*Muhammad Irfan-ul-Hassan, Assistant professor, Civil Engg. Department, U.E.T. Lahore. Pakistan. *Ali Ahmed, Lecturer, Civil Engg. Department, U.E.T. Lahore.
DESIGN COMPARISON OF FLAT SLAB SYSTEM BY DIRECT DESIGN METHOD AND EQUIVALENT FRAME METHOD M.I.Hassan* and A. Ahmed* Department of Civil Engineering U.E.T. Lahore-Pakistan.
Abstract The trend of construction of flat slab system is very common in multistorey buildings. There are different design methods used for designing these slabs system. This research work is an effort to see the difference between two methods, First method is DDM having some limitations and based on coefficients and second is EFM having no limitations and it is an exact method based on stiffness. For this purpose A detailed research study has been carried out to see the design comparison of a slab system by these two methods given by ACI 318-05 code. A flat slab system is selected for the comparison of the design by the two specified methods. The design software are developed by Visual BASIC and MS Excel interaction for each method to carry out the design. The software designed is manually verified. Various slab systems are designed to see the design difference and It is also concluded that in case of flat slab with edge beams the exterior negative moment calculated by Equivalent Frame Method is larger than that calculated by Direct Design Method. It is also verified that in EFM the stiffness of each structural member is calculated. By using these values of stiffness and carry over factors the moment is distributed longitudinally In case of DDM the longitudinal distribution is done by the coefficients given in article 13.6.3 ( Negative and positive factored moments) in ACI 318-05 code. These coefficients do not change by changing beam thicknesses. Hence the distribution of moments in EFM is more realistic as it changes with the change in stiffness of the elements. From this research many other design arguments are concluded for the structural designers. This research work is very helpful for the practicing structural engineers.
Corresponding Authors *Muhammad Irfan-ul-Hassan, Assistant professor, Civil Engg. Department, U.E.T. Lahore. Pakistan *Ali Ahmed, Lecturer, Civil Engg. Department, U.E.T. Lahore. Pakistan
Keywords: Flat Slabs Design; DDM; EFM; Design Comparison
1. INTRODUCTION Reinforced concrete slab is a widely used structural element. It provides an economical and versatile method of supporting gravity loads. In addition, the slab also forms integral part of structural frames to resist lateral loads. These slabs, when combined with other elements (beams, drop panels or column capitals, etc.) are known as roofing system. Owing to the versatility of shapes in which concrete can be
cast and availability of local indigenous material a large number of concrete roofing systems are being employed in construction industry in Pakistan. According to ACI, a slab system may be designed by any method that satisfies equilibrium and geometric compatibility conditions. For this purpose ACI code provides only two methods i.e. Direct Design Method and Equivalent Frame Method. The procedure of these methods is different in the way moment is calculated and longitudinally transferred. There are different slab systems. Flat plate is the one in which there is no column capital, drop panel and there are no projected beams and the slab alone is directly resting on the columns. Flat slab is the one having either column capital or drop panel or both of these. To lighten the slab, reduce the slab moments and save material, the slab at mid-span can be replaced by intersecting ribs. However, near the columns, full depth is restrained to transmit loads from the slab to columns. This type of slab is known as waffle slab. Two way slabs with beams is a slab system in which slab is supported on all four sides on beams. These beams may monolith with the slab or may not be monolith. Flat slab system is selected for this research work and is designed by DDM and EFM. 2-REVIEW OF LITERATURE 2.1. Slab: 1. A reinforced slab is a broad, flat plate, usually horizontal, with top and bottom surfaces parallel or nearly so. 2. It may be supported by reinforced concrete beams (and is usually cast monolithically with such beams), by masonry or by reinforced concrete walls, by steel structural members, directly by columns, or continuously by ground (on grade). This research work is for two way slab system. 2.1.1. Two-Way Slab: Slab resting on walls or sufficiently deep and rigid beams on all sides. Other options are column supported slab e.g. Flat slab, waffle slab. Two way slabs have two way bending. 2.1.2. Design Methods: 3. ACI co-efficient method 4. Direct Design Method 2
5. Equivalent Frame Method 6. Finite Element Method 2.1.3. Minimum Depth of 2-Way Slab for Deflection Control:
7.
