6 Flexural Component Design

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Published on January 4, 2008

Author: Yuan

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PCI 6th Edition:  PCI 6th Edition Flexural Component Design Presentation Outline:  Presentation Outline What’s new to ACI 318 Gravity Loads Load Effects Concrete Stress Distribution Nominal Flexural Strength Flexural Strength Reduction Factors Shear Strength Torsion Serviceability Requirements New to ACI 318 – 02:  New to ACI 318 – 02 Load Combinations Stress limits Member Classification Strength Reduction factor is a function of reinforcement strain Minimum shear reinforcement requirements Torsion Design Method Load Combinations:  Load Combinations U = 1.4 (D + F) U = 1.2 (D + F + T) + 1.6 (L + H) + 0.5 (Lr or S or R) U = 1.2D + 1.6 (Lr or S or R) + (1.0L or 0.8W) U = 1.2D + 1.6W + 1.0L + 0.5(Lr or S or R) U = 1.2D + 1.0E + 1.0L + 0.2S U= 0.9D + 1.6W + 1.6H U= 0.9D + 1.0E + 1.6H Comparison of Load Combinations:  Comparison of Load Combinations U=1.2D + 1.6 L 2002 U= 1.4D + 1.7L 1999 If L=.75D i.e. a 10% reduction in required strength Classifications:  Classifications No Bottom Tensile Stress Limits Classify Members Strength Reduction Factor Tension-Controlled Transition Compression Controlled Three Tensile Stress Classifications Class U – Un-cracked Class T – Transition Class C – Cracked Slide7:  Copied from ACI 318 2002, ACI 318-02 table R18.3.3 Class C Members:  Class C Members Stress Analysis Based on Cracked Section Properties No Compression Stress limit No Tension Stress limit Increase awareness on serviceability Crack Control Displacements Side Skin Reinforcement Minimum Shear Reinforcing:  Minimum Shear Reinforcing 1999 2002 System Loads:  System Loads Gravity Load Systems Beams Columns Floor Member – Double Tees, Hollow Core Spandrels Tributary Area Floor members, actual top area Beams and spandrels Load distribution Load path Floor members  spandrels or beams  Columns Live Load Reduction:  Live Loads can be reduced based on: Where: KLL = 1 Lo = Unreduced live load and At = tributary area Live Load Reduction Live Load Reduction:  Live Load Reduction Or the alternative floor reduction shall not exceed or Where: R = % reduction ≤ 40% r = .08 Member Shear and Moment:  Member Shear and Moment Shear and moments on members can be found using statics methods and beam tables from Chapter 11 Strength Design:  Strength Design Strength design is based using the rectangular stress block The stress in the prestressing steel at nominal strength, fps, can be determined by strain compatibility or by an approximate empirical equation For elements with compression reinforcement, the nominal strength can be calculated by assuming that the compression reinforcement yields. Then verified. The designer will normally choose a section and reinforcement and then determine if it meets the basic design strength requirement: Concrete Stress Distribution:  Concrete Stress Distribution Parabolic distribution Equivalent rectangular distribution Stress Block Theory:  Stress Block Theory Stress-Strain relationship is not constant f’c=3,000 psi f’c=6,000 psi Stress Block Theory:  Stress Block Theory Stress-Strain relationship Stress-strain can be modeled by: Where :strain at max. stress and :max stress Stress Block Theory:  Stress Block Theory The Whitney stress block is a simplified stress distribution that shares the same centroid and total force as the real stress distribution = Equivalent Stress Block – b1 Definition:  Equivalent Stress Block – b1 Definition b1 = 0.85 when f’c < 3,000 psi b1 = 0.65 when f’c > 8,000 psi Design Strength:  Design Strength Mild Reinforcement – Non - Prestressed Prestress Reinforcement Strength Design Flowchart:  Strength Design Flowchart Figure 4.2.1.2 page 4-9 Non-Prestressed Path Prestressed Path Non-Prestressed Members:  Non-Prestressed Members Find depth of compression block Depth of Compression Block:  Depth of Compression Block Where: As is the area of tension steel A’s is the area of compression steel fy is the mild steel yield strength Assumes compression steel yields Flanged Sections:  Flanged Sections Checked to verify that the compression block is truly rectangular Compression Block Area:  Compression Block Area If compression block is rectangular, the flanged section can be