Inertial moment and torsion constant reduction of cracked sections

Inertial moment  and torsion constant reduction of cracked sections

The concept of the need to reduce the inertia moment  :

When the structural elements are loaded with vertical and horizontal loads and other loads, they have a curvature towards the load Since the sections of these elements have a certain hardness, at the beginning of the load is in the phase of the elastic deformation and then increased loads enter the plastic field ie cracking, and thus less effective height of the section (calculated on the basis of inertia) It is less resistant to increase deformations in the plastic phase, and therefore this plastic behavior (cracks) should be included in the structural calculations of the structure according to the requirements of the code and the theories of structural analysis known.

In the theory of elasticity, linear analysis of structural elements can be carried out in the elastic phase (ideal - without cracks), but taking into account the factors of reducing the inertia of the elements where cracks are expected to occur as a result of horizontal and vertical loads or by the impact of indirect acts such as heat, shrinkage and support cushions, etc. Investment during the design life of the structur.

With the effect of cracks, the effective depth of the structural section decreases and the inertial moment  of section (I) decreases and therefore the section 's rigidity (EI) is reduced due to cracking, which increases the value of the vertical deflections and horizontal Displacements. Therefore, we reduce the inertia (I) to approach the actual reality of the actual behavior of the constructional sections.

In non-linear analysis, cracks are automatically taken into account during the analysis and with great accuracy. This can be done with the help of large structural programs.

It can be concluded that linear and non-linear solutions are usually close together, depending on the type of structural element and its importance and the regularity and importance of origin and the type and value of loads applied, which controls the type of analysis is the cost and safety.The design considerations of an ordinary building differ from that of a very high building, and the experience of the structural engineer is of great importance.

Reducing inertia according to the American code:

According to ACI-11- 8.7.1, the hardness of the elements (EI & GJ) should be determined to reflect the degree of cracking of the elements and the non-linear action (inelastic action) that occurs to the elements before they reach the yielding stage. Therefore, when conducting a linear analysis of the structure, the code defines the coefficients of the hardness reduction of the structural elements.

When performing linear analysis of structure, the following characteristics  should be considered:

A) - Modulus of elasticity is used for concrete according to the following relationship:

                                            Ec = 4700*(f`c)^0.5

B)-Reducing inertia (I), as follows:

    - Elements subject to pressure forces mainly:

0.70 Ig	الاعمدة                                   Columns
0.70 Ig	جدران غير متشققة        Walls—Uncracked
0.35 Ig	- متشققة                        —Cracked

- Elements subject to  bending  :

0.35 Ig	الجوائز                                                Beams
0.25 Ig	البلاطات والبلاطات المسطحة Flat plates and flat slabs

C) - the total area of ​​the section taken in all previous calculations (Area = 0.1 Ag).

According to ACI-11- 10.10.4.2  when Lateral  loads (earthquakes or wind) are present or included in the analysis. The previous numbers of the pressure-prone elements are reduced again by the value (Bds + 1), where ( Bds) represents the ratio of shear due to constant loads (DL) to shear resulting from the sum of loads for the same design load structure and on the same floor, and ( βds) must be less than one.

If (βds) is calculated, the columns and shear walls are reduced in structural structures by taking values ​​within the range (0.6 → 0.8) depending on the type and value of loads, seismic zone, elemental spans, type of structure (uniformity) and other factors. Even beames can also be reduced by values ​​within the range (0.4 → 0.6) according to the previous indicators.

In order to perform the analysis using the service loads, the value of the inertia  moment (I) of the sections must be enlarged, ie, the increase of the previous reduction coefficients by 1.0 / 0.70 = 1.43 . Because the previous reduction figures are for Factored loads .

In the case of T-Beams, either the effective width of the wing is calculated, or, alternatively, the value of the inertial moment can be considered equal to twice the moment of the section without wings, ie, in both cases it is reduced by the same value for rectangular beames as Already. An approximate solution can be made by neglecting the work of the wings in the resistance, in which case the inertia moment of its inertia moment (as in the Arab code) will not be reduced.

Inserting the effect of cracks in the iTabs program:

In Etabs, consideration should be given to reducing the inertia moment for investment loads only. So the statice analises in Etabs depend on matrix method which depend on elastic method  (  F = K * Δ )  , Therefore, the introduction of reduction coefficients for Factored loads will give the program incorrect results and thus conclusions are reached Next   :

1- The inertial reduction factors are introduced for investment loads, ie the equations in the above code ACI are enlarged to 1.43 to read as follows:

    - Elements subject to pressure forces mainly:

1.0 Ig	الاعمدة                                   Columns
1.0 Ig	جدران غير متشققة        Walls—Uncracked
0. 5 Ig	- متشققة                        —Cracked

- Elements subject to bending :

0. 5 Ig	الجوائز                                                Beams
0.357 Ig	البلاطات والبلاطات المسطحة     Flat plates and flat slabs

When there are lateral loads (earthquakes or winds in the analysis), the previous figures are reduced again for the elements subjected to pressure by (Bds + 1), where βds represents the ratio of shear due to constant loads (DL) to shear resulting from the sum of loads for the same design load structure. With the same thoughtful floor, βds must be less than one.

In the case of T-Beams, for simplicity, they are analyzed without taking into account the wings' resistance, and then the impact of the cracks can be neglected .  case the inertia moment will not be reduced of T-sections and we considere them as rectangular its section  (  bw * h) .

Instead of calculating the value of (Bds) and the fact that the values ​​are very close to the values ​​in the table and the code FEMA_356 and the code ACI. The values ​​in FEMA can be directly adopted. Also, the cracked walls in weak and medium intensity seismic zones can be reduced to ( 0.5 → 0.7) .

Important note :

The ETABS program does not take T-Beams into consideration unless it is defined as Type T .