A beam is an element in engineering, which is a rod, which is loaded by forces acting in the direction perpendicular to the rod. The activities of engineers often include the need to calculate the deflection of the beam under load. This action is performed in order to limit the maximum deflection of the beam.
Types
Today, in construction, beams made of different materials can be used. It can be metal or wood. Each specific case implies different beams. In this case, the calculation of beams for deflection may have some differences that arise on the basis of the difference in structure and materials used.
Wooden beams
Today's individual construction involves the widespread use of beams made of wood. Almost every building contains wooden floors. Beams made of wood can be used as load-bearing elements, they are used in the manufacture of floors, and also as supports for floors between floors.
It's no secret that wooden, like steel beam, has the ability to bend under the influence of load forces. The deflection arrow depends on what material is used, the geometric characteristics of the structure in which the beam is used, and the nature of the loads.
The permissible deflection of the beam is formed from two factors:
- Correspondence of deflection and permissible values.
- Possibility of building operation taking into account deflection.
The calculations for strength and stiffness carried out during construction allow the most efficient assessment of what loads the building can withstand during operation. Also, these calculations make it possible to find out exactly what the deformation of structural elements will be in each particular case. Perhaps no one will argue that the detailed and most accurate calculations are part of the responsibilities of civil engineers, however, using several formulas and the skill of mathematical calculations, you can calculate all the necessary quantities yourself.
In order to make the correct calculation of the beam deflection, it is also necessary to take into account the fact that in construction the concepts of rigidity and strength are inextricable. Based on the strength calculation data, we can proceed with further calculations regarding stiffness. It is worth noting that the calculation of the deflection of the beam is one of the indispensable elements of the calculation of stiffness.
Please note that to conduct such calculations yourself it is best to use the enlarged calculations, resorting to fairly simple schemes. It is also recommended to make a small margin up. Especially if the calculation concerns the supporting elements.
Calculation of beams for deflection. Work algorithm
In fact, the algorithm by which such a calculation is made is quite simple. As an example, consider a slightly simplified calculation scheme, while omitting some specific terms and formulas. In order to calculate the beams for deflection, it is necessary to perform a series of actions in a certain order. The calculation algorithm is as follows:
- A design scheme is drawn up.
- The geometric characteristics of the beam are determined.
- The maximum load on this element is calculated.
- In case of need, the strength of the timber in bending moment is checked.
- The maximum deflection is calculated.
As you can see, all the actions are quite simple and quite doable.
Drawing up the design scheme of the beam
In order to draw up a design scheme, does not require much knowledge. To do this, it is enough to know the size and cross-sectional shape of the element, the span between the supports and the method of support. The span is the distance between the two supports. For example, you use beams as supporting floor beams for the supporting walls of the house, between which 4 m, then the span will be 4 m.
When calculating the deflection of a wooden beam, they are considered freely supported structural elements. In the case of a floor beam , a scheme with a load that is evenly distributed is adopted for calculation. It is indicated by q. If the load is concentrated, then a circuit with a concentrated load, denoted by F., is taken. The value of this load is equal to the weight that will exert pressure on the structure.
Moment of inertia
The geometric characteristic, called the moment of inertia, is important when calculating the beam deflection. The formula allows you to calculate this value, we will give it a little lower.
When calculating the moment of inertia, it is necessary to pay attention to the fact that the size of this characteristic depends on what is the orientation of the element in space. In this case, an inversely proportional relationship between the moment of inertia and the magnitude of the deflection is observed. The smaller the value of the moment of inertia, the greater the value of the deflection and vice versa. This dependence is quite easy to track in practice. Everyone knows that a board laid on an edge bends much less than a similar board in a normal position.
The calculation of the moment of inertia for a beam with a rectangular section is performed according to the formula:
J = b * h ^ 3/12, where:
b is the width of the section;
h is the beam height.
Maximum Load Calculation
Determination of the maximum load on a structural element is made taking into account a number of factors and indicators. Usually, when calculating the load level, the weight of 1 linear meter of the beam, the weight of 1 square meter of the floor, the temporary load of the floor and the load from the partitions per 1 square meter of floor are taken into account. The distance between the beams measured in meters is also taken into account. For an example of calculating the maximum load on a wooden beam, we take the averaged values according to which the weight of the floor is 60 kg / m², the temporary load on the floor is 250 kg / m², the partitions will weigh 75 kg / m². The weight of the beam itself is very simple to calculate, knowing its volume and density. Suppose that a wooden beam with a cross section of 0.15x0.2 m is used. In this case, its weight will be 18 kg / linear m. Also, for example, we take the distance between the floor beams equal to 600 mm. In this case, the coefficient we need will be 0.6.
