4Shear Strength of Slabs In two-way floor systems, the slab must have adequate thickness to resist both bending moments and shear forces at critical section. There are three cases to look at for shear.1.2.3.Two-way Slabs supported on beamsTwo-Way Slabs without beamsShear Reinforcement in two-way slabs without beams.
5Shear Strength of Slabs Two-way slabs supported on beamsThe critical location is found at d distance from the column, whereThe supporting beams are stiff and are capable of transmitting floor loads to the columns.
6Shear Strength of Slabs The shear force is calculated using the triangular and trapezoidal areas. If no shear reinforcement is provided, the shear force at a distance d from the beam must equalwhere,
7Shear Strength of Slabs Two-Way Slabs without beamsThere are two types of shear that need to be addressed1.2.One-way shear or beam shear at distance d from the columnTwo-way or punch out shear which occurs along a truncated cone.
8Shear Strength of Slabs One-way shear or beam shear at distance d from the columnTwo-way or punch out shear which occurs along a truncated cone.1.2.
9Shear Strength of Slabs One-way shear considers critical section a distance d from the column and the slab is considered as a wide beam spanning between supports.
10Shear Strength of Slabs Two-way shear fails along a a truncated cone or pyramid around the column. The critical section is located d/2 from the column face, column capital, or drop panel.
11Shear Strength of Slabs If shear reinforcement is not provided, the shear strength of concrete is the smaller of:perimeter of the critical section ratio of long side of column to short sidebo = bc =
12Shear Strength of Slabs If shear reinforcement is not provided, the shear strength of concrete is the smaller of:as is 40 for interior columns, 30 for edge columns, and 20 for corner columns.
13Shear Strength of Slabs Shear Reinforcement in two-way slabs without beams.For plates and flat slabs, which do not meet the condition for shear, one can either- Increase slab thickness - Add reinforcementReinforcement can be done by shearheads, anchor bars, conventional stirrup cages and studded steel strips.
14Shear Strength of Slabs Shearheadconsists of steel I-beams or channel welded into four cross arms to be placed in slab above a column. Does not apply to external columns due to lateral loads and torsion.
15Shear Strength of Slabs Anchor barsconsists of steel reinforcement rods or bent bar reinforcement
16Shear Strength of Slabs Conventional stirrup cages
18Shear Strength of Slabs The reinforced slab follows section in the ACI Code, where Vn can notThe spacing, s, can not exceed d/2.If a shearhead reinforcement is provided
19Example ProblemDetermine the shear reinforcement required for an interior flat panel considering the following: Vu= 195k, slab thickness = 9 in., d = 7.5 in., fc = 3 ksi, fy= 60 ksi, and column is 20 x 20 in.
20Example ProblemCompute the shear terms find b0 for
21Example Problem Compute the maximum allowable shear Vu =195 k > k Shear reinforcement is need!
22Example Problem Compute the maximum allowable shear So fVn >Vu Can use shear reinforcement
23Example ProblemUse a shear head or studs as in inexpensive spacing. Determine the a for
24Example Problem Determine the a for The depth = a+d = 41.8 in in. = 49.3 in. 50 in.
25Example Problem Determine shear reinforcement The fVs per side is fVs / 4 = k
26Example Problem Determine shear reinforcement Use a #3 stirrup Av = 2(0.11 in2) = 0.22 in2
27Example Problem Determine shear reinforcement spacing Maximum allowable spacing is
28Example Problem Use s = 3.5 in. The total distance is 15(3.5 in.)= 52.5 in.
29Example Problem The final result: 15 stirrups at total distance of 52.5 in. So that a = 45 in. and c = 20 in.
30Direct Design Method for Two-way Slab Method of dividing total static moment Mo into positive and negative moments.Limitations on use of Direct Design method1.2.Minimum of 3 continuous spans in each direction. (3 x 3 panel)Rectangular panels with long span/short span
31Direct Design Method for Two-way Slab Limitations on use of Direct Design method3.4.Successive span in each direction shall not differ by more than 1/3 the longer span.Columns may be offset from the basic rectangular grid of the building by up to 0.1 times the span parallel to the offset.
32Direct Design Method for Two-way Slab Limitations on use of Direct Design method5.6.All loads must be due to gravity only (N/A to unbraced laterally loaded frames, from mats or pre-stressed slabs)Service (unfactored) live load service dead load
33Direct Design Method for Two-way Slab Limitations on use of Direct Design method7.For panels with beams between supports on allsides, relative stiffness of the beams in the 2perpendicular directions.Shall not be less than 0.2 nor greater than 5.0
34Definition of Beam-to-Slab Stiffness Ratio, a Accounts for stiffness effect of beams located along slab edge reduces deflections of panel adjacent to beams.
35Definition of Beam-to-Slab Stiffness Ratio, a With width bounded laterally by centerline of adjacent panels on each side of the beam.
