EN 1993-1-1 Beam Design Worked Example — IPE400 Section Check

Full step-by-step worked example for EN 1993-1-1:2005 steel beam design using a European IPE400 section in S355 steel. This example covers cross-section classification (Clause 5.5), bending moment resistance Mc,Rd (Clause 6.2.5), shear resistance Vc,Rd (Clause 6.2.6), bending-shear interaction (Clause 6.2.8), lateral-torsional buckling resistance Mb,Rd (Clause 6.3.2), and serviceability deflection check. All calculations follow the UK National Annex values. This example is suitable for student learning, graduate engineer training, and independent design verification.

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Design Problem Statement

Problem: Verify a simply supported IPE400 beam in S355 steel spanning L = 8.0 m between supports. The beam supports a reinforced concrete floor slab that provides full lateral restraint to the top flange (compression flange under gravity loading). However, the beam also must be checked for the temporary condition where lateral restraint is present only at the supports (construction stage before slab hardening). The top flange is in compression in both cases.

Loading:

• Permanent action (dead): gk = 8.0 kN/m (slab + finishes + services + beam self-weight)
• Variable action (imposed): qk = 16.0 kN/m (office floor occupancy, Category B per EN 1991-1-1)

Design to EN 1993-1-1 with UK National Annex:

• gamma_M0 = 1.00
• gamma_M1 = 1.00 (UK NA)
• Steel grade: S355JR (EN 10025-2), fy = 355 MPa, fu = 470 MPa

Step 1 — IPE400 Section Properties

From the European section tables (ArcelorMittal Sections Handbook / SCI P363 equivalent for European sections):

Material: S355, fy = 355 N/mm^2 for tf <= 40 mm, E = 210,000 N/mm^2, G = 81,000 N/mm^2

Step 2 — Design Loads and Internal Forces

Ultimate Limit State (ULS)

UK NA Equation 6.10b (the standard UK combination):

w_Ed = 1.35 * gk + 1.5 * qk
w_Ed = 1.35 * 8.0 + 1.5 * 16.0 = 10.80 + 24.00 = 34.80 kN/m

Maximum design bending moment (simply supported, UDL):

M_y,Ed = w_Ed * L^2 / 8 = 34.80 * 8.0^2 / 8 = 34.80 * 8.0 = 278.4 kN.m

Maximum design shear force (simply supported, UDL):

V_Ed = w_Ed * L / 2 = 34.80 * 8.0 / 2 = 139.2 kN

Serviceability Limit State (SLS)

Imposed load only (deflection check):

w_SLS = qk = 16.0 kN/m

Step 3 — Cross-Section Classification (Clause 5.5)

Classify the flange (outstand in compression) and web (internal in bending) per Table 5.2.

Flange Classification

Outstand width c = (b - tw - 2*r) / 2 = (180 - 8.6 - 42) / 2 = 129.4 / 2 = 64.7 mm

epsilon = sqrt(235/fy) = sqrt(235/355) = sqrt(0.662) = 0.814

c/tf = 64.7 / 13.5 = 4.79

Limits for flange in compression (outstand):

• Class 1: c/tf <= 9*epsilon = 9 * 0.814 = 7.32

• Class 2: c/tf <= 10*epsilon = 10 * 0.814 = 8.14

• Class 3: c/tf <= 14*epsilon = 14 * 0.814 = 11.39

  4.79 <= 7.32 → Flange is Class 1.

Web Classification

Web depth between fillets: cw = h - 2tf - 2r = 400 - 27 - 42 = 331 mm

cw/tw = 331 / 8.6 = 38.49

For a web in pure bending, the limiting value for Class 1: cw/tw <= 72*epsilon = 72 * 0.814 = 58.61

38.49 <= 58.61 → Web is Class 1.

Result: IPE400 in S355 is Class 1 for pure bending. Plastic design may be used.

Step 4 — Bending Moment Resistance (Clause 6.2.5)

For Class 1 cross-sections:

Mc,Rd = Wpl,y * fy / gamma_M0
Mc,Rd = 1,307 * 10^3 * 355 / 1.00 = 464.0 * 10^6 N.mm = 464.0 kN.m

Utilisation check:

M_y,Ed / Mc,Rd = 278.4 / 464.0 = 0.600 — OK (60% utilised)

The IPE400 has substantial reserve moment capacity. A lighter IPE360 (Wpl,y = 1,019 cm^3, Mc,Rd = 361.7 kN.m) could also be adequate with 77% utilisation, but deflection and LTB checks may govern for the lighter section.

