Moment Redistribution

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Milo George
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1 TIME SAVING DESIGN AID Page 1 of 23 A 3-span continuous beam has center-to-center span lengths of 30 ft-0 in. The beam is 20 in. by 28 in. and all columns are 20 in. by 20 in. In this example, the beam is designed for flexure using moment redistribution. Design Data Materials Concrete: normal weight (150 pcf), f c = 4,000 psi Mild reinforcing steel: Grade 60 (f y = 60,000 psi) Loads Superimposed dead load = 3.5 kips/ft Live load = 2.75 kips/ft Determine factored loads U = 1.2D + 1.6L w D = = 4.1 kips/ft 144 w L = 2.75 kips/ft w U = 1.2w = ( ) + ( ) = 9.3 kips/ft D w L Determine bending moments for pattern loading Maximum negative moments at faces of supports and positive midspan moments were determined by computer analysis for the following 4 load cases: Case I: Live load applied to all 3 spans Case II: Live load applied to end span and interior span only Case III: Live load applied to end spans only Case IV: Live load applied to interior span only

2 TIME SAVING DESIGN AID Page 2 of 23 Factored Bending Moments Due to Pattern Loading Location Load Case M u (ft-kips) I II Exterior negative III IV I II End Span Positive III IV I II Interior negative III IV I II Negative III IV Interior Span I Positive IV II III Determine maximum allowable percentage increase or decrease in negative moment Use d = in. (assuming 1.5 in. clear cover, No. 4 stirrups, and No. 9 flexural bars) The required area of steel for the maximum bending moments due to pattern loading is summarized in the table below. Also given in the table is the corresponding net tensile strain in the extreme tension steel ε and the percent allowable adjustment in the negative moment 1000ε. t Moment Adjustments at Supports t Location M u (ft-kips) A s (in. 2 ) ε t (in./in.) 1000ε t (%)* End support Interior support *Permitted to increase or decrease negative moments by not more than 1000ε t 20% per ACI Redistribution of negative moments is permitted only when ε t (ACI 8.4.3).

3 TIME SAVING DESIGN AID Page 3 of 23 Adjustments of moments In this example, it is decided to reduce the negative moment at the first interior support and accept the corresponding increase in the positive moment in the end span. It is also decided that the negative moments at the exterior supports will not be adjusted. The following calculations illustrate the adjustment of the moments for Load Case II. Negative moment at first interior support = ft-kips with an allowable redistribution factor = 8.0% (see above table). Reduction to moment = = 54.0 ft-kips Adjusted negative moment = ( 54.0) = ft-kips Increase in positive moment in end span: For Load Case II, negative moment at exterior support = ft-kips Midspan ordinate on line from exterior support to interior support = 2 Moment due to uniform load = w 2 2 u ln /8 = / 8 = ft-kips Adjusted positive moment at midspan = ( 489.3) = ft-kips Similar calculations can be performed for the other load cases. ( 357 8) = ft-kips Determine flexural reinforcement based on adjusted moments The table below gives a summary of the required reinforcement at the negative and positive locations in both an exterior and interior span, based on the redistributed bending moments. Note that if an additional adjustment of the bending moments is warranted, further cycles of redistribution may be made. Summary of Flexural Reinforcement Location M u (ft-kips) Load Case Required Provided A s (in. 2 ) ρ Bars ρ Exterior negative III No End span Positive III No Interior negative II No Interior span Positive IV No ρ min = 200/f y = For comparison, output from pcabeam program is presented on the following pages.

