Steel Frame Analysis Guide — Direct Stiffness, P-Delta, and Portal Method

A comprehensive guide to structural analysis methods for steel frames. Covers the direct stiffness method (DSM), first-order vs second-order analysis, P-Delta and P-delta effects, approximate methods (portal and cantilever), and practical application in the Designer Hub pipeline.

Overview of Steel Frame Analysis

Structural analysis is the process of determining the internal forces (axial, shear, moment) and displacements in a frame given its geometry, member properties, support conditions, and applied loads. For steel design, the analysis output is the input to member design checks (S5) and connection design (S6).

The analysis method must be consistent with the design assumptions. If the frame is modelled with pinned connections (simple construction), the analysis should include member end releases (zero moment at pinned connections). If second-order effects are significant, a first-order analysis amplified by B1/B2 factors is not adequate — a full second-order analysis is required.

1. The Direct Stiffness Method (DSM)

The direct stiffness method is the computational engine behind virtually all modern structural analysis software. It is a matrix method that solves the system of equations:

[K] {u} = {F}

Where [K] is the global stiffness matrix, {u} is the vector of unknown nodal displacements, and {F} is the vector of applied nodal forces.

Step-by-Step DSM Procedure

Step 1: Discretise the structure. Divide the frame into elements connected at nodes. For a portal frame, each rafter segment and each column is an element. Nodes are at member intersections, support points, and at the location of concentrated loads or section changes.

Step 2: Form element stiffness matrices. For each element, compute the element stiffness matrix [k] in local coordinates. For a 2D frame element with axial and flexural stiffness:

        AE/L     0         0      -AE/L      0          0
         0    12EI/L^3  6EI/L^2     0    -12EI/L^3   6EI/L^2
         0    6EI/L^2    4EI/L      0    -6EI/L^2     2EI/L
[k] =  -AE/L     0         0       AE/L      0          0
         0   -12EI/L^3 -6EI/L^2     0     12EI/L^3  -6EI/L^2
         0    6EI/L^2    2EI/L      0    -6EI/L^2     4EI/L

Where A is the cross-sectional area, E is the modulus of elasticity, I is the moment of inertia, and L is the element length. The six degrees of freedom per element are: axial displacement and rotation at each node.

Step 3: Transform to global coordinates. Rotate each element stiffness matrix from local coordinates (aligned with the element axis) to global coordinates (horizontal and vertical):

[K_global] = [T]^T [k] [T]

Where [T] is the rotation matrix containing the direction cosines of the element axis.

Step 4: Assemble the global stiffness matrix. Sum the contributions of all element stiffness matrices into a single global stiffness matrix. Each element contributes terms to the rows and columns corresponding to the degrees of freedom at its end nodes. The assembly process uses the element connectivity table to map local node numbers to global node numbers.

Step 5: Apply boundary conditions. Modify the stiffness matrix and force vector to enforce support conditions. For a fixed support, the corresponding displacement degrees of freedom are zero. In practice, the rows and columns for restrained degrees of freedom are eliminated (the reduced stiffness method), or the diagonal term is set to a large number (the penalty method).

Step 6: Solve for displacements. Solve the linear system [K]{u} = {F}. For small to medium-sized frames (up to a few thousand degrees of freedom), Gaussian elimination or LU decomposition is used. The Designer Hub pipeline uses a sparse Cholesky solver for symmetric positive-definite stiffness matrices.

Step 7: Back-calculate element forces. Once nodal displacements are known, the element end forces are computed:

{f} = [k] [T] {u_element}

Where {u_element} is the displacement vector for the element's end nodes, extracted from the global displacement vector.

Application in the Designer Hub

The Designer Hub S4 FEA engine implements the full DSM procedure. The engine builds element stiffness matrices from the section properties assigned in S1, applies the loads generated in S2 (converted to equivalent nodal forces), enforces the boundary conditions specified in S0, and solves for displacements and member forces. The results drive the member design checks in S5.

2. First-Order vs Second-Order Analysis

First-Order (Linear) Analysis

A first-order analysis assumes:

• Equilibrium is satisfied on the undeformed geometry
• Displacements are small (infinitesimal strain theory)
• Material behaviour is linear elastic
• Superposition is valid

For most building frames under gravity loads with adequate lateral bracing, first-order analysis is sufficient and conservative (it underestimates moments at restrained nodes and overestimates moments at nodes free to sway, but the differences are typically within 10 percent).