designed as a rectangular beam = = Compression Block Area:  Compression Block Area If the compression block is not rectangular (a> hf), = To find “a” Determine Neutral Axis:  Determine Neutral Axis From statics and strain compatibility Check Compression Steel:  Check Compression Steel Verify that compression steel has reached yield using strain compatibility Compression Comments:  Compression Comments By strain compatibility, compression steel yields if: If compression steel has not yielded, calculation for “a” must be revised by substituting actual stress for yield stress Non prestressed members should always be tension controlled, therefore c / dt < 0.375 Add compression reinforcement to create tesnion controlled secions Moment Capacity:  Moment Capacity 2 equations rectangular stress block in the flange section rectangular stress block in flange and stem section Slide31:  Strength Design Flowchart Figure 4.2.1.2 page 4-9 Non- Prestressed Path Prestressed Path Stress in Strand:  Stress in Strand fse - stress in the strand after losses fpu - is the ultimate strength of the strand fps - stress in the strand at nominal strength Stress in Strand:  Stress in Strand Typically the jacking force is 65% or greater The short term losses at midspan are about 10% or less The long term losses at midspan are about 20% or less Stress in Strand:  Stress in Strand Nearly all prestressed concrete is bonded Stress in Strand :  Stress in Strand Prestressed Bonded reinforcement gp = factor for type of prestressing strand, see ACI 18.0 = .55 for fpy/fpu not less than .80 = .45 for fpy/fpu not less than .85 = .28 for fpy/fpu not less than .90 (Low Relaxation Strand) rp = prestressing reinforcement ratio Determine Compression Block:  Determine Compression Block Compression Block Height:  Compression Block Height Where Aps - area of prestressing steel fps - prestressing steel strength Flange Sections Check:  Flange Sections Check Compression Steel Check:  Compression Steel Check Verify that compression steel has reached yield using strain compatibility Moment Capacity:  Moment Capacity 2 Equations rectangular stress block in flange section rectangular stress block in flange and stem section Flexural Strength Reduction Factor:  Flexural Strength Reduction Factor Based on primary reinforcement strain Strain is an indication of failure mechanism Three Regions Member Classification:  Member Classification On figure 4.2.1.2 Compression Controlled:  Compression Controlled e < 0.002 at extreme steel tension fiber or c/dt > 0.600 = 0.70 with spiral ties = 0.65 with stirrups Tension Controlled:  Tension Controlled e > 0.005 at extreme steel tension fiber, or c/dt < 0.375 f = 0.90 with spiral ties or stirrups Transition Zone:  Transition Zone 0.002 < e < 0.005 at extreme steel tension fiber, or 0.375 < c/dt < 0.6 f = 0.57 + 67(e) or f = 0.48 + 83(e) with spiral ties f = 0.37 + 0.20/(c/dt) or f = 0.23 + 0.25/(c/dt) with stirrups Strand Slip Regions:  Strand Slip Regions ACI Section 9.3.2.7 ‘where the strand embedment length is less than the development length’ f =0.75 Limits of Reinforcement:  Limits of Reinforcement To prevent failure immediately upon cracking, Minimum As is determined by: As,min is allowed to be waived if tensile reinforcement is 1/3 greater than required by analysis Limits of Reinforcement:  Limits of Reinforcement The flexural member must also have adequate reinforcement to resist the cracking moment Where Section after composite has been applied, including prestress forces Correction for initial stresses on non-composite, prior to topping placement Critical Sections:  Critical Sections Horizontal Shear:  Horizontal Shear ACI requires that the interface between the composite and non-composite, be intentionally roughened, clean and free of laitance Experience and tests have shown that normal methods used for finishing precast components qualifies as “intentionally roughened” Horizontal Shear, Fh Positive Moment Region:  Horizontal Shear, Fh Positive Moment Region Based on the force transferred in topping (page 4-53) Horizontal Shear, Fh Negative Moment Region:  Horizontal Shear, Fh Negative Moment Region Based on the force transferred in topping (page 4-53) Unreinforced Horizontal Shear:  Unreinforced Horizontal Shear Where f – 0.75 bv – width of shear area lvh - length of the member subject