and consider the equilibrium of the left side of the truss (Fig. 2.39.

To the left side of the truss we apply the support reaction , the force acting in node 4, and the required forces in the truss rods , , . We direct these forces along the corresponding rods away from the node, that is, in the positive direction.

To determine the efforts , , you can use the following system of equations:

But in this case we will obtain a joint system of equations, which will include all the required efforts.

To simplify the solution of the problem, it is necessary to use equilibrium equations that include only one unknown.

To determine the force, such an equation is

i.e., the sum of the moments relative to node 3, at which the lines of action of forces and intersect, since the moments of these forces relative to node 3 are equal to zero. For effort, this equation is

i.e., the sum of moments relative to node 6 at which the lines of action of forces and intersect.

To determine the force, you should use the equation for the sum of moments relative to point O, at which the lines of action of forces and intersect, i.e.

When writing the above equations, math difficulties by determining the shoulders of forces relative to the corresponding points. To simplify the solution of this problem, it is recommended to expand the required force along the X, Y axes and use force projections when writing the equilibrium equation.

Let's show this using the example of effort (Fig. 2.40).

Let's write the equation:

Solving the equation, we get:

IN in this example the projection of the force on the X axis has a moment relative to point O equal to zero, since the line of its action passes through point O.

3) Determine the force in rod 3-4.

To determine the force, we cut 4 trusses into a node with a cross-section b-b(Fig. 2.41.a).

4) Determine the force in rod 7-8.

Cut out the node 8 section s-s(Fig. 2.41.b). We compose two equilibrium equations

To determine the force, we have two equations with three unknowns. Therefore, one of these unknowns ( or ) must be determined in advance.

If the force is known, then the equation can be used to determine the force:

the sum of the projections of the forces applied at the node onto the x-axis perpendicular to the line of action of the force.

It should be noted that the forces in the truss rods can be determined by considering the equilibrium of its nodes in turn and composing two equations for each node

It is necessary to start with a node in which only two rods converge, and then successively consider nodes in which there are only two unknown forces. Let's look at an example(Fig. 2.42).

1) We consider node 1, in which only two rods converge. Compose and solve equations

2) We consider node 2, in which 3 rods converge, but the force is known:

Solving the system of equations, we find:

Then node 4 is considered, etc.

This method of determining the forces in the truss rods has the following disadvantages:

· an error made during the calculation process applies to subsequent calculations;

· it is not rational for determining the forces only in individual truss rods.

The advantages of the method include the possibility of using it when compiling programs for calculations on a computer.

2.6.3. Checking the calculation results.

To check the calculation results, you need to use equilibrium equations, which include greatest number effort. So, for example, to check the efforts , , (Fig. 3.3) such equations are

A truss is a system of usually straight rods that are connected to each other by nodes. This is a geometrically unchangeable structure with hinged nodes (considered as hinged in a first approximation, since the rigidity of the nodes does not significantly affect the operation of the structure).

Due to the fact that the rods experience only tension or compression, the truss material is used more fully than in a solid beam. This makes such a system economical in terms of material costs, but labor-intensive to manufacture, so when designing it must be taken into account that the feasibility of using trusses increases in direct proportion to its span.

Trusses are widely used in industrial and civil construction. They are used in many construction industries: covering buildings, bridges, supports for power lines, transport overpasses, lifting cranes, etc.

Construction device

The main elements of trusses are the belts that make up the outline of the truss, as well as a lattice consisting of posts and braces. These elements are connected at nodes by abutment or by node gussets. The distance between the supports is called the span. Truss chords usually operate under longitudinal forces and bending moments (like solid beams); the truss lattice absorbs mainly the transverse force, just like the web in the beam.

According to the location of the rods, trusses are divided into flat (if everything is in the same plane) and spatial. Flat trusses capable of taking load only relative to their own plane. therefore, they must be secured from their plane using ties or other elements. Spatial farms are created to absorb load in any direction, as they create a rigid spatial system.

Classification by belts and grilles

For different types loads applied different kinds farms There are many classifications of them, depending on different characteristics.

Let's consider the types according to the outline of the belt:

a - segmental; b - polygonal; c - trapezoidal; g - with parallel arrangement of belts; d - i - triangular

The truss chords must correspond to the static load and the type of load that determines the bending moment diagram.

The outline of the belts largely determines the efficiency of the farm. In terms of the amount of steel used, the segmented truss is the most effective, but it is also the most difficult to manufacture.

According to the type of lattice system, trusses are divided into:

a - triangular; b - triangular with additional racks; c - braced with ascending braces; g - braced with descending braces; d - trussed; e - cross;

g - cross; z - rhombic; and - semi-diagonal

Features of calculation and design of tubular trusses

For production it uses steel with a thickness of 1.5 - 5 mm. The profile can be round or square.