According to ACI-318-2005
8.
9.
L n 0.8 f y 1500
Lx Ly 36 m 9 Ln = clear span in short direction
h min
m
2.2. Design Methodology: The design methods for DDM and EFM is given below. 2.2.1. Direct Design Method: Direct Design Method is a semi empirical method which is used to calculate moments in two way slabs. This design method has certain limitations which have to be satisfied before designing by this method. 2.2.1.1 Limitations of Direct Design Method: a. There must be a minimum of three continuous spans in each direction.
(ACI
13.6.1.1). b. Rectangular panels with long span/short span ≤ 2. (ACI 13.6.1.2) c. Successive span in each direction shall not differ by more than 1/3 times the longer span. (ACI 13.6.1.3) d. Columns may be offset from the basic rectangular grid of the building by up to 0.1 times the span parallel to the offset. (ACI 13.6.1.4) e. All loads must be due to gravity only (N/A to un-braced laterally loaded frames, from mats or pre-stressed slabs). (ACI 13.6.1.5) Service (un-factored) live load
≤
2 service dead load.(ACI 13.6.1.5)
f. For panels with beams between supports on all sides, relative stiffness of the beams in the 2 perpendicular directions. (ACI 13.6.1.6)
2.2.1.2 Design Steps in DDM: 3
These are the general steps for the design of two way slab system by DDM. The explanation of these steps has been given in the next articles. a. Determine whether the slab geometry and loading allow the use of the direct design method.
b. Select slab thickness to satisfy deflection and shear requirements. Such calculations require a knowledge of the supporting beam or column dimensions A reasonable value of such a dimension of columns or beams would be 8 to 15% of the average of the long and short span dimensions, namely (l1 +l2)/2. For shear check, the critical section is at a distance d/2 from the face of the support. If the thickness shown for deflection is not adequate to carry the shear, use one or more of the following:
(a) Increase the column dimension. (b) Increase concrete strength. (c) Increase slab thickness. (d) Use special shear reinforcement. (e) Use drop panels or column capitals to improve shear strength.
c. Divide the structure into equivalent design frames bound by centerlines of panels on each side of a line of columns.
d. Compute the total static factored moment
(ACI 13-4)
4
e. Select the longitudinal distribution factors for the design frame. The static moment is further multiplied by these factors to distribute the static moment in longitudinal direction. The distribution factors are selected from ACI 13.6.3.3. given in table 3.2.
f. Calculate transverse distribution factors. These are calculated as explained in sec. 3.2.4.3. g. Distribute the factored equivalent frame moments from step 5 to the column and middle strips according to the factors calculated in step 6.
h. Determine whether the trial slab thickness chosen is adequate for moment-shear transfer in the case of flat plates at the interior column junction computing that portion of the moment transferred by shear and the properties of the critical shear section at distance d/2 from column face.
i. Design the flexural reinforcement to resist the factored moments in step 7.
j. Select the size and spacing of the reinforcement to fulfill the requirements for crack control, bar development lengths, and shrinkage and temperature stresses.
2.2.2. Equivalent Frame Method Equivalent frame method has been described in ACI 13.7. It is a general method for design of two way column supported slab system based on stiffness and moment distribution technique, There is no restriction and no limitations in this method as in the direct design method. 2.2.2.1. General Steps For the Design of Two Way Slab System by EFM: There are general steps involved in this method. The explanation of these steps have been given in the later articles. a. The 3-D slab system is represented by four or more 2-D frames just like in DDM. These design strips (or design frames) are separately considered for analysis and design. b. The stiffness of frame elements is determined considering the fact that the slab is not supported along full width at the edge, torsion member is present but its effect cannot
5
be included directly in the 2-D analysis and the slab beam are non-prismatic members. Torsional stiffness of transverse beams and the slab edge condition at the junction with outer columns are included in the column stiffness and thus the concept of an equivalent column is used. ACI code allows the analysis of a particular floor of the building by considering free body of that floor with the columns below and the columns above (if present), with the far ends of these columns taken as fixed. This simplification is very useful for hand calculation. c. The 2-D frames obtained in the above steps are analyzed for full gravity loads (pattern loading is not considered if live load is with in certain percentage of the dead load). This is equivalent to the longitudinal distribution of total static moment in the direct design method. d. The negative and positive moment (M–ve and M+ve) are distributed laterally to column strips and middle strips using coefficients of DDM if the following limitation is satisfied.