Step 5 — Shear Resistance (Clause 6.2.6)

Shear Area

For an I-section loaded parallel to the web:

Av = A - 2*b*tf + (tw + 2*r)*tf but Av >= eta * hw * tw

Av = 8,450 - 2 _ 180 _ 13.5 + (8.6 + 2*21.0) * 13.5 = 8,450 - 4,860 + (8.6 + 42.0) _ 13.5 = 8,450 - 4,860 + 50.6 _ 13.5 = 3,590 + 683.1 = 4,273.1 mm^2

Minimum shear area: eta _ hw _ tw = 1.0 * (400 - 2*13.5) _ 8.6 = 1.0 _ 373 * 8.6 = 3,207.8 mm^2

Av = 4,273 mm^2 >= 3,208 mm^2 — OK.

Plastic Shear Resistance

Vpl,Rd = Av * fy / (sqrt(3) * gamma_M0)
Vpl,Rd = 4,273 * 355 / (1.732 * 1.00) = 4,273 * 204.9 = 875,600 N = 875.6 kN

Utilisation check:

V_Ed / Vpl,Rd = 139.2 / 875.6 = 0.159 < 0.50

V_Ed / Vpl,Rd < 0.50 → No bending-shear interaction required per Clause 6.2.8(2). The moment resistance does not need to be reduced for coincident shear.

Step 6 — Lateral-Torsional Buckling (Clause 6.3.2)

Two conditions are checked:

1. In-service (composite slab in place): Top flange continuously restrained by slab — no LTB check required. The full Mc,Rd = 464.0 kN.m applies.
2. Construction stage (steel beam alone): Lateral restraint only at supports, L = 8.0 m. LTB check required.

Construction Stage Loading

Construction load (EN 1991-1-6, during execution):

w_Ed,c = 1.35 * 8.0 + 1.5 * 0.75   (construction live load 0.75 kN/m^2, 1.0 m tributary, or simplified as minimal imposed)
      = 10.80 + 1.13 = 11.93 kN/m

Assume wet concrete + beam self-weight ~ 8.0 kN/m total (conservative, as slab is not yet composite). M_Ed,c = 11.93 * 8.0^2 / 8 = 95.4 kN.m

Elastic Critical Moment Mcr

For a simply supported beam under uniform moment with end warping prevented (fork supports, kz = kw = 1.0):

The general expression (NCCI SN003a-EN-EU) is:

Mcr = C1 * pi^2 * E * Iz / (kz*L)^2 *
      sqrt[(kz/kw)^2 * Iw/Iz + (kz*L)^2 * G * It / (pi^2 * E * Iz)]

For uniform moment (conservative, as UDL produces a parabolic moment diagram with lower LTB sensitivity): C1 = 1.00 (conservative for UDL — actual C1 ~ 1.13 would give higher Mcr and lower lambda_LT_bar)

Substituting:

Mcr = 1.00 * pi^2 * 210,000 * 1,318*10^4 / (8,000)^2 *
      sqrt[1.0 * 0.490*10^12 / 1,318*10^4 + (8,000)^2 * 81,000 * 46.4*10^4 / (pi^2 * 210,000 * 1,318*10^4)]

= 1.00 _ (2,069 _ 10^9) * 1,31810^4 / 6410^6 * sqrt[0.49010^12 / 13.1810^6 + 6410^6 * 81,000 * 0.46410^6 / (2,069 * 10^9 * 13.18*10^6)]

Note: pi^2 _ E = 9.8696 _ 210,000 = 2,072,600 ~ 2.073e6 N/mm^2

Mcr = 1.0 _ 2.073e6 _ 1.318e7 / 6.4e7 _ sqrt[0.490e12 / 1.318e7 + 6.4e7 _ 8.1e4 _ 4.64e5 / (2.073e6 _ 1.318e7)]

= 2.073e6 _ 1.318e7 / 6.4e7 _ sqrt[37,178 + 2.405e17 / 2.732e13] = 2.073e6 _ 0.2059 _ sqrt[37,178 + 8,803] = 426,800 _ sqrt[45,981] = 426,800 _ 214.4 = 91,520,000 N.mm = 91.5 kN.m

Non-Dimensional Slenderness lambda_LT_bar

lambda_LT_bar = sqrt(Wy * fy / Mcr)

For Class 1 section, Wy = Wpl,y = 1,307 * 10^3 mm^3:

lambda_LT_bar = sqrt(1,30710^3 * 355 / 91.510^6) = sqrt(464.0*10^6 / 91.5*10^6) = sqrt(5.07) = 2.25

Reduction Factor chi_LT

For hot-rolled I-sections, the LTB buckling curve is selected per Table 6.4 / Table 6.5. For h/b = 400/180 = 2.22 > 2 → buckling curve c (alpha_LT = 0.49).

