Finite element modelling of steel bridge structures exposed to ship collisions

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Department of Structural Engineering FACULTY OF ENGINEERING SCIENCE

NTNU – Norwegian University of Science and Technology

MASTER’S THESIS 2018

for

Kevin Mandt Ofstad

Finite element modelling of steel bridge structures exposed to ship collisions

The Norwegian Public Roads Administration is interested in developing a simulation method to predict the behaviour of steel bridge structures exposed to collision loads from ships. The simulation method shall be based on the finite element method and the structural parts shall be modelled with large shell elements to achieve reasonable calculation times. To ensure realistic results, both the plastic deformation and the failure of the structural parts must be calculated with sufficient accuracy, even when the size of the shell elements is several times the plate thickness. To this end, special techniques shall be applied to describe localized plastic deformation and failure in the shell elements.

In this project, numerical methods for simulating collisions between large steel structures, e.g. ship against bridge, will be tested and evaluated.

Two simulation methods shall be applied and evaluated. The first is based on the BWH criterion for plastic instability, but the failure criterion shall be modified to increase its robustness and to account for damage evolution in the post-necking region. The BWH criterion is often applied in ship collisions owing to its simplicity and ease of calibration. The second is based on the Cockcroft- Latham criterion with regularization to account for membrane- and bending-dominated deformation modes in the shell elements. This is a novel method recently developed at SIMLab. The two methods will be evaluated by comparison with existing experimental results from the literature.

The main tasks of the research project are as follows:

1. To perform a literature review on experimental and numerical studies of collisions between large steel structures, e.g. ships and bridges.

2. To perform a literature review on failure models for large-scale simulations of steel structures 3. To formulate mathematically, implement numerically and verify computationally (for simple

cases) the modified BWH model and the regularized Cockcroft-Latham model.

4. To perform a numerical study of a generic plane-stress problem, namely a sheet under biaxial stretching, in order to compare the accuracy and the mesh sensitivity of the two failure models.

A solid element model of the problem with the Cockcroft-Latham criterion will be used as the baseline case.

5. Perform simulations of existing experiments from the literature of collisions between large steel structures to assess the credibility and robustness of the two failure models.

6. Report the research work.

Supervisors:

Odd Sture Hopperstad, Torodd Berstad, David Morin (NTNU)

The report should be written in the style of a scientific article and submitted to the Department of Structural Engineering, NTNU, no later than June ??, 2018.

NTNU, January 15

^th

, 2018

Odd Sture Hopperstad

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Abstract

The behavior of the Bressan-Williams-Hill fracture criterion (BWH), which is often used in simulating ship collisions, was studied by modeling and studying existing experiments done on the subject. Three experiments are evaluated with the BWH criterion using both the middle integration point, as originally intended, and the first integration point through the thickness. A mesh scaling technique is also used and evaluated for both cases.

Numerical models are made for two of the experiments chosen, while the last model was received. All models are compared to the existing experiments, mainly by comparing the force-displacement curves.

The BWH criterion showed acceptable results for most of the cases, but was more conservative than initially expected. Using the first integration point instead of the middle, gives even more conservative results for these cases. The mesh scaling technique shows overall satisfactory results.

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Sammendrag

Oppførselen til Bressan-Williams-Hill bruddkriterie (BWH), som ofte brukes i simulering av skipskollisjoner, ble studert ved ˚a modellere og studere eksisterende eksperimenter gjort p˚a emnet. Tre eksperimenter evalueres med BWH-kriteriet ved ˚a bruke b˚ade det midtre integrasjonspunktet, som opprinnelig ment, og det første integrasjonspunktet gjennom tykkelsen.

En meshskaleringsteknikk er ogs˚a brukt og evaluert for begge tilfeller.

Numeriske modeller er laget for to av de valgte eksperimentene, mens den siste modellen ble mottatt. Alle modeller er sammenlignet med de eksisterende eksperimentene, hovedsake- lig ved ˚a sammenligne kraftforskyvningskurvene.

BWH-kriteriet viste akseptable resultater for de fleste tilfeller, men var mer konservativ enn forventet. Ved ˚a bruke det første integrasjonspunktet i stedet for midten, f˚ar du enda mer kon- servative resultater for disse tilfellene. Den anvendte meshskaleringsteknikken viser generelt tilfredsstillende resultater.

