Beyond Biggs – Review, Implementation and Comparisonof Modern SDOF Analysis

Henri LotvonenSurma Ltd, Helsinki, Finland, henri.lotvonen@survivability.fi

In naval vessels, as in any other ship hull structure, stiffened panels are used extensively as deck plating andbulkheads acting as blast walls. Blast wall resistance is often analyzed using a single-degree-of-freedom(SDOF) model. The most widely used SDOF model so far has been the Biggs model [1] which covers only rate-independent bending response of the structure. Therefore, the Biggs model ignores two major componentsinherent in the structural response of blast walls: material rate sensitivity and membrane action. Material ratesensitivity is an important factor when ductile materials are subjected to high deformation rates and membraneaction can have a significant influence on the member response in the presence of axial restraints and largedisplacements. Disregarding these phenomena can result in overly conservative estimates of structural response.The Fire and Blast Information Group (FABIG) presented in References [2] and [3] an extension to the originalBiggs model including the following features:

• effect of support stiffness• effect of different moment capacities at the supports• effect of axial restraints• material strain rate sensitivity• approximate method to calculate plastic strains in plastic hinges

This model was originally validated using only a simple beam geometry. While the SDOF model showed goodcorrespondence with results from finite element analysis, it is not clear that similar correspondence would beachieved with more complex structures such as blast walls.

The purpose of this paper is to present a simplified approach to model structural response to air blast usingmethods that can be easily implemented and automated to computer software. We have implemented thesemethods to SURMA (Survivability Manager Application) resulting in robust and efficient blast responseanalysis which requires minimal user effort. Performing a FE analysis of a blast scenario requires extensiveknowledge from the engineer and it takes a considerable amount of time to even set up the model for theanalysis. Therefore, it is not feasible to perform several detailed FE analyses for different ship compartments.The herein presented SDOF blast analysis method can be used to quickly calculate countless of different blastscenarios without the need to create complicated models.

SDOF analysis

Here only the main features of the SDOF model are presented. For more detailed discussion of the method andthe necessary equations the reader should look into References [2] and [3].

Behavior of the SDOF model is characterized by the resistance function. The response consists of differentstages which are shown in Fig. 1. The slope of each curve equals the equivalent stiffness coefficient during thecorresponding stage. The only response stages in Biggs model were elastic stage, one elasto-plastic stage andplastic bending stage. For any realistic structure the displacements in elastic stage are small and the influence ofbending on stretching can be ignored. The elastic stage ends in the formation of first plastic hinge which can belocated in either support or in the midspan of the structure. The order and number of the plastic hinges isdetermined by the relative bending moment capacities and degrees of bending stiffness of the supports and ofthe beam. Plastic hinge does not occur for a pinned support. Therefore, one to three plastic hinges are possible.

Plastic bending stage is initiated after the formation of a rigid plastic mechanism. The response is assumedperfectly plastic, meaning that there is no strain hardening. Therefore, in plastic bending stage the staticresistance remains constant. Since displacements can become large during this stage, the influence of bendingon stretching, and hence on the development of axial forces, should be considered. This is handled in thesubsequent membrane stages.

In the first membrane stage the axial force begins to affect. The axial force varies quadratically withdisplacement, leading to cubic variation in the static resistance. The plastic interaction between bending momentand axial force at each plastic hinge is assumed linear. This approximation gives a conservative representation ofthe cross-sectional response and keeps the model simple, as it stays independent of the cross-sectional shape.

As the axial force increases the bending moment decreases. When the axial force reaches the overall plasticaxial limit, a second membrane stage is initiated. In this stage the full plastic axial capacity is sustained with azero bending moment. Therefore, the resistance-displacement response becomes independent of the bendingmoment capacities of the supports and the beam. In this stage the static resistance continues to grow linearlyuntil unloading or structural failure occurs.

The first step of the solution procedure is to calculate the response history of the SDOF system using staticstrength properties. From the obtained response history a displacement rate is approximated. We use the averagedisplacement rate till the first peak displacement. This displacement rate is then used to update all modelparameters to corresponding dynamic strength properties. The update to dynamic strength properties is based onCowper-Symonds model:

σd=σ y(1+( ϵ̇ p

D )1n) (1)

where n and D are material dependent parameters and σ y is the static yield strength. Using the dynamicstrength properties new response history and new approximate displacement rate can be calculated. This processis repeated until displacement rate has converged to acceptable tolerance.

