Box-girder bridge, prestress losses and serviceability to SIA 262

Follows the force of the tendons along a continuous box girder to SIA 262:2025, from stressing to time infinity, then the equivalent loads, the isostatic and hyperstatic moments of the prestress, the stresses, the crack control and the deflections. 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.

See the Basic plan Already subscribed? Sign in

About this check

clauses, worked example and questions

What it checks

Friction and slip: P_max = σ_p,max·A_p, P_max·e^(−μ·(φ_x + Δφ·x)), clause 4.1.5.3.3, equation (24), and the length l_s affected by the slip, from the area between the friction curve and its mirror equal to Δl_s·E_p·A_p (clause 4.1.5.3.1). The elongations Δl_L and Δl_R = ∫ P(x)/(E_p·A_p) dx.
Elastic shortening: Δσ_el = j·E_p/E_cm(t)·σ_c,p, with j = (n_P − 1)/(2·n_P), and σ_p0 ≤ min(0.75·f_pk; 0.85·f_p0,1k), clause 4.1.5.2.2, equation (22).
Long-term losses at the key sections: the notional size h₀ = 2·A_c/u (clause 3.1.2.7.8), the relaxation Δσ_pr = 3·ρ₁₀₀₀·σ_p0 (clause 3.3.2.7.1), creep, shrinkage and relaxation together by EN 1992-1-1 5.10.6, equation (5.46), and σ_p∞ ≥ 0.45·f_pk, clause 4.1.5.2.3.
Equivalent loads at time infinity: the upward loads P·|y″| along the low parabolas, the downward load P/R_min over the reverse curves, the forces F_H, F_V and the moment M_E at the anchors, and the share of g₁ + g₂ balanced.
Moments of the prestress, after stressing and at time infinity: the isostatic M_iso,0 = −P₀·e and M_iso,∞ = −P_∞·e, the hyperstatic M_hyp,0 and M_hyp,∞ from the support reactions of the continuous girder, their sums M_P,0 and M_P,∞, and M_QP = M_g + M_P,∞.
Stresses at stressing: |σ_c| ≤ 0.6·f_ck(t), clause 4.1.5.2.4, equation (23), under P₀ and the self-weight.
Stresses in service at both faces, quasi-permanent and frequent.
Crack control: σ_s,adm = √(9·E_s·f_ctm·w_nom/Ø_s) ≤ f_yk, clause 4.4.2.3.4, equation (102), with w_nom from Table 17, and f_ct,eff = k_t·f_ctm for each slab, clause 4.4.1.3, equations (99) and (100).
Deflections at the low points: w_g, w_P, w_c, w_cφ = w_c·(1 + φ), clause 4.4.3.2.4, equation (103), and w_fr. Checks: |w_cφ| and w_fr against the limits you enter.

What you enter and what you get

You enter the spans and the two cross-sections, the concrete at 28 days and at stressing, the permanent loads, load model 1 of SIA 261 with ψ_1,q and ψ_1,Q, the prestressing steel, the tendons in the bridge and the profile of the resultant tendon. For the stressing: σ_p,max, one or both ends, μ, Δφ, Δl_s and n_P. Then φ, ε_cs,∞ and ρ₁₀₀₀, the requirements for cracking, the bars of the slabs, and the deflection limits l/700 and l/500 or your own.

You get the force along the bridge at each stage, the losses at the key sections, the moments of the prestress, every stress and deflection check with its utilisation and clause. With the Basic plan, it prints as an A4 calculation note.

Assumptions and limits

The profile is the one of the layout module, from the same inputs.
The further permanent loads act after stressing, on the girder already prestressed.
The quasi-permanent case holds the permanent loads and the prestress only: ψ₂ = 0 for road traffic.
Creep stays proportional to the stress below 0.4·f_cm (clause 3.1.2.6.3).
The hyperstatic moments are linear between the supports, computed with the stiffness of each section. They act on the resistance as well.
Uncracked girder: equation (103) holds while the moments stay below the cracking moment. The deflections are relative to the supports.
With no tension at either face under the quasi-permanent and frequent cases, the minimum reinforcement against cracking may be omitted (clause 4.4.2.3.5).

Worked example

The three-span girder of 36, 45 and 36 m with ten tendons of 19 strands, stressed from both ends to 1,394 N/mm², μ = 0.2, Δl_s = 6 mm, φ = 2.0, ε_cs,∞ = 0.30 ‰ and increased requirements for cracking. Every check passes. These are the module's own results for its worked example, computed when this page was built.

