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Post-tensioned slab design to SANS 10100
In a post-tensioned slab, steel tendons are stressed after the concrete hardens. A tendon draped low in the spans and high over the supports pushes up on the slab wherever it curves: that upward load cancels part of the floor load (load balancing). The tendon also squeezes the whole slab (precompression), which keeps tension cracks closed. The design checks the stresses at transfer and in service, the strength at ultimate, punching, deflection and the anchorage zones. Every number below comes from the free STATIX post-tensioned slab calculator with the inputs in step 1.
1. The problem
| Input | Value |
|---|---|
| Spans column centres, 0 = no third | 8000 mm |
| Span 2 mm | 8000 mm |
| Depth h | 250 mm |
| Panel width l2 | 8000 mm |
| Strand | 12.9 mm (100 mm²) |
| Strands per metre | 4 /m |
| Stressed from | One end |
| Superimposed dead | 1.5 kN/m² |
| Imposed qk | 2.5 kN/m² |
| fcu / at transfer fci | 35 MPa |
| Transfer strength MPa | 25 MPa |
| Stress class | Class 2 (limited tension) |
| Code | SANS 10100-1 + ACI 318-19 |
| Top / bottom to the strand centre | 50 mm |
| Bottom cover to strand centre mm | 50 mm |
| Quasi-permanent factor ψ on the imposed load | 0.3 |
| Relative humidity for shrinkage | 60 % |
| Column size along / across the strip | 600 mm |
| Column size across mm | 600 mm |
| Storey heights below / above | 3000 mm |
| Storey height above mm | 3000 mm |
| Partitions | Brittle |
| Fire rating | 1 h |
| Design across too the strip at right angles, its own tendons | No |
2. Load balancing and precompression
A parabolic tendon with sag e over a span L at force P gives an upward load of 8 P e / L² per metre. The number of strands is chosen so this balances a good part of the dead load. P / A is the average squeeze on the concrete; it has a minimum.
Balanced load (strand count given: 4), governing span 1:
[design choice (0,8 x self weight - TR43 practice value - not verified on disk; user can change it)]
4 strands x 115.58 kN x 8 x 112.5 / 8000^2 over 1000 mm = 6.501 kN/m2 (1.084 x self weight); the SLS stress and deflection rows decide
4.800 ≤ 6.501 kN/m2 OK
Average precompression P/A >= 0.862 MPa:
[ACI 318-19 cl.8.6.2.1 (p.112): 125 psi = 0,862 MPa]
P/A = 4 x 115.58 kN / (250 x 1000) = 1.849 MPa. SANS 10100-1 gives no flat-slab minimum (5.4 refers to specialist literature).
0.862 ≤ 1.849 MPa OK
3. Losses: from jacking to final force
The jack applies P0. Friction along the duct and the wedges seating at the anchor reduce it at once (P_i, at transfer). Over time, shrinkage and creep of the concrete and relaxation of the steel reduce it further (P_e, effective).
jacking P0 = 139.5 kN per strand
at transfer P_i = 127.4 kN
effective P_e = 115.6 kN
total loss 17.1 %
4. Layout, materials and the jacking force
Tendons must be close enough to spread the force and far enough apart to place concrete; the concrete must be strong enough when the tendons are stressed; the jacking force has an upper limit.
