Checks a reinforced or prestressed concrete beam section of any shape under a bending moment with an axial compression or tension, to SIA 262:2025: axial and bending resistance, cracking moment, service stresses and the bar stress that limits the crack width. Every step shows its formula and the clause it comes from.

What it checks

Section: A_c, y_c and I_c of the gross concrete, A_s,tot of the bars and their smallest clear cover c. Idealised section: α_s = E_s/E_cm, A_i = A_c + (α_s − 1) · A_s and I_i.
Design strengths: η_fc = (40/f_ck)^(1/3) ≤ 1, 4.2.1.2 (26); f_cd = η_fc · η_t · f_ck/γ_c, 2.4.2.3 (2); f_yd = f_yk/γ_s, 2.4.2.5 (4), and f_pd = f_p0,1k/γ_s for prestressing steel, equation (5).
Check 1, axial force: |N_d| ≤ N_Rd, with N_Rd,c = f_cd · (A_c − A_s) + f_yd · A_s in compression, the whole section at ε_c2d (4.2.1.5), and N_Rd,t = A_s · f_yd in tension, all the steel yielding and no concrete.
Check 2, bending under N_d: the neutral axis x from equilibrium with N = N_d, a 0.85 · x stress block at f_cd and ε_c2d at the compression face (4.2.1.5). M_Rd is the moment of the internal forces about the axis of the member, 2.4.2.1 (1), and M_d ≤ [M_Rd].
Cracking moment, 4.4.1.3: k_t = 1/(1 + 0.5 · t), equation (100), with t = h/3; f_ct,eff = k_t · f_ctm, equation (99); M_cr = (f_ct,eff + N/A_i) · I_i/v_t − N · (y_G − y_i).
Bar stress at first cracking: σ_s,r just after the first crack, concrete in tension ignored (4.1.4.2.1), against σ_s,adm = √(9 · E_s · f_ctm · w_nom/Ø_s) ≤ f_yk, 4.4.2.3.4 (102) and Table 17.
Service state under N and M: cracked or not, the concrete stresses σ_c,c and σ_c,t and the bar stresses σ_s and σ_s′.

What you enter and what you get

For the beam you choose a rectangle, a T-section, a circle or a polygon drawn by clicking its corners. You choose sagging or hogging, and an axial compression or tension. You place each reinforcing bar or prestressing unit by clicking in the drawing, drag it, remove it, or edit the table; a bar layer places a row at once. You enter f_ck, ε_c2d, f_ctm, E_cm, f_yk, E_s, γ_c, γ_s, η_t, N_d and M_d. For serviceability: the tensile strength for the cracking moment, the requirement level and w_nom from Table 17, N, M and the load case.

You get the section values, the resistances and each check with its utilisation and clause, the cracking moment and the stresses. With the Basic plan, it prints as an A4 calculation note.

Assumptions and limits

N acts at the centroid of the gross concrete section, the axis of the member, and M is taken about that axis.
Prestress is not an axial force here: place the units in the section.
Sagging puts the top of the drawn section in compression; hogging the reverse, for example a T-beam over a support with its flange in tension.
The upper fractile f_ctk0,95 = 1.3 · f_ctm, equation (8), may replace the mean value when the uncracked state is unfavourable (4.4.1.4).
Ø_s is the largest bar.
The limits on x/d of 4.1.4.2.5 and 4.1.4.2.6 are written for members in bending; with an axial compression the compression zone is deeper by nature.

Worked example

A 400 × 600 mm rectangle with eight Ø20 bars, three at the top, three at the bottom and one on each side at mid-depth, in f_ck = 30 N/mm², sagging, under N_d = 1,500 kN in compression and M_d = 300 kNm, with increased crack requirements, w_nom = 0.5 mm and a quasi-permanent load case of N = 1,000 kN and M = 150 kNm. These are the module's own results for its worked example, computed when this page was built.