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List of Figures

2.1 Von Mises yield surface . . . 3

2.2 Von Mises yield surface for plane stress with σ₃ = 0 . . . 3

2.3 Normality rule for the von Mises yield surface . . . 4

2.4 Isotropic hardening . . . 5

2.5 Tensile specimen with local neck along the direction of zero extension . . . 6

2.6 Orientation of the inclined plane in the sheet and definition of the Cartesian coordinate systems (x1, x2, x3) and (¯x1,x¯2,x¯3) where ¯x2is parallel to x2 . . . 7

2.7 Failure strain vs strain-increment ratio for the BWH criterion withK = 500 MPa andn= 0.5 . . . 9

3.1 (a) Geometry of the test specimens (b) Geometry of the impactor . . . 11

3.2 Mesh with 10 mm elements of Broekhuijsen’s formability tests . . . 11

3.3 Setup for formability test 3 with 20 mm elements . . . 12

3.4 Results of Broekhuijsen’s formability tests without BWH . . . 12

3.5 Formability test 1 - Broekhuijsen . . . 13

4.1 Geometry of components - Alsos: (a) shows the transverse cross section of the component and (b) shows a longitudinal cross section with and without stiffeners . 19 4.2 5 mm mesh for 2 flat bar stiffeners . . . 20

4.3 Cross section of the weld elements . . . 21

4.4 Setup for 2-HP 20 mm elements . . . 21

4.5 Results - Unstiffened plate . . . 22

4.6 Results - 1-FB stiffener . . . 23

4.7 Results - 2-FB stiffeners . . . 24

4.9 Experiment 2-FB after fracture . . . 25

4.10 Results - 1-HP stiffener . . . 26

4.11 Results - 2-HP stiffeners . . . 27

4.12 Plastic equivalent strain concentration in weld elements 2-HP 30 mm mesh size - middle integration point . . . 28

5.1 (a) Stiffened steel plate seen from the bottom, (b) cross section of test specimen, (c) stiffener cross-section, and (d) length and center-to-center distance of the fillet welds. . . 29

5.2 (a) Specimen clamped between bottom and top frame, (b) cross-section in the longitudinal direction, (c) lower support frame side, (d) profile in the width direction, (e) cross section in longitudinal direction, and (f) cross section in width direction. 30 5.3 Geometry of the indenters . . . 31

5.5 Results - Indenter A . . . 33

5.6 Plastic equivalent strain before fracture and for 200 mm indentation for 3 mm mesh with BWH using first integration point . . . 34

5.7 Plastic equivalent stress at 150 mm indentation for 12 mm mesh first integration point with mesh scaling . . . 35

5.8 Results - Indenter B . . . 36

List of Tables

3.1 Material parameters for the formability tests . . . 10

4.1 Material parameters for the different components . . . 20

5.1 Material parameters for the stiffened steel plates . . . 31

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1 Introduction

1.1 Background

Collisions for ships may occur wherever there is ship traffic. Fortunately, the events are rare, but the ship traffic continues to increase, which also increases the probability of an accident happening. The consequences of a collision can be severe, and it is important to predict these events quite accurate to design good structures that are able to withstand the collision as good as possible.

There is an interest in developing a simulation method for collision between ships and steel bridges.

The method shall be based on the finite element method. To achieve reasonable calculation times, large shell elements have to be used. The method should be able to predict the deformation and failure of the structure with sufficient accuracy, even when the shell elements are very large compared to the thickness of the component.

1.2 Literature review

Over the last decades, a lot of research has been performed that are relevant for this subject. This part summarizes some of the work that are relevant for steel bridges exposed to ship collisions.

Martin Storheim investigated the structural response in ship-platform and ship-ice collisions in his doctoral thesis in 2016 [1]. He proposes e.g. an extension of the BWH fracture criterion where post-necking effects is included. With a combination of coupled damage and a strain-state dependent erosion criterion, it is shown that a more robust fracture prediction is achieved, and has reduced mesh dependence.

S¨oren Ehlers et al. [2] looked at the response of three different ship side structures using the finite element method and LS-DYNA with different fracture criteria. The study shows that han- dling the mesh size sensitivity is often more important than the fracture criterion for the three cases investigated.

