UR S35 Plate Buckling Benchmark: Hand Calculations vs. SDC Verifier

This benchmark evaluates the implementation of the UR S35 Plate Buckling standard (February 2023, Corr. 1 Sep. 2024) using a finite element model built and checked in SDC Verifier. A steel plate model with dimensions 10.2 × 5.4 × 1.1 meters was created and loaded with realistic force cases to simulate buckling conditions. The goal of this benchmark is to validate SDC Verifier’s implementation against calculations performed manually, based on UR S35 methodology.

The model setup includes accurate material properties, realistic boundary conditions, and a clear application of loads across the top plate. Hand calculations were carried out step-by-step, applying all relevant code equations and intermediate factors to determine the ultimate buckling stresses and utilization factor. These results were then compared with the automatic check in SDC Verifier.

A test plate model with 10.2 × 5.4 × 1.1 m dimensions was designed for the purpose of this benchmark:

The model was constrained at four bottom corners where side plates are connected.
Forces were applied on the edges of the top plate with the following values:

• |𝐹ₗ⁺| = |𝐹ₗ⁻| = 3000 kN
• |𝐹ₛ⁺| = |𝐹ₛ⁻| = 2550 kN
• |𝐹ₚ⁺| = |𝐹ₚ⁻| = 2500 kN

One of the top plates was chosen for all the calculations included in the check.

The plate dimension:

• Length: a = 3.400 m

• Width: b = 1.350 m

• Thickness: tₚ = 0.012 m

The plate material – mild steel properties:

• Young’s Modulus: E = 210 GPa

• Poisson’s Ratio: ν = 0.3

• Mass Density: ρ = 7850 kg/m³

• Tensile Strength: Rₘ = 360 MPa

• Yield Stress: RₑH,P = 235 MPa

Due to the complexity of the model, all required stress values were obtained with the help of FEM.

Obtained values:

• σₓ = 37.14 MPa

• σᵧ = 25.12 MPa

• τ = 16.34 MPa

Calculations

In order to check the results, analytical calculations were first carried out.
Final equations for limit states according to code (Sec. 5 / [2.2.1]):

I.

$${\left(\frac{{\gamma }_1{\sigma }_x^S}{{\sigma }_x}\right)}^{e_0}-B{\left(\frac{{\gamma }_1{\sigma }_x^S}{{\sigma }_x}\right)}^{e_0\mathrm{/}2}{\left(\frac{{\gamma }_1{\sigma }_y^S}{{\sigma }_y}\right)}^{e_0\mathrm{/}2}+{\left(\frac{{\gamma }_1{\sigma }_y^S}{{\sigma }_y}\right)}^{e_0}+{\left(\frac{{\gamma }_1\mathrm{\mid }\tau {\mathrm{\mid }}^S}{{\tau }_c}\right)}^{e_0}=1$$

II. (when $sigma_x geq 0$)

$${\left(\frac{{\gamma }_2{\sigma }_x^S}{{\sigma }_x}\right)}^{2\mathrm{/}{\beta }_p^{2.5}}+{\left(\frac{{\gamma }_2\mathrm{\mid }\tau {\mathrm{\mid }}^S}{{\tau }_c}\right)}^{2\mathrm{/}{\beta }_p^{2.5}}=1$$

III. (when $sigma_y geq 0$)

$${\left(\frac{{\gamma }_3{\sigma }_y^S}{{\sigma }_y}\right)}^{2\mathrm{/}{\beta }_p^{2.5}}+{\left(\frac{{\gamma }_3\mathrm{\mid }\tau {\mathrm{\mid }}^S}{{\tau }_c}\right)}^{2\mathrm{/}{\beta }_p^{2.5}}=1$$

IV.

$$\frac{{\gamma }_4\mathrm{\mid }\tau {\mathrm{\mid }}^S}{{\tau }_c}=1$$

---

Aspect Ratio of the Plate Panel (Sec. 5 / Symbols):

$$\alpha =\frac{a}{b}=\frac{3.40}{1.35}=\alpha =2.519$$

---

Elastic Buckling Reference Stress (Sec. 5 / Symbols):

$$sigma_E = frac{pi^2 E}{12(1 – nu^2)} left( frac{t_p}{b} right)^2$$

$$$$$${\sigma }_E=\frac{{\pi }^2\cdot 210\cdot {10}^9}{12\cdot (1-{0.3}^2)}{\left(\frac{0.012}{1.35}\right)}^2={\sigma }_E=1.499\times {10}^{6}Pa$$