Tendon spacing (distributed direction) <= lesser of 8h and 1 524 mm:
[ACI 318-19 cl.8.7.2.3 (p.114): lesser of 8h and 5 ft = 1 524 mm]
spacing = 1000 / 4 = 250 mm; 8h = 2000 mm
250.0 ≤ 1,524.0 mm OK
Clear gap between sheaths >= hagg + 5 mm:
[SANS 10100-1 cl.5.9.4.1 (p.137), cl.4.11.8.1.2.1 hagg + 5 (p.106)]
gap = 250 - sheath 15.9 mm (sheath OD strand + 3 mm, TR43 practice value - not verified on disk; user can change it); evenly spaced - bundling not modelled
25.0 ≤ 234.1 mm OK
Concrete grade >= 30 MPa for post-tensioning (recommended):
[SANS 10100-1 cl.5.1.5.1 (p.114)]
30.0 ≤ 35.0 MPa OK
Transfer strength >= 18 MPa (unbonded):
[SANS 10100-1 cl.5.1.5.1 (p.114)]
18.0 ≤ 25.0 MPa OK
Jacking force <= 0,75 fpk Ap (up to 0,80 with additional consideration):
[SANS 10100-1 cl.5.8.1.1 (p.127)]
P0 = 0.75 x 1860 x 100 = 139.5 kN
139.5 ≤ 139.5 kN OK
Nominal cover to the sheath >= table 38 floors, continuous, 1 h:
[SANS 10100-1 cl.5.9.3.2, table 38 floors (p.135-136)]
cover = strand centroid cover - sheath OD/2 (least of top and bottom); durability cover (SABS 0100-2) not checked
20.0 ≤ 42.0 mm OK
5. Stresses at transfer
Right after stressing, the full initial force acts but only the self-weight is on the slab, and the concrete is still young. Tension and compression at the top and bottom faces are checked against limits at the transfer strength.
Transfer: flexural tension <= table 30 at fci (class 2):
[SANS 10100-1 cl.5.3.2.3.2 (p.118): class 1 1 MPa; class 2 table 30 at fci]
governing at x = 7600 mm, bottom fibre; P = n Pi = 4 x 127.405 kN; M = M_sw + M_PT(Pi) (self weight 1,0 only, formwork struck); sigma = P/A +- M/Z; table 30 at fci 25 MPa = 1.8 MPa (0,36 sqrt(fci) below grade 30). Class 2 post-tensioned: unstressed bars near the tension face (the bonded steel below).
0.00 ≤ 1.80 MPa OK
Transfer: compression <= 0,45 fci:
[SANS 10100-1 cl.5.3.2.3.1 (p.118): 0,45 fci (triangular distribution)]
governing at x = 7600 mm, top fibre; 0,45 x 25 = 11.25 MPa
3.06 ≤ 11.25 MPa OK
6. Stresses in service
In service the force has dropped to P_e and the full service load acts. The face stresses must stay within the class limits (tension) and the compression limits.
Service: flexural tension (class 2: 2.2 MPa (table 30 at fcu 35)):
[SANS 10100-1 cl.5.3.2.2.2 a)/b) table 30 (p.116-117): class 1 none; class 2 post-tensioned 0,36 sqrt(fcu); loads SANS 10160-1:2019 cl.8.3.1.1 eq.(10) (p.47): 1,1 G (1,0 favourable) + 1,0 Q]
governing at x = 8000 mm, top fibre; P = n Pe = 4 x 115.58 kN; M = gG (M_sw + M_sdl) + M_Q (none / all / odd / even) + M_PT(Pe), gG 1,0 and 1,1; uncracked section SANS 10100-1 cl.5.3.2.1 (p.116): elastic, uncracked (class 1 and 2)
0.78 ≤ 2.20 MPa OK
Service: compression <= 0,33 fcu (0,4 fcu at supports):
[SANS 10100-1 cl.5.3.2.2.1 table 29 (p.116): 0,33 fcu; 0,4 fcu within the range of support moments]
governing at x = 8000 mm, bottom, support-moment range: 0,4 fcu fibre; 0,33 fcu = 11.55, 0,4 fcu = 14 MPa
4.48 ≤ 14.00 MPa OK
7. Strength at ultimate
At ultimate the tendons and any bonded bars work like reinforcement. The design moment includes the secondary moment from the prestress in a continuous strip. Minimum bonded steel controls cracking where unbonded tendons are used.
