Calculation and manufacture of a metal truss for a canopy

Recommendation points

• General calculation methodology
• Determination of combined actions and support reactions
• Differential effort calculation
• Determination of the section of elements
• Manufacturing of parts for the farm
• Assembly on hardware or welding?

The calculation of steel structures has become a stumbling block for many builders. Using the example of the simplest trusses for an outdoor shed, we will tell you how to correctly calculate the loads, and also share simple methods of self-assembly without using expensive equipment.

General calculation methodology

Trusses are used where it is impractical to use a solid bearing beam. These structures are characterized by a lower spatial density, while maintaining stability to perceive influences without deformation due to the correct arrangement of parts.

Structurally, the truss consists of an outer chord and filling elements. The essence of the operation of such a lattice is quite simple: since each horizontal (conditionally) element cannot withstand the full load due to an insufficiently large section, two elements are located on the axis of the main impact (gravity) in such a way that the distance between them provides a sufficiently large cross section of the entire structure … It can be explained even more simply as follows: from the point of view of the perception of loads, the truss is considered as if it is made of solid material, while the filling provides sufficient strength, based only on the calculated applied weight.

The structure of the truss from a shaped pipe: 1 – lower belt; 2 – braces; 3 – racks; 4 – side belt; 5 – upper belt

This approach is extremely simple and often more than enough for the construction of simple metal structures, however, the material consumption is extremely high with a rough calculation. A more detailed consideration of the existing impacts helps to reduce metal consumption by 2 or more times, this approach will be the most useful for our task – to design a light and fairly rigid truss, and then assemble it.

The main profiles of trusses for the canopy: 1 – trapezoidal; 2 – with parallel belts; 3 – triangular; 4 – arched

You start by defining the overall configuration for your farm. It usually has a triangular or trapezoidal profile. The lower element of the belt is placed mainly horizontally, the upper one – at an angle, which ensures the correct slope of the roofing system. In this case, the cross-section and strength of the chord elements should be chosen close to such that the structure can support its own weight with the existing support system. Next, you add vertical bridges and oblique ties in any amount. The structure must be displayed on a sketch to visualize the mechanics of interaction, indicating the real dimensions of all elements. Then Her Majesty Physicist comes into play.

Determination of combined actions and support reactions

From the statics section of the school mechanics course, we will take two key equations: the balance of forces and moments. We will use them to calculate the response of the supports on which the beam is placed. For simplicity of calculations, the supports will be considered articulated, that is, they do not have rigid connections (embedments) at the point of contact with the beam.

An example of a metal farm: 1 – farm; 2 – lathing beams; 3 – roofing

On the sketch, you must first mark the pitch of the roofing system, because it is in these places that the points of concentration of the applied load should be located. Usually, it is at the points of application of the load that the nodes of convergence of the braces are located, so it is easier to calculate the load. Knowing the total weight of the roof and the number of trusses in the canopy, it is not difficult to calculate the load on one truss, and the factor of uniformity of the coverage will determine whether the applied forces at the points of concentration are equal, or they will differ. The latter, by the way, is possible if in a certain part of the canopy one coating material is replaced by another, there is a gangway or, for example, an area with an unevenly distributed snow load. Also, the effect on different points of the truss will be uneven if its upper beam has a rounding, in this case the points of application of the force must be connected by segments and the arc should be considered as a broken line.

When all the acting forces are marked on the truss sketch, we proceed to calculating the support reaction. With respect to each of them, the farm can be represented as nothing more than a lever with the corresponding sum of influences on it. To calculate the moment of force at the fulcrum, you need to multiply the load at each point in kilograms by the length of the arm of the application of this load in meters. The first equation says that the sum of the actions at each point is equal to the reaction of the support:

• 200 1.5 + 200 3 + 200 4.5 + 100 6 = R₂ 6 – the equation of equilibrium of moments relative to the node and, where 6 m is the shoulder length)
• R₂ = (200 1.5 + 200 3 + 200 4.5 + 100 6) / 6 = 400 kg

The second equation determines equilibrium: the sum of the reactions of the two supports will be exactly equal to the applied weight, that is, knowing the reaction of one support, you can easily find the value for the other:

• R₁ + R₂ = 100 + 200 + 200 + 200 + 100
• R1 = 800 – 400 = 400 kg

But make no mistake: the leverage rule also applies here, so if the truss has a significant extension beyond one of the supports, then the load in this place will be higher in proportion to the difference in the distances from the center of mass to the supports.

Differential effort calculation

We pass from the general to the particular: now it is necessary to establish the quantitative value of the efforts acting on each element of the farm. To do this, we list each belt segment and filling inserts with a list, then we consider each of them as a balanced flat system.

For the convenience of calculations, each connecting node of the truss can be represented as a vector diagram, where the action vectors run along the longitudinal axes of the elements. All that is needed for calculations is to know the length of the segments converging at the node and the angles between them..

You need to start from the node for which the maximum possible number of known quantities was established during the calculation of the support reaction. Let’s start with the extreme vertical element: the equilibrium equation for it says that the sum of the vectors of converging loads is equal to zero, respectively, the counteraction to the force of gravity acting along the vertical axis is equivalent to the reaction of the support, equal in magnitude, but opposite in sign. Note that the obtained value is only a part of the overall reaction of the support acting for a given node, the rest of the load will fall on the horizontal parts of the belt.