Reduction of real structure to SDOF representation

The conversion tables and equations used to derive the SDOF model properties in References [2] and [3] aregiven for beams. However, blast walls and ship structures are often attached to surrounding structures from allouter edges. In this case it is not always clear how to reduce the structural strength properties of the realstructure to one-dimensional SDOF model properties. One-way stiffened plates and corrugated plates canreadily be reduced to beams by accounting only one corrugation or one stiffener with the attached plate. Fortwo-way stiffened plates with girders and supporting stiffeners it is suggested in Ref. [4] to first calculate thereaction forces of the stiffeners and apply them as a load to the girder. However, in Ref. [5] it was found that theresistance of the stiffeners is not critical with respect to the explosion actions. Therefore, two-way stiffened platecan be reduced to a beam by omitting the stiffeners and taking into account the girder and the attached plate. Weadopted this approach because of it's simplicity and because it can give equally accurate results compared to themore complicated method.

In Ref. [6] cross-sections are classified as follows:• Class 1: Plastic cross-sections which can form a plastic hinge with the rotation capacity required for

plastic analysis.• Class 2: Compact cross-sections which can develop their plastic moment resistance but have limited

rotation capacity.

Fig. 1. Response stages of the SDOF model.

• Class 3: Semi-compact cross-sections where the calculated stress in the extreme compression fiber ofthe steel member can reach its yield strength, but local buckling is liable to prevent the development ofthe plastic moment resistance.

• Class 4: Slender cross-sections where it is necessary to make explicit allowances for the effects of localbuckling when determining their moment resistance or compression resistance.

When the structural element falls into class 4, effective section properties need to be calculated iteratively forthe structure. For the attached plate we calculate the effective plate width as recommended in Ref. [4] with smallsimplifications: for continuous stiffeners the mean value between effective width at the midspan and the fullwidth is used and when the stiffeners are not attached to surrounding structures the effective width at themidspan is used.

Comparison with finite element analysis

Three different blast wall configurations were examined: two T-stiffened plates and one corrugated plate. Thewidth and height of each plate were 3m x 2m. For stiffened plates the stiffener spacing and web height werevaried. The profile properties of these structures are presented in Table 1. The finite element models are shownin Fig. 2. These models were meshed using 3-node and 4-node under-integrated shell elements with averageelement size of 12.5mm and five through-the-thickness integration points.

Table 1. Profile details of the blast walls used in test cases. All dimensions are in millimetres.

Platethickness

Webthickness

Flangethickness

Plate width betweenstiffeners/corrugations

Web height Flange width

Plate with smallstiffeners

5 6 8 500 100 100

Plate with largestiffeners

5 6 8 750 140 100

Corrugated plate 5 - - 240 125 100

Boundary conditions were chosen to represent realistic structural connections in ship structures. For stiffenedplates all plate edges and stiffener ends were fully fixed (U_X = U_Y = U_Z = R_X = R_Y = R_Z = 0) to

Fig. 2. Finite element models used for validation cases.

represent structure where the stiffeners are continuous or fully welded to adjacent structures. In the followingdiscussion we call this boundary condition “fixed” for simplicity. Only the plate edges were fixed to representstructure where the stiffeners are not attached to adjacent structures. We call this boundary condition “pinned”for simplicity. For corrugated plate all end plate edges and the outer edges of the corrugated plate were fixed torepresent structure where the corrugated plate is fully welded to adjacent structures (fixed boundary condition).The connection of corrugated plate to primary framework using angles was modeled by setting U_X = U_Y =U_Z = 0 for the other edge of the top end plate and UX = UZ = 0 for the other edge of the bottom end plate.This allows axial movement of the corrugated plate. We call this boundary condition “pinned” for simplicity.Similarly for the corresponding SDOF models the boundary conditions were set to either fixed or pinned.