ULS flexure, hogging (unbonded tendons + bonded top bars):
[SANS 10100-1 cl.5.3.3.2.6 eq.(15)/(16) (p.120-121), cl.5.3.3.2.1/.2 (p.119-120); SANS 10100-1 cl.5.3.3.1 b) (p.119): 0,45 fcu over the compression zone (depth 0,9 x), eps_cu 0,0035; cl.3.3.3.2 gamma_m 1,15 steel (p.9); SANS 10100-1 cl.4.6.5.1.2 (p.66): 1,2 Gn + 1,6 Qn; SANS 10100-1 cl.5.1.4.2 (p.113); ACI 318-19 cl.5.3.11 (p.65): secondary (hyperstatic) prestress effects, load factor 1,0]
x = 8000 mm: Mu = 1,2 G + 1,6 Q + M2 = 74.956 (M2 secondary -0.314 = M_PT 34.36 - Pe e 34.674); dp 200, le 16000 mm (whole tendon), fpe 1155.8, fpb = fpe + 7 000/(le/d) (1 - 1,7 fpu Aps/(fcu b d)) = 1227.49 MPa (<= 0,7 fpu 1302), bonded As 197.92 mm2 at 391.5 MPa (d 219), x 40.105 mm (x/d 0.201)
75.0 ≤ 104.9 kNm OK
ULS flexure, sagging (unbonded tendons + bonded bottom bars):
[SANS 10100-1 cl.5.3.3.2.6 eq.(15)/(16) (p.120-121), cl.5.3.3.2.1/.2 (p.119-120); SANS 10100-1 cl.5.3.3.1 b) (p.119): 0,45 fcu over the compression zone (depth 0,9 x), eps_cu 0,0035; cl.3.3.3.2 gamma_m 1,15 steel (p.9); SANS 10100-1 cl.4.6.5.1.2 (p.66): 1,2 Gn + 1,6 Qn; SANS 10100-1 cl.5.1.4.2 (p.113); ACI 318-19 cl.5.3.11 (p.65): secondary (hyperstatic) prestress effects, load factor 1,0]
x = 3648.333 mm: Mu = 1,2 G + 1,6 Q + M2 = 52.864 (M2 secondary 15.004 = M_PT -19.402 - Pe e -34.406); dp 199.42, le 16000 mm (whole tendon), fpe 1155.8, fpb = fpe + 7 000/(le/d) (1 - 1,7 fpu Aps/(fcu b d)) = 1227.236 MPa (<= 0,7 fpu 1302), bonded As 0 mm2 at 0 MPa (d 219), x 34.631 mm (x/d 0.174)
52.9 ≤ 90.2 kNm OK
Bonded top steel at the columns >= 0,000 75 Acf (unbonded tendons):
[ACI 318-19 cl.8.6.2.3 table 8.6.2.3 (c) (p.112): 0,000 75 Acf at columns; cl.8.7.5.3 (p.119): within 1,5h outside the column faces, >= 4 bars, s <= 12 in (305 mm); cl.8.7.5.5.1 (p.119): >= ln/6 past the face]
support 1: 0,000 75 x 250 x 8000 = 1500 mm2 per column vs 1583.363 provided; Acf = h x max(l1, l2); placed within 1350 mm, >= 4 bars, s <= 305 mm, extending ln/6 past the face
PASS
Bonded bottom steel in positive-moment areas (ACI 8.6.2.3 a/b):
[ACI 318-19 cl.8.6.2.3 table 8.6.2.3 (a)/(b) (p.112): none if ft <= 2 sqrt(fc') psi, else Nc / (0,5 fy), fy <= 60 000 psi]
span 2: ft 0 <= 2 sqrt(fc') = 0.888 MPa: not required (a); fc' = fck 28.571 MPa (EN 1992-1-1:2004 table 3.1 (p.29): fck from the cube strength fcu (linear between classes))
PASS
Tendons + bonded bars (as As fy/fpu) >= 0,15 % of b h (cracking precedes failure):
[SANS 10100-1 cl.5.9.2.2 (p.135), cl.5.3.3.4 (p.122) As fy/fpu]
least of span 0.16 % and support 0.179 %; tendons alone 0.16 %
0.1 ≤ 0.2 % OK
8. Punching at the columns
Each column is checked for punching. At interior columns the precompression helps; at edge columns the prestress benefit is not taken.