to shear, 1/2 the span for simply supported members Reinforced Horizontal Shear:  Reinforced Horizontal Shear Where f – 0.75 rv - shear reinforcement ratio Acs - Area of shear reinforcement me - Effective shear friction coefficient Shear Friction Coefficient:  Shear Friction Coefficient Shear Resistance by Non-Prestressed Concrete:  Shear Resistance by Non-Prestressed Concrete Shear strength for non-prestressed sections Prestress Concrete Shear Capacity:  Prestress Concrete Shear Capacity Where: ACI Eq 11-9 Effective prestress must be 0.4fpu Accounts for shear combined with moment May be used unless more detail is required Prestress Concrete Shear Capacity:  Prestress Concrete Shear Capacity Concrete shear strength is minimum is Maximum allowed shear resistance from concrete is: Shear Capacity, Prestressed:  Shear Capacity, Prestressed Resistance by concrete when diagonal cracking is a result of combined shear and moment Where: Vi and Mmax - factored externally applied loads e.g. no self weight Vd - is un-factored dead load shear Shear Capacity, Prestressed:  Shear Capacity, Prestressed Resistance by concrete when diagonal cracking is a result of principal tensile stress in the web is in excess of cracking stress. Where: Vp = the vertical component of effective prestress force (harped or draped strand only) Vcmax:  Vcmax Shear capacity is the minimum of Vc, or if a detailed analysis is used the minimum of Vci or Vcw Shear Steel:  Shear Steel If: Then: Shear Steel Minimum Requirements:  Shear Steel Minimum Requirements Non-prestressed members Prestressed members Remember both legs of a stirrup count for Av Torsion:  Torsion Current ACI Based on compact sections Greater degree of fixity than PC can provide Provision for alternate solution Zia, Paul and Hsu, T.C., “Design for Torsion and Shear in Prestressed Concrete,” Preprint 3424, American Society of Civil Engineers, October, 1978. Reprinted in revised form in PCI JOURNAL, V. 49, No. 3, May-June 2004. Torsion:  Torsion For members loaded two sides, such as inverted tee beams, find the worst case condition with full load on one side, and dead load on the other Torsion:  Torsion In order to neglect Torsion Where: Tu(min) – minimum torsional strength provided by concrete Minimum Torsional Strength:  Minimum Torsional Strength Where: x and y - are short and long side, respectively of a component rectangle g - is the prestress factor Prestress Factor, g:  Prestress Factor, g For Prestressed Members Where: fpc – level of prestress after losses Maximum Torsional Strength:  Maximum Torsional Strength Avoid compression failures due to over reinforcing Where: Maximum Shear Strength:  Maximum Shear Strength Avoid compression failures due to over reinforcing Torsion/Shear Relationship:  Torsion/Shear Relationship Determine the torsion carried by the concrete Where: T’c and V’c - concrete resistance under pure torsion and shear respectively Tc and Vc - portions of the concrete resistance of torsion and shear Torsion/Shear Relationship:  Torsion/Shear Relationship Determine the shear carried by the concrete Torsion Steel Design:  Torsion Steel Design Provide stirrups for torsion moment - in addition to shear Where x and y - short and long dimensions of the closed stirrup Torsion Steel Design:  Torsion Steel Design Minimum area of closed stirrups is limited by Longitudinal Torsion Steel:  Longitudinal Torsion Steel Provide longitudinal steel for torsion based on equation or Whichever greater Longitudinal Steel limits:  Longitudinal Steel limits The factor in the second equation need not exceed Detailing Requirements, Stirrups:  Detailing Requirements, Stirrups 135 degree hooks are required unless sufficient cover is supplied The 135 degree stirrup hooks are to be anchored around a longitudinal bar Torsion steel is in addition to shear steel Detailing Requirements, Longitudinal Steel:  Detailing Requirements, Longitudinal Steel Placement of the bars should be around the perimeter Spacing should spaced at no more than 12 inches Longitudinal torsion steel must be in addition to required flexural steel (note at ends flexural demand reduces) Prestressing strand is permitted (@ 60ksi) The critical section is at the end of simply supported members, therefore U-bars may be required to meet bar development requirements Serviceability Requirements:  