The tubular profile for compressed bars is the most efficient in terms of steel consumption due to the favorable distribution of the material relative to the center of gravity. With the same cross-sectional area, it has the largest radius of gyration compared to other types of rolled products. This allows you to design rods with the least flexibility and reduce steel consumption by 20%. Also, a significant advantage of pipes is their streamlining. Due to this, the wind pressure on such farms is less. Pipes are easy to clean and paint. all this makes the tubular profile advantageous for use in farms.

When designing trusses, you should try to center the elements at the nodes along the axes. This is done to avoid additional stress. Nodal connections of pipe trusses must ensure hermetic connection(it is necessary to prevent the occurrence of corrosion in the internal cavity of the farm).

The most rational for tubular trusses are unshaped units with the lattice rods connecting directly to the chords. Such nodes are performed using a special figure cutting ends, which allows you to minimize the cost of labor and material. The rods are centered along the geometric axes. In the absence of a mechanism for such cutting, the ends of the lattice are flattened.

Such units are not permissible for all types of steel (only low-carbon steel or others with high ductility). If the grid and belt pipes are of the same diameter, then it is advisable to connect them on a ring.

Calculation of trusses depending on the angle of inclination of the roof

Construction at a roof angle of 22-30 degrees

The roof angle is considered optimal for gable roof 20-45 degrees, for a single slope 20-30 degrees.

The roof structure of buildings usually consists of roof trusses placed side by side. If they are connected to each other only by runs, then the system becomes variable and may lose stability.

To ensure the immutability of the structure, designers provide several spatial blocks of adjacent trusses, which are held together by ties in the planes of the chords and vertical transverse ties. Other trusses are attached to such rigid blocks using horizontal elements, which ensures the stability of the structure.

To calculate the roofing of a building, it is necessary to determine the angle of inclination of the roof. This parameter depends on several factors:

• type of rafter system
• roofing pie
• sheathing
• roofing material

If the angle of inclination is significant, then I use triangular-type trusses. But they have some disadvantages. This is a complex support assembly that requires a hinge connection, which makes the entire structure less rigid in the transverse direction.

Load collection

Typically, the load acting on the structure is applied at the locations of the nodes to which the elements of the transverse structures are attached (for example, suspended ceiling or roof purlins). For each type of load, it is advisable to determine the forces in the rods separately. Types of loads for roof trusses:

• constant (own weight of the structure and the entire supported system);
• temporary (load from suspended equipment, payload);
• short-term (atmospheric, including snow and wind);

To determine the constant design load, you must first find the load area from which it will be collected.

Formula for determining roof load:

F = (g + g1/cos a)*b ,

where g is the own mass of the truss and its connections, horizontal projection, g1 is the mass of the roof, a is the angle of inclination of the upper chord relative to the horizon, b is the distance between the trusses

Based on this formula, the greater the angle of inclination, the less the load acting on the roof. However, it should be borne in mind that an increase in the angle also entails a significant increase in price due to an increase in the volume of building materials.

Also, when designing the roof, the construction region is taken into account. If a significant wind load is expected, then the angle of inclination is set to a minimum and the roof is made pitched.

Snow is a temporary load and only partially loads the farm. Loading half the truss can be very unprofitable for medium frames.

Full snow load for the roof is calculated according to the formula:

Sp – calculated value of snow weight per 1 m2 of horizontal surface;

μ – calculated coefficient to take into account the slope of the roof (according to SNiP, equals one if the angle of inclination is less than 25 degrees and 0.7 if the angle is from 25 to 60 degrees)

Wind pressure is considered significant only for vertical surfaces and surfaces if their angle of inclination to the horizon is more than 30 degrees (relevant for masts, towers and steep trusses). Wind load like the rest it comes down to a nodal point.

Definition of effort

When designing tubular trusses, one should take into account their increased bending rigidity and the significant influence of the stiffness of the connections in the nodes. Therefore, for tubular profiles, calculation of trusses using a hinged scheme is allowed with a ratio of sectional height to length of no more than 1/10 for structures that will be operated at a design temperature below -40 degrees.

In other cases, it is necessary to calculate the bending moments in the rods that arise due to the rigidity of the nodes. In this case, the axial forces can be calculated using the hinge diagram, and additional moments can be found approximately.