Other studies relevant for this thesis is presented when it becomes relevant.

1.3 Objective and scope

The present study is looking at different existing experiments relevant to ship collisions against steel. In communication with the supervisors, it is agreed that more emphasis on the Bressan- William-Hill criterion while other parts, like the Cockroft-Latham criterion is omitted. The Bressan-Williams-Hill (BWH) failure criterion is evaluated and compared when using both the first integration point, and the middle integration point through the thickness of the plate. This is also compared with and without the mesh scaling technique proposed by Alsos et al. [3]. Abaqus 2016 [4] is used for the numerical simulations.

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2 Modelling of plasticity and ductile failure

This section will first take a short look at the yield criterions in general and then go closer into the von Mises yield criterion and some of its properties, which is often used for ductile materials e.g.

metals. We will then go into the Bressan-Williams-Hill (BWH) failure criterion. The equations from the first part of this chapter are taken from [5], while the equations from the BWH part are taken from [6].

2.1 Plasticity theory

2.1.1 Yield criteria in general

When a material is exposed to forces, it will first be elastic deformations which is reversible. The relationship between stresses and strains is typically linear here for metals. If the stresses become high enough, the material reaches a limit called the yield limit. This is where plastic or irreversible deformations starts and is mathematically stated as

f(σ) = 0. (2–1)

The yield function, f, is a continuous function of the stress tensorσ. This function is negative when the material is in the elastic domain (f(σ) <0). It is also assumed that f(σ) >0 is not allowed.

The yield function can be written as

f(σ) =ϕ(σ)−σ_Y (2–2)

whereσY is the yield stress andσeq=ϕ(σ) is the equivalent stress which is the magnitude of the stresses caused by external loads on the material. σeq is here a positive homogeneous function of order one (ϕ(aσ) =aϕ(σ)). This leads to the following property of the equivalent stress:

σij

∂ϕ(σ)

∂σij

=ϕ(σ) (2–3)

2.1.2 Von Mises yield criterion

Von Mises yield criterion is assuming pressure insensitivity and isotropy. The criterion is assuming that yielding occurs when J2, the second deviatoric stress invariant, reaches a critical value k² as shown in (2–4). Theories based on the von Mises criterion are therefore often calledJ₂ flow theories since it only depend on this invariant.

J₂=k² (2–4)

This criterion is independent of the first stress invariant,J₁, and is therefore often used on ductile materials such as metals.

The stress deviator is defined as

σ_ij⁰ =σij−1

3σkkδij (2–5)

We can then expressJ2 by using (2–5) J₂=1

2σ_ij⁰ σ_ij⁰ =1

2(σ_ij−1

3σ_kkδ_ij)(σ_ij−1

3σ_mmδ_ij) = 1

2σ_ijσ_ij−1

6σ_kkσ_mm (2–6) Uniaxial tension tests are often used to determine the yield stress. In uniaxial tensionσ=σ₁₁= σ_Y at yielding, and the rest of the stress components are zero. This implies that (2–6) is reduced to

J₂=1

3σ₁₁² =1

3σ_Y² =k² (2–7)

by combining it with (2–4).

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It is now easy to show that the von Mises yield function can be written on this form f(σ) =p

3J2−σY = r3

2σ⁰_ijσ⁰_ij−σY = 0 (2–8) It can also be shown that it can be expressed in terms of the principal stresses as follows

f(σ₁, σ₂, σ₃) = r1

2((σ₁−σ₂)²+ (σ₂−σ₃)²+ (σ₃−σ₁)²)−σ_Y = 0 (2–9) This is an equation for a circular cylinder with radiusq

2

3σY and is shown in Figure 2.1

Figure 2.1: Von Mises yield surface [7]

The yield surface on this figure is shown together with the hydrostatic axis and the deviatoric plane(also called Π-plane) which is a plane perpendicular to the hydrostatic axis. Also seen on the figure, this criterion is independent of the hydrostatic stresses.

If we setσ₃ = 0 in (2–9), we get

σ₁²+σ₂²−σ1σ2=σ²_Y (2–10)

This is an ellipse which is shown in Figure 2.2

Figure 2.2: Von Mises yueld surface for plane stress withσ3 = 0 [7]

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2.1.3 Plastic flow rule

The plastic flow rule is defined by (2–11) in the most general case, and ensures that the dissipation is non-negative.

