Two-way slabs. Flat plate with or without drop panels / capitals

Two-way slabs Two-way slab behavior is described by plate bending theory which is a complex extension of beam bending. Codes of practice allow use of simplified methods for analysis and design of two-way slabs. This chapter will first cover the flat plate which is a slab supported directly by columns (without beams). Slabs with beams will be studied after. Columns in a flat plate may have drop panels or / and capitals. Flat plate with or without drop panels / capitals

Flat Plate: Span not exceeding 6.0 to 7.5 m and live load not exceeding 3.5 to 4.5 kn/m Advantages ow cost formwork Exposed flat ceilings Fast Disadvantages ow shear capacity ow Stiffness (notable deflection) Need of special formwork for drop panels and capitals Waffle Slab: Span up to 4 m and live load up to 7.5 kn/m Advantages Carries heavy loads Attractive exposed ceilings Disadvantages Formwork with panels is expensive Slab with beams: Span up to 0 m Advantages Versatile Framing of beams with columns Disadvantages Visibility of drop beams in ceilings Methods of analysis of slabs Direct Design Method DDM (object of the course) Equivalent Frame Method EFM Yield ine Method YM Finite Element Method FEM (most powerful)

Part A: Flat plate Thickness of two way slabs Table 3.: Minimum thickness for slabs without interior beams Without drop panels With drop panels Internal Internal f y Exterior panel Exterior panel panel panel (MPa) No edge With edge No edge With edge beams beams beams beams 300 n / 33 n / 36 n / 36 n / 36 n / 40 n / 40 40 n / 30 n / 33 n / 33 n / 33 n / 36 n / 36 50 n / 8 n / 3 n / 3 n / 3 n / 34 n / 34 n is the maximum clear length of the panel. inear interpolation must be performed for intermediate values of steel grade f y. The thickness of slabs with interior beams is given by equations to be seen later. ACI, SBC and other codes of practice allow the use of simplified methods for the analysis and design of two way slabs. Among these is the Direct Design Method. Direct Design Method DDM The procedure consists first in dividing the slab in each direction into frames using mid-lines, then compute in each span, the static moment, as well as positive and negative moments using appropriate coefficients. These moments are then distributed across the frame width, based on geometry and stiffness. The various parts (column strips, middle strips and possible beams) are then designed. The next figure shows the frames in X-direction. The frames in Y-direction are obtained similarly. X4 Frame X4 X3 Frame X3 X Frame X X Y Y Y3 Y4 3 Frame X

Subdivision in frames in both directions 4

Panel dimensions, Column strip CS and middle strip MS is the panel dimension in the studied direction. is the dimension in the other direction (transverse width of the frame). n is the clear length in the studied direction. Moments vary across the width of the frame in each span. Each span of the frame is composed of a column strip (containing the column line) and a middle strip The column strip width is equal to half the minimum length. Half middle Strip min / Column Strip n Half middle Strip min = Min(, ) When computing the clear length n with respect to a circular column, the latter is replaced by an equivalent square section with the same area. The moment distribution along the frame is deduced from the values of the static moments. For each span, the positive and negative moments are function of the static moment equal to: w M n 0 where w is the uniform line load (kn/m) 8 If w s is the slab area load (kn/m ), the beam load would be: w ws The static moment for the panel is therefore: M w 0 s n 8 M 0 M + M - M o Positive and negative moments in each span are deduced according to the following table. Table 3.: Distribution of factored static moment M 0 (a) (b) (c) (d) (e) Int. negative M 0.75 0.70 0.70 0.70 0.65 Positive M 0.63 0.57 0.5 0.50 0.35 Ext. negative M 0 0.6 0.6 0.30 0.65 (a): Exterior edge unrestrained (b): Slab with beams between all supports (c): Slab with no beams at all (d): Slab with edge beams only (e): Slab with exterior edge fully restrained (interior span) 5

In general: M M R M M 0 w s n 8 For a typical interior panel, the total static moment is divided into positive moment 0.35M o and negative moment of 0.65M o. 6

Distribution of moments over column strips and middle strips The following table shows the portions (%) of moments carried by column strips and middle strips. Moments in column strip and middle strip (%) Interior negative moment Positive moment Exterior negative moment Column strip 75 60 00 Middle strip 5 40 0 Conditions of the DDM. Minimum of three spans in each direction. Minimum of 3 x 3 = 9 panels. xi. Rectangular panels with aspect ratio between 0.5 and.0 0.5. 0 3. Successive spans in each direction must not differ by more than one third of the largest span. i 3 i Max i, i 4. Column offset from basic rectangular grid must not exceed 0 % of span in offset direction. 5. Gravity loading 6. ive load less or equal to twice the dead load D ( ) 7. For slabs with beams, the relative beam stiffness must be such: 0. 5. 0 ( ) Beam relative stiffness to be defined later. yi Column offset limited to 0% 7

Example : Thickness check and computation of static moments A 50 mm slab is subjected to a live load of 4.5 kn/m and a super imposed dead load of.0 kn/m. The rectangular column section is 600 x 300 mm. The 300 mm diameter circular columns of line X have 600 mm diameter capitals. Check the slab thickness and determine the static moment for span XY3- XY4 of frame X in X-direction and for span Y3X-Y3X for frame Y3 in Y-direction. X4 7.0 m 7.5 m 7.5 m 6.0 m X3 6.5 m X 6.0 m X Y Y Y3 Y4 The ultimate slab load is w s.4 4 x0.50.0.7x 4.5 7.45kN/ m For columns with capitals, the clear length (used for minimum thickness and static moment) must be computed between capital faces and by replacing the circular capital by an equivalent square one with the same area. D c 4 Thus c D 0.886D 0.866 x600 53 mm Thickness check n The minimum thickness is 30 for the eight external panels and n 33 for the internal panel. For each panel, n is the maximum clear length. The maximum clear length for the external panels with rectangular columns is: 0.6 0.6 6900 n 7.5 6.9 m 6900 mm This gives a minimum thickness of 30 mm 30 The maximum clear length for the external panels with equivalent square capitals is: 0.53 0.53 6968 n 7.5 6.968 m 6968 mm This gives a minimum thickness of 3.7 mm 30 8