36Two-Way Slab Design Static Equilibrium of Two-Way Slabs Analogy of two-way slab to plank and beam floorSection A-A:Moment per ft width in planksTotal Moment
37Two-Way Slab Design Static Equilibrium of Two-Way Slabs Analogy of two-way slab to plank and beam floorUniform load on each beamMoment in one beam (Sec: B-B)
38Two-Way Slab Design Static Equilibrium of Two-Way Slabs Total Moment in both beamsFull load was transferred east-west by the planks and then was transferred north-south by the beams;The same is true for a two-way slab or any other floor system.
39Basic Steps in Two-way Slab Design 1.2.3.Choose layout and type of slab.Choose slab thickness to control deflection. Also, check if thickness is adequate for shear.Choose Design methodEquivalent Frame Method- use elastic frame analysis to compute positive and negative momentsDirect Design Method - uses coefficients to compute positive and negative slab moments
40Basic Steps in Two-way Slab Design 188.8.131.52.8.Calculate positive and negative moments in the slab.Determine distribution of moments across the width of the slab. - Based on geometry and beam stiffness.Assign a portion of moment to beams, if present.Design reinforcement for moments from steps 5 and 6.Check shear strengths at the columns
41Minimum Slab Thickness for two-way construction Maximum Spacing of ReinforcementAt points of max. +/- M:Min Reinforcement Requirements
42Distribution of Moments Slab is considered to be a series of frames in two directions:
43Distribution of Moments Slab is considered to be a series of frames in two directions:
44Distribution of Moments Total static Moment, Mowhere
45Column Strips and Middle Strips Moments vary across width of slab panelDesign moments are averaged over the width of column strips over the columns & middle strips between column strips.
46Column Strips and Middle Strips Column strips Design w/width on either side of a column centerline equal to smaller ofl1= length of span in direction moments are being determined.l2= length of span transverse to l1
47Column Strips and Middle Strips Middle strips: Design strip bounded by two column strips.
48Positive and Negative Moments in Panels M0 is divided into + M and -M Rules given in ACI sec
50Positive and Negative Moments in Panels M0 is divided into + M and -M Rules given in ACI sec
51Longitudinal Distribution of Moments in Slabs For a typical interior panel, the total static moment is divided into positive moment 0.35 Mo and negative moment of Mo.For an exterior panel, the total static moment is dependent on the type of reinforcement at the outside edge.
53Moment DistributionThe factored components of the moment for the beam.
54Transverse Distribution of Moments The longitudinal moment values mentioned are for the entire width of the equivalent building frame. The width of two half column strips and two half-middle stripes of adjacent panels.
55Transverse Distribution of Moments Transverse distribution of the longitudinal moments to middle and column strips is a function of the ratio of length l2/l1,a1, and bt.
56Transverse Distribution of Moments Transverse distribution of the longitudinal moments to middle and column strips is a function of the ratio of length l2/l1,a1, and bt.torsional constant
57Distribution of M0 ACI Sec 184.108.40.206 For spans framing into a common support negative moment sections shall be designed to resist the larger of the 2 interior Mu’sACI SecEdge beams or edges of slab shall be proportioned to resist in torsion their share of exterior negative factored moments
58Factored Moment in Column Strip Ratio of flexural stiffness of beam to stiffness of slab in direction l1.Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)bt=
62Factored Moment in Column Strip Ratio of flexural stiffness of beam to stiffness of slab in direction l1.Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)bt=
63Factored Moment in Column Strip Ratio of flexural stiffness of beam to stiffness of slab in direction l1.Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)bt=
64Factored Moment in Column Strip Ratio of flexural stiffness of beam to stiffness of slab in direction l1.Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)bt=
65Factored Moments Factored Moments in beams (ACI Sec. 13.6.3) Resist a percentage of column strip moment plus moments due to loads applied directly to beams.
66Factored Moments Factored Moments in Middle strips (ACI Sec. 13.6.3) The portion of the + Mu and - Mu 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 2 half middle strips.
67ACI Provisions for Effects of Pattern Loads The maximum and minimum bending moments at the critical sections are obtained by placing the live load in specific patterns to produce the extreme values. Placing the live load on all spans will not produce either the maximum positive or negative bending moments.
68ACI Provisions for Effects of Pattern Loads 1.2.3.The ratio of live to dead load. A high ratio will increase the effect of pattern loadings.The ratio of column to beam stiffness. A low ratio will increase the effect of pattern loadings.Pattern loadings. Maximum positive moments within the spans are less affected by pattern loadings.
69Reinforcement Details Loads After all percentages of the static moments in the column and middle strip are determined, the steel reinforcement can be calculated for negative and positive moments in each strip.
70Reinforcement Details Loads Calculate Ru and determine the steel ratio r, where f =0.9. As = rbd. Calculate the minimum As from ACI codes. Figure is used to determine the minimum development length of the bars.
71Minimum extension for reinforcement in slabs without beams(Fig. 13. 3
72Moment DistributionThe factored components of the moment for the beam.
73Transverse Distribution of Moments The longitudinal moment values mentioned are for the entire width of the equivalent building frame. The width of two half column strips and two half-middle stripes of adjacent panels.