A kosour in a staircase is called an inclined metal beam on which the steps rest.
This calculation concerns metal stringers from rolled channels.
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Initial data.
The width of the flight of stairs is 1.05 m (prefabricated stairs LS11, weight of 1 step is 105 kg). The number of stringers - 2. H \u003d 1.65 m - half the height of the floor; l 1 \u003d 3.7 m - the length of the stringer. Stringer angle α = 27°, cosα = 0.892.
Collection of loads.
As a result, the current standard load on the inclined stringer is q 1 n \u003d 449 kg / m 2, and the calculated q 1 p \u003d 584 kg / m 2.
Calculation (selection of the section of the stringer).
The first thing to do in this calculation is to bring the load per 1 sq. m of the march area to the horizontal and find the horizontal projection of the kosour. Those. in fact, with the actual length of the kosour l 1 and the load per 1 sq.m of the march q 1, we translate these values into the horizontal plane through cosα so that the relationship between q and l remained in effect.
For this we have two formulas:
1) the load per 1 m 2 of the horizontal projection of the march is:
q = q 1 / cos 2 α;
2) the horizontal projection of the march is:
l = l 1 cosα.
Please note that the steeper the angle of inclination of the kosour, the shorter the length of the march projection, but the greater the load per 1 m 2 of this horizontal projection. This just preserves the dependence between q and l to which we aspire.
As proof, consider two stringers of the same length 3 m with the same load of 600 kg / m 2, but the first is located at an angle of 60 degrees, and the second - 30. It can be seen from the figure that for these stringers the projections of the load and the length of the stringer are very different from each other, but the bending moment is the same for both cases.
Let us determine the normative and calculated value of q, as well as l for our example:
q n \u003d q n 1 / cos 2 α \u003d 449 / 0.892 2 \u003d 564 kg / m 2 \u003d 0.0564 kg / cm 2;
q p \u003d q p 1 / cos 2 α \u003d 584 / 0.892 2 \u003d 734 kg / m 2 \u003d 0.0734 kg / cm 2;
l = l 1 cosα \u003d 3.7 * 0.892 \u003d 3.3 m.
In order to select the cross section of the stringer, it is necessary to determine its moment of resistance W and moment of inertia I.
The moment of resistance is found by the formula W \u003d q p a l 2 /(2*8mR), where
q p \u003d 0.0734 kg / cm 2;
l\u003d 3.3 m \u003d 330 cm - the length of the horizontal projection of the stringer;
m = 0.9 is the coefficient of operating conditions of the stringer;
R \u003d 2100 kg / cm 2 - design resistance of steel grade St3;
8 - part of the notorious formula for determining the bending moment (M \u003d ql 2 / 8).
So, W \u003d 0.0734 * 105 * 330 2 / (2 * 8 * 0.9 * 2100) \u003d 27.8 cm 3.
The moment of inertia is found by the formula I \u003d 150 * 5 * aq n l 3 /(384*2Ecos?) , where
E \u003d 2100000 kg / cm 2 - the modulus of elasticity of steel;
150 - from the condition of maximum deflection f = l/150;
a \u003d 1.05 m \u003d 105 cm - march width;
2 - the number of stringers in the march;
5/348 is a dimensionless coefficient.
For those who want to understand in more detail the definition of the moment of inertia, let's turn to Linovich and derive the above formula (it is somewhat different from the original source, but the result of the calculations will be the same).
The moment of inertia can be determined from the formula for the allowable relative deflection of the element. The deflection of the stringer is calculated by the formula: f = 5q l 4 /348EI, whence I = 5q l 4/348Ef.
In our case:
q \u003d aq n 1 / 2 \u003d aq n cos 2 ? / 2 - distributed load on the stringer from half the march (in the comments they often ask why the kosour is considered for the entire load from the march, and not for half - and so, the deuce in this formula just gives half the load);
l 4 = l 1 4 = (l/cos?) 4 = l 4 / cos? 4 ;
f= l 1 /150 = l/150cos? - relative deflection (according to DSTU "Deflections and displacements" for a span of 3 m).
Plugging everything into the formula, we get:
I \u003d 150 * cos? * 5aq n cos 2 ? l 4 / (348 * 2E l cos 4 ?) = 150*5*aq n l 3 /(348*2Ecos?).
Linovich has, in fact, the same thing, only all the numbers in the formula are reduced to the "coefficient With, depending on the deflection. But since in modern standards the requirements for deflections are more stringent (we need to limit ourselves to 1/150 instead of 1/200), then for ease of understanding, all numbers are left in the formula, without any abbreviations.
So, I \u003d 150 * 5 * 105 * 0.0564 * 330 3 / (384 * 2 * 2100000 * 0.892) \u003d 110.9 cm 4.
We select a rolling element from the table below. Channel number 10 suits us.
This calculation was made according to the recommendations of the book Linovich L.E. "Calculation and design of parts of civil buildings" and provides only for the selection of the section of a metal element. For those who want to understand in more detail the calculation of the metal stringer, as well as the design of the elements of the stairs, you must refer to the following regulatory documents:
SNiP III-18-75 "Metal structures";
DBN V.2.6-163:2010 "Steel structures".
In addition to calculating the kosour using the above formulas, you also need to calculate the fluctuation. What it is? Kosour can be strong and reliable, but when walking up the stairs, it seems that she shudders with every step. The feeling is not pleasant, so the standards provide for the following condition: if you load the stringer with a concentrated load of 100 kg in the middle of the span, it should bend no more than 0.7 mm (see DSTU B.V.1.2-3: 2006, table 1, item 4).
The table below shows the results of the calculation for fluctuation for stairs with steps 300x150 (h), this is the most convenient size of steps for a person, with different floor heights, and hence different lengths of the stringer. As a result, even if the above calculation gives a smaller section of the element, you need to finally select the kosour by checking the data in the table.
|
March projection length Lx, m |
March height H, m |
Stringer length L, m |
Rolled channel number GOST 8240-97, DSTU 3436-96 |
Number of bent channel GOST 8278-83 |
I-beam number GOST 8239-89 |
Dimensions of bent square pipe GOST 30245-94, DSTU B.V.2-6-8-95 |
In order to properly design the stairs, you can use the typical series:
1.450-1 "Stairs from prefabricated reinforced concrete steps on steel stringers";
1.450-3 "Steel stairs, platforms, ladders and railings".
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