Structures were subjected to uniform blast pressure waves from 5kg and 15kg TNT charges. The blast wavecharacteristics were calculated using empirical equations implemented in SURMA and the blast load curves arepresented in Fig. 3. Two different blast load magnitudes were used in order to excite different structuralresponses: small blast load is enough to keep the structural response mainly in plastic bending stage, whereaslarge blast load forces the structures into membrane deformation range. The pressure loads were applied oneither side of the plates in order to see the different behaviors when the plate is in compression or in tension,depending on the load direction.

Strain rate dependent Cowper-Symonds model was used to simulate material response. The used material wassteel with the following properties: E = 210GPa, ρ = 7850kg/m3, υ = 0.3, σy = 280MPa, D = 40 and n = 5. Thenumerical solution method was explicit time integration and the used finite element software was IMPACT [7].

The displacement results are presented in Figures 4-15. The results clearly show the improved performance ofthe advanced SDOF model against the original Biggs' bending model. Considering the complex structuralbehavior, the results from the advanced SDOF model are mostly in good accordance with the FE results,whereas all of the results from the original BIGGS model are over conservative.

Fig. 4. Small stiffeners, fixed constraints, 5kg TNT.

Fig. 3. Blast load curves used for validation cases.

Fig. 5. Small stiffeners, fixed constraints, 15kg TNT.

Fig. 6. Small stiffeners, pinned constraints, 5kg TNT.

Fig. 7. Small stiffeners, pinned constraints, 15kg TNT.

Fig. 8. Large stiffeners, fixed constraints, 5kg TNT.

Fig. 9. Large stiffeners, fixed constraints, 15kg TNT.

Fig. 10. Large stiffeners, pinned constraints, 5kg TNT.

Fig. 11. Large stiffeners, pinned constraints, 15kg TNT.

Fig. 12. Corrugated plate, fixed constraints, 5kg TNT.

Fig. 13. Corrugated plate, fixed constraints, 15kg TNT.

For stiffened plates with fixed boundary conditions the results are consistent. The membrane effect does notinitiate for the 5kg blast case as the displacements stay low enough. Therefore, for these cases, the displacementhistories for original Biggs model and SDOF model with membrane effect are similar. The discrepancy betweenFE results and SDOF results grows as the displacements increase. This is caused by the assumption of linearplastic interaction between bending moment and axial force, which is a conservative approach. Comparison ofresults for the stiffened plates with different sized stiffeners shows that as the bending rigidity of the stiffenercompared to the plate flange increases, the accuracy of the SDOF representation decreases.

For stiffened plates with pinned boundary conditions the results are not fully consistent. It can be seen that whenthe blast pressure occurs on the plate side of the structure, the displacements are greatly overestimated by theSDOF model. The reason for this is that the neutral axis of the beam does not pass through the support in thesecases. This offset between support reaction and neutral axis creates a bending moment at the support, effectivelystiffening the response. Therefore, the real boundary conditions are something between pinned and fixedconstraints. However, determination of the correct support stiffness values is difficult and, therefore, the resultsfor the blast on stiffener side could be used as a conservative estimate. For pinned constraints the discrepancybetween FE results and SDOF results decreases as the displacements increase. This is caused by buckling andloss of structural integrity which is captured by FE model but not by SDOF model. For fixed boundaries thisphenomenon does not occur as the boundary conditions effectively prevent the onset of buckling.

The FE results show that the corrugated plate is liable to buckling. This is caused by the modeling of the endplates which allow some axial displacements for the corrugated plate profile edges. As shown in Fig. 12 theSDOF results for small blast load are very good compared to FE results because no buckling occurs. In Figures14-15 the results are less consistent as the SDOF model is not capable of predicting buckling. However, the

Fig. 14. Corrugated plate, pinned constraints, 5kg TNT.

Fig. 15. Corrugated plate, pinned constraints, 15kg TNT.

structure can be considered failed after excessive buckling has occurred and choosing an appropriate failurecriteria, such as maximum displacement or ductility ratio, for SDOF model could give reasonable prediction ofstructural response until failure.

Application to confined explosion in ship compartment

We applied the SDOF response calculation to confined explosion in a naval vessel and compared the resultswith finite element simulations and experimental results from tests conducted on Ex-Helsinki-class fast attackcraft by the Finnish Navy.

