Prestressed concrete beam to SIA 262

Checks a post-tensioned (bonded or unbonded) or pre-tensioned beam over one or several spans, to SIA 262:2025, in design order: the tendon force with every loss, the stresses at transfer and in service, the bending and shear resistance and the anchorage of the prestress. Every step shows its formula and the clause it comes from.

Basic SIA 262:2025

Part of the Basic plan

This module is included in the Basic plan and above, with its calculation note.

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About this check

clauses, worked example and questions

What it checks

Section and tendon: A_c, y_G, I_c, W_t, W_b, the core points k_t and k_b, g₀ = γ · A_c, A_p = n_p · A_p1 and f_pd = f_p0,1k/γ_s, equation (5).
Steel stresses: σ_p,max ≤ min(0.8 · f_pk ; 0.9 · f_p0,1k), clause 4.1.5.2.1, equation (21); after anchoring σ_p ≤ min(0.75 · f_pk ; 0.85 · f_p0,1k), clause 4.1.5.2.2, equation (22); after all losses σ_p∞ ≥ 0.45 · f_pk, clause 4.1.5.2.3.
Losses: friction P_max · e^(−μ · (φ_x + Δφ · x)), clause 4.1.5.3.3, equation (24); anchor slip over the length l_s (clause 4.1.5.3.1); elastic shortening Δσ_p,el for tendons tensioned one after the other (clause 4.1.5.3.1); relaxation Δσ_pr = 3 · ρ₁₀₀₀ · σ_p0 (clause 3.3.2.7.1); creep, shrinkage and relaxation together by EN 1992-1-1 equation (5.46) (clause 4.1.5.3.4).
Equivalent load u = 8 · P_∞ · f/l², clause 4.1.5.1.1, and the share of g₀ + g₂ it balances.
Concrete at transfer: f_ck(t) ≥ 0.3 · f_ck and |σ_c| ≤ 0.6 · f_ck(t), clause 4.1.5.2.4, equation (23), outside the anchorage zones.
Service: stresses under P_∞ and g₀ + g₂ + ψ₂ · q, f_ct,eff (clause 4.4.1.3), and at mid-span σ_s,r ≤ σ_s,adm, Table 17, with σ_s the increase from zero concrete strain (clause 4.4.2.3.10).
Bending (clause 4.2.1.5): M_d ≤ M_Rd with the bars and the bonded tendon, M_cr ≤ M_Rd (clause 4.3.1.2) and x/d ≤ 0.5 · 435/f_yd (clause 4.1.4.2.6).
Shear near each support: ρ_w ≥ ρ_w,min, equation (116), and s ≤ 25 · Ø_sw, clause 5.5.2.2; V_d ≤ V_Rd,s + ΔV_Rd,p, clause 4.3.3.4.3, equation (45); V_d ≤ V_Rd,c + ΔV_Rd,p, clause 4.3.3.4.6, equation (47); the tension chord, clause 4.3.3.4.10.
Anchorage: P_d = γ_P · P_max with γ_P = 1.2 (clause 4.1.5.5.3), local compression with k_c = √(A_c1/A_c0) ≤ 3.0, clause 4.2.1.11, equation (32), and the bursting forces T_v, T_v,1 and T_h against the links, clause 5.3.3.

What you enter and what you get

You enter the static system and the spans of the beam, the section (rectangle, T or I), the bottom and top bars, the loads g₂, q and ψ₂, the concrete at 28 days and at transfer, the steels and the resistance factors. For the prestress: the type, the tendons, the ducts, the heights y_E and y_M of the profile, σ_p,max, the jacking ends, μ, Δφ, Δl_s, φ(∞,t₀) and ε_cs,∞. Then the crack requirements, γ_G and γ_Q, the stirrups, the anchor plates and the links of the end block.

You get the forces P_μ, P_s, P₀ and P_∞ at the key sections, every stress and check with its utilisation and clause, and the reinforcement for the bursting forces. With the Basic plan, it prints as an A4 calculation note.

Assumptions and limits

The section is constant along the beam; the loads are uniform on each span.
On a continuous beam the prestress creates secondary moments (clause 4.1.5.1.2). A simply supported beam has none.
Pre-tensioned strands are straight, in a single span.
At the ultimate limit state the bonded tendon starts from its pre-strain and can reach f_pd (clauses 4.1.5.1.5 and 4.1.5.5.1).
Shear is checked at a distance h from the support axis.
The bursting forces come from a strut-and-tie model, concrete tension neglected. The approval of the prestressing system governs edge distances, spacing and f_ck(t) at the anchors (clause 4.1.5.2.5).

