Portal Frame Types and Analysis Methods

Elastic Global Analysis (Clause 5.2)

The frame is analysed assuming linear-elastic material behaviour throughout. Member design checks (cross-section resistance and member buckling) are performed using the internal forces from the elastic analysis. This is the most common method in European practice for portal frames up to approximately 40 m span.

Advantages: Simpler to perform, conservative, compatible with all section classes (1-4), no rotation capacity checking required. Disadvantages: Does not exploit plastic redistribution — heavier sections than plastic design.

Plastic Global Analysis (Clause 5.4)

The frame is analysed assuming plastic hinge formation at discrete locations, with moment redistribution between hinges. Per Clause 5.4.1, plastic global analysis is permitted provided:

1. All sections where plastic hinges form are Class 1 (plastic) — capable of forming plastic hinges with sufficient rotation capacity
2. Lateral-torsional buckling is prevented at plastic hinge locations by adequate lateral restraint
3. The frame is not subject to repeated plastic deformation (low-cycle fatigue)

Plastic hinges typically form at:

• Eaves connections (rafter-to-column)
• Rafter apex (ridge)
• Rafter haunch toes (first restraint from eaves)
• Column bases (if fixed, not pinned)

Plastic Hinge Formation and Rotation Capacity

Hinge Locations in a Typical Portal Frame

Under gravity load (dead + snow), the typical plastic hinge sequence in a pinned-base portal frame is:

1. Rafter at eaves haunch toe — first hinge, as the rafter moment reaches M_pl,Rd at the point of maximum bending adjacent to the haunch
2. Rafter at ridge — second hinge, as the mid-span moment develops under increasing vertical deflection
3. Eaves connection — may remain elastic if the haunch is adequately proportioned to over-strengthen the rafter-to-column joint

Under wind uplift, the hinge pattern reverses: the bottom flange of the rafter near the eaves goes into compression, and hinges may form on the underside of the rafter.

Rotation Capacity Requirements

EN 1993-1-1 Clause 5.4.1(3) requires that plastic hinges have sufficient rotation capacity to allow the assumed moment redistribution. For Class 1 sections in uniform bending (typical rafter condition away from the haunch), the required rotation capacity is typically 3-5 times the elastic rotation at first yield. This is generally satisfied by standard I-sections with b/t_f ratios within the Class 1 limits.

For sections where the plastic neutral axis is close to the flange (deep haunch sections), the rotation capacity may be limited by local buckling of the compression flange. Clause 5.4.1(4) requires a more rigorous check — either by testing or by limiting the flange slenderness to 0.8 × the Class 1 limit.

Snap-Through Buckling of Pitched Rafters

Mechanism

Snap-through buckling is a stability failure mode unique to pitched portal frames where the rafter snaps through the horizontal plane under vertical load. It occurs when:

1. The rafter is relatively shallow compared to its span (low arching stiffness)
2. The rafter slope is low (shallow pitch, typically < 6°)
3. Rafter-to-column connections provide insufficient rotational restraint

The buckling mode involves a sudden, large-displacement snap of the rafter from a concave-up to concave-down configuration (or vice versa), with a corresponding large vertical displacement at the ridge.

Snap-Through Check — Clause 5.2.4

EN 1993-1-1 does not provide an explicit snap-through formula, but Annex BB of EN 1993-1-1 (informative) gives guidance on in-plane stability of triangulated and portal frames. In practice, the snap-through stability is verified by:

1. Second-order elastic analysis with initial imperfection (Clause 5.3.2) — including the rafter imperfection in the shape of the lowest buckling mode
2. Load factor at first snap-through: α_cr,snap ≥ 10 for elastic analysis to be valid without geometric nonlinearity

For typical portal frames with rafter pitch above 6° (approximately 1:10 slope), snap-through buckling is not critical. For low-pitch portal frames (3° to 6°), a second-order analysis should be performed with imperfections modelled as a half-sine wave on the rafter with amplitude L/300.

Frame Imperfections — Clause 5.3.2

Sway Imperfection φ

EN 1993-1-1 Clause 5.3.2(3) requires an equivalent geometric imperfection for the global frame:

φ = φ_0 × α_h × α_m

Where:

• φ_0 = 1/200 (basic sway imperfection)
• α_h = 2 / sqrt(h) with 2/3 ≤ α_h ≤ 1.0 (height factor, h in metres)
• α_m = sqrt(0.5 × (1 + 1/m)) (number of columns factor, m = columns in a row)

For a single-bay 8 m tall portal frame with 2 columns: α_h = 2/sqrt(8) = 0.707, α_m = sqrt(0.5 × (1 + 0.5)) = 0.866. φ = 1/200 × 0.707 × 0.866 = 1/327.