Inputs
Input	Value	Ref.
Number of spans n	3
Span lengths	End spans alike, interior spans alike
End spans l_e	36 m
Interior spans l_i	45 m
Cross-section along the bridge	A thicker section over the interior supports
Width of the deck B	12.5 m
Depth of the girder h	2400 mm
Thickness of the top slab t_t	280 mm
Length of each cantilever b_c	3.0 m
Thickness of a web b_w	450 mm
Width of the bottom slab b_inf	5.0 m
Thickness of the bottom slab t_b	250 mm
Thickness of the top slab t_t,s	280 mm
Thickness of a web b_w,s	550 mm
Thickness of the bottom slab t_b,s	450 mm
Length of this section on each side of a support l_s	8 m
Characteristic compressive strength f_ck	40 N/mm²
Mean compressive strength f_cm	48 N/mm²
Mean tensile strength f_ctm	3.5 N/mm²
Mean modulus of elasticity E_cm	36000 N/mm²
Compressive strength at stressing f_ck(t)	30 N/mm²	§ 4.1.5.2.4
Modulus of elasticity at stressing E_cm(t)	33000 N/mm²
Unit weight of the reinforced concrete γ	25 kN/m³
Edge beams and parapets, both sides g_2,e	10 kN/m
Safety barriers, both sides g_2,b	2 kN/m
Width of the surfacing B_r	10.5 m
Thickness of the surfacing e_r	100 mm
Unit weight of the surfacing γ_r	24 kN/m³
Width of a notional lane w_l	3 m
Number of notional lanes n_l	3
Lane 1: uniform load q_k1	9 kN/m²
Lane 1: load of one axle of the tandem Q_k1	300 kN
Lane 2: uniform load q_k2	2.5 kN/m²
Lane 2: load of one axle of the tandem Q_k2	200 kN
Lane 3: uniform load q_k3	2.5 kN/m²
Lane 3: load of one axle of the tandem Q_k3	100 kN
Uniform load on the remaining area q_kr	2.5 kN/m²
Adjustment factor of the uniform loads α_q	0.9
Adjustment factor of the tandems α_Q	0.9
Factor ψ₁ of the uniform traffic loads ψ_1,q	0.4
Factor ψ₁ of the tandems ψ_1,Q	0.75
Tensile strength of the prestressing steel f_pk	1860 N/mm²
Proof stress of the prestressing steel f_p0,1k	1640 N/mm²
Modulus of elasticity of the prestressing steel E_p	195000 N/mm²
Area of one strand A_p1	150 mm²
Strands per tendon n_s	19
Tendons in the bridge n_T	10
Height of the anchors at the ends y_E	1300 mm
Height of the tendons at the low points y_p,f	250 mm
Height of the tendons over the interior supports y_p,s	2150 mm
Low point in the end spans, from the end support ξ	0.4
Minimum radius of the tendon R_min	6.5 m
Stress at the jack σ_p,max	1394 N/mm²
Tendons stressed	From both ends of the bridge
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
Tendons stressed one after the other n_P	10
Final creep coefficient φ	2.0	§ 3.1.2.6.2
Final shrinkage strain ε_cs,∞	0.30 ‰	§ 3.1.2.7.6
Relaxation loss after 1000 hours ρ₁₀₀₀	2.5 %	§ 3.3.2.7.1
Requirements for cracking	Increased	§ 4.4.2.2.3
Modulus of elasticity of the reinforcing steel E_s	205000 N/mm²
Yield strength of the reinforcing steel f_yk	500 N/mm²
Diameter of the bars of the slabs Ø_s	16 mm
Long-term deflection under quasi-permanent loads: limit l/… l/w_qp	700
Deflection under the frequent traffic loads: limit l/… l/w_fr	500

Calculation: The bridge
Step	Value	Ref.
Total length2·l_e + (n − 2)·l_i	L = 117.00 m

Calculation: The two cross-sections
Step	1In the spans	2Over the supportsl_s = 8.0 m each side
Area A_c m²outline minus the cell	6.455	7.659
Height of the centroid y_G mmabove the bottom face	1,559	1,410
Second moment of area I_c m⁴	5.089	6.294
Notional size h₀ mm2·A_c/u, u outside and cell	304	367

Calculation: Permanent loads
Step	Value	Ref.
Self-weight in the spansγ·A_c	g_1,f = 161.4 kN/m
Self-weight over the supportsγ·A_c,s	g_1,s = 191.5 kN/m
Further permanent loadsg_2,e + g_2,b + B_r·e_r·γ_rthey act after stressing, on the girder already prestressed	g₂ = 37.2 kN/m

Calculation: Road traffic
Step	Value	Ref.
Uniform traffic load on the girderα_q·(Σ q_k,i·w_l + q_kr·b_r)	q_k = 41.2 kN/m
All the tandems togetherα_Q·Σ 2·Q_k,itwo axles 1.2 m apart	Q_k = 1,080 kN
Frequent uniform load	ψ_1,q·q_k = 16.5 kN/m
Frequent tandems	ψ_1,Q·Q_k = 810 kN

