1 ENCV600103 Peranc. Struktur Beton 2: Two-way slab: Equivalent Frame Sjahril A. Rahim Departemen Teknik Sipil FTUI 2013 Equivalent Frame: • The EFM of analysis for gravity loading converts a three-dimensional frame system with two-way slab into a series of two-dimensional frame (slab-beams and columns), with each frame extending the full height of the building, as illustrated in Figure 1. • The width of the equivalent frame extends to mid-span between column centerlines. • The complete analysis of the system consists of analyzing a series of equivalent interior and exterior frames spanning longitudinally and transversally through the building. • For gravity loading, the slab-beam of each floor or roof (level) may be analyzed separately, with far ends of attached columns considered fixed. • For lateral load analysis, the stiffness of frame members are modified to account for cracking and other relevant factors.
Transcript
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ENCV600103 Peranc. Struktur Beton 2:Two-way slab: Equivalent Frame
Sjahril A. Rahim
Departemen Teknik Sipil FTUI
2013
Equivalent Frame:
• The EFM of analysis for gravity loading converts a three-dimensional frame system with two-way slab into a series of two-dimensional frame (slab-beams and columns), with each frame extending the full height of the building, as illustrated in Figure 1.
• The width of the equivalent frame extends to mid-span between column centerlines.
• The complete analysis of the system consists of analyzing a series of equivalent interior and exterior frames spanning longitudinally and transversally through the building.
• For gravity loading, the slab-beam of each floor or roof (level) may be analyzed separately, with far ends of attached columns considered fixed.
• For lateral load analysis, the stiffness of frame members are modified to account for cracking and other relevant factors.
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Equivalent Frame:
Design strip of Equivalent Frame:
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Preliminary Design:
• Preliminary slab thickness h needs to be determined for control of deflection, according to minimum slab thickness requirement of Section 11.5 SNI (Table 8).
• For slab systems without beams, its advisable to check the shear strength of the slab in the vicinity of columns or other support locations.
Members of Equivalent Frame:
• Slab-beam
• Torsional members (horizontal members)
• Columns (vertical members)
Equivalent Column
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Members of Equivalent Frame:
Slab-Beams:
Equivalent Slab-Beam Stiffness diagram
Equivalent Slab-Beam Stiffness diagram
Slab system without beams
Slab system with drop
panel
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Slab-Beams:
Slab system with column
capitals
Slab system with
beams
Equivalent Slab-Beam Stiffness diagram
Equivalent Slab-Beam Stiffness diagram
Columns:
Slab system without beams
Slab system with column
capitals
Slab system with drop
panel
Slab system with
beams
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Torsional members: The stiffness of torsional members:
3//63.01
/1
9
3
3222
yxyxC
lcl
CEK cs
t
x= the shorter dimension of rectangular part
y= the longer dimension of rectangular part
If beams frame into the support in direction moments are being determined, the torsional stiffness Kt needs to increased as follows:
ssbtta IIKK /
Torsional members:
If beams frame into the support in direction moments are being determined, the torsional stiffness Kt needs to increased as follows:
12/
/3
2hlI
IIKK
s
ssbtta
Isb=moment of inertia of the slab section specified for Is including that portion of the beam stem extending above and below the slab.
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Cross Sectional Constants C of Torsional members:
Equivalent Columns:
A single-stiffness element consisting of the actual columns above and below the slab-beams plus an attached transverse torsional members into single-stiffness.The flexural stiffness of the equivalent column Kec is given in the term of its inverse, or flexibility, as follows:
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Equivalent Columns:
Arrangement of Live Load:
Case 1: when L ¾ D, only loading pattern (1) with full factored live load on all spans need be analyzed for negative and positive factored moments.
Case 2: when L ¾ D, the five loading patterns shown need to be analyzed to determined all factored moments in the slab-beam members.
Note: D= service dead loadL= service live load
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Partial Frame Analysis for Vertical LoadingLoading
pattern for L3/4D
Loading pattern for
L3/4D
Factor Moments:
Moment distribution are used for analyzing partial frames involving severalcontinuous span with the far ends of upper and lower columns fixed.
(1) The use of the equivalent column concept to determined joint distribution factors;
(2) The proper procedure to distribute the equivalent column moment obtained in the frame analysis to the actual column above and below the slab-beam joint.
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Moment Distribution Factor:
Moment Distribution Factor:
Equivalent column stiffness:
)()(
))((
ttcbct
ttcbctec
tc
tcec
KKKK
KKKKK
KK
KKK
Slab-beam distribution factor:
ecbb
b
ecbb
b
KKK
KspanDF
KKK
KspanDF
21
2
21
1
)32(
)12(
Equivalent column distribution factor (unbalanced moment from slab-beam):
ecbb
ec
KKK
KDF
21
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Moment Distribution Factor:
Portion of unbalanced moment to upper column:
ctcb
cb
KK
KDF
Portion of unbalanced moment to lower column:
ctcb
ct
KK
KDF
The actual column are then designed for these moments.
Critical Section for Negative Factor Moments:
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Moment Redistribution:
M0
Slab system that meets the limitations of the DDM, analyzed with EFM, the factored moment may be reduced so that the total static moment need not exceed M0 computed by Eqs 13.3.
Permissible reduction
Eqs 13.3.
Factor Moments in Column Strips:
Definition:scs
bcb
IE
IE Ratio of flexural stiffness of beam section to
flexural stiffness of a width of slab bounded laterally by centerlines of adjacent panels (if any) on each side of the beam.
1 = in direction of l1
2 = in direction of l2
scs
cbt IE
CE
2 Ratio of torsional stiffness of edge beam section
to flexural stiffness of a width of slab equal to span length of beam, center-to-center of supports.
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Factor Moments in Column Strips:
457590(1l2/l1)1.0
757575(1l2/l1)=0
2.01.00.50l2/l1
Interior negative moments:
Note berlaku bilamana: 0.52.02
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221 l
l
Factor Moments in Column Strips:
t2.5
t2.5
t=0
t=0
457590
757575
100100100(1l2/l1)1.0
100100100(1l2/l1)=0
2.01.00.50l2/l1
Exterior negative moments:
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Factor Moments in Column Strips:
457590(1l2/l1)1.0
606060(1l2/l1)=0
2.01.00.50l2/l1
Positive Factor Moments:
Factor Moments in Middle Strips:
That portion of negative moment and positive factored moments not resisted by column strips shall be proportionately assigned to corresponding half middle strips.
Each middle strip shall be proportioned to resist the sum of the moments assigned to its two half middle strips.
A middle strip adjacent to and parallel with a wall-supported edge shall be proportioned to resist twice the moment assigned to the half middle strip corresponding to the first row of interior supports.