4 TIME SAVING DESIGN AID Page 4 of 23 X Z Y pcabeam v2.00. Licensed to: PCA. License ID: D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Moment redistribution example.slb Project: Time Saving Design Aid Frame: Engineer: DAA

5 TIME SAVING DESIGN AID Page 5 of 23 CASE: EQ CASE: Wind CASE: Live CASE: Dead CASE: SELF pcabeam v2.00. Licensed to: PCA. License ID: D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Moment redistribution example.slb Project: Time Saving Design Aid Frame: Engineer: DAA

6 TIME SAVING DESIGN AID Page 6 of Shear Diagram - kip Moment Diagram - k-ft LEGEND: Envelope (Non-Redistributed) pcabeam v2.00. Licensed to: PCA. License ID: D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Moment redistribution example.slb Project: Time Saving Design Aid Frame: Engineer: DAA

7 TIME SAVING DESIGN AID Page 7 of Shear Diagram - kip - kip Moment Diagram - k-ft - k-ft LEGEND: Envelope (Redist.) pcabeam v2.00. Licensed to: PCA. License ID: D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Moment redistribution example.slb Project: Time Saving Design Aid Frame: Engineer: DAA

8 TIME SAVING DESIGN AID Page 8 of 23 3-#9(360.0)c 2-#9(229.7) 3-#9(360.0)c 2-#9(212.7) 3-#9(360.0)c 2-#9(229.7) 2-#9(60.2) 2-#9(80.1) 3-#9(114.1) 3-#9(87.6) 3-#9(81.4) 3-#9(110.7) 3-#9(110.7) 3-#9(81.4) 3-#9(87.6) 3-#9(114.1) 2-#9(80.1) 2-#9(60.2) Flexural Reinforcement pcabeam v2.00. Licensed to: PCA. License ID: D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Moment redistribution example.slb Project: Time Saving Design Aid Frame: Engineer: DAA

9 TIME SAVING DESIGN AID Page 9 of Moment Moment Capacity Capacity - k-ft - k-ft LEGEND: Envelope Curve Capacity Curve Support Centere Face of Support pcabeam v2.00. Licensed to: PCA. License ID: D2DE-2C8D0 File: C:\Data\Time Saving Design Aid\Moment redistribution example.slb Project: Time Saving Design Aid Frame: Engineer: DAA

10 TIME SAVING DESIGN AID Page 10 of 23 C:\Data\Time Saving Design Aid\Moment redistribution example.slb Page 1 ooooooo oooooo ooooo oooooooo oooooooo ooooooo oo oo oo oo oo oo oo oo oo oo oo oooooooo oo ooooooo ooooo ooooooo oo oo ooooooo ooooo oo oo oo oo oo oo oooooooo oo oo oo oooooo oo oo OOOOO OOOOOO OOOOO OO OO OO OO OO OO OO OOO OOO OO OO OO OO OO OO O O OO OO OO OO OO OO OO O O OO OOOOO OOOOO OOOOOOO OO O OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OOOOO OOOOOO OO OO OO OO ============================================================================= pcabeam v2.00 (TM) A Computer Program for Analysis, Design, and Investigation of Reinforced Concrete Beams and One-way Slab Systems ============================================================================= Copyright , Portland Cement Association All rights reserved Licensee stated above acknowledges that Portland Cement Association (PCA) is not and cannot be responsible for either the accuracy or adequacy of the material supplied as input for processing by the pcabeam computer program. Furthermore, PCA neither makes any warranty expressed nor implied with respect to the correctness of the output prepared by the pcabeam program. Although PCA has endeavored to produce pcabeam error free the program is not and cannot be certified infallible. The final and only responsibility for analysis, design and engineering documents is the licensees. Accordingly, PCA disclaims all responsibility in contract, negligence or other tort for any analysis, design or engineering documents prepared in connection with the use of the pcabeam program. ============================================================================================= [1] INPUT ECHO ============================================================================================= General Information: ==================== File name: C:\Data\Time Saving Design Aid\Moment redistribution example.slb Project: Time Saving Design Aid Frame: Engineer: DAA Code: ACI Reinforcement Database: ASTM A615 Mode: Design Number of supports = 4 Floor System: One-Way/Beam Live load pattern ratio = 100% Deflections are based on cracked section properties. In negative moment regions, Ig and Mcr DO NOT include flange/slab contribution (if available) Long-term deflections are calculated for load duration of 60 months. 0% of live load is sustained. Compression reinforcement calculations NOT selected. Moment redistribution selected. Effective flange width calculations NOT selected. Rigid beam-column joint NOT selected. Torsion analysis and design NOT selected. Material Properties: ==================== Slabs Beams Columns wc = lb/ft3 f'c = 4 4 ksi Ec = ksi fr = ksi fy = 60 ksi, Bars are not epoxy-coated fyt = 60 ksi Es = ksi Reinforcement Database: ===============