3. DESIGN BY DDM:
Interior Design Frame (E~W) Wu = ln = l2 = Mo =
14.021 5.857 6 360.737
2
Wu = ln = l2 = Mo =
KN/m m m KN-m
Interior Pannel
Exterior Pannel Description Longitudinal Dist. Factor
Longitudinal Dist. % CS Dist. Factor Mu For C.S
Ex. M- Mid Span M+
Ist M-
(KN-m) 0.3 108.221 92.66 100.278
(KN-m) 0.7 252.516 75 189.387
(KN-m) 0.5 180.369 60 108.221
14.021 5.238 6 288.517
Int. M+
M-
(KN-m) 0.35 100.981 60 60.589
(KN-m) 0.65 187.536 75 140.652
6
% Beam Dist. Factor Mu For Beam Net Mu For C.S % MS Dist. Factor Mu For M.S
4.
0 0.000 100.278 7.34 7.943
DESIGN BY EFM
0 0.000 108.221 40 72.147
0 0.000 189.387 25 63.129
0 0.000 60.589 40 40.392
0 0.000 140.652 25 46.884
:
The same slab system is designed by EFM also, for this purpose a universal software is developed using the Visual BASIC and Excel interaction. This software is also verified with the solved example. This was an additional part of the research work. The detailed calculations for this method are given below.
7
8
9
C=
1260878400
mm4 3
Kt
" = Σ ( 9 x Ecs x C / l 2 x (1- c2/l 2) )" = Σ ( 9 x Ecs x 1260878400 / 6000 x (1- 0.127 )^3 ) / 1000 = 5685.278 Ecs kN-mm/rad
Ic
" = b x h / 12 " = 762 x 762 ^3 / 12 4 mm = 28095621228.
3
1 762 mm 3.6 m 3.6 m 3,420 mm 3,420 mm 90 mm 90 mm 90 mm 90 mm
C1 = l (upper) = l (lower) = lu (upper) = lu (lower) = ta (upper) = ta (lower) = tb (upper) = tb (lower) = Upper Storey l / lu = ta / tb =
tb C1
lu (upper)
l c(upper)
ta
1.053 1
ta
From Table A-23 ka = 4.594 Ca = 0.526
C1
lu (lower)
l c(lower)
tb
Kc (upper) = ka x Ec x Ic / Lc = 4.594 x Ec x 28095621228 / 3600 x 1000 = 35854.395 Ec kN-mm/rad Lower Storey l / lu = ta / tb =
1.053 1
Fig. 5. General vertical cross section of flat slab system
From Table A-23 ka = 4.594 Ca = 0.526 Kc (lower) = ka x Ec x Ic / Lc = 4.594 x Ec x 28095621228 / 3600 x 1000 = 35854.395 Ec kN-mm/rad Σ Kc =
71,708.79
kN-mm/rad
1/Kec = 1/Σ Kc + 1/Kt 1/Kec = 1/71708.79 + 1/5685.28 Kec = 5267.64 E kN-mm/rad
10
11
5- RESULTS AND DISCUSSIONS Table 4.1 Comparison of Transverse Moments of interior frame in east West Direction
Panel Designation
Location
Strip
Exterior – Column Strip ve Middle Strip Column Strip Exterior +ve Panel Middle Strip st 1 Interior Column Strip –ve Middle Strip Column Strip - ve st Middle Strip 1 Interior Panel Column Strip + ve Middle Strip Column Strip - ve Middle Strip Interior Panel Column Strip + ve Middle Strip
By DDM
By EFM
kN-m
kN-m
Percent Difference (%)
100.278 7.943 108.221 72.147 189.387 63.129 140.652 46.884 60.589 40.392 140.652 46.884 60.589 40.392
172.111 13.634 96.885 64.590 159.585 53.195 120.485 40.162 70.757 47.171 125.247 41.749 73.697 49.132
41.74 41.74 10.47 10.47 15.74 15.74 14.34 14.34 14.37 14.37 10.95 10.95 17.79 17.79
Table 4.2 Comparison of Transverse Moments of exterior frame in east West Direction
Panel Designation
Location
Strip
- Column Strip Middle Strip Column Strip Exterior +ve Panel Middle Strip st 1 Interior Column Strip -ve Middle Strip Column Strip - ve st Middle Strip 1 Interior Panel Column Strip + ve Middle Strip Column Strip - ve Middle Strip Interior Panel Column Strip + ve Middle Strip Exterior ve
By DDM
By EFM
kN-m
kN-m
Percent Difference (%)
7.042 7.164 12.125 9.352 16.975 13.093 12.607 9.724 6.788 5.236 12.607 9.724 6.788 5.236
13.173 13.401 10.100 7.790 14.691 11.331 11.263 8.687 7.995 6.167 11.352 8.756 8.047 6.207
46.54 46.54 16.70 16.70 13.46 13.46 10.66 10.66 15.10 15.10 9.95 9.95 15.64 15.64
12
Table 4.3 Comparison of Transverse Moments of Interior Frame In North South Direction