Worked example

A simply supported I-section beam of 24 m, h = 1,300 mm, f_ck = 45 N/mm² and f_ck(t) = 35 N/mm², with two grouted tendons of 1,950 mm² from y_E = 650 mm to y_M = 150 mm, tensioned from the left end to 1,395 N/mm², under g₂ = 20 kN/m and q = 25 kN/m. Every check passes; the module warns that the plain concrete could not carry the pressure under the anchor plates, so the anchorage relies on the spiral of the system. These are the module's own results for its worked example, computed when this page was built.

Inputs
Input	Value	Ref.
Static system	Simply supported — one span
Span l	24 m
Shape of the section	I-section
Total depth h	1300 mm
Web width b_w	320 mm
Width of the top flange b_f	1200 mm
Thickness of the top flange h_f	180 mm
Width of the bottom flange b_b	700 mm
Thickness of the bottom flange h_b	220 mm
Bottom bars n_b	4
Diameter of the bottom bars Ø_b	20 mm
Height of their centre above the bottom face y_b	60 mm
Top bars n_t	6
Diameter of the top bars Ø_t	12 mm
Depth of their centre below the top face y_t	50 mm
Additional permanent load g₂	20 kN/m
Imposed load q	25 kN/m
Quasi-permanent factor of the imposed load ψ₂	0.3
Characteristic compressive strength f_ck	45 N/mm²	§ 3.1.2.2.2
Ultimate compressive strain ε_c2d	3.0 ‰	§ 4.2.1.5
Mean tensile strength f_ctm	3.8 N/mm²	§ 3.1.2.2.4
Modulus of elasticity E_cm	36500 N/mm²	§ 3.1.2.3.3
Unit weight of the reinforced concrete γ	25 kN/m³
Compressive strength at transfer f_ck(t)	35 N/mm²	§ 4.1.5.2.4
Mean tensile strength at transfer f_ctm(t)	3.2 N/mm²
Modulus of elasticity at transfer E_cm(t)	34000 N/mm²
Characteristic yield strength f_yk	500 N/mm²
Modulus of elasticity E_s	205000 N/mm²
Resistance factor for concrete γ_c	1.5	§ 2.4.2.6
Resistance factor for steel γ_s	1.15	§ 2.4.2.6
Duration factor η_t	1.0 — both conditions of 4.2.1.3 met	§ 4.2.1.3
Type of prestressing	Post-tensioned, grouted (bonded)
Number of tendons n_p	2
Area of one tendon A_p1	1950 mm²
Characteristic tensile strength f_pk	1860 N/mm²	§ 3.3.2.2
Characteristic 0.1 % proof stress f_p0,1k	1600 N/mm²	§ 3.3.2.2
Modulus of elasticity E_p	195000 N/mm²	§ 3.3.2.4
Relaxation loss after 1000 hours ρ₁₀₀₀	2.5 %	§ 3.3.2.7.1
Ducts	Grouted steel ducts (k_H = 0.5)	§ 4.3.3.3.5
Outer diameter of a duct Ø_H	85 mm
Ducts side by side across the web n_H	1
Height at the end anchors y_E	650 mm
Height at the low points y_M	150 mm
Stress at the jack σ_p,max	1395 N/mm²	§ 4.1.5.2.1
Tendons tensioned	From the left end
Friction coefficient μ	0.2	§ 4.1.5.3.3
Unintentional angular deviation Δφ	0.005 rad/m	§ 4.1.5.3.3
Slip at the anchor (wedge draw-in) Δl_s	6 mm	§ 4.1.5.3.1
Final creep coefficient φ(∞,t₀)	2.0	§ 3.1.2.6.2
Final shrinkage strain ε_cs,∞	0.3 ‰	§ 3.1.2.7
Requirement level	Increased	§ 4.4.2.2.3
Tensile strength for cracking	Mean tensile strength (4.4.1.3)	§ 4.4.1.3
Load factor for the permanent loads γ_G	1.35
Load factor for the imposed load γ_Q	1.5
Diameter of the stirrups Ø_w	12 mm
Legs of a stirrup n_w	2
Spacing of the stirrups near the supports s	150 mm
Yield strength of the stirrups f_yk,w	500 N/mm²
Anchors at one end n_a	2
Side of a square anchor plate a₀	280 mm
Vertical spacing of the anchors s_a	340 mm
Diameter of the links Ø_l	12 mm
Vertical legs of a link n_l	2
Spacing of the links s_l	100 mm