The internal panel has a maximum clear length of 0.6 0.6 n 7.5 6.9 m 6900 mm The minimum thickness is 6900 09.09 mm 33 The slab minimum thickness is therefore 3.7 mm. The actual thickness of 50 mm is thus OK. The next figure shows the minimum thickness check using RC-SAB software. The value of the minimum thickness is shown for each panel. Minimum thickness check using RC-SAB software showing minimum thickness for each panel 9

Static moment for span XY3-XY4 Panel dimensions and static moment are: 6 6.5 0.6 0.6 6. 5 m n 7.5 6. 9m n M n 6.9 ws 7.45 x6.5 649.06 knm. 8 8 0 Static moment for span Y3X-Y3X 6. 0 m 7. 5m The clear length must be computed by replacing the circular capital by an equivalent square one with the same area. = 7.5 m n D c 4 Thus c D 0.886D 0.866 x600 53 mm = 7.5 m 0.3 0.53 n 5.584 n 6.0 5. 584 m M 0 ws 7.45 x7.5 50.0 kn. m 8 8 Example : Moment distribution over column strip and middle strip Determine moments along strips of frame X of the previous example. The figure shows frame X and with the column strips shaded. 7.0 m 7.5 m 7.5 m 6.5 m X 6.0 m Y Y Y3 Y4 The following table sums all the results including the panel dimensions, static moment, column strip width, positive and negative moments, as well as CS and MS moments. 0

Span XY- XY Span XY- XY3 Span XY3- XY4 (m) 7.0 7.5 7.5 (m) 6.5 6.5 6.5 n (m) 6.4 6.9 6.9 558.4 649.06 649.06 n M 0 ws (kn.m) 8 min (m) 6.5 6.5 6.5 CS width (m) 0.5 min 3.5 3.5 3.5 Moment coefficients -0.6 0.5-0.70-0.65 0.35-0.65-0.70 0.5-0.6 -ve and +ve moments -45. 90.4-390.9-4.9 7. -4.9-454.3 337.5-68.8 CS moment (%) 00 60 75 75 60 75 75 60 00 CS moments (kn.m) -45. 74. -93. -36.4 36.3-36.4-340.7 0.5-68.8 MS moment (%) 0 40 5 5 40 5 5 40 0 MS moments (kn.m) 0 6. -97.7-05.5 90.9-05.5-3.6 35.0 0 The next figure shows RC-SAB output in tabular and graphical forms.

Analysis of frame X using RC-SAB software (Tabular and graphical output) RC design of strips In DDM, strips have known widths. The column and middle strips are designed using the actual strip width b. RC design must therefore deliver the total required bar number and not the bar spacing as in one way slabs. Both minimum steel and maximum spacing requirements must be met.

Minimum steel in slabs: A s min 0.00 bh 0.008 bh 0.008 bh 40 f y if if if f f y f y y 300 to 350 MPa 40 MPa 40 MPa Maximum spacing in slabs: Min h, 300 S. max s For a known bar diameter d b with bar area minimum steel area given by: A s min S A A b b max b d 4 b, maximum spacing is equivalent to another The actual reinforcement must be greater than or equal to for both values of minimum steel. The minimum bar number is therefore N b min Max A s min A b, A s min RC design of the column strip under maximum negative moment: The maximum internal negative moment in the column strip is 340.7 kn.m The section dimensions for the column strip are b = 35 mm and h = 50 mm Using 8-mm diameter bars, the bar area is Steel depth is computed as: Assuming A b db d h cover 8 4 54.47 mm 8 50 0 mm ' f c 5 MPa, f y 40 MPa, the required steel area is A s 4349.0mm Minimum steel is A s min.008bh 0.008 x35 x50 406. 5 0 mm The maximum spacing is S Min h,300 300 mm max. Maximum spacing is equivalent to another minimum steel area: A A b 54.47 x35 b s min 650. 73 Smax 300 The minimum bar number is mm Max As min, As min 650.73 N b min 0.4 Thus N bmin = A 54.47 b The actual reinforcement must be greater than or equal to for both values of minimum steel. We thus adopt a steel area A s 4349.0mm 3

As 4349.0 This gives a number of bars equal to N b 7. A 54.47 b 8 bars of 8-mm diameter are therefore required at the maximum internal negative moment. This is equivalent to a spacing S b N b 8 73.6mm The next figure shows RC-SAB output for design of the column strip X. RC design of the column strip X using RC-SAB software showing capacity moment diagram with bar cutoff 4

RC design of middle strips As seen previously, a middle strip has two parts belonging to two adjacent frames. Design of a middle strip requires therefore the analysis of the two adjacent frames and summing both contributions to find the total middle strip moments and widths in all spans. Middle strip X in X-direction takes contributions from frames X and X3. Analysis of middle strip X using RC-SAB software 5

RC design of the middle strip X using RC-SAB software showing capacity moment diagram with bar cutoff The same steps must be performed for the Y-direction. 6