The sway imperfection is applied as equivalent horizontal forces (EHF) at each storey level:

H_EHF = φ × N_Ed (at each column)

Where N_Ed is the total vertical reaction from the roof at the column top.

Rafter Imperfection

For rafter stability, a bow imperfection of e_0/L between restraints is applied per Clause 5.3.2(6). For buckling curve a (hot-rolled IPE/HE sections): e_0/L = 1/350. For buckling curve b (welded sections): e_0/L = 1/300.

Frame Stability Classification

Sway versus Non-Sway Frames — Clause 5.2.1

A frame may be classified as non-sway (braced against sidesway) if:

α_cr = F_cr / F_Ed ≥ 10 for elastic analysis

Where α_cr is the factor by which the design loads must be increased to cause elastic instability in a global sway mode. This is calculated by elastic critical load analysis (eigenvalue buckling analysis) of the complete frame.

Typical portal frames are sway-sensitive (α_cr < 10), requiring either:

1. Second-order analysis (P-delta effects included explicitly)
2. Amplified first-order moments using the amplification factor: 1 / (1 − 1/α_cr)

Per Clause 5.2.2(3), when α_cr ≥ 3, first-order analysis with amplified sway moments is acceptable. When α_cr < 3, a full second-order analysis is required.

Second-Order Effects — Clause 5.2.2

Amplified Sway Moment Method (α_cr ≥ 3)

The design sway moments from first-order analysis are multiplied by:

k_amp = 1 / (1 − 1/α_cr)

Non-sway moments (those caused by member imperfections and loads between restraints) are not amplified.

Full Second-Order Analysis (α_cr < 3)

A P-delta analysis is performed where the equilibrium equations include the effect of axial forces on the deformed geometry. This can be implemented by:

• Geometric stiffness matrix in finite element analysis
• Iterative P-delta analysis (stepwise updating of geometry under load)
• Notional horizontal forces equivalent to the P-delta effect

For typical single-storey portal frames with α_cr ≈ 5-8, the amplified sway moment method is adequate. For very slender frames (span > 40 m, column height > 15 m) or frames with crane runways, full second-order analysis is recommended.

Serviceability Limits — Clause 7.2

Vertical Deflection Limits

EN 1993-1-1 Clause 7.2.1 recommends the following deflection limits for portal frame rafters. The exact values should be confirmed against the applicable National Annex:

For portal frames with crane runways, stricter limits may apply per EN 1993-6.

Horizontal Deflection (Sway) Limits

At column top: δ_h ≤ h / 150 for portal frames without cranes, h / 300 for frames with overhead cranes.

Where h is the column height from the base to the eaves.

Worked Example — 30 m Span Portal Frame

Step 1 — Design Loads (ULS)

Snow-dominant combination (EN 1990, Eq. 6.10b):

q_Ed,snow = 1.35 × g_k + 1.50 × q_k,snow = 1.35 × 0.40 + 1.50 × 0.60 = 1.44 kN/m²

Line load on rafter (per 6 m bay): w_Ed = 1.44 × 6.0 = 8.64 kN/m

Step 2 — Elastic Global Analysis

First-order elastic analysis yields:

• Rafter maximum moment (eaves region): M_Ed = 486 kN·m
• Apex moment: M_Ed = 312 kN·m
• Column base moment (pinned): 0
• Column top moment: M_Ed = 486 kN·m (same as eaves — moment continuity)

Step 3 — Stability Check (α_cr)

Eigenvalue buckling analysis gives α_cr = 7.2 for the first sway mode.

Since 3 ≤ α_cr = 7.2 < 10: first-order analysis with amplified sway moments is acceptable.

Sway amplification factor: k_amp = 1 / (1 − 1/7.2) = 1 / 0.861 = 1.16

Amplified rafter moment: M_Ed,amp = 486 × 1.16 = 564 kN·m

Step 4 — Rafter Cross-Section Check (IPE 500, Class 1)

M_pl,Rd = W_pl × f_y / γ_M0 = 2,194 × 10³ × 355 / 1.00 = 778.9 kN·m > 564 kN·m — OK. Utilisation = 0.72.

Step 5 — Snap-Through Check

Rafter pitch = 6° > 6° threshold — snap-through not critical for this frame. Nevertheless, verify with second-order analysis including rafter imperfection e_0 = L/350 = 30,000/350 = 85.7 mm (half-sine wave).