Calculation: The beam
Step	Value	Ref.
Span	l = 24.00 m
Slendernesslargest span over the depth; prestressed beams commonly reach 20 to 30	l/h = 18.5

Calculation: Properties of the concrete section
Step	Value	Ref.
Area	A_c = 658,000 mm²
Height of the centroidabove the bottom face	y_G = 716 mm
Second moment of area	I_c = 130,519,834,853 mm⁴
Section modulus, topI_c/(h − y_G)	W_t = 223,569,666 mm³
Section modulus, bottomI_c/y_G	W_b = 182,239,213 mm³
Upper core pointW_b/A_c above Ga force N inside the core leaves the whole section compressed	k_t = 277 mm
Lower core pointW_t/A_c below G	k_b = 340 mm

Calculation: Ordinary reinforcement
Step	Value	Ref.
Bottom bars4Ø20	A_s,b = 1,257 mm²
Top bars6Ø12	A_s,t = 679 mm²
Self-weightγ·A_c	g₀ = 16.45 kN/m

Calculation: Prestressing steel
Step	Value	Ref.
Area of the prestressing steeln_p·A_p1	A_p = 3,900 mm²
Design strengthf_p0,1k/γ_s	f_pd = 1,391 N/mm²	§ 2.4.2.5 (5)

Calculation: Tendon profile
Step	Value	Ref.
Eccentricity at the low pointsy_G − y_M	e_M = 566 mm
Eccentricity at the anchorsy_G − y_Enear the centroid: no moment from the anchors	e_E = 66 mm
Low pointmid-span	x_M = 12.00 m

Calculation: Check 1 — stress at tensioning
Step	Value	Ref.
Stress in the prestressing steel during tensioningσ_p,max/lim	η = 0.969 · Pass97 % of the capacity used	§ 4.1.5.2.1 (21)
Force at the jackσ_p,max·A_p	P_max = 5,441 kN

Calculation: Friction along the tendon
Step	Value	Ref.
Sum of the angular deviationsΣ\|Δθ\| over the whole tendon9.5°	Σφ = 0.166 rad
Force at the far end after frictionP_max·e^(−μ·(φ_x + Δφ·x))	P(L) = 5,138 kN	§ 4.1.5.3.3 (24)
Largest friction loss	ΔP_μ/P_max = 5.6 %

Calculation: Slip at the anchors
Step	Value	Ref.
Length affected by the slip∫ 2·(P(x) − P(l_s)) dx = Δl_s·E_p·A_pfrom the jacked end	l_s = 19.03 m	§ 4.1.5.3.1
Force at the anchor after the slip2·P(l_s) − P(0)	P_A = 4,957 kN

Calculation: Check 2 — stress after anchoring
Step	Value	Ref.
Largest steel stress after anchoringσ_p/lim	η = 0.980 · Pass98 % of the capacity used	§ 4.1.5.2.2 (22)

Calculation: Shortening of the concrete at transfer
Step	Value	Ref.
Modular ratio at transferE_p/E_cm(t)	α_p(t) = 5.74
Concrete stress at the tendon, at x = 12.00 mP/A_c + P·e²/I_c − M_g0·e/I_ccompression positive here	σ_c,p = 15.18 N/mm²
Loss of steel stress(n_p − 1)/(2·n_p)·α_p(t)·σ_c,p — tendons tensioned one after the other	Δσ_p,el = 21.8 N/mm²	§ 4.1.5.3.1

Calculation: Time-dependent losses at x = 12.00 m
Step	Value	Ref.
Relaxation loss, long term3·ρ₁₀₀₀·σ_p0	Δσ_pr = 96.7 N/mm²	§ 3.3.2.7.1
Concrete stress at the tendon, quasi-permanentP₀ with g₀ + g₂ + ψ₂·qcompression positive here	σ_c,QP = 6.26 N/mm²
Loss from creep, shrinkage and relaxation(ε_cs·E_p + 0.8·Δσ_pr + α_p·φ·σ_c,QP)/(1 + α_p·A_p/A_c·(1 + A_c·e²/I_c)·(1 + 0.8·φ))the material laws of 3.1.2.5 to 3.1.2.7 and 3.3.2.7 (4.1.5.3.4) combined	Δσ_p,c+s+r = 166.8 N/mm²	EN 1992-1-1 (5.46)
Force after all losses / force at the jack	P_∞/P_max = 80.4 %