Serviceability Requirements Three classifications for prestressed components Class U: Uncracked Class T: Transition Class C: Cracked Stress Uncracked Section:  Uncracked Section Table 4.2.2.1 (Page 4.24) Easiest computation Use traditional mechanics of materials methods to determine stresses, gross section and deflection. No crack control or side skin reinforcement requirements Transition Section:  Transition Section Table 4.2.2.1 (Page 4.24) Use traditional mechanics of materials methods to determine stresses only. Use bilinear cracked section to determine deflection No crack control or side skin reinforcement requirements Cracked Section:  Cracked Section Table 4.2.2.1 (Page 4.24) Iterative process Use bilinear cracked section to determine deflection and to determine member stresses Must use crack control steel per ACI 10.6.4 modified by ACI 18.4.4.1 and ACI 10.6.7 Cracked Section Stress Calculation:  Cracked Section Stress Calculation Class C member require stress to be check using a Cracked Transformed Section The reinforcement spacing requirements must be adhered to Cracked Transformed Section Property Calculation Steps:  Cracked Transformed Section Property Calculation Steps Step 1 – Determine if section is cracked Step 2 – Estimate Decompression Force in Strand Step 3 – Estimate Decompression Force in mild reinforcement (if any) Step 4 – Create an equivalent force in topping if present Step 5 – Calculate transformed section of all elements and modular ratios Step 6 – Iterate the location of the neutral axis until the normal stress at this level is zero Step 7 – Check Results with a a moment and force equilibrium set of equations Steel Stress:  Steel Stress fdc – decompression stress stress in the strand when the surrounding concrete stress is zero – Conservative to use, fse (stress after losses) when no additional mild steel is present. Simple Example:  Simple Example Page 4-31 Deflection Calculation – Bilinear Cracked Section:  Deflection Calculation – Bilinear Cracked Section Deflection before the member has cracked is calculated using the gross (uncracked) moment of inertia, Ig Additional deflection after cracking is calculated using the moment of inertia of the cracked section Icr Effective Moment of Inertia:  Effective Moment of Inertia Alternative method Where: ftl = final stress fl = stress due to live load fr = modulus of rupture Prestress Losses:  Prestress Losses Prestressing losses Sources of total prestress loss (TL) TL = ES + CR + SH + RE Elastic Shortening (SH) Creep (CR) Shrinkage (SH) Relaxation of tendons (RE) Elastic Shortening:  Elastic Shortening Caused by the prestressed force in the precast member Where: Kes = 1.0 for pre-tensioned members Eps = modulus of elasticity of prestressing tendons (about 28,500 ksi) Eci = modulus of elasticity of concrete at time prestress is applied fcir = net compressive stress in concrete at center of gravity of prestressing force immediately after the prestress has been applied to the concrete fcir:  fcir Where: Pi = initial prestress force (after anchorage seating loss) e = eccentricity of center of gravity of tendons with respect to center of gravity of concrete at the cross section considered Mg = bending moment due to dead weight of prestressed member and any other permanent loads in place at time of prestressing Kcir = 0.9 for pretensioned members Creep:  Creep Creep (CR) Caused by stress in the concrete Where: Kcr = 2.0 normal weight concrete = 1.6 sand-lightweight concrete fcds = stress in concrete at center of gravity of prestressing force due to all uperimposed permanent dead loads that are applied to the member after it has been prestressed fcds:  fcds Where: Msd = moment due to all superimposed permanent dead and sustained loads applied after prestressing Shrinkage:  Shrinkage Volume change determined by section and environment Where: Ksh = 1.0 for pretensioned members V/S = volume-to-surface ratio R.H. = average ambient relative humidity from map Relative Humidity:  Relative Humidity Page 3-114 Figure 3.10.12 Relaxation:  Relaxation Relaxation of prestressing tendons is based on the strand properties Where: Kre and J - Tabulated in the PCI handbook C - Tabulated or by empirical equations in the PCI handbook Relaxation Table:  Relaxation Table Values for Kre and J for given strand Table 4.7.3.1 page 4-85 