Plastic Cross-Section Checks Using Partial Internal Forces Method

In steel structures, cross-sections that meet certain criteria (in EC3, these requirements are linked to cross-section classes 1 and 2, for example) may be designed plastically. This means that stresses can be redistributed by the yielding of the base material in the cross-section. While the formulas for the calculation of plastic limit section forces in common steel construction standards apply only to selected cross-section types and section force combinations, if at all, the partial internal forces method (PSFM) is almost universally applicable. For example, structural components that are subjected to axial force, bending, and mixed torsion (including warping torsion) can also be designed efficiently. The partial internal forces method is now also available to users in RFEM 6 and RSTAB 9 under the “extended plastic design rules” (see ultimate limit settings of the “Steel Design” add-on).

Extended Plastic Design Checks in Ultimate Limit State Settings

The partial internal forces method (TSV) was developed by Kindmann and Frickel at Ruhr University Bochum (Germany) and is described in detail in [1]. Two different variants are implemented in the program:

1. Partial Internal Forces Method with Redistribution

The redistribution method is suitable for 2- and 3-metal sheet cross-sections with orthogonally oriented cross-section parts and thus covers the most important open cross-section shapes in steel structures. Additional solutions are implemented for rectangular and circular hollow cross-sections, so that the following cross-section types can be designed using this method:

• Double / single / unsymmetric I-sections
• Channel / T- / Z- / L-sections
• PI sections (Type A)
• Double-symmetric rectangular (RHS) / square (SHS) hollow and box sections
• Circular hollow sections (CHS)

The procedure for plastic design using the partial internal forces method with redistribution is as follows:

1. Transformation of the internal forces from the structural analysis into a special (ȳ-z̄) reference system (the origin for I-sections is set in the web center, for example)
2. Distribution and design of internal forces that cause shear stresses (transverse forces and torsional moments) at the cross-section level
3. Distribution and design of internal forces that cause local bending in cross-section parts orthogonal to the reference cross-section part (for example, a web for I-section). The yield strength is reduced due to the acting shear stresses from 2.
4. Design of the residual load-bearing capacity of the cross-section for internal forces that cause bending in cross-section parts parallel to the reference cross-section part (for example, a web for I-section) (with reduced yield strength due to shear) plus axial force.

Please note that the cross-section resistance design is not carried out using the fully plastic state of the cross-section. Instead, in Step 4, a case distinction is used to check whether the internal forces are within a certain range of values and can be absorbed by the cross-section. The resulting design ratio of the cross-section check is therefore generally not proportional to the action and only provides information about the success (design ratio less than or equal to 1) or failure (design ratio greater than 1) of the cross-section checks.

2. Partial Internal Forces Method Without Redistribution

The partial internal forces method without redistribution [1] is generally suitable for all thin-walled cross-section types. The procedure for this design variant is as follows:

1. Division of the cross-section into its elements. Limit values for the length-to-width ratio can be entered, above which an element is to be taken into account for the design.
2. Determination of internal forces in each cross-section part, based on the elastic stresses at the ends of the cross-section parts
3. Design of the determined internal forces against the plastic limit internal forces of the cross-section part

The partial internal forces are therefore calculated depending on the elastic stress distribution in each cross-section part. A plastic redistribution of the stress is only taken into account within and not between the cross-section parts. Nevertheless, significantly more efficient results can often be achieved compared to a purely elastic design.

In order to avoid overloading the output, only the design result for the cross-section part with the highest design ratio is displayed at each design point in Steel Design.

Example of Cross-Section Check with PIFM

Design Example: Unsymmetric I-Section

The given example is also described in [1] under Section 10.7.6 and clearly shows the efficiency of the partial internal forces method. Even for unsymmetric cross-sections (here, IU 322/0/208/234/74/12/25/19/0/0/0/0, f_y = 240 N/mm²) with general stress (axial force + double bending + mixed torsion), it is possible to perform a plastic cross-section design check:

1. PIFM with Redistribution

Based on slight deviations in load and cross-section geometry, the bending design of the lower flange is slightly exceeded in Steel Design, while in [1], it results in a design ratio of 100%. In order to fully explain the design concept at this point, the internal forces from Table 1 are reduced by 2.5% and calculated with a load factor of 97.5%.

In the first step, the internal forces from the (u-v) principal axis system are transformed into the (ȳ-z̄) reference system. The reference system has its origin at the center of gravity of the web plate and also corresponds to the orientation of the global (Y-Z) coordinate system in image 2. The main axis inclination α is 35.5°:

V_ȳ = V_u * cos(α) - V_v * sin(α) = -430.8 kN

V_z̄ = V_v * cos(α) + V_u * sin(α) = -67.6 kN

M_x̄s = M_xs - V_u * v_M-D + V_v * u_M-D = 70.4 kNm

M_ȳ = M_u * cos(α) - M_v * sin(α) + N * z̄_S-D = 217.4 kNm

M_z̄ = M_v * cos(α) + M_u * sin(α) - N * ȳ_S-D = 199.3 kNm

M_ω̄ = - M_ω + M_u * u_M-D + M_v * v_M-D + N * ω̄ _k = 3.15 kNm²

In the second step, the shear stresses in the separate cross-section parts are designed. To do this, the relevant internal forces (shear forces and primary and secondary torsional moments) are first distributed to the flange and web plates (as an example here and reduced for the lower flange):