Second-Order (P-Delta) Analysis

A second-order analysis accounts for the effect of axial forces acting through displaced geometry. Two distinct effects are included:

P-Delta (capital delta). The additional overturning moment caused by gravity loads acting through the lateral drift of the frame. For a column carrying axial load P and undergoing story drift Delta, the additional moment at the column base is P x Delta. This is a global effect that affects the entire frame.

P-delta (lowercase delta). The additional moment caused by axial load acting through the member curvature (deflection between nodes). For a beam-column with axial load P and midspan deflection delta, the additional midspan moment is P x delta. This is a local effect that affects individual members.

When Second-Order Effects Matter

The significance of second-order effects depends on the ratio of axial load to the elastic buckling load. Define:

theta = (P x Delta) / (V x h)

Where P is the total gravity load on the storey, Delta is the first-order story drift, V is the story shear, and h is the story height. This is the stability coefficient per ASCE 7-22 Section 12.8.7.

• theta <= 0.10: second-order effects are negligible; first-order analysis is acceptable
• 0.10 < theta <= theta_max: second-order effects are significant; P-Delta analysis or amplified first-order analysis is required
• theta > theta_max: the structure is potentially unstable; redesign (increase stiffness) is required

Per AISC 360 Chapter C, if the ratio of second-order drift to first-order drift exceeds 1.5 (i.e., second-order amplification is more than 50 percent), the frame is too flexible and must be stiffened.

Per EN 1993-1-1 Clause 5.2

The Eurocode uses alpha_cr, the elastic critical buckling load factor:

alpha_cr = H_Ed / V_Ed x h / delta_H_Ed

Where H_Ed is the horizontal design load, V_Ed is the vertical design load, h is the storey height, and delta_H_Ed is the first-order horizontal displacement at the top of the storey under H_Ed.

• alpha_cr >= 10: first-order analysis is sufficient
• 3 <= alpha_cr < 10: amplified sway moments (first-order + amplification factor) are required
• alpha_cr < 3: full second-order analysis is required
• alpha_cr < 2.5: the frame is too flexible; increase stiffness

For typical industrial portal frames with spans of 15-30 metres, alpha_cr typically falls between 3 and 8, meaning amplified sway moments or second-order analysis is required.

3. P-Delta Analysis Methods

Three approaches are used in practice for incorporating P-Delta effects:

Method 1: Geometric Stiffness Matrix (Rigorous)

The element stiffness matrix is augmented with a geometric stiffness matrix that depends on the axial force:

[K_total] = [K_elastic] - [K_geometric]

Where [K_geometric] is a function of the axial load in the element. For a straight element with constant axial force:

       0     0        0      0     0       0
       0   6/5L    1/10     0   -6/5L   1/10
       0   1/10    2L/15    0   -1/10  -L/30
[Kg] = P x ...

Because the geometric stiffness depends on the axial force, which depends on the displacements (through second-order effects), the analysis is nonlinear. The solution is iterative: assume axial forces (from first-order analysis), compute [Kg], solve [K_total]{u} = {F}, update axial forces from the new displacements, recompute [Kg], and repeat until convergence.

The Designer Hub pipeline uses this method. Convergence is typically achieved in 3 to 5 iterations for frames that are not approaching their buckling load.

Method 2: B1/B2 Amplifier Method (AISC 360 Appendix 8)

This is a simplified approach that applies amplification factors to first-order results:

B1 factor (P-delta, member-level amplification):

B1 = Cm / (1 - alpha x Pr / Pe1) >= 1.0

Where Cm accounts for the moment gradient along the member, alpha = 1.0 for LRFD, Pr is the required axial strength, and Pe1 is the Euler buckling load for the member considering the member slenderness and end restraint (not the story sway).

B2 factor (P-Delta, story-level amplification):

B2 = 1 / (1 - alpha x Sigma_Pr / Sigma_Pe_story) >= 1.0

Where Sigma_Pr is the total vertical load on the story, and Sigma_Pe_story is the total elastic story buckling load:

Sigma_Pe_story = R_M x Sigma (HL / Delta_H)

Where R_M accounts for the influence of P-delta on the column, HL is the story shear, and Delta_H is the first-order story drift.