Relaxation Table Values for C:  Relaxation Table Values for C fpi = initial stress in prestress strand fpu = ultimate stress for prestress strand Table 4.7.3.2 (Page 4-86) Prestress Transfer Length:  Prestress Transfer Length Transfer length – Length when the stress in the strand is applied to the concrete Transfer length is not used to calculate capacity Prestress Development Length:  Prestress Development Length Development length - length required to develop ultimate strand capacity Development length is not used to calculate stresses in the member Beam Ledge Geometry:  Beam Ledge Geometry Beam Ledge Design:  Beam Ledge Design For Concentrated loads where s > bt + hl, find the lesser of: Beam Ledge Design:  Beam Ledge Design For Concentrated loads where s < bt + hl, find the lesser of: Beam Ledge Reinforcement:  Beam Ledge Reinforcement For continuous loads or closely spaced concentrated loads: Ledge reinforcement should be provided by 3 checks As, cantilevered bending of ledge Al, longitudinal bending of ledge Ash, shear of ledge Beam Ledge Reinforcement:  Beam Ledge Reinforcement Transverse (cantilever) bending reinforcement, As Uniformly spaced over width of 6hl on either side of the bearing Not to exceed half the distance to the next load Bar spacing should not exceed the ledge depth, hl, or 18 in Longitudinal Ledge Reinforcement:  Longitudinal Ledge Reinforcement Placed in both the top and bottom of the ledge portion of the beam: Where: dl - is the depth of steel U-bars or hooked bars may be required to develop reinforcement at the end of the ledge Hanger Reinforcement:  Hanger Reinforcement Required for attachment of the ledge to the web Distribution and spacing of Ash reinforcement should follow the same guidelines as for As Hanger (Shear) Ledge Reinforcement:  Hanger (Shear) Ledge Reinforcement Ash is not additive to shear and torsion reinforcement “m” is a modification factor which can be derived, and is dependent on beam section geometry. PCI 6th edition has design aids on table 4.5.4.1 Dap Design:  Dap Design (1) Flexure (cantilever bending) and axial tension in the extended end. Provide flexural reinforcement, Af, plus axial tension reinforcement, An. Dap Design:  Dap Design (2) Direct shear at the junction of the dap and the main body of the member. Provide shear friction steel, composed of Avf + Ah, plus axial tension reinforcement, An Dap Design:  Dap Design (3) Diagonal tension emanating from the re-entrant corner. Provide shear reinforcement, Ash Dap Design:  Dap Design (4) Diagonal tension in the extended end. Provide shear reinforcement composed of Ah and Av Dap Design:  Dap Design (5) Diagonal tension in the undapped portion. This is resisted by providing a full development length for As beyond the potential crack. Dap Reinforcement:  Dap Reinforcement 5 Main Areas of Steel Tension - As Shear steel - Ah Diagonal cracking – Ash, A’sh Dap Shear Steel - Av Tension Steel – As:  Tension Steel – As The horizontal reinforcement is determined in a manner similar to that for column corbels: Shear Steel – Ah:  Shear Steel – Ah The potential vertical crack (2) is resisted by a combination of As and Ah Shear Steel – Ah:  Shear Steel – Ah Note the development ld of Ah beyond the assumed crack plane. Ah is usually a U-bar such that the bar is developed in the dap Diagonal Cracking Steel – Ash:  Diagonal Cracking Steel – Ash The reinforcement required to resist diagonal tension cracking starting from the re-entrant corner (3) can be calculated from: Dap Shear Steel – Av:  Dap Shear Steel – Av Additional reinforcement for Crack (4) is required in the extended end, such that: Dap Shear Steel – Av:  Dap Shear Steel – Av At least one-half of the reinforcement required in this area should be placed vertically. Thus: Dap Limitations and Considerations:  Dap Limitations and Considerations Design Condition as a dap if any of the following apply The depth of the recess exceeds 0.2H or 8 in. The width of the recess (lp) exceeds 12 in. For members less than 8 in. wide, less than one-half of the main flexural reinforcement extends to the end of the member above the dap For members 8 in. or more wide, less than one-third of the main flexural reinforcement extends to the end of the member above the dap Questions?:  Questions?

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