Shear strength of two way slabs (flat plate) There are two possible shear failure mechanisms: One-way shear or beam shear at distance d from the column Two-way or punching shear which occurs along a truncated cone at distance d/ from the column. Two-way shear fails along a truncated cone or pyramid around the column. The critical section is located d/ from the column face, column capital, or drop panel. Forces are transferred between the slab and the column through the conic region as shown. 0.44 Corner Edge The tributary areas for shear in a flat plate are shown in the figure with the critical transfer perimeter shown in broken lines for internal, edge and corner columns. The shear panel dimensions are obtained by 0.56 0.5 Internal dividing the first (and last) span in two unequal Critical two-way shear zones parts (0.44 and 0.56), The other spans are halved. For two-way shear mechanism, the critical perimeter is located at a distance d/ from the column faces, d/ where d is the average steel depth in the slab. For an internal column the critical perimeter is as shown. d/ 7

For one-way shear the critical distance is d. Edge Corner 0.44 Usually two-way shear is most critical for internal and edge columns and one-way shear controls corner columns. It is convenient to check both two-way and one-way shear. We must check in both cases that: V V c u 0.56 0.5 Internal Critical one-way shear zones If shear is not OK, either we increase the slab thickness or provide drop panels / capitals. Special shear reinforcement may also be provided, such the stirrup cages shown. Stirrup cages for slab shear reinforcement 8

Concrete shear strength in two-way slabs For two-way shear, the concrete nominal shear strength is given by: V c Min ' f c 3 c 6 sd f b0 b d 0 f ' c b d ' c 0 b d 0 () () (3) d : average steel depth, : length of critical perimeter : column aspect ratio (long side / short side) c s b 0 : column location factor (40 : Internal, 30 : Edge, 0 : Corner) The ultimate shear is computed over the loaded area defined as the total shear panel area minus the critical area: V w A A w l l p u u 0 c u p l and l are the shear panel dimensions and p and p are the critical perimeter dimensions If steel data is not available, the average steel depth may be estimated by: d h 40 For one-way shear, concrete nominal shear strength is as in beams given by: V c ' fc bwd (4) 6 Example 3: Shear check in a flat plate A 50 mm flat plate slab with a concrete grade of ' f c 5 MPa, is subjected to a live load of 3.0 kn/m and a super imposed dead load of.0 kn/m. The steel depths in both directions are Check shear for the shown internal shear panel. The ultimate slab load is: d 5 mm d 5 mm 350 650 5.5 m w u 4 x0.50.0.7x 3.0.54 kn/.4 m a) Two-way shear The shear panel dimensions are: The column dimensions are: l l 5. 5m c 350 mm 0. 35 m c 650 mm 0. 65 m d d The average steel depth is: d 0. 0mm The critical perimeter dimensions are thus: p c d 350 0 470 mm 0. 47 m p c d 650 0 770 mm 0. 77 m The ultimate shear is obtained from the loaded area: 9 5.5 m p p l l

A A w l l p p.545.5 0.77 x0.47 344. kn Vu wu 0 c u 9 For two-way shear, the concrete nominal shear strength is given by the minimum of equations () to (3). V c Min ' f c 3 c 6 sd f b0 b d 0 f ' c b d ' c 0 b d 0 () () (3) d 0 mm c 650 350.857 40 (Internal column) s b 0 p p 470 770 480 mm 5 480 x0 55097.5 N 55.kN.857 6 40 x0 5 Vc Min 480 x0 488000 N 488.0 kn Vc 488. 0 kn 480 5 480 x0 496000 N 496.0 kn 3 V c 0.75 x488.0 366. 0kN this is greater than V u. Two-way shear is thus OK. Vu 344.9 Two-way shear ratio is: 0. 94 V 366.0 c b) One-way shear For one-way shear, there are two possibilities: p Case : l c 5.5 0.35 p d 0.5. 45m d l = 5.5 m The ultimate shear given by the shaded loaded area is: Vu wu pl.54 x.45 x5.5 55. 5kN 5.5 m Case : l c 5.5 0.65 p d 0.5. 3m The ultimate shear given by the shaded loaded area is: p d 5.5 m Vu wu pl.54 x.3x5.5 46. 6kN l = 5.5 m 0

The first case is the controlling one as it gives the highest ultimate shear force. The nominal concrete shear strength given by equation (4) is: ' ' fc fc 5 Vc bwd ld 5500 x5 5797 N 57. 9kN 6 6 6 V 0.75 x57.9 49.7 knv 55. kn One way shear is also OK. c u 5 Vu 55.5 One-way shear ratio is: 0. 36 V 49.7 c Two-way shear (with a greater ratio) is more critical for this internal column. It must be pointed out that in case of presence of drop panels, punching shear must be checked around the drop panel using the slab thickness and around the column using the drop panel thickness. Steps for analysis and design of two-way slabs: It is usually preferable to check the slab thickness for shear before performing analysis and design. The recommended order is therefore:. Slab thickness. Shear check 3. Slab loading 4. Static moment M 0 5. Positive and negative moments M + and M - 6. Distribution of moments over column strip and middle strip 7. Design of strips 8. Detailing