˙

ε^p_ij= ˙λ ∂g

∂σ_ij (2–11)

λ˙ ≥0 is a scalar called the plastic multiplier andg =g(σ)≥0 is the plastic potential function.

The plastic multiplier is determined from the consistency condition in the theory of plasticity and therefore must the stress be located on the yield surface during plastic flow. Since the plastic flow rule ensures non-negative dissipation, the following restriction applies for the plastic potential function

σij

∂g

∂σij

≥0 (2–12)

One choice for the plastic potential function is to associate it to the yield function,f, so that we get

˙

ε^p_ij= ˙λ ∂f

∂σij

(2–13) Equation (2–13) is called the associated flow rule and it can be easily seen that this satisfies non- negative dissipation by using (2–2) and (2–3).

It is implied from the associated flow rule that the plastic strain increment vector dε^p = ε˙^pdt is parallel to the gradient of the yield surface atσ is pointing outwards from the surface at this point along the normal. The normality rule is therefore another name for the associated flow rule and is visualized in Figure 2.3 for the von Mises yield surface in the principal stress state.

Figure 2.3: Normality rule for the von Mises yield surface [6]

For the von Mises yield criterion, the normality rule gives

˙

ε^p_ij= ˙λ ∂f

∂σij

= ˙λ3 2

σ_ij⁰ σeq

(2–14) It can also be shown that the equivalent plastic strain rate for the von Mises yield criterion and the associated flow rule is

˙ p=

r2

3ε˙^p_ijε˙^p_ij (2–15)

by using equation (2–14), calculating the plastic dissipation for this model (σijε˙^p_ij) and using the relation, ˙p= ˙λ

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2.1.4 Isotropic hardening

Materials work-harden in general when they undergo plastic deformations and this increases the strength of the material. Isotropic hardening is one of the most common ways to account for this and implies that the yield surface in stress space expands during plastic deformation. This is shown in Figure 2.4 and it can be seen that the yield surface expands as the material work-hardens.

Figure 2.4: Isotropic hardening [5]

Isotropic hardening can be introduced by defining a variable,R, that is representing the expansion of the yield surface. This leads to the following form of the yield function

f(σ, R) =ϕ(σ)−σY(R)≤0 (2–16)

σeq =ϕ(σ) is the equivalent stress andσY =σ0+Ris the flow stress, whereσ0is the initial yield stress. This implies that when the material work-hardens,R andσY is increasing and leads to a larger elastic region.

In general, the isotropic hardening rule is defined by

R˙ =h_Rλ˙ (2–17)

whereh_R is the hardening modulus and depends on the state of the material. The power law is one of many isotropic hardening rules and is formulated as

R(p) =Kpⁿ (2–18)

Kandnis parameters determined from material tests. If we assume the associated flow rule, the plastic strain rate ˙p= ˙λand the hardening modulus is

hR=Knpⁿ⁻¹ (2–19)

The choice of which hardening law to use depends on the applications and what material you are using.

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2.2 Bressan-Williams-Hill failure criterion

2.2.1 Hill’s criterion for local necking

It is shown in that a local neck in a sheet metal loaded in tension, develops in a way similar to diffuse necking in a tensile bar [8]. The width of the local neck in the sheet is approximately equal to the thickness. The global strain field is not affected by the local necking, but since the strains increase rapidly, local necking is a precursor to failure. One condition for a necking process to be considered local, is that the boundary conditions is unaffected by the neck. Since the neck is so narrow, kinematic compatibility requires that the strain increment parallel to the local neck is zero inside and just outside the necking area (see Figure 2.5).

Figure 2.5: Tensile specimen with local neck along the direction of zero extension [6]

We can then use this criterion to determine the angleθ, and the transformation rule for strains gives

d¯ε^p₂₂=dε^p₁cos²θ+dε^p₂sin²θ= 0 (2–20) which leads to

tanθ= 1

√−β (2–21)

If we now consider a sheet material in plane stress with σ3 = 0 in the thickness direction, the principal tractions in the sheet are

T1=σ1t, T2=σ2t (2–22)

where t is the sheet thickness.