Punching at edge column 1 (SANS reinforced-concrete route, prestress ignored):
[SANS 10100-1 cl.4.6.5 / 4.4.5.2 via punching-engine.js (reinforced-concrete route, prestress ignored)]
edge column: ACI 22.6.5.5 excludes columns within 4h of a discontinuous edge (22.6.5.4 b); V = 376.181 kN, beta 1.4 (preset), u1 = 3078 mm at 1,5d, d 213, 100As/bd 0.551; Punching reinforcement required — see design below: 669.848 mm2 per perimeter, 3 perimeters. Slab edge taken flush with the column outer face.
0.80 ≥ 0.57 MPa OK
Punching at interior column 2 (prestressed two-way, ACI 22.6.5.5):
[ACI 318-19 cl.22.6.5.4/22.6.5.5 (p.416-417), cl.22.6.2.2 d >= 0,8h and cl.22.6.4.1 b0 at d/2 (p.412), cl.8.4.2.2.2 / 8.4.4.2.2 gamma (p.104, 107), R8.4.4.2.3 Jc (p.107), table 21.2.1 phi 0,75 (p.391)]
d = max(0,8h, h - c - dia) = 213, b0 = 2(600 + d) + 2(600 + d) = 3252 mm; Vu = R 787.275 (odd spans, 1,2G + 1,6Q + secondary 1,0) - wu 13 kPa x (c1 + d)(c2 + d) = 778.682 kN; Msc 136.318 kNm, gamma_v 0.4; vu = Vu/(b0 d) + gamma_v Msc (b1/2)/Jc = 1.41 MPa; vc = min(3,5; 1,5 + 40 d/b0) sqrt(fc') + 0,3 fpc = 1.96 (fc' 28.571, fpc 1.356, Vp taken 0: conservative); phi 0,75
1.41 ≤ 1.47 MPa OK
Column 2: SANS reinforced-concrete punching, prestress ignored (cross-check):
[SANS 10100-1 cl.4.6.5 / 4.4.5.2 via punching-engine.js (reinforced-concrete route, prestress ignored)]
Punching reinforcement required — see design below; beta 1.15, u1 4956 mm
0.86 ≥ 0.57 MPa OK
Punching at edge column 3 (SANS reinforced-concrete route, prestress ignored):
[SANS 10100-1 cl.4.6.5 / 4.4.5.2 via punching-engine.js (reinforced-concrete route, prestress ignored)]
edge column: ACI 22.6.5.5 excludes columns within 4h of a discontinuous edge (22.6.5.4 b); V = 376.181 kN, beta 1.4 (preset), u1 = 3078 mm at 1,5d, d 213, 100As/bd 0.551; Punching reinforcement required — see design below: 669.848 mm2 per perimeter, 3 perimeters. Slab edge taken flush with the column outer face.
0.80 ≥ 0.57 MPa OK
9. Deflection and vibration
The long-term deflection uses the effective force and creep; the camber at transfer must not be too large. A long, light slab is also checked for its natural frequency.
Final deflection <= span/250:
[SANS 10100-1 cl.3.2.3.2.1.1 (p.4): span/250 final; cl.3.2.3.2.1.2 (p.5): after partitions span/500 or 10 mm (brittle), span/350 or 20 mm (flexible); cl.3.2.3.2.1.3 (p.5): upward span/300 with finishes; SANS 10100-1 cl.5.3.6.1 (p.126): elastic, concrete section, Ec table 1 (p.10); long-term: effective modulus from the specific creep cl.5.8.2.5.3 (p.129); SANS 10160-1:2019 cl.8.3.3.1 eq.(11) (p.47) with psi from table 2 (p.34): 1,1 G + psi Q]
span 1: (1,1 G + psi Q + PT(Pe)) Ec/Eeff + (1 - psi) Q_max; span/250 = 32; Ec 29500 MPa (fcu), Eci 26000 (fci), Eeff = 1/(1/Ec + 57.6e-6) = 10929.164 MPa; gross section; strip deflection along the column line (the panel-centre value adds the orthogonal strip - from the app 3-D model)
2.4 ≤ 32.0 mm OK
Deflection after the partitions:
[SANS 10100-1 cl.3.2.3.2.1.1 (p.4): span/250 final; cl.3.2.3.2.1.2 (p.5): after partitions span/500 or 10 mm (brittle), span/350 or 20 mm (flexible); cl.3.2.3.2.1.3 (p.5): upward span/300 with finishes]