Inputs
Input	Value	Ref.
Shape of the section	Rectangle
Width b	400 mm
Depth h	600 mm
Direction of the moment	Sagging — compression at the top
Axial force N	Compression
Steel in the section	{"b":[[50,50,20],[200,50,20],[350,50,20],[50,550,20],[200,550,20],[350,550,20],[50,300,20],[350,300,20]],"p":[],"v":[]}
Characteristic compressive strength f_ck	30 N/mm²	§ 3.1.2.2.2
Ultimate compressive strain ε_c2d	3.5 ‰	§ 4.2.1.5
Mean tensile strength f_ctm	2.9 N/mm²	§ 3.1.2.2.4
Modulus of elasticity E_cm	33600 N/mm²	§ 3.1.2.3.3
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
Design axial force N_d	1500 kN
Design bending moment acting with N_d M_d	300 kNm
Tensile strength for the cracking moment	Mean tensile strength (4.4.1.3)	§ 4.4.1.3
Requirement level	Increased	§ 4.4.2.2.3
Nominal crack width w_nom	0.5 mm	Table 17
Axial force in the load case N	1000 kN
Bending moment in the load case M	150 kNm
Load case of N and M	Quasi-permanent	Table 17

Calculation: Concrete section
Step	Value	Ref.
Gross concrete area	A_c = 240,000 mm²
Depth of the section	h = 600 mm
Height of the concrete centroidmeasured from the bottom face	y_c = 300 mm
Second moment of area of the concreteabout the horizontal axis through the centroid	I_c = 7,200,000,000 mm⁴

Calculation: Steel in the section
Step	Value	Ref.
Bars	n_s = 8Ø20
Total area of the barsΣ π·Ø²/4	A_s,tot = 2,513 mm²
Smallest clear cover of the barsdistance of the bar edge to the nearest face	c = 40 mm

Calculation: Idealised sections
Step	Value	Ref.
Modular ratio of the barsE_s/E_cm	α_s = 6.10
Idealised section for the momentA_c + (α_s − 1)·A_s	A_i = 252,821 mm²
Second moment of area, uncracked	I_i = 7,800,969,845 mm⁴

Calculation: Design strengths
Step	Value	Ref.
Factor for the brittleness of strong concrete(40/f_ck)^(1/3) ≤ 1	η_fc = 1.000	§ 4.2.1.2 (26)
Duration factor	η_t = 1.00	§ 4.2.1.3
Design compressive strength of the concreteη_fc·η_t·f_ck/γ_c	f_cd = 20.0 N/mm²	§ 2.4.2.3 (2)
Design yield strength of the barsf_yk/γ_s	f_yd = 434.8 N/mm²	§ 2.4.2.5 (4)
Design yield strain of the barsf_yd/E_s	ε_yd = 2.12 ‰

Calculation: Axial resistance of the section
Step	Value	Ref.
Resistance in centred compressionf_cd·(A_c − A_s) + f_yd·A_sthe whole section at ε_c2d	N_Rd,c = 5,842 kN	§ 4.2.1.5
Resistance in centred tensionA_s·f_ydall the steel yields, no concrete	N_Rd,t = 1,093 kN

Calculation: Check 1 — axial force: |N_d| ≤ N_Rd
Step	Value	Ref.
Axial compression against the resistance\|N_d\|/N_Rd	η = 0.257 · Pass26 % of the capacity used	§ 4.2.1.5

Calculation: Section at the ultimate limit state
Step	Value	Ref.
Depth of the neutral axisfrom equilibrium, N = N_d, with the 0.85·x block at f_cd and ε_c2d at the compression face	x = 240.0 mm	§ 4.2.1.5
Depth of the stress block	0.85·x = 204.0 mm	§ 4.2.1.5
Concrete compression forcef_cd·A_c,comp	F_c = 1,632 kN
Force in the compression barsΣ A_s,i·(σ_s,i − f_cd), bars inside the block	F_s′ = 391 kN
Force in the tension barsΣ A_s,i·σ_s,i	F_s = 523 kN
Effective depthcompression face to the resultant of the tension forces in the tension half of the section	d = 550 mm
Largest bar strainε_c2d·(z_s − x)/x	ε_s,max = 4.52 ‰
Bending resistance under N_dmoment of the internal forces about the axis of the member	M_Rd = 523.3 kNm	§ 2.4.2.1 (1)