---

Edge Stress Ratio

As defined in Sec. 5 / Symbols, the edge stress ratio was set in both directions as 1.
Stresses are calculated using a weighted average approach (App. 1 / [2.2.1]).

$$\psi =1$$

Correction factor F_long (Sec. 5 / [2.2.4]) was set as 1 (Sec. 5 / Table 2):

$$F_{long}=1$$

Correction factor F_tran (Sec. 5 / [2.2.5]) was set as 1:

$$F_{tran}=1$$

Ultimate Buckling Stresses – Case 1

(Calculated according to Sec. 5 / Table 3)

---

Compression Setup:

Top plate compressed in x-direction with edge stress ratio:

$$\psi =1$$

---

Intermediate Parameters:

Effective width factor

$$c=\min ((1.25-0.12\cdot \psi ),1.25)=\min (1.13,1.25)=1.13$$

---

Slenderness parameter

$$lambda_c = frac{c}{2} left(1 + sqrt{1 – frac{0.88}{c}}right) = frac{1.13}{2} left(1 + sqrt{1 – frac{0.88}{1.13}}right)$$$${\lambda }_c=0.831$$

---

Buckling coefficient in x-direction

$$K_x=\frac{F_{long}\cdot 8.4}{\psi +1.1}=\frac{1\cdot 8.4}{1+1.1}=4$$

---

Reference Degree of Slenderness in x-direction

(Sec. 5 / [2.2.2])

$${\lambda }_x=\sqrt{\frac{R_{eH,P}}{K_x\cdot {\sigma }_E}}=\sqrt{\frac{235\cdot {10}^6}{4\cdot 1.4996549\cdot {10}^6}}=1.979$$

---

Reduction Factor for Stress in x-direction

(Sec. 5 / Table 3)

$$C_x=c(\frac{1}{{\lambda }_x}-\frac{0.22}{{\lambda }_x^2})=1.13\cdot (\frac{1}{1.979}-\frac{0.22}{(1.979{)}^2})=0.507$$

---

 

Ultimate Buckling Stresses – Case 2

(Calculated according to Sec. 5 / Table 3)

---

Compression Setup:

Top plate compressed in y-direction with edge stress ratio:

$$\psi =1$$

---

Intermediate Parameters:

Effective width factor

$$c=\min ((1.25-0.12\cdot \psi ),1.25)=\min (1.13,1.25)=1.13$$

---

Slenderness parameter

$$lambda_c = frac{c}{2} left(1 + sqrt{1 – frac{0.88}{c}}right) = frac{1.13}{2} left(1 + sqrt{1 – frac{0.88}{1.13}}right)$$

$$$$$${\lambda }_c=0.831$$

---

Parameter

$$f_1=(1-\psi )(\alpha -1)=(1-1)(2.519-1)=0$$

---

Buckling coefficient in y-direction

$$K_y = F_{tran} cdot frac{2 left(1 + frac{1}{alpha^2}right)^2}{1 + psi + frac{1 – psi}{100} cdot left( frac{2.4}{alpha^2 + 6.9 f_1} right)}$$

$$$$$$K_y=1\cdot \frac{2{(1+\frac{1}{{2.519}^2})}^2}{1+1+\frac{1-1}{100}\cdot \left(\frac{2.4}{{2.519}^2+6.9\cdot 0}\right)}=1.340$$

---

Reference Degree of Slenderness in y-direction

(Sec. 5 / [2.2.2])

$${\lambda }_y=\sqrt{\frac{R_{eH,P}}{K_y\cdot {\sigma }_E}}=\sqrt{\frac{235\cdot {10}^6}{1.340\cdot 1.4996549\cdot {10}^6}}=3.419$$

---

Factor $c_1$

(Sec. 5 / [2.2.3], based on SP-A assessment method)

$$c_1 = left(1 – frac{1}{alpha}right), quad c_1 geq 0$$

$$$$$$c_1=1-\frac{1}{2.519}=0.603$$

---

$$R=0.220$$

---

Conditions for $lambda_p^2$ Based on Slenderness:

1.  

$${\lambda }_p^2={\lambda }_y^2-0.5\text{and}1\le {\lambda }_p^2\le 3$$

2.  

$${\lambda }_p^2={3.419}^2-0.5\text{and}1\le {\lambda }_p^2\le 3$$

3.  