In [9] it is postulated that local necking occurs when T1, the major principal traction, reaches a maximum value, i.e. when

dT₁=d(σ₁t) =dσ₁t+σ₁dt= 0 (2–23) If we neglect the elastic strains and using thatdε^p₃= ^dt_t, we get

dσ₁ σ1

+dt

t = 0 (2–24)

Now, we can use plastic incompressibility (dε^p₁+dε^p₂+dε^p₃ = 0) and the strain increment ratio,β, which is defined as

β =dε^p₂

dε^p₁ (2–25)

This leads to the following form of the criterion for local necking:

1 σ₁

dσ1

dε^p₁ = 1 +β (2–26)

Ifβ <−1, we get thickening of the sheet sincedε^p₃=−(1 +β)dε^p₁>0 forβ <−1, andT₁will not reach a maximum. This means the expressions shown above are only valid forβ >−1.

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It is also postulated in [9] that the loading is proportional at incipient local necking. This implies

that dσ₁

σ1

=dσ₂ σ2

= dσ_eq σeq

(2–27) If we now use that, and the equivalent plastic strain increment in von Mises plasticity (see equation (2–15)), we get

1 σeq

dσ_eq

dp − 1 +β q4

3(1 +β+β²)

= 0 for β∈(−1,0] (2–28)

Assuming the power law hardening rule,σeq=Kpⁿ, we get the form

pf =n q4

3(1 +β+β²)

1 +β for β∈(−1,0] (2–29)

Here, pf is the plastic strain at failure by local necking. We can see from equation (2–21) that we only get real solutions for θ if β ≤ 0. That means that local necking is not possible in an uniformly loaded homogeneous sheet ifβ >0. For biaxial tension, i.e. β ∈(0,1], imperfections are necessary to trigger local necking, and we need to use another analysis.

2.2.2 Bressan-William’s criterion for shear failure

One approach for biaxial tension is the Bressan-William’s criterion for shear failure [10]. Exper- iments show that sheets subjected to biaxial tension often show shear failure along an inclined plane through the thickness of the sheet with an angleϕ(see Figure 2.6). An assumption used in [10] is that shear failure occurs when the shear stress on the plane,τ, reaches a critical value,τcr. The angle,ϕ, is defined such that a material element oriented along ¯x3 in Figure 2.6 should not experience any change in length.

Figure 2.6: Orientation of the inclined plane in the sheet and definition of the Cartesian coordinate systems (x1, x2, x3) and (¯x1,x¯2,x¯3) where ¯x2 is parallel tox2[6]

Using the figure above, the transformation rules given in [6] and plastic incompressibility (dε^p₃ =

−(dε^p₁+dε^p₂) =−dε^p₁(1 +β), we get

d¯ε^p₃₃=dε^p₁sin²ϕ+dε^p₃cos²ϕ=dε^p₁(sin²ϕ−(1 +β) cos²ϕ) = 0 (2–30) This implies that

sin²ϕ−(1 +β) cos²ϕ= 0 (2–31)

sincedε^p₁ is assumed to be positive.

Now, we can use the relations cos 2ϕ= cos²ϕ−sin²ϕand cos²ϕ=¹₂(1 + cos 2ϕ), and get cos 2ϕ=− β

2 +β (2–32)

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Using the transformation formulas, we get the following expression for the resolved shear stress on the plane defined byϕ:

τ ≡σ¯₁₂= (σ₃−σ₁) cosϕsinϕ=−σ₁

2 sin 2ϕ (2–33)

In the latter equality, it is used thatσ₃ = 0 in the sheet, and the relation sin 2ϕ= 2 cosϕsinϕ.

This is, as mentioned, assumed to reach shear failure when|τ| reaches a critical valueτ_cr. Now we can use that sin²2ϕ= 1−cos²2ϕ= 1− _β

2+β

²

and get

|τ|= σ1

2 s

1− β

2 +β ²

=τcr (2–34)

This can then be rearranged to get the Bressan-Williams shear failure criterion:

σ1= 2τcr

r

1− _β

2+β

²

for β∈[0,1] (2–35)

It is possible to show that

σeq =√ 3σ1

p1 +β+β²

2 +β (2–36)

by using the normality rule and the definition of the equivalent stress for von Mises. This leads to an alternative form of this shear failure criterion

σeq−√ 3τcr

p1 +β+β²

√1 +β = 0 for β∈[0,1] (2–37)

2.2.3 Bressan-Williams-Hill failure criterion

The Bressan-Williams-Hill failure criterion (BWH) [11] is a useful criterion when large shell elements are used in FE simulations of thin-walled structures made of plates and sheet materials.

