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 .
