This page shows the calculation of a wooden beam for deflection and permissible load in accordance with the requirements of the science of resistance of materials (strength).
According to the text of the article, I will try to put each aspect on the shelves as clearly as possible in simple words. When calculating the parameters, I take the calculated data of wood, based on the 3rd grade, because other varieties are very difficult to find, and, unfortunately, 90% of them are exported from the country.
The calculations take a little time and they all ultimately boil down to the calculation for the action of a bending moment (determination of the moment of resistance + permissible deflection).
Below is the main table of the dependence of the dimensions of your beam and the moment of resistance, just to which the whole calculation is reduced.
As an example for the calculation, I take the standard length of lumber - 6 meters and the step between the beams - 60 cm. (Of course, these parameters will be different for everyone)
Basic concepts:
- Beam pitch (a) - the distance between the axes (centers) of the beams;
- Beam length (L) - lumber length;
- Reference length (Loп) - the length of the part of the beam, supported on the supporting structure;
- Effective length (Lo) - the length of the beam between the centers of the bearing pads;
- Clear length (Lw) - the width of the room (from support to support).
The calculation begins with the functional purpose of the room. If our floor is living quarters, the average load temporarily created by people while living is equal to 150 kg / sq.m. or 1.5 kPa (P1). A mandatory parameter in the calculation is a reliability factor equal to - 1.2 (K1), which intentionally increases the design margin by 20%.
Now, we calculate the load from the own weight of the floor (P2). It is equal to the weight of the beams themselves + sheathing from the bottom + insulation + rough and final floors. On average, this value is the same 150 kg / m2, which we take into account. At this stage, we set a safety factor of 1.3, i.e. thirty% (K2). The factor is decent, since in the future the floor can be replaced with a heavier one or we decide to hang a heavy ceiling.
We consider the total load: Psumm = P1 * K1 + P2 * K2 = 1.5 * 1.2 + 1.5 * 1.3 = 3.75 kPa
We consider the regulatory load: Rnorm = P1 + P2 = 1.5 + 1.5 = 3 kPa
The next step, calculating the estimated length (Lo). As an example, we take the support area of the beam on the wall Lop = 120 mm, so the calculated length is:
Lo = L - 2 (Lop / 2) = L - Lop = 6 - 0.12 = 5.88 m.
Next, consider the load on the beam: Qcalculated = Ptot * a = 3.75 * 0.6 = 2.25 or 225 kg / m. (the larger the step of the beams, the higher the load on the beam)
Further, the normative load: Qnorm = Pnorm * a = 3 * 0.6 = 1.8 or 180 kg / m.
Determine the design effort:
Maximum lateral force: Q = (Qcalc * Lo) / 2 = 6.6
Maximum bending moment: M = (Qcal * Lo ^ 2) / 8 = 9.72
Above, we have identified the main components of the beam, now the calculation itself:
Bending moment action:
M / W
W is the moment of resistance of the cross section,
Ri - design resistance of wood to bending (For the 3rd grade of wood = 10 MPa.)
From the above formula, we obtain the required moment of resistance W = M / Ri,
W = 9.72 / 10 = 0.972 = 972 cc.
We return to the above plate (given at the very beginning of the article), where the values of the moments of resistance are already presented in the finished form and select the section, rounding up.
P.S. If you have a non-standard beam, then the moment of your beam can be obtained by the formula: W = (b * h ^ 2) / 6, like all the values in the given plate.
As you can see, there are a lot of cross sections that satisfy our calculation. So, we choose a beam (1056> 972) with a width of b = 110 mm. and height h = 240 mm.
When we have chosen a beam, we do a check - we calculate the permissible deflection, and if it does not satisfy us in aesthetic parameters (strong sag, despite the reliability of the structure), choose a section with a higher moment of resistance of the cross section beams.
Deflection calculation:
We calculate the moment of inertia: I = (b * h ^ 3) / 12 = 110 * 240 ^ 3/12 = 12672 cm ^ 4
Determine the deflection by the formula: f = 5/384 * (Qnorm * Lo ^ 4) / (E * I), where:
E - modulus of elasticity for wood, taken as 10,000 MPa.
So, f = 0.0130208 * (1.8 * 1195.389) / (10,000 * 12672) = 2.21 cm.
Having received the deflection (sag) along the vertical central axis - 2.21 cm, we need to compare it with the table value in terms of aesthetic and psychological parameters (see. Table E.1)
According to the table, we have vertical limit deflections L / xxx. To compare our value with this characteristic, you need to get the parameter of the maximum permissible values, therefore we divide the calculated length by the deflection Lo / f = 5.88 / 2.21 = 266. This parameter is inversely proportional to the length, so it should be higher, not lower, than the tabular one.
Since we used a 6 m long beam in the calculation, we find the corresponding row and its value in table E1:
The parameter we received 266 < 200 (less than the tabular), therefore, the deflection of our beam will be less, since it fits freely into the condition.
Selected beam - goes through all calculations! That's all! Please use it!
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Further, a series of materials is planned for the canal on ways to eliminate the deflection of beams without supports and columns.
Also in the following articles I will describe the calculations of channels and I-beams. Let's talk about wide-flange I-beams, where and what types are more optimal to use, reducing the height of the floors and increasing the strength.
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