span 1: final - immediate at transfer (SW + PT(Pi)) Ec/Eci; brittle partitions: least of span/500 and 10 mm = 10
3.0 ≤ 10.0 mm OK
Upward deflection (camber) <= span/300 with finishes:
[SANS 10100-1 cl.3.2.3.2.1.1 (p.4): span/250 final; cl.3.2.3.2.1.2 (p.5): after partitions span/500 or 10 mm (brittle), span/350 or 20 mm (flexible); cl.3.2.3.2.1.3 (p.5): upward span/300 with finishes]
span 1: at transfer (SW + PT(Pi)) Ec/Eci and long-term (G + PT(Pe)) Ec/Eeff; span/300 = 26.667
0.7 ≤ 26.7 mm OK
Floor natural frequency of the strip (Rayleigh, own analysis, alternate spans):
[SANS 10160-1 cl.8.2.5: limit set by the designer]
Rayleigh quotient w^2 = g Int(p y)/Int(w y^2) on the static deflection of the strip (own analysis: gross section, Ec table 1, column springs) under the vibrating load SW + SDL + psi Q = 8.25 kN/m2, every span the same way and alternate spans opposite (the first mode of a continuous strip); f1 = the lower: f1 = 9.83 Hz (every span the same way 11.09, alternate spans 9.83 Hz); give a limit (Hz) to check it (cl.8.2.5.1)
INFO
10. Anchorage zones
Behind each anchor the large concentrated force spreads into the slab and causes bursting (splitting) tension, which needs bursting steel through the depth and along the edge.
Anchorage zone bursting, vertical (through the depth):
[SANS 10100-1 cl.5.8.5.2.1 table 37, cl.5.8.5.2.2 (p.134): Fbst/Pk by Ypo/Yo, in 0,2 Yo..2 Yo, steel at 200 MPa (SLS) or 0,87 fy (ULS), strain 0,001 when cover < 50 mm]
per anchor: Ypo/Yo = 27.5/125 = 0.22 -> Fbst/Pk 0.23; SLS Pk = P0 139.5 kN at 200 MPa, ULS Pk = fpk Ap 186 kN at 200 MPa (cover 25 < 50 mm: strain 0,001); As = 213.9 mm2 -> 3 Y10 legs (closed links / hairpins) between 25 and 250 mm from the anchor face. Anchor plate 130 x 55 mm (typical monostrand anchor casting (supplier catalogue value - not on disk; user can change it))
213.9 ≤ 235.6 mm² OK
Anchorage zone bursting, horizontal (along the slab edge):
[SANS 10100-1 cl.5.8.5.2.1 table 37, cl.5.8.5.2.2 (p.134): Fbst/Pk by Ypo/Yo, in 0,2 Yo..2 Yo, steel at 200 MPa (SLS) or 0,87 fy (ULS), strain 0,001 when cover < 50 mm]
per anchor: Ypo/Yo = 65/125 = 0.52 -> Fbst/Pk 0.164; SLS Pk = P0 139.5 kN at 200 MPa, ULS Pk = fpk Ap 186 kN at 200 MPa (cover 25 < 50 mm: strain 0,001); As = 152.52 mm2 -> 2 Y10 legs (closed links / hairpins) between 25 and 250 mm from the anchor face. Anchor plate 130 x 55 mm (typical monostrand anchor casting (supplier catalogue value - not on disk; user can change it))
152.5 ≤ 157.1 mm² OK
Anchorage local zone (bearing under the anchor):
[SANS 10100-1 cl.5.8.5.1 c)/d) (p.133-134); ACI 318-19 cl.25.9.3.1: local zone = the anchorage device (supplier)]
by the anchorage supplier: the device and its confining spiral are proprietary (not designed here)
INFO
The tendon profile and the strip plan are drawn on the server (paid plan); the calc sheet repeats each step with every formula.
Run this example yourselfOpen the free STATIX post-tensioned slab calculator and press Load the example (two 8 m spans, 250 mm slab, 4 strands) under "How it is calculated". Change any value and every step updates, with a calc sheet.
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