Two way slab example 4: Flat plate The figure shows a flat plate floor. The slab extends 00 mm offset past the exterior column face in both directions. The slab thickness is 85 mm. Corner columns have a square section 300 x 300 mm whereas other columns have the same rectangular section 500 x 300 mm with the orientations as shown. Superimposed dead load SD =. kn/m ive load =.0 kn/m Materials: ' fc 5 MPa f y 40 MPa c 4 kn/ m () Check if conditions of DDM are satisfied. 3. Minimum of three spans in each direction: OK. Fore each panel, ratio of longer span to shorter span less than : OK 3. Successive spans differ by not more than one third of longer span: OK 4. Column offsets up to 0%: OK 5. Uniform gravity load: OK 6. ive load not exceeding twice dead load. =.0 D = 4 x 0.85 +. = 5.64 : OK 7. There are no beams. The condition on relative beam stiffness does not apply All conditions are therefore satisfied. 00 mm () Check the slab thickness for deflection control Check all panels using the minimum thickness Table 3. and take the maximum. There are no beams and no drop panels. For the steel grade of 40 MPa, in all corner and edge panels, the minimum thickness is l n /30 and for the interior panel it is l n /33. l n is the longer clear length of the panel. ln 500 Corner panel: l n, max 5500 (50 50) 500 hmin 70.0 mm 30 30 ln 5500 Edge panel: l n, max 6000 (50 50) 5500 hmin 83.33mm 30 30 ln 5500 Interior panel: l n, max 6000 (50 50) 5500 hmin 66.67 mm 33 33 X5 X4 X3 X 5.5 m 6 m 5.5 m X Y Y Y3 Y4 4.5 m 4.5 m 4.5 m 4.5 m

The minimum thickness is therefore 83.33 mm and is just less than the actual thickness of 85 mm: OK f y (MPa) Table 3.: Minimum thickness for slabs without interior beams No edge beams Without drop panels With drop panels Exterior panel Int. panel Exterior panel Int. panel With edge beams No edge beams With edge beams 300 n / 33 n / 36 n / 36 n / 36 n / 40 n / 40 40 n / 30 n / 33 n / 33 n / 33 n / 36 n / 36 50 n / 8 n / 3 n / 3 n / 3 n / 34 n / 34 Minimum thickness check using RC-SAB software 3

(3) Check the thickness for two-way shear at edge / internal columns The factored slab load is: w u.4(4 x0.85.).7 x.0.3kn/ m The tributary areas for shear in a flat plate are shown in the next figure: Any offset must be added to the corresponding span. 0.44 Corner Edge 0.56 For the two columns, shear areas are as shown 0.5 Internal 0.44 x 4.5 =.98 m Edge 00 + 50 = 50 mm. 0.56 x 4.5 =.5 m 0.5 x 4.5 =.5 m Internal 0.56 x 5.5 = 3.08 m 0.5 x 6.0 = 3.0 m The concrete nominal shear strength is given by: 4

V c Min ' f c 3 c 6 sd f b0 b d 0 f ' c b d ' c 0 b d 0 () () (3) d : average steel depth, : length of critical perimeter : column aspect ratio (long side / short side) s c b 0 : column location factor (40 : Internal, 30 : Edge, 0 : Corner) The ultimate shear is computed over the loaded area defined as the total panel area minus the critical area: V u w u A A w l l p 0 c u p The average steel depth is estimated by: d h 40 85 40 45 mm Internal column: Critical perimeter dimensions are: p c d 500 45 645 mm p c d 300 45 445 mm The perimeter length is therefore b (645 445) 80 mm 0 500 The column coefficients are c. 667 and s 40 (internal column) 300 V c as determined by () to (3) is: V c = Min (579.5, 63.83, 56.83) = 56.83 kn With shear strength reduction factor 0. 75, we find: V c 395. 5kN The ultimate shear is: V w ( 3.08 3.0) x(.5.5) 0.645 x0.445 34. kn u u 5 Vc V u The thickness is therefore OK Two-way shear ratio = 0.8 Edge column: Critical perimeter: p c d 500 45 645 mm p c 0.5d 00 300 7.5 00 47. 5mm The perimeter length with three sides is therefore b p p x47.5 645 590 mm 0 500 The column coefficients are c. 667 and s 30 (edge column) 300 V c is determined in the same manner as for a beam. We find that V c 88. kn The ultimate shear is obtained as: V w ( 3.08 3.0) x(.98 0.5) 0.645 x0.475 49. kn u u 8 Vc V u The thickness is therefore OK Two-way shear ratio = 0.5 5

RC-SAB output for shear check: RC-SAB software performs one-way and two-way shear checks for all columns and delivers the ratios Vu of demand to capacity. Ratio Vc A safe shear corresponds to a ratio less than or equal to unity. It also delivers detailed calculations for any user-chosen column. The following listing gives shear ratios for all columns. One and Two way shear check with following data Ultimate slab load (kn/m) =.960 Steel depth in X-direction dx (mm) = 55.0 Steel depth in Y-direction dy (mm) = 35.0 Average steel depth d (mm) = 45.0 Concrete strength f'c (MPa) = 5.00 One / Two way shear ratios for column X-Y = 0.4778 0.3779 One / Two way shear ratios for column X-Y = 0.466 0.4873 One / Two way shear ratios for column X3-Y = 0.466 0.4590 One / Two way shear ratios for column X4-Y = 0.466 0.4873 One / Two way shear ratios for column X5-Y = 0.4778 0.3779 One / Two way shear ratios for column X-Y = 0.69 0.595 One / Two way shear ratios for column X-Y = 0.99 0.809 One / Two way shear ratios for column X3-Y = 0.39 0.7740 One / Two way shear ratios for column X4-Y = 0.99 0.809 One / Two way shear ratios for column X5-Y = 0.69 0.595 One / Two way shear ratios for column X-Y3 = 0.69 0.595 One / Two way shear ratios for column X-Y3 = 0.99 0.809 One / Two way shear ratios for column X3-Y3 = 0.39 0.7740 One / Two way shear ratios for column X4-Y3 = 0.99 0.809 One / Two way shear ratios for column X5-Y3 = 0.69 0.595 One / Two way shear ratios for column X-Y4 = 0.4778 0.3779 One / Two way shear ratios for column X-Y4 = 0.466 0.4873 One / Two way shear ratios for column X3-Y4 = 0.466 0.4590 One / Two way shear ratios for column X4-Y4 = 0.466 0.4873 One / Two way shear ratios for column X5-Y4 = 0.4778 0.3779 Maximum shear ratio is: 0.8 Shear is OK It can be noted that for corner columns, one way-shear is more important. 6