Calculation: Check 2 — bending resistance: M_d ≤ M_Rd under N_d
Step	Value	Ref.
Design moment against the resistance under N_dM_d/M_Rd	η = 0.573 · Pass57 % of the capacity used

Calculation: Depth of the compression zone
Step	Value	Ref.
Relative depth of the compression zone	x/d = 0.436

Calculation: Common values
Step	Value	Ref.
Distance from the centroid to the tension face	v_t = 300.0 mm
Axial force in the load caseat the axis of the membercompression	N = 1,000.0 kN
Dimension for the size factorh/3, member in bending	t = 200 mm	§ 4.4.1.3
Size factor1/(1 + 0.5·t), t in m	k_t = 0.909	§ 4.4.1.3 (100)
Effective tensile strength of the concretek_t·f_ctm	f_ct,eff = 2.64 N/mm²	§ 4.4.1.3 (99)
Cracking moment(f_ct,eff + N/A_i)·I_i/v_t − N·(y_G − y_i)about the axis of the member, with N	M_cr = 171.4 kNm	§ 4.4.1.3

Calculation: Just after the first crack (at the crack)
Step	Value	Ref.
Neutral axis depth just after crackingN_int = N and M_int = M_cr, concrete in tension ignored	x_r = 419.2 mm	§ 4.1.4.2.1
Stress of the outer tension bar just after crackingE_s·κ·(z_s − x_r)	σ_s,r = 21.8 N/mm²
Nominal crack width at first crackingσ_s,r²·Ø_s/(9·E_s·f_ctm)eq. (102) solved for w_nom, with Ø_s the largest bar (20 mm)	w_nom,r = 0.002 mm	§ 4.4.2.3.4 (102)

Calculation: Check 1 — bar stress when the section cracks (restrained deformation)
Step	Value	Ref.
Admissible bar stress√(9·E_s·f_ctm·w_nom/Ø_s) ≤ f_ykØ_s = largest bar, 20 mm	σ_s,adm = 366 N/mm²	§ 4.4.2.3.4 (102)
Bar stress at first crackingσ_s,r/σ_s,adm	η = 0.060 · Pass6 % of the capacity used	Table 17

Calculation: State under N = 1,000.0 kN (compression) and M = 150.0 kNm
Step	Value	Ref.
State of the section	uncracked (M ≤ M_cr)
Concrete stress at the compression face−N/A_i − M_i·v_c/I_icompression is negative	σ_c,c = -9.72 N/mm²
Concrete stress at the tension face−N/A_i + M_i·v_t/I_ibelow f_ct,eff	σ_c,t = 1.81 N/mm²
Moment about the centroid of the idealised sectionM + N·(y_G − y_i)N acts at the centroid of the gross section	M_i = 150.0 kNm
Stress of the outer tension barα_s·σ_c at the bar	σ_s = 5.2 N/mm²
Stress of the outer compression bar	σ_s′ = -53.5 N/mm²

How this module is verified

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

Questions

Does the module include second-order effects?

No. Check 1 is without second-order effects and without the minimum eccentricity of a column (4.3.7). For a slender member, add them to M_d or use the columns module.

Why is the beam uncracked in the worked example?

With the axial compression, M_cr = 171.4 kNm. The service moment M = 150 kNm stays below it, so the whole section works.

Can I print the calculation?

The full calculation is shown on screen, free and without an account. A print-ready A4 calculation note is part of the Basic plan.

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

Concrete beam in bending with axial force to SIA 262

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