Step 6 — Serviceability

Vertical deflection at apex under frequent combination: δ_z = 48 mm Limit = L/250 = 30,000/250 = 120 mm. Utilisation = 48/120 = 0.40 — OK.

Horizontal sway at eaves: δ_h = 32 mm. Limit = 8,000/150 = 53.3 mm. Utilisation = 0.60 — OK.

Frequently Asked Questions

When is plastic global analysis permitted for portal frames per EN 1993-1-1?

EN 1993-1-1 Clause 5.4 permits plastic global analysis when: (a) all sections where plastic hinges form are Class 1 (able to develop and sustain a plastic hinge with sufficient rotation capacity); (b) lateral-torsional buckling is prevented at hinge locations by adequate restraints at 1.5 d spacing; (c) the frame is not subject to repeated plastic loading (low-cycle fatigue from crane or seismic loading); and (d) the structure has adequate redundancy such that local hinge formation does not cause progressive collapse. For typical single-storey portal frames without cranes, plastic design is the dominant method in UK and Irish practice as it achieves 15-25% steel weight savings over elastic design.

What is snap-through buckling and when does it govern portal frame design?

Snap-through buckling is a geometric instability where a pitched rafter suddenly inverts (snaps through) under vertical load, transitioning from one equilibrium configuration to another with a large displacement jump. It is analogous to the snap-through of a shallow arch. It governs when: (1) the rafter pitch is shallow (< 6° or 1:10); (2) the rafter is slender (large span-to-depth ratio, typically L/d > 60); (3) the frame provides low resistance to rafter spread at eaves level (pinned column bases, flexible columns). For frames with rafter pitch ≥ 6°, snap-through is generally not critical and can be ruled out by inspection. For low-pitch frames, a second-order analysis with rafter imperfection is required.

How are equivalent horizontal forces (EHF) applied for frame imperfections?

Per EN 1993-1-1 Clause 5.3.2(7), the frame sway imperfection φ is converted to equivalent horizontal forces at each column top: H_EHF = φ × N_Ed. For multi-bay frames, the EHF is applied in the most unfavourable direction for each load combination. For a portal frame with 8 m column height, N_Ed = 260 kN per column (vertical reaction), and φ = 1/327: H_EHF = 260/327 = 0.79 kN per column — a relatively small force representing the notional lateral load from imperfections. The EHF replaces the explicit modelling of frame out-of-plumb in the analysis.

What determines the α_cr value for portal frame stability classification?

The elastic critical load factor α_cr is determined by eigenvalue buckling analysis (linearised stability analysis) of the full frame model. It represents the factor by which the design loads must be multiplied to cause elastic sway instability. Key parameters: column stiffness (EI/L³), rafter stiffness (contributes to rotational restraint at the column top), connection rigidity (nominally pinned vs full moment continuity), and column base fixity. Typical α_cr values: 5-10 for pinned-base portal frames with standard I-sections, 10-20 for fixed-base frames, 3-5 for very slender industrial frames with large spans and light columns. α_cr < 3 indicates a sway-critical frame requiring full second-order analysis.

What are the key differences between elastic and plastic portal frame design per EN 1993?

Elastic design: Uses linear-elastic analysis with stiffness based on gross cross-sections. Member checks use elastic section modulus W_el and buckling curves. Simpler, more conservative, works for all section classes. Used when: Class 2-4 sections, fatigue loading, or when plastic rotation capacity cannot be guaranteed.

Plastic design: Uses plastic analysis with moment redistribution through hinge formation. Member checks use plastic modulus W_pl at hinge locations. Requires Class 1 sections, lateral restraints at hinge locations, and additional rotation capacity checks per Clause 5.4.1. Achieves 15-25% weight savings. Used for: typical portal frames without cranes, frames where serviceability governs rather than ULS, and structures where hinge formation enables the structure to carry load beyond first yield.

Design Resources

• EN 1993 Column Design — Buckling per Clause 6.3
• EN 1993 Beam Design — Flexure and LTB per Clause 6.3
• EN 1993 Moment Frame Design — Sway and Non-Sway
• EN 1993 Lateral-Torsional Buckling — M_cr and χ_LT
• EN 1993 Combined Loading — Axial + Bending
• EN 1993 Load Combinations — EN 1990 Expressions
• All European Reference Guides →

Reference only. Verify all values against the current edition of EN 1993-1-1:2005 Clause 5.2-5.4 and the applicable National Annex. Portal frame design must be independently verified by a licensed Structural Engineer. This guide is for educational purposes only and does not constitute professional engineering advice.