V_y,u = - (V_ȳ * z̄_o + M_x̄s) / (z̄_u - z̄_o) = -452.3 kN

M_xp,u = M_xp * I_T,u / I_T = 1.46 kNm

where I_T,u / I_T describes the ratio of the torsional stiffness of the lower flange to the torsional stiffness of the entire cross-section (here 37.6%). The relevant plastic load-bearing capacities (V_pl,y,u and M_pl,xp,u) of the cross-section part are then determined, and the design ratio is obtained:

η_τ,u = |M_xp,u| / (2 * M_pl,xp,u) + √((M_xp,u / (2 * M_pl,xp,u))² + (V_y,u / V_y,u)²) = 0.64

In the third step, the local bending moments of the flanges are checked. Here, the partial stress consists of the bending moment M_z̄ and the warping bimoment M_ω̄ . Again, only the lower flange is considered as an example:

M_Sa,z,u = (- M_z̄ * z̄_o + M_ω̄ ) / (z̄_u - z̄_o) = 111.2 kNm

The design is performed with a reduced yield stress due to shear stress (see above) and under consideration of an eccentricity parameter δ:

M_pl,z,u,τ = M_pl,z,u * f_y,d,u * √(1 - (τ_u / τ_u,Rd)²) = 89.8 kNm

η_Mz̄ = (|M_Sa,z,u| / M_pl,z,u,τ) / (1 + δ_u²) =0.99

Finally, it is checked whether the effective axial force N and the bending moment M_ȳ can be absorbed by the “remaining” cross-section. A closed analytical solution is not available for this last step. Instead, a 2-dimensional solution space is determined and it is checked whether the acting N-M_ȳ combination lies within or outside the limit (= interaction diagram) of this solution space. The limit curve is described for the positive and negative moment range using two linear and one parabolic equation. The case differentiation is used to check which sections of the limit curve are relevant for design for the given axial force. The exact calculation steps can be found in [1] or in the results details of the steel design. The resulting limit curve with the different sections for the example is displayed in the following:

Limit Curve for Interaction Design of Axial Force–Bending

Image 3 shows the limit curve and the N-M_ȳ combination acting in the example (red diamond). It is immediately apparent that the applied load is within the solution space of the limit curve, meaning that the cross-section check is fulfilled. However, it is unclear how large the remaining “true” capacity of the cross-section is, that is, what increase in the applied internal force combination would be possible until the ultimate limit state is reached. The proportionality between stress and design ratio is violated by the nonlinear interaction conditions (already in Step 2). The “true” design ratio can therefore only be determined iteratively (that is, in several calculation steps with varying loading).

2. PIFM Without Redistribution

For comparison purposes, the cross-section is also designed using the PIFM without redistribution. First, the elastic normal and shear stresses at the initial, middle, and final nodes are determined in each cross-section part (each thin-walled element is considered a separate cross-section part). Here, the calculation (as in Steel Design) is only presented for the governing cross-section part (Element 5 in Image 2):

The plastic partial internal forces of the cross-section parts (here, cross-section part i = 5) are then calculated from the stresses, taking into account the dimensions:

N₅ = t * l * (σ_x,A + σ_x,E) / 2 = 800.0 kN

M₅ = t * l² * (σ_x,A - σ_x,E) / 12 = 19.0 kNm

V₅ = t * l * (τ_A + 4 * τ_M + τ_E) / 6 = 402.4 kN

M_xp,5 = M_xp * I_T,5 / I_T = 1.1 kNm

Then, the shear capacity of the cross-section part is designed:

η_τ,5 = |M_xp,5| / (2 * M_pl,xp,5) + √((M_xp,5 / (2 * M_pl,xp,5))² + (V₅ / V_pl,5)²) = 0.74

Finally, the axial force-moment interaction is checked. As in the case of PIFM with redistribution, the resistances are calculated with a reduced yield stress:

f_y,5,red = f_y,5 * √(1 - (τ₅ / τ_Rd,5)²) = 161.4 N/mm²

η_N+M,5 = (N₅ / N_pl,τ,5)² + |M₅| / M_pl,τ,5 = 1.611

With the initial loading displayed in Table 1, the cross-section check is therefore not fulfilled. An iterative calculation shows that the design can just be fulfilled if the loading is reduced to 86%.

3. Elastic Cross-Section Check

The elastic cross-section check is also significantly exceeded with a maximum design ratio of 129% in Element 5. In this case, the maximum load factor can be determined directly from the reciprocal of the maximum design ratio as 77.5%.

Conclusion

Plastic design using the partial internal forces method allows for significantly more efficient design compared to elastic cross-section check, provided that this is allowable. In the example, a limit load increase of 11% (PIFM without redistribution) or 25.8% (PIFM with redistribution) can be achieved.