The pressure loads on cabin walls were calculated with SURMA, which uses empirical equations and the so-called mirror model to approximate initial blast wave and reflections from adjacent walls. Quasi-static pressureis also accounted for. The shape of the pressure-time history for each wall is presented in Fig. 16. The pressureswere set as uniform pressures acting on each wall, which might be a slightly conservative approach.

The structural material was aluminium and the used material model in the FE analysis was Cowper-Symondswith linear strain hardening. The FE model was meshed using 3-node and 4-node under-integrated shellelements with five through-the-thickness integration points. SURMA was used to calculate the SDOF responseanalysis and IMPACT was used for FE analysis.

Fig. 16. Pressure-time history for cabin walls.

Fig. 17. Contour plot of displacements.

Fig. 18. Contour plot of displacements.

Fig. 19. Photograph of the ruptured connection.

The experimental results showed moderate deflections in the cabin walls. Many weld tears were observed andthe upper connection of the corrugated panel presented in Fig. 19 ruptured. The finite element results were ingood accordance with the experimental results. In Fig. 20 is shown a high plastic strain concentration in thesame area where the rupture occurred. Figures 17 and 18 show the displacements in the structure.

Fig. 21 Shows comparison of results from FE and SDOF model. For two-way stiffened plate the peakdisplacements agree well. For one-way stiffened plate the peak displacement is slightly overestimated by theSDOF model and the rebound is also overestimated. This happens because the rebound is assumed to be elasticin the SDOF model. For corrugated plate the SDOF model underestimates the peak displacement. This happensbecause the SDOF model is not capable of accurately representing the cross-section flattening and buckling.However, the corrugated plate can be considered to be failed before it reaches the peak displacement and settingan appropriate failure criterion would result in better agreement with FE and SDOF results.

Development and future work

It is now clear that this SDOF model gives good results in terms of displacements, but a reasonable failurecriterion is also needed. In References [4] and [8] some failure criteria are given as ductility ratio limits.However, these limits can be overly conservative, because in addition to post-blast load-bearing capacity of theblast wall we may also be interested in the ability of the blast wall to contain the pressure loads inside the shipcompartment. Even if the structure has failed according to these existing failure criteria in a way that it nolonger contributes to the global structural strength, it may still be able to contain the quasi-static pressure and

Fig. 20. Plastic strains in the upper connection of corrugated plate.

Fig. 21. Comparison of results from FEM and SDOF analysis.

prevent damage propagation. Therefore, an additional failure criterion based on plastic strain could be set inaddition to ductility ratios and two different damage states could be used in the analysis.

Conclusions

This paper presented a simple method of conducting a blast response analysis for ductile structural memberswhich can easily be implemented to existing software. As can be seen, the level of accuracy is in many casescomparable with finite element analysis. We have implemented the proposed scheme to SURMA resulting infully automated blast analysis method where the only obligatory input from the user is the ship geometry.Therefore, the user does not need to worry about effective section properties of the structures or define boundaryconditions. This allows the automated simulation of thousands of blast scenarios in minimal time using one shipmodel.

References

1. Biggs J.M., Introduction to Structural Dynamics, McGraw Hill, 1964

2. Fire and Blast Information Group, TN7, Simplified Methods for Analysis of Response to Dynamic Loading,The Steel Construction Institute, 2002

3. Fire and Blast Information Group, TN10, An Advanced SDOF Model for Steel Members Subject to ExplosionLoading: Material Rate Sensitivity, The Steel Construction Institute, 2007

4. Det Norske Veritas, Recommended Practice DNV-RP-C204: Design Against Accidental Loads, 2010

5. Amdahl J., Resistance to Accidental and Very Extreme Explosions, 2004,http://www.usfos.no/publications/explosion/documents/2004-ResistanceAccidental&ExtremeExplosions.pdf

6. ENV 1993-1-1, Eurocode 3: Design of steel structures - Part 1.1: General rules and rules for buildings, BSI,1993

7. http://www.impact-fem.org/

8. Fire and Blast Information Group, TN4, Explosion Resistant Design of Offshore Structures, The SteelConstruction Institute, 1996