11 TIME SAVING DESIGN AID Page 11 of 23 C:\Data\Time Saving Design Aid\Moment redistribution example.slb Page 2 Units: Db (in), Ab (in^2), Wb (lb/ft) Size Db Ab Wb Size Db Ab Wb # # # # # # # # # # # Span Data: ========== Slabs: L1, wl, wr (ft); t, Hmin (in) Span Loc L1 t wl wr Hmin Int Int Int Ribs and Longitudinal Beams: b, h, Sp (in) Ribs Beams Span Span b h Sp b h Hmin Support Data: ============= Columns: c1a, c2a, c1b, c2b (in); Ha, Hb (ft) Supp c1a c2a Ha c1b c2b Hb Red% Limits Supp Left[%] Right[%] Boundary Conditions: Kz (kip/in); Kry (kip-in/rad) Supp Spring Kz Spring Kry Far End A Far End B Fixed Fixed Fixed Fixed Fixed Fixed Fixed Fixed Load Data: ========== Load Cases and Combinations: Case SELF Dead Live Wind EQ Type DEAD DEAD LIVE LATERAL LATERAL U Span Loads: Span Case Wa La Wb Lb Line Loads - Wa Wb (lb/ft), La Lb (ft): 1 Dead Dead Dead Live Live Live Support Loads - Fz (kip), My (k-ft): Supp Case Fz My SELF SELF SELF SELF Live Live Live Live 0 0 Support Displacements - D (in), R (rad): Supp Case D R SELF 0 0

12 TIME SAVING DESIGN AID Page 12 of 23 C:\Data\Time Saving Design Aid\Moment redistribution example.slb Page 3 2 SELF SELF SELF Live Live Live Live 0 0 Lateral Load Effects - M (k-ft): Span Case Mleft Mright EQ EQ EQ Wind Wind Wind 0 0 Reinforcement Criteria: ======================= Top bars Bottom bars Stirrups Min Max Min Max Min Max Slabs and Ribs: Bar Size #5 #8 #5 #8 Bar spacing in Reinf ratio % Cover in Top bars have 12 in of concrete below them. Beams: Bar Size #9 #9 #9 #9 #3 #5 Bar spacing in Reinf ratio % Cover in Side cover in Layer dist in No. of legs 2 6 Top bars have 12 in of concrete below them. ============================================================================================= [2] DESIGN RESULTS ============================================================================================= Factors: ============================== Units: Org.Mu (k-ft) Calculated User Applied_ Supp Side Org.Mu Iter.# EpsilonT Factor[%] Limit[%] Factor[%] Right Left Right Left Right Left Top Reinforcement: ================== Units: Width (ft), Mmax (k-ft), Xmax (ft), As (in^2), Sp (in) Span Zone Width Mmax Xmax AsMin AsMax SpReq AsReq Bars Left #9 Middle Right #9 2 Left #9 Middle Right #9 3 Left #9 Middle Right #9 Top Bar Details: ================ Units: Length (ft) Left Continuous Right Span Bars Length Bars Length Bars Length Bars Length Bars Length # # # # # # # # # # # #9 5.02

13 TIME SAVING DESIGN AID Page 13 of 23 C:\Data\Time Saving Design Aid\Moment redistribution example.slb Page 4 Bottom Reinforcement: ===================== Units: Width (ft), Mmax (k-ft), Xmax (ft), As (in^2), Sp (in) Span Width Mmax Xmax AsMin AsMax SpReq AsReq Bars # # #9 Bottom Bar Details: =================== Units: Start (ft), Length (ft) Long Bars Short Bars Span Bars Start Length Bars Start Length # # # # # # Flexural Capacity: ================== Units: x (ft), As (in^2), PhiMn (k-ft) Span x AsTop AsBot PhiMn- PhiMn