The total second-order moment is then:

Mr = B1 x M_nt + B2 x M_lt

Where M_nt is the first-order moment from loads that cause no translation (no-sway), and M_lt is the first-order moment from loads that cause lateral translation (sway).

Method 3: Notional Load Method (AISC 360 Chapter C, Direct Analysis Method)

In addition to the applied loads, notional lateral loads are applied to represent initial out-of-plumbness:

Ni = 0.002 x Yi

Where Yi is the gravity load applied at level i. The notional loads are applied in the direction that produces the most unfavourable effect. This approach is simpler than the B1/B2 method because it does not require classification of moments into sway and non-sway components, but it requires a second-order analysis (or the B2 factor) to capture the P-Delta amplification of the notional loads.

4. Approximate Methods — Portal and Cantilever

Portal Method

The portal method is an approximate hand analysis method for multi-storey, multi-bay frames under lateral load. It was developed before computer analysis became standard and remains useful for preliminary sizing and for checking computer results.

Assumptions:

1. Points of contraflexure (zero bending moment) occur at the mid-height of all columns and the mid-span of all beams.
2. The total horizontal shear at each storey level is distributed among columns in proportion to their tributary width: interior columns carry twice the shear of exterior columns.

Procedure:

1. Draw the frame and label all storeys and bays.
2. For each storey, sum the horizontal loads above that storey. This is the total storey shear.
3. Distribute storey shear to columns: each exterior column carries V/n (where n is an appropriate distribution factor), and each interior column carries 2V/n.
4. With the column shear and the column height known, compute the column moment at each end: M = (V_column) x (h/2), where h is the storey height (since the contraflexure point is at mid-height).
5. Working from top to bottom, compute beam moments at each joint from equilibrium: the sum of column moments at a joint equals the sum of beam moments.
6. Compute beam shears and column axial forces from equilibrium.

Accuracy: Within 15-20 percent of a rigorous stiffness analysis for regular frames with uniform storey heights and bay widths. Errors increase for frames with significant variations in stiffness between storeys or for frames with very stiff or very flexible beams relative to columns.

Example — Two-storey, three-bay frame:

Consider a two-storey, three-bay frame with 4-metre storey heights, 6-metre bay widths, and a 50 kN lateral load at each floor level.

Storey 2 shear = 50 kN (load at roof only). Distribute to columns: exterior columns = 50/(1+2+2+1) x 1 = 8.33 kN each, interior columns = 16.67 kN each. Column moments = 8.33 x 2.0 = 16.67 kN-m (exterior), 16.67 x 2.0 = 33.33 kN-m (interior).

Storey 1 shear = 50 + 50 = 100 kN. Exterior columns = 16.67 kN each, interior = 33.33 kN each. Column moments = 16.67 x 2.0 = 33.33 kN-m (exterior), 33.33 x 2.0 = 66.67 kN-m (interior).

Beam moments at the first-floor level are found from joint equilibrium: at an exterior joint, the column moment in the upper column (16.67 kN-m) plus the column moment in the lower column (33.33 kN-m) equals the beam moment (50.0 kN-m).

These approximate moments can be used to select trial member sizes in S1 before running the full FEA in S4.

Cantilever Method

The cantilever method is an alternative approximate method suitable for tall, slender frames where column axial deformations dominate the lateral response (as opposed to the portal method, where bending deformations dominate).

Assumptions:

1. Points of contraflexure at mid-height of columns and mid-span of beams (same as portal method).
2. The axial stress in columns varies linearly with distance from the centroid of the column group on each storey (plane sections remain plane assumption, treating the frame as a cantilever beam).

The cantilever method is more accurate than the portal method for tall frames (height-to-width ratio greater than about 4:1) and less accurate for low-rise frames where bending deformations dominate. For portal frame buildings (typically single-storey), the cantilever method is not applicable.

5. FE Modelling Considerations for Portal Frames

Member Modelling

For a 2D portal frame analysis, each rafter and column is modelled as a line element with 6 degrees of freedom per node (3 translations, 3 rotations) in 3D, or 3 degrees of freedom per node (2 translations, 1 rotation) in a 2D plane frame analysis. The latter is sufficient for most portal frames where out-of-plane behaviour (lateral-torsional buckling, purlin restraint) is handled through member design checks rather than frame analysis.