Panel Designation
Location
Strip
Column Strip Middle Strip Column Strip Exterior +ve Panel Middle Strip st 1 Interior - Column Strip ve Middle Strip Column Strip - ve st Middle Strip 1 Interior Panel Column Strip + ve Middle Strip Column Strip - ve Middle Strip Interior Panel Column Strip + ve Middle Strip Exterior -ve
By DDM
By EFM
kN-m
kN-m
Percent Difference (%)
68.959 5.463 74.421 49.614 151.471 50.490 140.652 46.884 60.589 40.392 140.652 46.884 60.589 40.392
105.109 8.326 67.943 45.295 117.174 39.058 125.483 41.828 73.988 49.325 124.483 41.494 73.357 48.905
34.39 34.39 8.71 8.71 22.64 22.64 10.78 10.78 18.11 18.11 11.50 11.50 17.41 17.41
Table 4.4 Comparison of Transverse Moments of Exterior Frame In North South Direction
Panel Designation
Location Exterior -ve
Exterior Panel
+ve 1st Interior -ve - ve
st
1 Interior Panel + ve - ve Interior Panel + ve
Strip Column Strip Middle Strip Column Strip Middle Strip Column Strip Middle Strip Column Strip Middle Strip Column Strip Middle Strip Column Strip Middle Strip Column Strip Middle Strip
By DDM
By EFM
(kN-m )
(kN-m)
Percent Difference (%)
5.738 5.157 9.049 12.025 14.735 19.580 13.682 18.182 7.367 9.790 13.682 18.182 7.367 9.790
10.067 9.048 7.882 10.475 10.907 14.493 12.312 16.361 8.797 11.691 12.273 16.309 8.775 11.660
43.00 43.00 12.89 12.89 25.98 25.98 10.02 10.02 16.26 16.26 10.30 10.30 16.04 16.04
13
161.48 kN-m
167 kN-m
160.65 kN-m
185.74 kN-m
212.78 kN-m 180.37 kN-m
108.221 kN-m
122.83 kN-m
117.93 kN-m
By EFM
100.98 kN-m
100.98 kN-m
187.54 kN-m
187.54 kN-m
252.516 By DDM
Fig .6. Comparison of Bending Moment Diagram for interior frame in east west direction.
Fig .2. Comparison of steel for interior design frame in east west direction 14
Fig .3. Comparison of steel for exterior design frame in east west direction
Fig .4. Comparison of steel for interior design frame in north south direction
15
Fig .5. Comparison of steel for exterior design frame in north south direction
6. CONCLUSIONS a. The moments calculated by EFM are more accurate, as EFM takes into account the stiffness of the members and based on those stiffnesses moments are distributed longitudinally. In DDM the longitudinal distribution of moments is carried out using coefficients given by the code in article 13.6.3 (Negative and positive factored moments in ACI 318-05). b. In DDM the static moment, (Mo) is calculated which is distributed longitudinally according to the coefficients given in ACI code in article 13.6.3 (Negative and positive factored moments in ACI 318-05) but in EFM the whole structure has to be analyzed for the calculation of moments. c. Negative steel being calculated by EFM on the exterior edge is more than that calculated by DDM because in EFM the increased stiffness at the exterior edge due to edge beam is being directly considered while DDM considers some coefficients which do not change with the change in the beam dimensions. The main reason for this difference is that EFM directly takes into account the stiffness of a member and based on the stiffness it distributes the moment longitudinally. When there is a beam on the outer edge and the cross section of beam is large, it increases the stiffness on the outer edge of the slab. 16
d. With increase in the slab thickness the stiffness increases and this increase in stiffness do not effect the moment in DDM while the moments in EFM get changed. Hence the distribution of moments in EFM is more realistic as it changes with the change in slab thickness. e.