The following listing includes details for the two studied columns One / Two way shear for column X4-Y Column section dimensions (mm) Cx = 500.0 Cy = 300.0 Shear panel dimensions (m) x = 6.0800 y = 4.7700 Critical perimeter dimensions (mm) Px = 645.0 Py = 445.0 Critical perimeter length (mm) b0 = 80.0 Ultimate two way shear force (kn) Vu = 34.3598 Nominal concrete shear strength = Min of following 3 equations Two way equations () to (3) = 579.567 63.8333 56.8333 Nominal two way shear strength = 56.8333 Design two way shear strength = 395.50 Two way shear ratio = 0.809 One way shear loaded area dimensions (m) =.350 6.0800 Ultimate one way shear force (kn) Vu = 53.499 Nominal one way shear strength (kn) Vc = 684.0000 Design one way shear strength (kn) = 53.0000 One way shear ratio = 0.99 One / Two way shear for column X5-Y Column section dimensions (mm) Cx = 500.0 Cy = 300.0 Shear panel dimensions (m) x = 6.0800 y =.300 Critical perimeter dimensions (mm) Px = 645.0 Py = 47.5 Critical perimeter length (mm) b0 = 590.0 Ultimate two way shear force (kn) Vu = 49.73 Nominal concrete shear strength = Min of following 3 equations Two way equations () to (3) = 4.6750 454.9375 384.500 Nominal two way shear strength = 384.500 Design two way shear strength = 88.875 Two way shear ratio = 0.595 One way shear loaded area dimensions (m) = 6.0800.6950 Ultimate one way shear force (kn) Vu = 6.4 Nominal one way shear strength (kn) Vc = 684.0000 Design one way shear strength (kn) = 53.0000 One way shear ratio = 0.69. The results are identical to those obtained previously. These numerical results are similar to the previous analytical ones. RC-SAB also displays frames in both directions highlighting column and middle strips. It also displays critical shear perimeters in one-way and two-way shear. 7

RC-SAB display of X-frames with column strips highlighted 8

RC-SAB display of one-way shear critical zones 9

(4) Compute the moments in the slab strips along column line X (Frame X) Frame X (in X-direction) has three spans and includes four columns (supports). The panel layout is determined by lines mid-way between column lines. is the length of the panel (X-direction) and is its width (Y-direction). n n is the clear length. The static moment is each span is given by: M 0 wu. 8 Negative and positive moments in each span as well as portions of moments in column strips, are deduced using appropriate coefficients. The following Table gives all the results. Span XY- XY Span XY- XY3 Span XY3- XY4 (m) 5.5 6.0 5.5 (m) 4.5 4.5 4.5 n (m) 5. 5.5 5. M 0 (kn.m) 65.3 9. 65.3 CS width (m) 0.5 min.5.5.5 Moment coefficients -0.6 0.5-0.70-0.65 0.35-0.65-0.70 0.5-0.6 -ve and +ve moments -43.0 86.0-5.7-4.9 67.3-4.9-5.7 86.0-43.0 CS moment (%) 00 60 75 75 60 75 75 60 00 CS moments (kn.m) -43.0 5.6-86.8-93.7 40.4-93.7-86.8 5.6-43.0 MS moments (kn.m) 0 34.4-8.9-3. 6.9-3. -8.9 34.4 0 Distribution of factored static moment (a) (b) (c) (d) (e) Int. negative M 0.75 0.70 0.70 0.70 0.65 Positive M 0.63 0.57 0.5 0.50 0.35 Ext. negative M 0 0.6 0.6 0.30 0.65 (a): Exterior edge unrestrained (b): Slab with beams between all supports (c): Slab with no beams at all (d): Slab with edge beams only (e): Slab with exterior edge fully restrained (interior span) Column (c) was used for the external spans and column (e) was used for the internal span. 30

RC-SAB analysis output 3

(5) Design column strip for the maximum interior negative moment The moment is 93.75 kn.m and the section dimensions are: b = 50 mm, and h = 85 mm Using 4-mm bars, bar area and steel depth are: 4 4 A b 53.9 mm d = h cover (d b /) = 85 0 7 = 58 mm We find that the required steel area is A s = 645 mm It is greater than the minimum steel area given by: A smin = 0.008 bh = 0.008 x 50 x 85 = 749.3 mm Maximum spacing is S Minh, 300 300 mm max The corresponding minimum steel is: Minimum bar number A A b 53.9 x 50 b s min 54. 5 mm. Smax 300 Max As min, As min 54.5 N b min 7.5 That is N bmin = 8 bars A 53.9 As 645.0 Required bar number N b 0. 7 That is N b = bars A 53.9 b b RC-SAB design output 3

Part B: Two way slabs with beams X4 X3 CS Frame X3 X X Y Y Y3 Y4 Y5 The direct design method can again be used provided its conditions are satisfied. Moment distribution is affected by beam presence. In each frame, the longitudinal beam is part of the column strip and contributes to the flexural rigidity. The frame is also affected by the torsion rigidity of the transverse edge beams. The effective beam section is a T-section for internal beams and -section for edge beams. -Section T-Section Flexural effect of longitudinal beams in edge / internal frame Torsion effect of transverse edge beams 33