Haunch Modelling

If haunches are used at the eaves, they are modelled as tapered elements with variable section properties along their length. The section properties at discrete points along the haunch (typically 3 to 5 stations) are computed and the element stiffness is integrated numerically. The Designer Hub pipeline supports prismatic and linearly tapered elements.

Purlin and Girt Restraint

Purlins and girts provide lateral and torsional restraint to the rafter and column compression flanges. In the frame analysis, this restraint is represented as:

• Lateral supports (preventing out-of-plane translation) at purlin and girt attachment points
• Torsional restraints (preventing twist) if fly bracing is provided
• These restraints affect the effective length factors for LTB checks in S5, not the in-plane frame analysis

Base Fixity Modelling

• Pinned base: release the rotational degree of freedom at the column base (moment = 0). The column base node has zero translation.
• Fixed base: all degrees of freedom are restrained at the column base node.
• Semi-rigid base: a rotational spring with stiffness K_theta is applied at the column base. The pipeline adds a spring element to the global stiffness matrix. Typical rotational stiffness values range from 0.2EI/L (pinned equivalent) to 20EI/L (fixed equivalent) for the column.

6. Verification and Validation of Analysis Results

Before using FEA results for member design, verify:

Equilibrium check. Sum of vertical reactions should equal the sum of applied vertical loads. Sum of horizontal reactions should equal the sum of applied horizontal loads. The check should close to within 1 percent. If it does not, there may be an error in load application or boundary condition definition.

Deflection check. The deflected shape should be physically reasonable. Rafters under downward load should sag (downward deflection at midspan). The frame under lateral load should sway in the direction of the load. If the deflected shape looks wrong, check the load direction, support conditions, and section orientation.

Moment diagram check. The moment diagram should be consistent with the deflected shape and the support conditions. Moment at a pinned support should be zero. Moment at a fixed support should be non-zero. The sign of the moment should change at points of contraflexure.

Comparison with approximate methods. For regular frames, approximate moments from the portal method should be within 20 percent of the FEA results. Large discrepancies may indicate modelling errors (wrong section properties, wrong load magnitude, or incorrect support conditions).

The Designer Hub S4 analysis page provides visualisation tools for the deflected shape, moment diagram, shear diagram, and axial force diagram, making these verification checks straightforward.

Frequently Asked Questions

What is the direct stiffness method and how is it used in frame analysis?

The DSM is a matrix structural analysis technique that assembles element stiffness matrices into a global stiffness matrix [K], applies loads {F}, and solves [K]{u} = {F} for displacements. Element forces are then computed from displacements. It is the foundation of all modern structural FEA software, including the Designer Hub analysis engine, and handles arbitrary geometry, member properties, and load patterns.

When is second-order (P-Delta) analysis required for steel frames?

Second-order analysis is required when P-Delta effects increase forces or displacements by more than 10 percent over first-order results. Per AISC 360 Chapter C, if the second-order drift ratio exceeds 1.5, second-order analysis is mandatory. Per EN 1993-1-1, if alpha_cr < 10, second-order effects must be considered. For typical industrial portal frames (spans over 15 metres, column heights over 6 metres), P-Delta effects are typically 5 to 15 percent, so the check is mandatory.

What is the portal method and when is it used?

The portal method is an approximate hand analysis method for multi-storey frames under lateral load. It assumes points of contraflexure at mid-height of columns and mid-span of beams, and distributes shear with interior columns carrying twice the exterior column shear. It yields moments within 15-20 percent of rigorous analysis for regular frames and is useful for preliminary sizing before running the full FEA.

How does the Designer Hub pipeline perform frame analysis?

The S4 analysis stage implements full DSM with 2D or 3D frame elements, automatic load generation from S2, P-Delta iteration using geometric stiffness, and visualisation of deflected shape, moment, shear, and axial force diagrams. The engine runs in-browser via WebAssembly and reports displacements, member end forces, and support reactions for every load combination.

Related Resources

• Designer Hub — Frame Analysis (S4)
• Complete Portal Frame Design Tutorial
• Portal Frame Design Guide
• Steel Building Geometry Guide
• Steel Member Design Checks Explained
• Steel Design Verification Guide

Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.