The magnitude of moment in DDM is not affected by the change in beams and columns dimensions except span length while in EFM by changing the beam dimensions the stiffness changes which in turn cause the moments to change.
f. DDM is a conventional method which is limited to its application. It is applicable to two way slabs and gives moment for slabs only while EFM gives moment both in columns and slab. 7. Recommendations a. EFM is more reliable as this method involves moment calculation by exact elastic analysis by stiffness and moment distribution method unlikely DDM. b. EFM should be used for more accurate results as it takes into account every change in dimensions of Slab, Beam and Column etc while in DDM the longitudinal distribution factors are used which do not change with the changing dimensions of beam. c. EFM is more computer applicable, as we can develop softwares/packages for calculations of stiffness and moment distribution which reduces the time required for the analysis and chances of error are accordingly reduced. d. The research work should be extended for the slab system design comparison by software SAFE based on finite element method and EFM. 8. Acknowledgement All thanks are due to Almighty Allah who enable us always to think and to search. Peace be upon the Prophet Muhammad who have been a source of inspiration to us. The authors are thankful to Prof. Dr. Muhammad Ashraf and Prof. Dr. Zahid Siddiqi, Civil Engineering Department, U.E.T. Lahore for their valuable guidance.
REFERENCES 1. ACI 318-05, Building Code Requirements for Structural Concrete, American Concrete Institute, USA 2005. 17
2. MacGregor, J.G.; Design of reinforced concrete structures, McGraw Hill, 1997. 3. Winter and Nilson; Design of concrete structures , McGraw Hill, 12th edition ,1997. 4. Leet K. & Bernal D.; Reinforced concrete design, McGraw Hill, 1997. 5. Wang, C.K. and Salmon, C.G.; Reinforced concrete Designers handbook, ELBS, 1992. 6. Gamble, W. L.; Moments in Beam-Supported Slabs,” J. ACI. Vol 69, no. 3, 1972, pp. 149-157 7. Peabody. D. Jr.; “Continuous Frame Analysis of Flat Slab,” J. Boston Society Civ. Eng., January 1948. 8. Corley W. G. and Jirsa J.O.; “Equivalent Frame Analysis for Slab Design,” J. ACI, Vol 67, no. 11, 1970, pp. 875-884 9. Nawy, E.G.; Reinforced Concrete, 5th edition,2005 pp.440-500 List of Abbreviations and Symbols Ac = Area of concrete section resisting shear transfer ( mm2 ). Ag = Gross area of concrete section ( mm2 ). As,min=Minimum area of flexural reinforcement ( mm2 ). Ast = Total area of nonprestressed longitudinal reinforcement (bars or steel shapes) b = Width of compression face of member (mm). DDM = Direct Design Method EFM = Equivalent Frame Method fc′ = Specified compressive strength of concrete, psi, = square root of specified compressive strength of concrete. fs = Calculated tensile stress in reinforcement at service loads h = Overall thickness or height of member (mm). Ib = Moment of inertia of gross section of beam about centroidal axis (mm4). Is = Moment of inertia of gross section of slab about centroidal axis defined for calculating αf and βt, (mm4). lc = Length of compression member in a frame, measured center-to-center of the joints in the frame. ln = Length of clear span measured face-to-face of of supports. Vu = Factored shear force at section (kN). αfm = Average value of αf for all beams on edges of a panel. β = Ratio of long to short dimensions: clear spans for two-way slabs. Kt = Torsional stiffness of torsional member; moment per unit rotation. 18