Calculation: Prestressing steel
Step	Value	Ref.
Prestressing steel of all the tendonsn_T·n_s·A_p1	A_p = 28,500 mm²
Largest stress after stressingmin(0.75·f_pk; 0.85·f_p0,1k)	σ_p,lim = 1,394 N/mm²	§ 4.1.5.2.2 (22)

Calculation: Friction and slip at stressing
Step	Value	Ref.
Force at the jackσ_p,max·A_p	P_max = 39.73 MN
Angular deviation from one end to the otherfrom the profile	Σφ = 1.620 rad
Force where the friction curves of the two jacks meetP_max·e^(−μ·(φ_x + Δφ·x))at x = 58.5 m	P_min = 31.87 MN	§ 4.1.5.3.3 (24)
Friction loss there	ΔP_μ/P_max = 19.8 %
Length affected by the slip at the left anchorarea between the friction curve and its mirror = Δl_s·E_p·A_p	l_s = 17.22 m	§ 4.1.5.3.1
Force at the left anchor after the slip	P_a = 35.78 MN
Elongation at the left jack∫ P(x)/(E_p·A_p) dxmeasured on site to control the stressing	Δl_L = 377 mm
Elongation at the right jack	Δl_R = 377 mm

Calculation: Force after anchoring and elastic shortening
Step	Value	Ref.
Loss factor of elastic shortening(n_P − 1)/(2·n_P)	j = 0.450
Largest loss of elastic shorteningj·E_p/E_cm(t)·σ_c,p	Δσ_el = 32.5 N/mm²
Stress in the tendons after stressingσ_p0 ≤ σ_p,lim	η = 0.928 · Pass93 % of the capacity used	§ 4.1.5.2.2 (22)

Calculation: Notional size and coefficients
Step	Value	Ref.
Notional size of the girder2·A_c/uto read φ in Figure 2 and β(t − t_s) in Figure 3	h₀ = 304 mm	§ 3.1.2.7.8

Calculation: Losses at the key sections
Step	Value	Ref.
Final stress in the tendons not below 0.45·f_pk0.45·f_pk ≤ σ_p∞	η = 0.877 · Pass88 % of the capacity used	§ 4.1.5.2.3

Step	1Span 1, low pointx = 14.3 m	2Support 1x = 36.0 m	3Span 2, low pointx = 58.5 m	4Support 2x = 81.0 m	5Span 3, low pointx = 102.8 m	Ref.
Stress after anchoring and elastic shortening σ_p0 N/mm²	1,281	1,201	1,103	1,201	1,281
Concrete stress at the tendons, quasi-permanent σ_c,QP N/mm²compression positive	10.98	6.25	4.89	6.25	10.98
Relaxation, long term Δσ_pr N/mm²3·ρ₁₀₀₀·σ_p0	96.1	90.1	82.7	90.1	96.1	§ 3.3.2.7.1
Creep, shrinkage and relaxation together Δσ_p,c+s+r N/mm²(ε_cs·E_p + 0.8·Δσ_pr + E_p/E_cm·φ·σ_c,QP)/(1 + E_p/E_cm·A_p/A_c·(1 + A_c·e²/I_c)·(1 + 0.8·φ))	212.4	182.3	148.4	182.3	212.4	EN 1992-1-1 5.10.6 (5.46)
Stress at time infinity σ_p∞ N/mm²	1,069	1,018	954	1,018	1,069
Loss from the stress at the jack Δσ/σ_p,max %	23.3	26.9	31.5	26.9	23.3

Calculation: Equivalent loads of the prestress at time infinity
Step	1Span 1	2Span 2	3Span 3
Mean force of the tendons P_∞,m MN	30.50	27.89	30.50
Upward load, left of the low point u_L kN/mP·\|y″\|: 2·P·Δy/(D·(D − a)), or 2·P·Δy/D² towards an end	308.8	220.1	262.3
Upward load, right of the low point u_R kN/m	262.3	220.1	308.8
Downward load over the left support u_s,L kN/mP/R_min over the length a	·	4,290	4,692
Downward load over the right support u_s,R kN/m	4,692	4,290	·
Load balanced u/(g₁ + g₂) %mean upward load over the permanent load	144	111	144

Calculation: Forces at the anchors
Step	Value	Ref.
Horizontal force at the left anchor, time infinityP·cos θ_E	F_H = 29.74 MN
Vertical force at the left anchorP·sin θ_Eslope of the tendon θ_E = 8.30°	F_V = 4.337 MN
Moment of the anchor force about the centroid−P·e_E	M_E = -7.79 MNm