Instructions for calculating a truss

• the design load is determined (using SNiP “Loads and impacts”)

• the forces are located in the truss rods (you should decide on the design scheme)

• the estimated length of the rod is calculated (equal to the product of the length reduction coefficient (0.8) and the distance between the centers of the nodes)

• testing compressed rods for flexibility

• Having specified the flexibility of the rods, select the cross-section according to the area

During preliminary selection for belts, the flexibility value is taken from 60 to 80, for gratings 100-120.

Let's sum it up

With proper design of the rafter system, you can significantly reduce the amount of material used and make roof construction much cheaper. For correct calculation it is necessary to know the construction region and decide on the type of profile based on the purpose and type of object. By applying the right technique to find the calculated data, you can achieve optimal ratio between the cost of constructing a structure and its performance characteristics.

Ministry of Science and Education Russian Federation Federal Agency for Education State educational institution

higher vocational education"Rostov State Construction University"

CALCULATION OF FLAT TRUSSES

Guidelines And control tasks for students correspondence department

Rostov-on-Don

Calculation of flat trusses: Guidelines and tests for correspondence students. - Rostov-on-Don: Rost. state builds. univ., 2006 - 23 p.

Designed for correspondence students of all specialties. Are given various methods calculations of flat trusses and solutions to typical examples are analyzed.

Compiled by: T.V. Vilenskaya S.S. Savchenkova

Reviewer: npof. I.F. Khrdzhiyants

Editor N.E. Gladkikh Templan 2006, pos. 171

Signed for publication on 05/24/06. Format 60x84/16. Writing paper. Risograph. Academic ed. l.. 1.4. Circulation 100 copies. Order Editorial – publishing center RGSU

344022, Rostov n/D, st. Socialist, 162

© Rostov State University of Civil Engineering, 2006

INTRODUCTION

When building bridges, cranes and other structures, structures called trusses are used.

A truss is a structure consisting of rods connected to each other at the ends by hinges and forming a geometrically unchangeable system.

The hinged connections of the truss rods are called its nodes. If the axes of all the rods of a truss lie in the same plane, then the truss is called flat.

We will only consider flat trusses. We assume that the following conditions are met:

1) all truss rods are straight;

2) there is no friction in the hinges;

3) all specified forces are applied only at the truss nodes;

4) the weight of the rods can be neglected.

In this case, each truss rod is under the influence of only two forces, which will cause it to stretch or compress.

Let the truss have “m” rods and “n” nodes. Let's find the relationship between m and n, ensuring the rigidity of the structure (Fig. 1).

To connect the first three nodes, three rods are needed; to rigidly connect each of the remaining (n-3) nodes, 2 rods are needed, that is

or m = 2n-3. (1)

If m< 2n - 3, то конструкция не будет геометрически неизменяемой, если m >2n - 3, the truss will have an “extra” rod.

Equality (1) is called the rigidity condition.

The farm shown in Fig. 1, is a rigid structure

Rice. 1 Calculation of a truss comes down to determining the support reactions and forces in

rods, that is, the forces acting from the nodes on the rods adjacent to it.

Let us find out at what ratio between the number of rods and nodes the truss will be statically determinate. If all unknown forces can be determined from the equilibrium equations, that is, the number of independent equations is equal to the number of unknowns, then the structure is statically determinate.

Since each truss node is acted upon by a plane system of converging forces, it is always possible to construct 2n equilibrium equations. Total unknowns - m + 3, (where m forces in the rods and 3 support reactions).

Condition for static definability of the truss m + 3 = 2n

or m = 2n - 3 (2)

Comparing (2) with (1), we see that the condition of static definability coincides with the condition of rigidity. Therefore, a rigid truss without extra rods is statically determinate.

DETERMINATION OF SUPPORT REACTIONS

To determine the support reactions, we consider the equilibrium of the entire truss as a whole under the action of an arbitrary plane system of forces. We compose three equilibrium equations. After finding the support reactions, it is necessary to check.

DETERMINATION OF FORCES IN THE RODS OF THE TRUSS The forces in the stubbles of the farm can be determined in two ways: the method

cutting out nodes and the section method (Ritter method).

The knot cutting method is as follows:

the equilibrium of all truss nodes under the influence of external forces and reactions of cut rods is sequentially considered. A plane system of converging forces is applied to each node, for which two equilibrium equations can be constructed. It is advisable to start the calculation from the node where the two rods meet. In this case, one equilibrium equation of the penultimate node and two equations of the last node are verification ones.

The Ritter method is as follows:

farm to which are attached external forces, including support reactions, is cut into two parts along three rods, if possible. The number of bars cut must include the forces that need to be determined.

One of the truss parts is discarded. The action of the discarded part on the remaining one is replaced by unknown reactions.