Phi_LT = 0.5 * [1 + alpha_LT * (lambda_LT_bar - lambda_LT,0) + beta * lambda_LT_bar^2]

But using the standard expression (Cl. 6.3.2.3, method in 6.3.2.2):

Using EN 1993-1-1 Clause 6.3.2.2(1):

Phi_LT = 0.5 * [1 + alpha_LT * (lambda_LT_bar - 0.2) + lambda_LT_bar^2]
Phi_LT = 0.5 * [1 + 0.49 * (2.25 - 0.2) + 2.25^2]
       = 0.5 * [1 + 0.49 * 2.05 + 5.063]
       = 0.5 * [1 + 1.005 + 5.063]
       = 0.5 * 7.068
       = 3.534

chi_LT = 1 / (Phi_LT + sqrt(Phi_LT^2 - lambda_LT_bar^2))
       = 1 / (3.534 + sqrt(12.488 - 5.063))
       = 1 / (3.534 + sqrt(7.425))
       = 1 / (3.534 + 2.725)
       = 1 / 6.259
       = 0.160

Buckling Resistance Moment Mb,Rd

Mb,Rd = chi_LT * Wy * fy / gamma_M1

Using gamma_M1 = 1.00 (UK NA):

Mb,Rd = 0.160 * 1,307*10^3 * 355 / 1.00 = 0.160 * 464.0*10^6 = 74.1 kN.m

Utilisation: M_Ed,c / Mb,Rd = 95.4 / 74.1 = 1.29 — FAIL.

Conclusion: The IPE400 beam without intermediate lateral restraint WILL FAIL in LTB during the construction stage. The construction stage therefore requires either:

1. Temporary propping at midspan (reduces L_cr to 4.0 m → lambda_LT_bar ~ 1.13 → chi_LT ~ 0.54 → Mb,Rd ~ 251 kN.m > 95.4 — OK)
2. Intermediate lateral restraints at third points (L_cr = 2.67 m)
3. A stockier section with greater lateral stiffness (e.g., HEA360)

In typical UK practice, primary beams of this span would be laterally restrained by incoming secondary beams at 2.5-3.0 m centres, providing adequate restraint for the construction stage. If the beam is a secondary beam with the slab cast on top, the flange is not continuously restrained until the concrete hardens, and intermediate restraint must be provided.

For the in-service condition with the slab providing continuous restraint to the top (compression) flange: the beam is fully restrained against LTB. Utilisation: 278.4 / 464.0 = 0.600 — OK.

Step 7 — Serviceability Deflection Check

Deflection under imposed (live) load only (EN 1990 Annex A1.4 recommendation for floors):

The recommended limit for total deflection is L/200 and for imposed load deflection is L/360. UK NA adopts these.

SLS imposed load (unfactored): w_SLS = qk = 16.0 kN/m

Deflection formula for simply supported beam under UDL:

delta = 5 * w * L^4 / (384 * E * Iy)

delta = 5 _ 16.0 _ 8,000^4 / (384 _ 210,000 _ 23,13010^4) = 5 * 16.0 * 4.09610^15 / (384 _ 210,000 _ 231.310^6) = 3.27710^17 / (384 * 4.85710^13) = 3.27710^17 / 1.865*10^16 = 17.6 mm

Limits:

• Imposed load (L/360): 8,000/360 = 22.2 mm → 17.6 mm < 22.2 mm — OK (79% utilised)
• Total load (w = gk + qk = 24.0 kN/m): delta_total = 17.6 * (24.0/16.0) = 26.4 mm. Limit L/200 = 40.0 mm — OK (66% utilised)

Deflection is satisfactory. No pre-camber required.

Step 8 — Additional Checks

Web Bearing and Buckling at Supports (Clause 6.2.6.2)

At the support, the reaction introduces a localised compression into the web. For unstiffened webs:

Effective loaded length for bearing: ss = stiff bearing length (assumed 100 mm for typical seated connection)

ly = ss + 2tf(1 + (beff/ss)^0.5) ... simplified to ss + 5*(tf + r) for a conservative check.

The web bearing resistance F_Rd should be >> V_Ed = 139.2 kN. For the IPE400 web (tw = 8.6 mm, fy = 355 MPa), even a 50 mm bearing length provides several hundred kN capacity — not critical for this lightly loaded beam.

Flange-Induced Buckling (Clause 6.2.6.3)

Not required for rolled I-sections in S355 — this limit state applies primarily to plate girders.

Torsional Buckling

Not relevant for doubly-symmetric I-section with primary loading in the plane of the web.