As a result of calculating the maximum load, we obtain the following result: q = (60 + 250 + 75) * 0.6 + 18 = 249 kg / m.
When the value is obtained, we can proceed to the calculation of the maximum deflection.
Calculation of the maximum deflection value
When the beam is calculated, the formula displays all the necessary elements. It should be borne in mind that the formula used for calculations may have a slightly different look if the calculation is carried out for different types of loads that will affect the beam.
First, we bring to your attention the formula used to calculate the maximum deflection of a wooden beam with a distributed load.
f = -5 * q * l ^ 4/384 * E * J.
Note that in this formula, E is a constant value, which is called the modulus of elasticity of the material. For wood, this value is 100,000 kgf / m².
Continuing the calculations with our data used as an example, we get that for a timber beam with a cross section of 0.15x0.2 m and a length of 4 m, the maximum deflection when exposed to a distributed load is 0.83 cm.
Please note that when the calculation of the deflection is carried out taking into account the scheme with a concentrated load, the formula takes the following form:
f = -F * l ^ 3/48 * E * J, where:
F - pressure force on the beam.
We also draw attention to the fact that the value of the elastic modulus used in the calculations may vary for different types of wood. The influence is exerted not only by the species of wood, but also by the type of timber. Therefore, a solid beam of wood, glued beams or logs will have different moduli of elasticity, and therefore different values of maximum deflection.
You can pursue different goals by calculating the beams for deflection. If you want to know the limits of deformation of structural elements, then at the end of the calculation of the deflection arrows, you can stop. If your goal is to establish the level of compliance of the found indicators with building standards, then they need to be compared with the data that are placed in special documents of a regulatory nature.
I-beam
Please note that I-beam beams are used less frequently due to their shape. However, one should not forget that such a structural element can withstand much greater loads than a corner or channel, an alternative to which can be an I-beam.
Calculation of the deflection of an I-beam should be done if you intend to use it as a powerful structural element.
We also draw your attention to the fact that it is not possible to calculate the deflection for all types of I-beam beams. In what cases is it allowed to calculate the deflection of an I-beam? There are 6 such cases, which correspond to six types of I-beams. These types are as follows:
- Single-span beam with uniformly distributed load.
- Console with rigid termination at one end and evenly distributed load.
- A beam from one span with a console on one side, to which a uniformly distributed load is applied.
- Single span beam with articulated bearing type with concentrated force.
- Single-span articulated beam with two concentrated forces.
- Console with tight fit and concentrated force.
Metal beams
The calculation of maximum deflection is the same, whether it is a steel beam or an element of another material. The main thing is to remember those quantities that are specific and constant, such as the elastic modulus of a material. When working with metal beams, it is important to remember that they can be made of steel or I-beams.
The deflection of a metal beam made of steel is calculated taking into account that the constant E in this case is 2 · 105 MPa. All other elements, such as the moment of inertia, are calculated according to the algorithms described above.
Calculation of the maximum deflection for a beam with two supports
As an example, we consider a scheme in which the beam is on two supports, and a concentrated force is applied to it at an arbitrary point. Before the application of force, the beam was a straight line, however, under the influence of force, it changed its appearance and, due to deformation, became a curve.
Suppose that the XY plane is the plane of symmetry of the beam on two supports. All loads act on the beam in this plane. In this case, it will be a fact that the curve obtained as a result of the force will also be in this plane. This curve is called the elastic line of the beam or the deflection line of the beam. Algebraically solve the elastic line of the beam and calculate the deflection of the beam, the formula of which will be constant for beams with two supports, as follows.
Deflection at a distance z from the left beam support at 0 ≤ z ≤ a
F (z) = (P * a 2 * b 2 ) / (6E * J * l) * (2 * z / a + z / bz 3 / a 2 * b)
Beam deflection on two supports at a distance z from the left support for a ≤ z ≤l
f (z) = (- P * a 2 * b 2 ) / (6E * J * l) * (2 * (lz) / b + (lz) / a- (lz) 3 / a + b 2 ), where P is the applied force, E is the elastic modulus of the material, J is the axial moment of inertia.
In the case of a beam with two supports, the moment of inertia is calculated as follows:
J = b 1 h 1 3/12, where b 1 and h 1 are the values of the width and height of the section of the beam used, respectively.
Conclusion
In conclusion, we can conclude that it is quite simple to independently calculate the maximum deflection of a beam of different types. As shown in this article, the main thing is to know some characteristics that depend on the material and its geometric characteristics, as well as perform calculations using several formulas in which each parameter has its own explanation and is not taken out of nowhere.