Knot b

• -100 + S₁ = 0
• S₁ = 100 kg

Next, we move on to the extreme lower corner node, in which the vertical and horizontal segments of the chord converge, as well as the inclined brace. The force acting on the vertical segment, calculated in the previous paragraph, is the pressing weight and the reaction of the support. The force acting on the inclined element is calculated from the projection of the axis of this element onto the vertical axis: subtract the action of gravity from the support reaction, then divide the “pure” result by the sin of the angle at which the brace is inclined to the horizontal. The load on a horizontal element is also found by projection, but already on the horizontal axis. We multiply the newly obtained load on the inclined element by the cos of the angle of inclination of the brace and obtain the value of the impact on the extreme horizontal segment of the chord.

Knot a

• -100 + 400 – sin (33.69) S₃ = 0 – equilibrium equation per axis at
• S₃ = 300 / sin (33.69) = 540.83 kg – rod 3compressed
• -S₃ Cos (33.69) + S₄ = 0 – equilibrium equation per axis x
• S₄ = 540.83 cos (33.69) = 450 kg – rod 4stretched

Thus, successively passing from node to node, it is necessary to calculate the forces acting in each of them. Note that counter-directed action vectors compress the bar and vice versa – stretch it if directed oppositely from each other.

Determination of the section of elements

When all the acting loads are known for the truss, it is time to determine the section of the elements. It does not have to be equal for all parts: the belt is traditionally made of rolled products with a larger section than the filling parts. This ensures a safety margin of the design.

Where: F_tr – cross-sectional area of ​​the stretched part; N – effort from the design loads; R_y – design material resistance; ?_from – coefficient of working conditions.

If everything is relatively simple with breaking loads for steel parts, then the calculation of compressed bars is performed not for strength, but for stability, since the final result is quantitatively less and, accordingly, is considered a critical value. It can be calculated using an online calculator, or it can be done manually, having previously determined the length reduction factor, which determines at what part of the total length the rod is capable of bending. This coefficient depends on the method of fastening the edges of the bar: for butt welding it is a unit, and in the presence of “ideally” rigid gussets it can approach 0.5.

Where: F_tr – cross-sectional area of ​​the compressed part; N – effort from the design loads; ? – coefficient of buckling of compressed members (determined from the table); R_y – design material resistance; ?_from – coefficient of working conditions.

You also need to know the minimum radius of gyration, defined as the square root of the quotient of dividing the axial moment of inertia by the cross-sectional area. The axial moment is determined by the shape and symmetry of the section, it is better to take this value from the table.

Where: i_x – radius of inertia of the section; J_x – axial moment of inertia; F_tr – cross-sectional area.

Thus, if you divide the length (taking into account the coefficient of reduction) by the minimum radius of gyration, you can get a quantitative value of the flexibility. For a stable bar, the condition is met that the quotient from dividing the load by the cross-sectional area should not be less than the product of the permissible compressive load and the buckling coefficient, which is determined by the value of the flexibility of a particular bar and the material of its manufacture.

Where: l_x – estimated length in the plane of the truss; i_x – the minimum radius of inertia of the section along the x axis; l_y – estimated length from the plane of the truss; i_y – the minimum radius of gyration of the section along the y-axis.

Please note that it is in the compressed bar stability analysis that the whole essence of the truss operation is displayed. If the section of the element is insufficient, which does not allow ensuring its stability, we have the right to add thinner connections by changing the fastening system. This complicates the configuration of the truss, but allows for greater stability with less weight..

Manufacturing of parts for the farm

The accuracy of the assembly of the truss is extremely important, because we carried out all the calculations by the method of vector diagrams, and the vector, as you know, can only be absolutely straight. Therefore, the slightest stresses arising from curvatures due to improper fit of the elements will make the truss extremely unstable..

First you need to decide on the dimensions of the parts of the outer belt. If everything is quite simple with the lower beam, then to find the length of the upper one, you can use either the Pythagorean theorem or the trigonometric ratio of sides and angles. The latter is preferable when working with materials such as angle steel and shaped tube. If the angle of the truss slope is known, it can be made as a correction when trimming the edges of parts. The right angles of the belt are connected by trimming at 45 °, inclined – by adding to the 45 ° the angle of inclination on one side of the joint and subtracting it from the other.

The filling details are cut out by analogy with the belt elements. The main catch is that the farm is a strictly unified product, and therefore accurate detailing is required for its manufacture. As in the calculation of actions, each element must be considered individually, determining the angles of convergence and, accordingly, the angles of undercut edges..

Farms are often made with radius ones. Such structures have a more complex calculation method, but greater structural strength, due to a more uniform perception of loads. It makes no sense to make filler elements with rounded elements, but for belt parts it is quite applicable. Typically, arched trusses consist of several segments that are connected at the points of convergence of filling braces, which must be taken into account when designing.

Assembly on hardware or welding?

In conclusion, it would be nice to outline the practical difference between the methods of assembling a truss by welding and using detachable joints. To begin with, drilling holes for bolts or rivets in the body of an element practically does not affect its flexibility, and therefore in practice is not taken into account.

When it came to the method of fastening the elements of the truss, we found that in the presence of gussets, the length of the section of the rod that is capable of bending is significantly reduced, due to which its cross-section can be reduced. This is the advantage of assembling the truss on gussets, which are attached to the side of the truss elements. In this case, there is no particular difference in the assembly method: the length of the welds will be guaranteed to be sufficient to withstand the concentrated stresses in the nodes..

If the truss is assembled by joining elements without gussets, special skills are needed here. The strength of the entire truss is determined by its least strong unit, and therefore a marriage in welding at least one of the elements can lead to the destruction of the entire structure. If you do not have enough welding skills, it is recommended to assemble with bolts or rivets using clamps, angle brackets or cover plates. In this case, the fastening of each element to the node must be carried out at least at two points.