Relative flexural stiffness of longitudinal beams The flexural effect of the beam is related to its relative beam stiffness compared to slab stiffness. The relative beam stiffness is defined as: E E b s I I b s Using the same concrete for beams and slabs leads to: I I b s The beam moment of inertia is computed by considering the effective beam section which is a T-section for internal beams and -section for edge beams. The slab moment of inertia is computed by considering a rectangular section defined by the panel (frame) width and slab thickness. For the beam section, the extra distance to be added (on one side for the -section and on both sides of the T-section), is equal to Min (h w, 4h s ). Min(h w, 4h s ) Min(h w, 4h s ) Min(h w, 4h s ) h s h w h h s h w b w b w Example 5: Compute for the edge beam shown. 6.0 m 400 mm 00 mm 300 mm Beam section and inertia The extra distance to be added to the -section is equal to Min (h w, 4h s ) = Min (00, 800) = 00 mm The -section dimensions are therefore as shown. The centroid Y-coordinate (from bottom base) is: AY AY 60000 x00 00000 x300 Y g 5. 0mm A A 60000 00000 The centroid moment of inertia is therefore: I b 300 00 00 400 00 mm 3 3 300 x 00 500 x 00 mm I b I b 60000 x(5 00) 00000 x(300 5) 300 I b I 5 5 8 4 b I b 375 x0 8958.333 x0 0.333 x0 mm 34

Slab section and inertia The slab panel (frame) width is equal to half distance between the beams plus the offset (half beam section): The slab inertia is therefore: I s 350 x 00 3.0 x0 8 mm I b 0.333 The beam relative stiffness is: 0. 968 I.0 4 s The beam relative stiffness may also be obtained using the chart in Figure 3. of the Textbook. It is given as: a = 400 mm (beam thickness) h = 00 mm (slab thickness) a h.0 3 b a a b f The chart gives f in function of and l h h h b and. 5 h 3 b = 300 mm (beam width) l = 350 mm (slab width) 3000 + 50 = 350 mm we read from the chart that f =.7 300 400 Thus.7 0. 968 we find the same result 350 00 00 mm Relative torsion stiffness of transverse edge beams The distribution of external negative moment (in external spans) depends also on the relative torsion stiffness of the transverse edge beams. This is defined as: Using the same concrete for beams and slabs leads to: t EbC t E I C is the torsional constant of the edge beam, roughly equal to the polar moment of inertia. It is determined x x by dividing the cross section (-section) in rectangles as: 3 y C 0.63 y 3 where x and y are the shorter and longer sides respectively of the rectangular section. The subdivision leading to the largest value of C must be used. The slab moment of inertia I s is computed as before. C I s s s 35

Two possible subdivisions of -section in rectangles for torsion constant C Example: Compute torsion constant C for the previous edge beam 500 300 00 00 00 mm 400 00 300 mm (a) (b) Subdivision (a): Subdivision (b): C C 00 00 300 300 3 00 00 500 500 3 3 3 6 4 0.63 0.63 46.333 x0 mm 300 300 400 400 3 00 00 00 00 3 3 3 6 4 0.63 0.63 096.333 x0 mm Therefore C 6 4 4 096.333 x0 mm 0.00096333 m Beam relative torsion stiffness is then obtained by: A typical frame would have a constant flexural stiffness ( ) in all its spans and two different torsion relative stiffnesses ( ) at its two end spans (resulting from the two transverse edge beams). t An edge beam with a given torsion constant C generates different values of the relation torsion stiffness in the various frames it crosses. t C I s 36

Thickness of slabs with beams The thickness for each slab panel depends on the average beam relative stiffness m which is the average of the values for the four beams of the panel. The minimum thickness is determined as follows: m 3 4 4 (a) 0. : Use minimum thickness Table 3. for flat plate (and slabs without interior beams) m (b) 0.. 0 : m h h min min f y n 0.8 500 36 5 ( 0.) 0 mm m Equation (3.0) (c). 0 : m h h min min n 90 mm f y 0.8 500 36 9 Equation (3.) n is the maximum clear length of the panel and is the clear length ratio (Max n / Min n ) n Max n, n Max Min n n,, n n Column strip moments With the presence of beams, the column strip portions of the moments change as compared to flat plates. The CS portion of moments depends on and is the panel length in the studied direction and is the panel length in the other direction. is the value of in direction of. For exterior negative moments, the distribution depends also on the torsion parameter t.. 37

Portion (%) of column strip moment in slabs with beams Interior negative moment Positive moment Exterior negative moment 0.5.0.0 0.5.0.0 0.5.0.0 0.0 75 75 75 60 60 60 90 75 45 90 75 45 00 00 00 t 0 75 75 75 t. 5 00 00 00 t 0 90 75 45 t. 5 inear interpolation must be performed at intermediate points It can be noted the flat plate portions are retrieved if beam stiffness is equal to zero. Distribution of column strip moments over the beam and slab Column strip moments are divided between the beam and the slab according to the value of : If. 0 then the beam takes 85 % of the CS moment and the slab takes 5 %. If. 0 a linear interpolation is performed of (between 0 and 85 % for beam part). Beam direct loading It must be pointed out that the beam moments must be added to those caused by direct loading on beams. Beam loading causes moments in the beam only. These are determined using the same approach, by computing the static moment, then deducing positive and negative moments using the same coefficients. Beam direct loading includes the weight of the beam web (not considered in the slab load) and any possible wall loading. It should lastly be signaled that beam presence cancels the risk of punching shear. 38