The equilibrium of the remaining part is considered. Equilibrium equations are constructed so that each of them includes only one unknown. This is achieved by a special choice of equations: when drawing up the moment equation, the moment point is selected where the lines of action of two unknown forces intersect, which in this moment are not determined. When composing the projection equation, the projection axis is chosen perpendicular to

two parallel efforts.

When constructing equilibrium equations using both methods, it is assumed that all the rods are in tension. If the result is a minus sign, the rod is compressed.

Typical example: Determine the support reactions and forces in the truss rods if F=20 kH, P=20 kH, α=60°, Q=30 kN. (Fig. 2, 3).

We determine the support reactions by considering the equilibrium of the system as a whole (Fig. 3).

∑ X = 0:Х А –F · сos α + Q = 0;

∑ Н = 0:Y А + YВ – Р – F · sin α = 0;

∑ M A = 0:-Q · а – Р · 2а – F · sin α · 3а + F · сos α · а + YВ · 4а = 0.

Solving these equations, we find:

XA = -20 kH; YA = 9.33 kH; YВ = 28 kH.

Let's check the correctness of the results obtained. To do this, let’s compile the sum of the moments of forces relative to point C.

∑ MC = ХА · а – YA · а – Р · а – F · sin α · 2а + YВ · 3а = = (-20 – 9.33 – 20 - 20·1.73 + 28 · 3) а = 0.

Let's move on to determining the forces in the truss rods.

Knot cutting method.

We start the calculation from node A, where two rods meet.

The node whose equilibrium is being considered should be drawn (Fig. 4). Since we assume that all the rods are stretched, we direct the reactions of the rods from the node (S 1 and S 5 ). Then the forces in the rods (reactions

For node A we compose two equilibrium equations:

∑ X = 0:+X A + S5 + S1 cos 45° = 0;

∑ Y = 0:Y A + S1 cos 45° = 0.

We get: S 1 13.2 kH ;

S 5 29.32kH.

∑ X = 0:Q + S 2 + S6 · cos 45° - S1 · cos 45°= 0;

∑ Y = 0:- S 1 · cos 45° - S6 · cos 45° = 0.

When substituting the value of S1, we take into account that the force is negative.

We get: S 6 13.2 kH ;

S 2 48.7kH.

The remaining nodes are calculated similarly (Fig. 6,7).

∑ X = 0:- S 2 – S7 · cos 45° - S3 · cos 45° - F · cos α= 0;

∑ Y = 0:- S 7 · cos 45° - S3 · cos 45° - F · sin α = 0.

Hence: S 3 39.6 kH;

S 7 15.13kH.

∑ X = 0:- S 4 – S3 cos 45° = 0;

The second verification equation:

∑ Y = +Y B + S3 cos 45° = 28-39.6 0.71 =0. S4 = 28.0kH.

To check, consider the equilibrium of node E. (Fig. 8)

∑ Х = - S 5 + S4 – S6 · cos 45° + S7 · cos 45° = 0;

∑ Y = S 6 cos 45° + S7 cos 45° - P = 0.

Since the equations turned into identities, the calculation was done correctly.

Section method (Ritter method).

The Ritter method is convenient to use if you need to determine the forces not in all rods, and as a test method, since it allows you to determine each force independently of the others.

Let us determine the forces in rods 2, 6, 5. We cut the truss into two parts along rods 2, 6, 5. We discard the right part and consider the equilibrium of the left

To determine the force S5, we create an equation of moments about the point where the forces S2 and S6 intersect (point C).

∑ MC = 0: ХА · а – YA · а + S5 · a = 0;. S5 = 29.32 kH.

To determine the force S2, we create an equation of moments about point E:

∑ ME = 0:- Q · а – S2 · а – YA · 2а =0; S2 = 48.64kH.

To determine the force S6, an equation of projections on the Y axis should be drawn up:

∑ Y = 0:-S6 cos 45° + YA = 0; S6 = 13.2kH.

The results should be entered in the table. 1.

Forces in truss rods, kN

Rod number, method
	cutting
Ritter method

CALCULATION OF A TRUSS USING THE PRINCIPLE OF POSSIBLE MOVEMENTS

Principle possible movements is the basic principle analytical mechanics. He gives the most general methods solving problems of statics and allows you to determine each unknown force independently of all others, creating one equilibrium equation for it.

The principle of possible movements (Lagrange-Ostrogradsky theorem):

For balance mechanical system, subject to ideal, geometric and stationary connections, it is necessary and sufficient that the sum of the work of the active forces acting on the system is equal to zero for any possible movement of the system:

A k (a ) 0 . k 1

Landline communications- connections that are clearly independent of time.

Ideal connections are connections whose sum of reaction work on any possible movement of the system is equal to zero.