Calculation: Forces at the key sections
Step	1Left anchorx = 0.00 m	2Low pointx = 12.00 m	3Right anchorx = 24.00 m	Ref.
Height of the tendon y_p mm	650	150	650
Eccentricity e mmy_G − y_p, positive below G	66	566	66
After friction P_μ kN	5,441	5,287	5,138	§ 4.1.5.3.3 (24)
After the slip at the anchors P_s kN	4,957	5,111	5,138	§ 4.1.5.3.1
At transfer P₀ kNafter the shortening of the concrete	4,914	5,026	5,093
After all losses P_∞ kN	4,139	4,376	4,297	§ 4.1.5.3.4
Final steel stress σ_p∞ N/mm²P_∞/A_p	1,061	1,122	1,102

Calculation: Check 3 — final stress in the prestressing steel
Step	Value	Ref.
Smallest stress after all losses at least 0.45·f_pk0.45·f_pk/σ_p∞	η = 0.789 · Pass79 % of the capacity used	§ 4.1.5.2.3

Calculation: Equivalent loads of the prestress
Step	Value	Ref.
Upward load from the tendon8·P_∞·f/l², f = y_E − y_M	u = 30.4 kN/m	§ 4.1.5.1.1
Share of the permanent load it balancesabout 70 to 100 % keeps the beam nearly straight under the permanent load	u/(g₀ + g₂) = 83 %

Calculation: Stresses at transfer: P_0 and the self-weight
Step	Value	Ref.
Largest compression, outside the anchorage zonesbottom face, x = 12.50 m; compression negative	σ_c,0 = -16.77 N/mm²
Largest tensionnone: the whole beam is compressed	σ_ct,0 = -0.21 N/mm²

Calculation: Check 1 — concrete at transfer
Step	Value	Ref.
Strength at transfer at least 0.3·f_ck0.3·f_ck/f_ck(t)	η = 0.386 · Pass39 % of the capacity used	§ 4.1.5.2.4
Largest concrete compression at transfer\|σ_c,0\|/(0.6·f_ck(t))	η = 0.799 · Pass80 % of the capacity used	§ 4.1.5.2.4 (23)

Calculation: Stresses in service: P_∞, g_0 + g_2 + ψ_2·q
Step	Value	Ref.
Largest compressiontop face, x = 12.00 m	σ_c,QP = -9.72 N/mm²
Largest tensionnone: decompressed nowhere	σ_ct,QP = -2.87 N/mm²
Effective tensile strengthk_t·f_ctm, k_t = 1/(1 + 0.5·h/3)	f_ct,eff = 3.12 N/mm²	§ 4.4.1.3

Calculation: Cracks at mid-span, x = 12.00 m
Step	Value	Ref.
Concrete stress at the bottom facecompressed	σ_ct = -2.88 N/mm²
Bar stress at first cracking — at mid-spanσ_s,r/σ_s,adm	η = 0.069 · Pass7 % of the capacity used	Table 17

Calculation: Design actions
Step	Value	Ref.
Largest sagging design momentγ_G·M_g + γ_Q·M_qa simply supported beam has no secondary moment	M_d,max = 6,243 kNm	§ 4.1.5.5.2
Largest design shearγ_G·V_g + γ_Q·V_qat a support axis	V_d,max = 1,040 kN

Calculation: Check 1 — bending at mid-span, x = 12.00 m
Step	Value	Ref.
Design momentsagging	M_d = 6,243 kNm
Bending resistancesection with the bars and the bonded tendon	M_Rd = 6,435 kNm	§ 4.2.1.5
Depth of the compression zone	x = 194 mm
Design moment against the bending resistance — at mid-spanM_d/M_Rd	η = 0.970 · Pass97 % of the capacity used
Resistance above the cracking moment (minimum reinforcement) — at mid-spanM_cr/M_Rd	η = 0.715 · Pass71 % of the capacity used	§ 4.3.1.2
Compression zone shallow enough for a ductile section — at mid-span(x/d)/(0.5·435/f_yd)	η = 0.334 · Pass33 % of the capacity used	§ 4.1.4.2.6

Calculation: Check 2 — shear near the left support, x = 1.30 m
Step	Value	Ref.
Design shearγ_G·V_g + γ_Q·V_qat a distance h from the support axis	V_d = 928 kN
Inclination of the tendonrising towards the support: it carries part of the shear	β_p = 4.2 °
Shear reinforcement ratio — near the left supportρ_w	η = 0.260 · Pass26 % of the capacity used	§ 5.5.2.2 (116)
Spacing of the stirrups — near the left supports	η = 0.500 · Pass50 % of the capacity used	§ 5.5.2.2
Design shear against the shear reinforcement — near the left supportV_d/V_Rd,s	η = 0.774 · Pass77 % of the capacity used	§ 4.3.3.4.3 (45)
Design shear against the compression field — near the left supportV_d/V_Rd,c	η = 0.515 · Pass51 % of the capacity used	§ 4.3.3.4.6 (47)
Tension chord — near the left supportF_t,d/F_t,Rd	η = 0.407 · Pass41 % of the capacity used	§ 4.3.3.4.10