Two way solid slab example 6: Slab with beams 5.5 m 5.75 7 m 7 m 5.5 m X5 5.775 6 m X4 Frame X4 7 m X3 7 m X Y Y Y3 Y4 Y5 The figure shows a two-way slab with beams. The slab thickness is 70 mm. All columns have the same square section 450 x 450 mm and the beams have the same section with a width of 450 mm and a total thickness of 450 mm. Concrete: f ' c 3 5 MPa c 4 kn/ m Steel: f y 40 MPa Superimposed dead load SD =.4 kn/m ive load = 3.8 kn/m All edge beams support a wall with a weight of 4.5 kn/m The span ratio ( max / min ) is less than two for all panels (two way action). Factored slab load w u.4(4 x0.70.4).7 x3.8 4.3kN/ m The slab has a vertical axis of symmetry (column line Y3) so only half of the floor panels are considered. / Check the slab thickness The minimum slab thickness depends on the average beam relative stiffness m. For each of the four beams of a panel, depends on the beam section (-section for edge beam and T-section for internal beam) and on the slab panel width. In all, there are four different panels to consider for the determination of the minimum slab thickness: (a): Corner panel, with two edge beams and two internal beams (b): Vertical edge panel, with one edge beam and three internal beams 39

(c): Horizontal edge panel, with one edge beam and three internal beams (d): Internal panel, with four internal beams Effective sections of beams The added offset from the basic rectangular shape is equal to Minh, 4h Min80,680 mm The cross sections of the internal beam and edge beams are shown. w s 80 80 450 80 450 80 70 80 mm 450 70 80 mm 450 mm 450 mm Internal beam Edge beam I b The beam relative stiffness is determined by computing the moments of inertia for both beam and I s slab sections (about centroid axis) or by using the chart in Fig. 3. of the Textbook. We find for beams: Internal beam: 9 4 I b 4.957 x0 mm Edge beam: I b 9 4 4.87 x0 mm The slab section inertia is 3 bshs I s where b s is the width of the panel and h s = 70 mm There are two different slab panel widths for the edge beams (lines Y and X5) and three slab panel widths for the internal beam (lines Y, Y3 and X4, line X3 is similar to line Y3) Slab panel width along edge line Y: This gives for edge beam along line Y: 3. 364 Slab panel width along edge line X5: This gives for edge beams along line X5: 3. 658 Slab panel width along internal line Y: This gives for internal beams along line Y:. 880 5775 450 b s 3. 5mm thus 575 450 b s 86. 5mm thus 5775 7000 b s 6387. 5mm thus I s I s I s 9 4.743 x0 mm 9 4.70 x0 mm 9 4.65 x0 mm Slab panel width along internal lines Y3 and X3: b s = 7000 mm thus I s 9 4.8659 x0 mm This gives for internal beams along lines Y3 and X3:. 75 Slab panel width along internal line X4: 575 7000 b s 637. 5mm thus I s 9 4.58 x0 mm 40

This gives for internal beams along lines X4:. 956 For example, for the internal panel, the average beam relative stiffness is: m.880.75.75.956.865 4 The minimum thickness is defined as follows: (a) 0. : Use minimum thickness Table 3. for flat plate m (b) 0.. 0 : m h h min min f y n 0.8 500 36 5 ( 0.) 0 mm m Equation (3.0) (c). 0 : m h h min min n 90 mm f y 0.8 500 36 9 Equation (3.) n is the maximum clear length of the panel and is the clear length ratio (Max n / Min n ) n Max n, n Min The next Table gives all the calculations leading to minimum thickness for the four panels Max n n,, n n Panel Corner Horizontal edge Vertical edge Internal n (mm) 575-450 = 485 n (mm) 5775-450 = 535 7000-450 = 6550 5775-450 = 535 575-450 = 485 7000-450 = 6550 7000-450 = 6550 7000-450 = 6550 n 535 6550 6550 6550.04.30.358.000.75.9.30.87 m h min equation 3., ACI 9.3 3., ACI 9.3 3., ACI 9.3 3.0, ACI 9. h min value (mm) 5.0 50.9 46.70 60.46 The minimum slab thickness is therefore 60.46 mm. The actual thickness of 70 mm is therefore OK. 4

The next figure shows the thickness check using RC-SAB software. Minimum thickness and average beam relative stiffness, are displayed for each panel Thickness check using RC-SAB software 4

RC-SAB delivers also values of beam relative stiffnesses as well as torsion constants for edge beams and performs thickness check for beams, as shown in the next listing. Relative beam flexural stiffnes Alpha = Ib/Is where Ib and Is are the beam and slab moment of inertia The slab width is that of the frame. Edge beams have torsional constants C Each frame has torsional relative parameters Beta-t = C/(Is) C is the torsion constant for the perpendicular edge beams X-Frames: Alpha Beta-t (left) Beta-t (right) Frame X: 3.364.095.095 Frame X:.880.0.0 Frame X3:.75 0.93 0.93 Frame X4:.880.0.0 Frame X5: 3.364.095.095 Y-Frames: Alpha Beta-t (bot) Beta-t (top) Frame Y: 3.658.78.78 Frame Y:.956.063.063 Frame Y3:.75 0.93 0.93 Frame Y4:.956.063.063 Frame Y5: 3.658.78.78 Edge beam torsion constant C (m4) Bottom edge X-beam: C = 0.00534059 Top edge X-beam: C = 0.00534059 eft edge Y-beam: C = 0.00534059 Rigth edge Y-beam: C = 0.00534059 X-Beams: Real thickness (mm) = 450.00 Minimum thickness (mm) = 333.33 Y-Beams: Real thickness (mm) = 450.00 Minimum thickness (mm) = 333.33 All Beam thicknesses OK 43