Geometric connections- connections that impose restrictions only on the coordinates of points in the system.

Active forces are forces acting on a system, except for coupling reactions.

Possible system movements

Possible movements of a mechanical system are infinitesimal movements of the system allowed by the constraints imposed on it.

The magnitudes of possible movements are indicated by symbols, for example - δ S, δφ, δХ.

Let us give examples of possible movements of systems (we will limit ourselves to consideration of flat systems):

1. The body is fixed with a fixed hinge, allowing the body to rotate around an axis passing through point O, perpendicular

drawing plane (Fig. 10).

Possible movement of the body is rotation around its axis by an angle δφ.

2. The body is secured by two movable hinges

These connections allow the body to move translationally parallel to the planes of the rollers.

Possible movement of the body - δХ.

3. The body is also secured by two movable hinges (the planes of the rollers are not parallel).

These connections allow the flat body to move only in the drawing plane. The possible movement of this body will be plane-parallel movement. And the plane-parallel movement of the body can be considered at the moment as rotational movement around an axis passing through

instantaneous body velocity center (m.c.s.) perpendicular to the drawing plane

Therefore, in order to see the possible movement of a given body, you need to know where the m.c.s. is located. this body. To construct an m.c.s., you need to know the directions of the velocities of two points of the body, draw perpendiculars to the velocities at these points, the point of intersection of the perpendiculars will be the m.c.s. bodies. In the example, we know the directions of the velocities of points A and B (they are parallel to the planes of the rollers). This means that the possible movement of this body is a rotation through an angle δφ around an axis passing through point A perpendicular to the plane of the drawing.

CONCLUSION: Since only flat systems, then in order to see the possible movement of a system consisting of flat solid bodies, it is necessary to see or construct for each rigid body

there will be a rotation around its m.c.s., or the body will move translationally if the m.c.s. absent. Possible movements of the system are determined only by the constraints imposed on the system and do not depend on the forces acting on the system. In the case of geometric and fixed connections the directions of possible movements of points in the system coincide with the directions of the velocities of these points during real movement.

Work of force on possible displacement

In the problems under consideration solids will have the ability to either move translationally or rotate around an axis perpendicular to the drawing plane. Let's write formulas to find possible work forces during such movements of bodies.

1. The body moves forward.

Then each point of the body moves by r. Consequently, the point of application of force F moves by r. Then A F r .

Special cases:

A 0.

2. The body rotates around an axis.

The work of force F is found as the elementary work of force applied to a rotating body. The body rotates through an angle δφ.

δА = Мz(F) δφ,

where Mz (F) is the moment of force F relative to the axis of rotation of the body (in our problems, the z axis is perpendicular to the plane of the drawing and finding Mz (F) is reduced to finding the moment of force F relative to the point of intersection of the axis with the plane).

δA > 0, if the force creates a moment directed in the direction of rotation of the body;.

δA< 0 , если сила создаёт момент, направленный в сторону, противоположную вращению тела.

Enter dimensions in millimeters:

X– Triangular length roof truss depends on the size of the span that needs to be covered and the method of attaching it to the walls. Wooden triangular trusses are used for spans with a length of 6000-12000 mm. When selecting a value X it is necessary to take into account the recommendations of SP 64.13330.2011 “Wooden structures” (updated edition of SNiP II-25-80).

Y– The height of a triangular truss is set by the ratio 1/5-1/6 of the length X.

Z– Thickness, W– Width of timber for making a truss. The required section of the beam depends on: loads (constant - the dead weight of the structure and roofing pie, as well as temporary ones - snow, wind), the quality of the material used, the length of the span to be covered. Detailed recommendations on the choice of beam cross-section for the manufacture of a truss, are outlined in SP 64.13330.2011 “Wooden Structures”; SP 20.13330.2011 “Loads and Impacts” should also be taken into account. Wood for load-bearing elements wooden structures must meet the requirements of grades 1, 2 and 3 according to GOST 8486-86 “Lumber coniferous species. Technical conditions".

S– Number of racks (internal vertical beams). The more racks, the higher the material consumption, weight and load-bearing capacity of the farm.

If struts for the truss are required (relevant for long trusses) and numbering of parts, mark the appropriate items.

By checking the “Black and white drawing” item, you will receive a drawing close to GOST requirements and will be able to print it without wasting colored paint or toner.

Triangular wooden trusses used mainly for roofs made of materials requiring a significant slope. Online calculator to calculate a wooden triangular truss will help determine required amount material, will make drawings of the truss indicating dimensions and numbering of parts to simplify the assembly process. Also, using this calculator you can find out the total length and volume of lumber for the roof truss.