Calculation: Check 3 — shear near the right support, x = 22.70 m
Step	Value	Ref.
Design shearγ_G·V_g + γ_Q·V_qat a distance h from the support axis	V_d = 928 kN
Inclination of the tendonrising towards the support: it carries part of the shear	β_p = 4.2 °
Shear reinforcement ratio — near the right supportρ_w	η = 0.260 · Pass26 % of the capacity used	§ 5.5.2.2 (116)
Spacing of the stirrups — near the right supports	η = 0.500 · Pass50 % of the capacity used	§ 5.5.2.2
Design shear against the shear reinforcement — near the right supportV_d/V_Rd,s	η = 0.767 · Pass77 % of the capacity used	§ 4.3.3.4.3 (45)
Design shear against the compression field — near the right supportV_d/V_Rd,c	η = 0.511 · Pass51 % of the capacity used	§ 4.3.3.4.6 (47)
Tension chord — near the right supportF_t,d/F_t,Rd	η = 0.407 · Pass41 % of the capacity used	§ 4.3.3.4.10

Calculation: Design force at the anchors
Step	Value	Ref.
Force at the jackthe characteristic force of the overtension	P_max = 5,441 kN
Design force for the anchorage zoneγ_P·P_max, γ_P = 1.2	P_d = 6,529 kN	§ 4.1.5.5.3
Design strength of the concrete at transferη_fc·η_t·f_ck(t)/γ_c	f_cd(t) = 23.3 N/mm²	§ 2.4.2.3 (2)

Calculation: Local compression under an anchor plate
Step	Value	Ref.
Loaded areaa₀²	A_c0 = 78,400 mm²
Spreading areaa₁·b_1, a₁ = min(a ; h), b₁ = b₀ + a₁ − a₀within b_w = 320 mm and 340 mm of depth per anchor	A_c1 = 102,400 mm²	§ 4.2.1.11
Enhancement for local compression√(A_c1/A_c0) ≤ 3.0	k_c = 1.14	§ 4.2.1.11 (32)
Bearing stress under the plateP_d/(n_a·a₀²)against k_c·f_cd(t) = 26.7 N/mm²	σ_c0 = 41.6 N/mm²

Calculation: Bursting forces behind the anchors
Step	Value	Ref.
Height of the spreading prism2·min(y_E ; h − y_E)the force spreads over a prism symmetric about the anchors	h_e = 1,300 mm
Height of the group of anchors(n_a − 1)·s_a + a₀	a_g = 620 mm
Vertical bursting force, whole group0.25·P_d·(1 − a_g/h_e)strut-and-tie model of the spreading force, concrete tension neglected	T_v = 854 kN	§ 5.3.3
Vertical bursting force between two anchors0.25·P_d/n_a·(1 − a₀/s_a)	T_v,1 = 144 kN	§ 5.3.3
Horizontal bursting force at each anchor0.25·P_d/n_a·(1 − a₀/b_w)	T_h = 102 kN	§ 5.3.3
Reinforcement for T_vmax(T_v ; T_v,1)/f_ydspread between 0.1·h_e and h_e behind the end face	A_s,v = 1,964 mm²

Calculation: Check 1 — links against the vertical bursting force
Step	Value	Ref.
Links within the spreading lengthn_l·π·Ø_l²/4·0.9·h_e/s_l	A_s,prov = 2,646 mm²
Bursting force against the linksT_v/(A_s·f_yd)	η = 0.742 · Pass74 % of the capacity used	§ 5.3.3

How this module is verified

Verified clause by clause against SIA 262:2025 and against independent calculations used on real projects, before release.

Questions

How much of the permanent load should the tendon balance?

About 70 to 100 % keeps the beam nearly straight under the permanent load.

Why is the force at the jack increased by 1.2 for the anchorage?

The anchorage zone is designed for P_d = γ_P · P_max, γ_P = 1.2, clause 4.1.5.5.3. P_max is the characteristic force of the overtension.

Which plan includes this module?

The Basic plan, with its calculation note.

Norm: SIA 262:2025 · Page updated: 5 November 2026