Calculation: Moments of the prestress
Step	1Span 1, low pointx = 14.3 m	2Support 1x = 36.0 m	3Span 2, low pointx = 58.5 m	4Support 2x = 81.0 m	5Span 3, low pointx = 102.8 m
Isostatic, after stressing M_iso,0 MNm−P₀·e	-47.8	25.3	-41.1	25.3	-47.8
Hyperstatic, after stressing M_hyp,0 MNmfrom the support reactions of the continuous girder	10.0	25.3	25.3	25.3	10.0
Total, after stressing M_P,0 MNmM_iso + M_hyp	-37.8	50.6	-15.9	50.6	-37.8
Isostatic, time infinity M_iso,∞ MNm−P_∞·e	-39.9	21.5	-35.6	21.5	-39.9
Hyperstatic, time infinity M_hyp,∞ MNm	8.5	21.6	21.6	21.6	8.5
Total, time infinity M_P,∞ MNm	-31.3	43.1	-14.0	43.1	-31.3
Permanent loads M_g MNmg₁ + g₂	17.1	-35.5	15.8	-35.5	17.1
Permanent loads and prestress, time infinity M_QP MNmM_g + M_P,∞	-14.2	7.6	1.7	7.6	-14.2

Calculation: Stresses at stressing
Step	Value	Ref.
Compression in the concrete at stressing\|σ_c\| ≤ 0.6·f_ck(t)	η = 0.724 · Pass72 % of the capacity used	§ 4.1.5.2.4 (23)
Largest tension at stressingbottom face at x = 36.9 m	σ_ct,0 = 0.63 N/mm²

Calculation: Stresses in service
Step	Value	Ref.
Largest compression, quasi-permanentbottom face at x = 12.8 m	σ_c,QP = -9.10 N/mm²
Largest tension, quasi-permanent, top faceat x = 11.3 m	σ_t,QP = -2.30 N/mm²
Largest tension, quasi-permanent, bottom faceat x = 80.1 m	σ_b,QP = -1.58 N/mm²
Largest tension, frequent, top faceat x = 104.3 m	σ_t,fr = -2.01 N/mm²
Largest tension, frequent, bottom faceat x = 58.5 m	σ_b,fr = -1.23 N/mm²
Largest compression, frequent	σ_c,fr = -9.72 N/mm²

Calculation: Crack control
Step	Value	Ref.
Allowed stress in the bars at cracking√(9·E_s·f_ctm·w_nom/Ø_s) ≤ f_ykw_nom = 0.5 mm for increased requirements (Table 17)	σ_s,adm = 449 N/mm²	§ 4.4.2.3.4 (102)
Effective tensile strength, top slabk_t·f_ctm, k_t = 1/(1 + 0.5·t)t = 280 mm, k_t = 0.877	f_ct,eff = 3.07 N/mm²	§ 4.4.1.3 (99) (100)
Effective tensile strength, bottom slabk_t·f_ctm, k_t = 1/(1 + 0.5·t)t = 250 mm, k_t = 0.889	f_ct,eff = 3.11 N/mm²	§ 4.4.1.3 (99) (100)

Calculation: Deflections at the low points (downwards positive)
Step	Value	Ref.
Long-term deflection under the quasi-permanent loads\|w_cφ\| ≤ l/700	η = 0.537 · Pass54 % of the capacity used
Deflection under the frequent traffic loadsw_fr ≤ l/500	η = 0.069 · Pass7 % of the capacity used

Step	1Span 1low point	2Span 2low point	3Span 3low point	Ref.
Permanent loads, elastic w_g mmE_cm·I_c of each section	9.1	10.6	9.1
Prestress at time infinity, elastic w_P mmnegative: upwards	-18.3	-7.3	-18.3
Quasi-permanent, elastic w_c mmw_g + w_P	-9.2	3.3	-9.2
Quasi-permanent, long term w_cφ mmw_c·(1 + φ)	-27.6	9.9	-27.6	§ 4.4.3.2.4 (103)
Frequent traffic w_fr mmψ_1,q·q_k on the spans that sag the point + ψ_1,Q·Q_k at its worst place	4.5	6.2	4.5
Degree of balancing \|w_P/w_g\| %upward deflection of the prestress over the downward one of the permanent loads	202	69	202

How this module is verified

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

Questions

What is the elongation at the jack for?

It is measured on site to control the stressing. In the worked example it is 377.2 mm at each jack.

How large is the friction loss in the worked example?

19.8 % where the friction curves of the two jacks meet, at x = 58.5 m, with P_min = 31.87 MN.

Which plan includes this module?

The Basic plan, with its calculation note.

Norm: SIA 262:2025 · Page updated: 8 November 2026