Summary — IPE400 Beam Design Results

Final Recommendation: IPE400 in S355 is acceptable for the 8.0 m span, provided that temporary propping or intermediate lateral restraint is provided during the construction stage before the slab hardens. If unpropped construction is required, either provide intermediate lateral bracing at 3.0 m centres, or select a section with greater lateral stiffness (e.g., HEA360 with Iz = 7,890 cm^4 vs IPE400 Iz = 1,318 cm^4 — a factor of 6 improvement in weak-axis stiffness).

Frequently Asked Questions

Why does the IPE400 fail LTB at the construction stage but pass when the slab is in place?

The IPE400 has a very low weak-axis second moment of area (Iz = 1,318 cm^4) relative to its strong-axis stiffness (Iy = 23,130 cm^4). This makes it inherently slender for lateral-torsional buckling. At the construction stage with no intermediate lateral restraint and an 8.0 m unbraced length, the elastic critical moment Mcr = 91.5 kN.m is far below the cross-section capacity, giving lambda_LT_bar = 2.25 and a severe reduction (chi_LT = 0.16). When the slab is cast and hardened, it continuously restrains the compression flange, eliminating the LTB failure mode entirely.

How is the LTB buckling curve selected for an IPE section?

EN 1993-1-1 Table 6.4 (as amended by the UK NA) selects the LTB buckling curve based on the depth-to-width ratio h/b. For h/b > 2.0: use curve c (alpha_LT = 0.49). For h/b <= 2.0: use curve b (alpha_LT = 0.34). The IPE400 has h/b = 400/180 = 2.22 > 2.0, so curve c applies. The higher imperfection factor for curve c reflects the greater LTB sensitivity of deep, narrow I-sections compared to stocky H-sections of similar weight.

Can using the actual C1 factor instead of C1 = 1.0 improve the LTB check?

Yes. Using C1 = 1.00 (uniform moment) is conservative for a UDL on a simply supported beam. The actual C1 factor depends on the moment distribution and end restraint conditions. For a simply supported beam under UDL with fork supports: C1 ≈ 1.13 per NCCI SN003. Using C1 = 1.13 increases Mcr by 13%, reducing lambda_LT_bar slightly. However, for lambda_LT_bar = 2.25, the improvement in chi_LT is marginal (~0.160 to ~0.175) and does not change the conclusion that LTB governs the construction stage. The C1 factor is most beneficial when lambda_LT_bar is in the 0.5-1.2 range where chi_LT is sensitive to small changes in slenderness.

What practical measures prevent LTB failure during construction?

Four practical measures are commonly used: (1) Temporary propping at midspan or third points during construction — props are removed after the slab reaches sufficient strength. (2) Plan bracing between adjacent beams using temporary steel angles or channels at third points. (3) Using the permanent secondary beams as lateral restraints if the steelwork erection sequence allows it. (4) Selecting a stockier section (HEA or UKC rather than IPE or UKB) with greater lateral stiffness — an HEA360 has Iz = 7,890 cm^4 vs IPE400 Iz = 1,318 cm^4, a 6x improvement. The choice depends on construction programme, access, and cost of temporary works vs increased steel tonnage.

How does the EN 1993-1-1 beam design method compare to AISC 360 for this IPE400?

For cross-section resistance: the approaches are nearly identical — both use Wpl * fy with partial factors (gamma_M0 = 1.00 vs phi_b = 0.90). The EN Mc,Rd = 464.0 kN.m while the AISC phi*Mn = 0.90 _ 1,307 _ 345 (A992 Gr 50) = 406 kN.m. The difference is primarily from the partial factor (1.00 vs 0.90) and the slightly higher fy of S355 vs A992 (355 vs 345 MPa). For LTB: EN 1993-1-1 uses chi_LT and buckling curves while AISC 360 uses the Cb-modified LTB formulae in Chapter F. With Cb = 1.14 (UDL), the AISC LTB capacity for the same unbraced length would be approximately 92 kN.m, comparable to the EN 1993-1-1 value of 91.5 kN.m (before chi_LT reduction). The methodologies differ in detail but produce broadly similar design outcomes.

Related Pages

• EN 1993-1-1 Beam Design — Flexure, Shear & LTB
• EN 1993 Column Design — Worked Example HEB240
• EN 1993 Column Buckling — Curves a0-d
• EN 1993 Steel Grades — S235, S275, S355, S460
• EN 1993-1-8 Connection Design — Bolts & Welds
• Beam Capacity Calculator — Free EN 1993 Tool
• Column Capacity Calculator — Free Online Tool

Educational reference only. This worked example is for training and verification purposes. Verify all design values against the current EN 1993-1-1 and the applicable National Annex for your jurisdiction. Section properties should be taken from the manufacturer's published data or the SCI/BCSA Blue Book. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent verification by a qualified structural engineer.