$${\lambda }_p^2=3$$

Correction Factor $F$ – Conditional Forms

1. General formula:

$$F=(1-\frac{(\frac{K_y}{0.91}-1)}{{\lambda }_p^2})\cdot c_1\text{and}F\ge 0$$

2. Substituted for Case 2:

$$F=(1-\frac{(\frac{1.340}{0.91}-1)}{3})\cdot 0.603\text{and}F\ge 0$$

3. Result:

$$F=0.508$$

Calculation of Parameter $T$

General Formula:

$$T={\lambda }_y+\frac{14}{15{\lambda }_y}+\frac{1}{3}$$

Substituted Values:

$$T=3.419+\frac{14}{15\cdot 3.419}+\frac{1}{3}$$

Result:

$$T=4.026$$

Calculation of Parameter $H$

General Formula:

$$H={\lambda }_y-\frac{2{\lambda }_y}{c(T+\sqrt{T^2-4})}\text{and}H\ge R$$

Substituted Values:

$$H=3.419-\frac{2\cdot 3.419}{1.13\cdot (4.026+\sqrt{{4.026}^2-4})}\text{and}H\ge 0.22$$

Result:

$$H=2.614$$

Reduction Factor for Stress in y-direction $C_y$

(Sec. 5 / Table 3)

General Formula:

$$C_y=c(\frac{1}{{\lambda }_y}-\frac{R+F^2(H-R)}{{\lambda }_y^2})$$

Substituted Values:

$$C_y=1.13\cdot (\frac{1}{3.419}-\frac{0.22+{0.508}^2\cdot (2.614-0.22)}{{3.419}^2})$$

Result:

$$C_y=0.250$$

Case 15: Shear Buckling in xy Direction

---

Shear Buckling Coefficient $K_tau$

General Formula:

$$K_{\tau }=\sqrt{3(5.34+\frac{4}{{\alpha }^2})}$$

Substituted:

$$K_{\tau }=\sqrt{3(5.34+\frac{4}{{2.519}^2})}=10.341$$

---

Reference Degree of Slenderness $lambda_tau$

(Sec. 5 / [2.2.2])

$$lambda_tau = sqrt{ frac{R_{eH,P}}{K_tau cdot sigma_E} }$$

$$$$$${\lambda }_{\tau }=\sqrt{\frac{235\cdot {10}^6}{10.341\cdot 1.4996549\cdot {10}^6}}=1.231$$

---

Reduction Factor for Shear Stress $C_tau$

(Sec. 5 / Table 3)

$$C_{\tau }=\frac{0.84}{{\lambda }_{\tau }}=\frac{0.84}{1.231}=0.682$$

Ultimate Buckling Stresses

(Calculated according to Sec. 5 / [2.2.3])

---

In the direction parallel to the longer edge of the buckling panel:

$${\sigma }_{cx}=C_x\cdot R_{eH,P}=0.507\cdot 235\text{MPa}=\mathbf{119.145}\text{MPa}$$

---

In the direction parallel to the shorter edge of the buckling panel:

$${\sigma }_{cy}=C_y\cdot R_{eH,P}=0.250\cdot 235\text{MPa}=\mathbf{58.750}\text{MPa}$$

---

Shear Buckling Stress:

$${\tau }_c=C_{\tau }\cdot \frac{R_{eH,P}}{\sqrt{3}}=0.682\cdot \frac{235\text{MPa}}{\sqrt{3}}=\mathbf{92.532}\text{MPa}$$

Calculation of Plate Slenderness Parameter $beta_p$

(Sec. 5 / [2.2.1])

General Formula:

$${\beta }_p=\frac{b}{t_p}\sqrt{\frac{R_{eH,P}}{E}}$$

Substituted Values:

$${\beta }_p=\frac{1.350}{0.012}\cdot \sqrt{\frac{235\cdot {10}^6}{210\cdot {10}^9}}$$

Result:

$${\beta }_p=\mathbf{3.763}$$

Coefficient $B$

(According to Sec. 5 / Table 1)

General Formula:

$$B=0.7-\frac{0.3\cdot {\beta }_p}{{\alpha }^2}$$

Substituted Values:

$$B=0.7-\frac{0.3\cdot 3.763}{{2.519}^2}$$

Result:

$$B=\mathbf{0.522}$$

Coefficient $e_0$

(According to Sec. 5 / Table 1)

General Formula:

$$e_0=\frac{2}{{\beta }_p^{0.25}}$$

Substituted Values:

$$e_0=\frac{2}{{3.763}^{0.25}}$$

Result:

$$e_0=\mathbf{1.436}$$

---

 

Final Equations for Limit States

(Transformed from Sec. 5 / [2.2.1] to calculate stress multiplier factors acting on loads $gamma$)

---

I.