/ Analysis of internal frame X4 Because of symmetry (about column line Y3), only two spans are considered. The figure below shows the panel dimensions as well the column strip widths for panel X4Y-X4Y and panel X4Y-X4Y3 5.75 m 7 m 7 m 5.75 m CS strip.6375 m CS strip 3.9375 m X4.8875 3.5 = 6.3875 n = 4.85 m n = 6.55 m Y Y Y3 Y4 Y5 For each panel, the dimensions, the clear length and the static moment are computed. is the length of the panel (parallel to line X4) and is its width. n is the clear length. n The static moment in each span is given by: M 0 wu. 8 Negative and positive moments in each span as well as portions of moments in column strips, are deduced using appropriate coefficients. The CS moments must then be distributed between the beam and the slab. The following Table gives all the results. Direct beam load includes the beam web self weight and any possible wall weight. For beam X4, there is no wall. The factored beam web weight is wbeam.4 cbhw.4 x4 x0.45 x0.8 4.34kN/ m Beam loading causes moments in the beam only. These are determined using the same approach, by computing the static moment, then deducing positive and negative moments using the same moment coefficients of row R0. For each span, the beam static moment is computed as: M w 0b beam n 8 44

Moments along Frame X4 R Span X3Y-X3Y Span X3Y-X3Y3 R (m) 5.75 7.0 R3 (m) 6.3875 6.3875 R4 n (m) 4.85 6.55 R5 M 0 (kn.m) 6.7 484.0 R6 CS width (m) 0.5 min.6375 3.9375 R7.88.88 R8.7.7 R9.0 t R0 Moment coefficients -0.6 0.57-0.70-0.65 0.35-0.65 R -ve and +ve moments -4.0 49.7-83.9-34.6 69.4-34.6 R CS moment (%) 87. 68.7 68.7 77.6 77.6 77.6 R3 CS moments (kn.m) -36.6 0.8-6.3-44. 3.5-44. R4 Slab portion (5 %) -5.5 5.4-8.9-36.6 9.7-36.6 R5 Beam portion (85 %) -3. 87.4-07.4-07.6.8-07.6 R6 Beam static moment.3.7 R7 -ve / +ve M add in beams -.0 7.0-8.6-4.8 8.0-4.8 R8 Total beam moments -33. 94.4-6.0 -.4 9.8 -.4 The moment coefficients in row R0 are determined using static moment distribution Table: Distribution of factored static moment (a) (b) (c) (d) (e) Int. negative M 0.75 0.70 0.70 0.70 0.65 Positive M 0.63 0.57 0.5 0.50 0.35 Ext. negative M 0 0.6 0.6 0.30 0.65 (a): Exterior edge unrestrained (b): Slab with beams between all supports (c): Slab with no beams at all (d): Slab with edge beams only (e): Slab with exterior edge fully restrained (interior span) For the first span, we use column (b) and for the internal span we use column (e). 45

The CS portion of moments is computed by interpolating using Tables below, according to and. For exterior negative moments, the distribution depends also on the torsion parameter t. For frame X4,. 0. t For the exterior moment for instance, the double interpolation gives a portion of 87. % for the CS moment. The CS moments are divided between the beam and the slab according to. This value is greater than.0 in all cases. Therefore, 85 % of the CS moments are assigned to the beam and 5 % to the slab. The total beam moments are obtained by adding the slab loading moments (R5) and the beam load moments (R7). Portion (%) of column strip moment Interior negative moment Positive moment Exterior negative moment 0.5.0.0 0.5.0.0 0.5.0.0 0.0 75 75 75 60 60 60 90 75 45 90 75 45 00 00 00 t 0 75 75 75 t. 5 00 00 00 t 0 90 75 45 t. 5 inear interpolation must be performed at intermediate points 46

RC-SAB analysis results of frame X4 47

The next figure shows RC design results of the slab part in the column strip using RC-SAB software. 48

The software displays the variable slab model width as well as bar cutoff. The minimum bar number is also variable (9 in end spans and in internal spans). Middle strips are designed as in flat plates by considering the contributions from the two adjacent frames. 3/ Analysis of edge frame X5 A similar procedure is used for this edge frame. X5 5.5 m 5.75 7 m 7 m 5.5 m 5.775 6 m X4 7 m X3 7 m X Y Y Y3 Y4 Y5 The first particularity is that when computing ratios and considered is equal to the full edge panel width, which is 5.775 mm., the panel width to be The second particularity is that beam loading includes the web weight as well as the wall weight: wbeam.4( bhw wwall ).4(4 x0.45 x0.8 4.5) 0.534 kn/ m For frame X5, the torsion relative stiffness is:. 0 Column strip width is equal to 4 min Offset The same steps are used and the results are presented in a tabular form. t For middle strips, the moments are determined by analyzing the two bounding frames and summing the contributions from both frames. 49

Moments in frame X5 R Span X5Y-X5Y Span X5Y-X5Y3 R (m) 5.75 7.0 R3 (m) 3.5 3.5 R4 ' (m) for / 5.775 5.775 R5 n (m) 4.85 6.55 R6 M 0 (kn.m) 8.0 35.9 R7 CS width (m).54375.66875 R8 3.364 3.364 R9 ' 3.68.78 R0.0.0 t R Moment coefficients -0.6 0.57-0.70-0.65 0.35-0.65 R -ve and +ve moments -0.5 73.0-89.6-53.3 8.6-53.3 R3 CS moment (%) 76.7 7. 7. 80.3 80.3 80.3 R4 CS moments (kn.m) -5.7 5.7-64.7-3. 66.3-3. R5 Slab portion (5 %) -.4 7.9-9.7-8.5 9.9-8.5 R6 Beam portion (85 %) -3.3 44.8-55.0-04.6 56.4-04.6 R7 Beam static moment 30.65 56.49 R8 -ve / +ve M add in beams -4.9 7.5 -.5-36.7 9.8-36.7 R9 Total beam moments -8. 6.3-76.5-4.3 76. -4.3 50

Analysis of frame X5 using RC-SAB software 5

RC design of beam X5 using RC-SAB (-section) 5