14 TIME SAVING DESIGN AID Page 14 of 23 C:\Data\Time Saving Design Aid\Moment redistribution example.slb Page 5 Longitudinal Beam Shear Reinforcement Required: =============================================== Units: d (in), Start, End, Xu (ft), PhiVc, Vu (kip), Av/s (in^2/in) Span d PhiVc Start End Vu Xu Av/s Longitudinal Beam Shear Reinforcement Details: ============================================== Units: spacing & distance (in). Span Size Stirrups (2 legs each unless otherwise noted) #3 6.4 [3L] [3L] 2 #3 5.3 [3L] < > [3L] 3 #3 5.3 [3L] [3L] Beam Shear Capacity: ==================== Units: d, Sp (in), Start, End, Xu (ft), PhiVc, PhiVn, Vu (kip), Av/s (in^2/in) Span d PhiVc Start End Av/s Sp PhiVn Vu Xu Slab Shear Capacity: ==================== Units: b, d (in), Xu (ft), PhiVc, Vu(kip) Span b d Vratio PhiVc Vu Xu Not checked Not checked Not checked --- Deflections: ============ Section properties Units: Ig, Icr, Ie (in^4), Mcr, Mmax (k-ft) Load Level

15 TIME SAVING DESIGN AID Page 15 of 23 C:\Data\Time Saving Design Aid\Moment redistribution example.slb Page 6 Ie,avg Dead Dead+Live Span Dead Dead+Live Zone Ig Icr Mcr Mmax Ie Mmax Ie Middle Right Left Middle Right Left Middle Maximum Instantaneous Deflections Units: D (in) Span Ddead Dlive Dtotal Maximum Long-term Deflections Time dependant factor for sustained loads = Units: D (in) Span Dsust Lambda Dcs Dcs+lu Dcs+l Dtotal Material Takeoff: ================= Reinforcement in the Direction of Analysis Top Bars: lb <=> 9.21 lb/ft <=> lb/ft^2 Bottom Bars: lb <=> lb/ft <=> lb/ft^2 Stirrups: lb <=> lb/ft <=> lb/ft^2 Total Steel: lb <=> lb/ft <=> lb/ft^2 Concrete: ft^3 <=> 3.89 ft^3/ft <=> ft^3/ft^2 ============================================================================================= [6] REDISTRIBUTED SEGMENTAL MOMENT AND SHEAR - ENVELOPES ============================================================================================= Span x (ft) M- (k-ft) Comb M+ (k-ft) Comb V- (kip) Comb V+ (kip) Comb U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U1

16 TIME SAVING DESIGN AID Page 16 of 23 C:\Data\Time Saving Design Aid\Moment redistribution example.slb Page U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U1

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19 TIME SAVING DESIGN AID Page 19 of 23 C:\Data\Time Saving Design Aid\Moment redistribution example.slb Page U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U1 ============================================================================================= [6a] NON-REDISTRIBUTED SEGMENTAL MOMENT AND SHEAR - ENVELOPES ============================================================================================= Span x (ft) M- (k-ft) Comb M+ (k-ft) Comb V- (kip) Comb V+ (kip) Comb U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U1

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spslab v3.11. Licensed to: STRUCTUREPOINT, LLC. License ID: D2DE-2175C File: C:\Data\CSA A Kt Revised.slb

$spslab v3.11. Licensed to: STRUCTUREPOINT, LLC. License ID: D2DE-2175C File: C:\Data\CSA A Kt Revised.slb$ X Z Y spslab v3.11. Licensed to: STRUCTUREPOINT, LLC. License ID: 00000-0000000-4-2D2DE-2175C File: C:\Data\CSA A23.3 - Kt Revised.slb Project: CSA A23.3 - Kt Torsional Stiffness Illustration Frame: Engineer: