# Two-Way Post-Tensioned Design

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1 Page 1 of 9 The following example illustrates the design methods presented in ACI and IBC Unless otherwise noted, all referenced table, figure, and equation numbers are from these books. The example presented here is for. Loads: Framing Dead Load = selfweight Superrimposed Dead Load = 25 psf partitions, M/E, misc. Live Load = 40 psf residential 2 hour fire-rating Materials: Concrete: Rebar: PT: Normal weight 150 pcf f' c = 5,000 psi f' ci = 3,000 psi f y = 60,000 psi Unbonded tendons 1 /2 ϕ, 7-wire strands, A = in 2 f pu = 270 ksi Estimated prestress losses = 15 ksi (ACI 18.6) f se = 0.7 (270 ksi) - 15 ksi = 174 ksi (ACI ) P eff = A*f se = (0.153)(174 ksi) = 26.6 kips/tendon Determine Preliminary Slab Thickness Start with L/h = 45 Longest span = 30 ft h = (30 ft)(12)/45 = 8.0 preliminary slab thickness Loading DL = Selfweight = (8in)(150 pcf) = 100 psf SIDL = 25 psf LL o = 40 psf IBC allows for LL reduction Exterior bay: A T = (25 ft)(27 ft) = 675 ft 2 K LL = 1 LL = 0.83 LL o = 33 psf Interior bay: A T = (25 ft)(30 ft) = 750 ft 2 K LL = 1 LL = 0.80 LL o = 32 psf

2 Page 2 of 9 DESIGN OF EAST-WEST INTERIOR FRAME Use Equivalent Frame Method, ACI 13.7 (excluding sections ) Total bay width between centerlines = 25 ft Ignore column stiffness in equations for simplicity of hand calculations No pattern loading required, since LL/DL < 3 /4 (ACI ) Calculate Section Properties Two-way slab must be designed as Class U (ACI ), Gross cross-sectional properties allowed (ACI ) A = bh = (300 in)(8 in) = 2,400 in 2 S = bh 2 /6 = (300 in)(8 in) 2 /6 = 3,200 in 3 Set Design Parameters Allowable stresses: Class U (ACI ) At time of jacking (ACI ) f' ci = 3,000 psi Compression = 0.60 f' ci = 0.6(3,000 psi) = 1,800 psi Tension = 3 f' ci = 3 3,000 = 164 psi At service loads (ACI (a) and ) f' c = 5,000 psi Compression = 0.45 f' c = 0.45(5,000 psi) = 2,250 psi Tension = 6 f' c = 6 5,000 = 424 psi Average precompression limits: P/A = 125 psi min. (ACI ) = 300 psi max. Target load balances: 60%-80% of DL(selfweight) for slabs (good approximation for hand calculation) For this example: 0.75 w DL = 0.75(100 psf) = 75 psf Cover Requirements (2-hour fire rating, assume carbonate aggregate) IBC 2003 Restrained slabs = 3 /4 bottom Unrestrained slabs = 1 1 /2 bottom = 3 /4 top

3 Page 3 of 9 Tendon profile: Parabolic shape; For a layout with spans of similar length, the tendons will be typically be located at the highest allowable point at the interior columns, the lowest possible point at the midspans, and the neutral axis at the anchor locations. This provides the maximum drape for load-balancing. a e a END INT PT Tendon Neutral Axis A L1 B L2 C L3 D Continuous Post-Tensioned Beam Tendon Ordinate Tendon (CG) Location* Exterior support - anchor 4.0 Interior support - top 7.0 Interior span - bottom 1.0 End span - bottom 1.75 (CG) = center of gravity *Measure from bottom of slab a INT = = 6.0 a END = ( )/ = 3.75 eccentricity, e, is the distance from the center to tendon to the neutral axis; varies along the span Prestress Force Required to Balance 75% of selfweight DL Since the spans are of similar length, the end span will typically govern the maximum required post-tensioning force. This is due to the significantly reduced tendon drape, a END. = 0.75 w DL = 0.75 (100 psf)(25 ft) = 1,875 plf = k/ft Force needed in tendons to counteract the load in the end bay: P = L 2 / 8a end = (1.875 k/ft)(27 ft) 2 / [8(3.75 in / 12)] = 547 k

4 Page 4 of 9 Check Precompression Allowance Determine number of tendons to achieve 547 k # tendons = (547 k) / (26.6 k/tendon) = Use 20 tendons Actual force for banded tendons = (20 tendons) (26.6 k) = 532 k P actual The balanced load for the end span is slightly adjusted = (532/547)(1.875 k/ft) = 1.82 k/ft Determine actual Precompression stress P actual /A = (532 k)(1000) / (2,400 in 2 ) = 221 psi > 125 psi min. ok < 300 psi max. ok Check Interior Span Force P = (1.875 k/ft)(30 ft) 2 / [8(6.0 in / 12)] = 421 k < 532 k Less force is required in the center bay For this example, continue the force required for the end spans into the interior span and check the amount of load that will be balanced: = (532 k)(8)(6.0 in /12) / (30 ft) 2 = 2.36 k/ft /w DL = 94%; This value is less than 100%; acceptable for this design. East-West interior frame: Effective prestress force, P eff = 532 kips

5 Page 5 of 9 Check Slab Stresses Separately calculate the maximum positive and negative moments in the frame for the dead, live, and balancing loads. A combination of these values will determine the slab stresses at the time of stressing and at service loads. Dead Load Moments w DL = (125 psf) (25 ft) / 1000 = plf Live Load Moments w LL = (33 psf) (25 ft) / 1000 = plf Total Balancing Moments, M bal = k/ft (average of 3 bays)

6 Page 6 of 9 Stage 1: Stresses immediately after jacking (DL + PT) (ACI ) Midspan Stresses = (-M DL + M bal = (+M DL - M bal Interior Span = [(-101ft-k + 65ft-k)(12)(1000)]/(3200 in3) - 221psi = = -356 psi compression < 0.60 f'ci = 1800 psi ok = [(101ft-k - 65ft-k)(12)(1000)]/(3200 in3) - 221psi = = -86 psi compression < 0.60 f'ci = 1800 psi ok End Span = [(-172ft-k + 110ft-k)(12)(1000)]/(3200 in 3 ) - 221psi = = -453 psi compression < 0.60 f' ci = 1800 psi ok = [(172ft-k - 110ft-k)(12)(1000)]/(3200 in 3 ) - 221psi = = 11 psi tension < 3 f' ci = 164 psi ok Support Stresses = (+M DL - M bal = (-M DL + M bal = [(240ft-k - 154ft-k)(12)(1000)]/(3200 in 3 ) - 221psi = = 102 psi tension < 3 f' ci = 164 psi ok = [(-240ft-k + 154ft-k)(12)(1000)]/(3200 in 3 ) - 221psi = = -544 psi compression < 0.60 f' ci = 1800 psi ok Stage 2: Stresses at service load (DL + LL + PT) ( and ) Midspan Stresses = (-M DL - M LL + M bal = (+M DL + M LL - M bal Interior Span = [(-101ft-k - 27ft-k+ 65ft-k)(1000)]/(3200 in 3 ) - 221psi = = -457 psi compression < 0.45 f' c = 2250 psi ok = [(101ft-k + 27ft-k - 65ft-k)(1000)]/(3200 in3) - 221psi = = 15 psi tension < 6 f' c = 424 psi ok

7 Page 7 of 9 End Span = [(-172ft-k - 45ft-k + 110ft-k)(12)(1000)]/(3200 in 3 ) - 221psi = = -622 psi compression < 0.45 f' c = 2250 psi ok = [(172ft-k + 45ft-k - 110ft-k)(12)(1000)]/(3200 in 3 ) - 221psi = = 180 psi tension < 6 f' c = 424 psi ok Support Stresses = (+M DL + M LL - M bal = (-M DL - M LL + M bal = [(240ft-k + 64ft-k - 154ft-k)(12)(1000)]/(3200 in 3 ) - 221psi = = 342 psi tension < 6 f' c = 424 psi ok = [(-240ft-k - 64 ft-k + 154ft-k)(12)(1000)]/(3200 in 3 ) - 221psi = = -784 psi compression < 0.45 f' c = 2250 psi ok All stresses are within the permissible code limits. Ultimate Strength Determine factored moments The primary post-tensioning moments, M 1, vary along the length of the span. M 1 = P * e e = 0 in. at the exterior support e = 3.0 in at the interior support (neutral axis to the center of tendon) M 1 = (532k)(3.0in) / (12) = 133ft-k The secondary post-tensioning moments, M sec, vary linearly between supports. M sec = M bal - M 1 = 154 ft-k ft-k = 21 ft-k at the interior supports The typical load combination for ultimate strength design is M u = 1.2 M DL M LL M sec At midspan: At support: M u = 1.2 (172ft-k) (45ft-k) (10.5 ft-k) = 289 ft-k M u = 1.2 (-240ft-k) (-64ft-k) (21 ft-k) = -370 ft-k

8 Page 8 of 9 Determine minimum bonded reinforcement: to see if acceptable for ultimate strength design. Positive moment region: Interior span: f t = 15 psi < 2 f' c = 2 5,000 = 141 psi No positive reinforcement required (ACI ) Exterior span: f t = 180 psi > 2 f' c = 2 5,000 = 141 psi Minimum positive moment reinforcement required (ACI ) y = f t /(f t + f c )h = [(180)/( )](8 in) = 1.80 in N c = M DL+LL /S * 0.5 * y * l 2 = [(172 ft-k + 45 ft-k)(12) / (3,200 in 3 )](0.5)(1.80 in)(25ft)(12) = 220 k = N c / 0.5 fy = (220 k) / [0.5(60ksi)] = 7.33 in 2 Distribute the positive moment reinforcement uniformly across the slab-beam width and as close as practicable to the extreme tension fiber. = (7.33 in 2 )/(25 ft) = 0.293in 2 /ft Use 12 in. oc Bottom = 0.31 in 2 /ft (or equivalent) Minimum length shall be 1/3 clear span and centered in positive moment region (ACI ) Negative moment region: = A cf (ACI ) Interior supports: A cf = max. (8in)[(30ft + 27ft)/2, 25ft]*12 = (2,736 in 2 ) = 2.05 in 2 = 11 - #4 Top (2.20 in 2 ) Exterior supports: A cf = max. (8in)[(27ft/2), 25ft]*12 = (2,400 in 2 ) = 1.80 in 2 = 9 - #4 Top (1.80 in 2 ) Must span a minimum of 1/6 the clear span on each side of support (ACI ) At least 4 bars required in each direction (ACI ) Place top bars within 1.5h away from the face of the support on each side (ACI ) = 1.5 (8 in) = 12 in Maximum bar spacing is 12 (ACI )

9 Page 9 of 9 Check minimum reinforcement if it is sufficient for ultimate strength M n d A ps = (A s f y + A ps ) (d-a/2) = effective depth = 0.153in 2 *(number of tendons) = 0.153in 2 *(20 tendons) = 3.06 in 2 = f se + 10,000 + (f' c bd)/(300a ps ) for slabs with L/h > 35 (ACI ) = 174,000psi + 10,000 + [(5,000psi)(25ft*12)d]/[(300)( 3.06 in 2 )] = 184,000psi d a = (A s f y + A ps ) / (0.85f' c b) At supports d = 8-3 /4-1 /4 = 7 = 184,000psi (7 ) = 195,438psi a = [(2.20 in 2 )(60 ksi) + (3.06 in 2 )(195ksi)]/[(0.85)(5ksi)(25ft*12)] = 0.57 ϕm n = 0.9 [(2.20 in 2 )(60 ksi) + (3.06 in 2 )(195ksi)][7 - (0.57)/2]/12 = 0.9 (728k)(6.72in)/12 = 367 ft-k < 370ft-k Reinforcement for ultimate strength requirements governs A s, reqd = 2.30in 12 - #4 Top at interior supports 9 - #4 Top at exterior supports When reinforcement is provided to meet ultimate strength requirements, the minimum lengths must also conform to the provision of ACI Chapter 12. (ACI ) At midspan (end span) d = /2-1 /4 = 6 1 /4 = 184,000psi (6.25 ) = 194,212psi a = [(7.33 in2)(60 ksi) + (3.06 in 2 )(194ksi)]/[(0.85)(5ksi)(25ft*12)] = 0.81 ϕm n = 0.9 [(7.33 in 2 )(60 ksi) + (3.06 in 2 )(194ksi)][ (0.81)/2]/12 = 0.9 (1033k)(5.85in)/12 = 453 ft-k > 289 ft-k Minimum reinforcement ok 12 oc Bottom at end spans This is a simplified hand calculation for a post-tensioned two-way plate design. A detailed example can be found in the PCA Notes on ACI Building Code Requirements for Structural Concrete.

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