$${\gamma }_{c1}={\left(\frac{1}{{\left(\frac{{\sigma }_x^S}{{\sigma }_{cx}}\right)}^{e_0}-B{\left(\frac{{\sigma }_x^S}{{\sigma }_{cx}}\right)}^{e_0\mathrm{/}2}{\left(\frac{{\sigma }_y^S}{{\sigma }_{cy}}\right)}^{e_0\mathrm{/}2}+{\left(\frac{{\sigma }_y^S}{{\sigma }_{cy}}\right)}^{e_0}+{\left(\frac{\mathrm{\mid }{\tau }^S\mathrm{\mid }}{{\tau }_c}\right)}^{e_0}}\right)}^{\frac{1}{e_0}}$$

---

II.

$${\gamma }_{c2}={\left(\frac{1}{{\left(\frac{{\sigma }_x^S}{{\sigma }_{cx}}\right)}^{2\mathrm{/}{\beta }_p^{0.25}}+{\left(\frac{\mathrm{\mid }{\tau }^S\mathrm{\mid }}{{\tau }_c}\right)}^{2\mathrm{/}{\beta }_p^{0.25}}}\right)}^{{\beta }_p^{0.25}\mathrm{/}2}$$

---

III.

$${\gamma }_{c3}={\left(\frac{1}{{\left(\frac{{\sigma }_y^S}{{\sigma }_{cy}}\right)}^{2\mathrm{/}{\beta }_p^{0.25}}+{\left(\frac{\mathrm{\mid }{\tau }^S\mathrm{\mid }}{{\tau }_c}\right)}^{2\mathrm{/}{\beta }_p^{0.25}}}\right)}^{{\beta }_p^{0.25}\mathrm{/}2}$$

---

IV.

$${\gamma }_{c4}=\frac{{\tau }_c}{\mathrm{\mid }{\tau }^S\mathrm{\mid }}$$

Partial Safety Factor

(Sec. 5 / Symbols)

$$S = 1$$

---

Calculated Values of Stress Multiplier Factors $gamma$:

I.

$${\gamma }_{c1}=1.763$$

II.

$${\gamma }_{c2}=2.486$$

III.

$${\gamma }_{c3}=1.968$$

IV.

$${\gamma }_{c4}=5.663$$

---

Failure Criterion

The minimum stress multiplier factor among all calculated values is used as the stress multiplier factor at failure:

$${\gamma }_c=\min ({\gamma }_{c1},{\gamma }_{c2},{\gamma }_{c3},{\gamma }_{c4})=\mathbf{1.763}$$

---

Utilization Factor $eta_{act}$

(Sec. 1 / [3.2.2])

Formula:

$${\eta }_{act}=\frac{1}{{\gamma }_c}$$

Substituted:

$${\eta }_{act}=\frac{1}{1.763}$$

Result:

$${\eta }_{act}=\mathbf{0.567}$$

Material Setup and SDC Verifier Check

In SDC Verifier, the standard was added using the same material assumptions, and the check was performed accordingly.

1. Mild Steel Properties

Top Plate Properties (T = 12 mm)

Properties Summary

Calculated for the CSys “0..Basic Rectangular”

FEM Loads – Long Edges

This section contains information about the applied loads to the model.

FEM Loads – Short Edges

This section contains information about the applied loads to the short edges of the model.

FEM Loads – Long Edges Parallel

This section describes the applied loads along the long edges in the Y-direction.

Constraints

This section provides information about constrained parts of the model.

Results: Job 1 – Load Set ‘1’

UR S35 Plate Buckling (2023)
Implementation according to UR S35 Buckling Strength Assessment of Ship Structural Elements, February 2023 (Corr. 1 Sep. 2024)

---

Unit System

• MKS (Meter / Kilogram / Second)

• Standards referenced: API RP 2A, ISO 19902, NORSOK N004, DIN 15018, FEM 1.001, Eurocode3.

Intermediate Results of Ultimate Buckling Stresses

Results Comparison: Hand Calculations vs. SDC Verifier Check

Ultimate Buckling Stresses [MPa]

---

Inverse of Stress Multiplier Factors Acting on Loads

---

Utilization Factor

η_act = 1 / γ_c1 | 0.567 (Hand) | 0.562 (SDC Verifier) |

---

Results from SDC Verifier are consistent with those obtained from hand calculations, validating the accuracy of the model and the implementation of UR S35 plate buckling checks.

Conclusion

The comparison demonstrates a high level of agreement between hand calculations and the results obtained using SDC Verifier. Deviations in computed stresses and utilization factors remained within a negligible margin, confirming both the correctness of the analytical approach and the integrity of the implemented standard in SDC Verifier. This benchmark provides strong confidence in using SDC Verifier for UR S35 plate buckling assessments in real-world structural analysis scenarios.