This criterion is a combination of Hill’s criterion and the Bressan-Williams criterion and is valid forβ ∈(−1,1]

1 σeq

dσ_eq

dp − 1 +β q4

3(1 +β+β²)

= 0 for β∈(−1,0]

σeq−√ 3τcr

p1 +β+β²

√1 +β = 0 for β∈[0,1]

(2–38)

Now, If we again assume, the hardening ruleσeq =Kpⁿ, we get

p_f =





 n

√4

3(1+β+β²)

1+β for β∈(−1,0]

√ 3^τ_K^cr

√

1+β+β²

√1+β

_n¹

for β∈[0,1]

(2–39)

The criterion is shown in Figure 2.7 withK= 500 MPa andn= 0.5.

To find the value for τ_cr, we can require that p_f is the same for Hill’s and Bressan-Williams criterion whenβ = 0 and get

τ_cr= K

√3 2n

√3 ⁿ

(2–40)

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Figure 2.7: Failure strain vs strain-increment ratio for the BWH criterion withK = 500 MPa andn = 0.5 [6]

It is also possible to show that an alternative form of the BWH criterion can be expressed as follows [11]:

σ1=











2K√ 3

1+¹₂β

√

1+β+β²

√2 3

ˆ ε1

1+β

p1 +β+β²ⁿ

for β ∈(−1,0]

√2 3K

√2 3εˆ1

n

q

1−(2+β^β )² for β ∈(0,1]

(2–41)

where ˆε1 is the critical strain and is assumed equal to n. This is the form that is used in the simulations described later.

The BWH failure criterion searches for instability in the middle integration point. When instability is reached, the load-carrying capacity is reduced in all integration points through the thickness.

Coarse meshes will often not detect the right stress concentration, and the BWH criterion will predict fracture too late. Alsos et al. [3] proposed to introduce a mesh scaling by multiplying ˆ

ε₁in equation 2–41 by a factor ¹₂(1 +^t_l^e

e) where t_e is the element thickness andl_e is the element length at initial configuration.

If strain concentrations are properly captured by the mesh used, the proposed mesh scaling will trigger fracture too early. Also suggested by Alsos et al [3], the mesh scaling could also be applied only to intersection elements so that the scaling is only used when needed.

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3 Formability tests of ship construction steel

These formability tests are from Broekhuijsens master thesis [12]. Six different 12 mm thick plates were tested with a 60 mm spherical indenter.

3.1 Material

The yield criterion is given by equation 2–16 whereσeq=ϕ(σ) is the von Mises equivalent stress andσY(R) =σY(εp) is a modified power law hardening rule, and is defined as

σ_Y(ε_p) =

(σ₀ if ε_p≤ε_plateau

K(ε0,ef f +εp)ⁿ if εp> εplateau

(3–1) whereKandnare hardening parameters, as in the normal power rule, andσ0 is the initial yield stress. As seen in the equation, the hardening is delayed until the plastic strainεpreachesεplateau, the plateau strain. We also have that

ε_{0,ef f} =ε₀−ε_plateau=σ₀ K

^1/n

−ε_plateau (3–2)

Here,ε0 is the strain at initial yield.

The parameters for these tests are given in the following table:

Table 3.1: Material parameters for the formability tests [13]

σ₀ [MPa] E [GPa] K [MPa] n ε_plateau

284 196.5 680 0.195 0.0

3.2 Numerical model

3.2.1 Geometry and boundary conditions

The geometries for the formability tests are shown in Figure 3.1. The holes are ignored in the numerical model. The plates are modelled as fixed outside a radius of 140 mm from the center of the plates. A similar assumption is made in [13]. The impactor is modelled as analytical rigid.

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(a)

(b)

Figure 3.1: (a) Geometry of the test specimen (b) Geometry of the impactor [12]

3.2.2 Mesh

The tests are modelled with 10 (le/te ≈ 0.8), 20 (le/te ≈ 1.6)and 28 mm (le/te ≈ 2.4) shell elements with element type S4R. The 10 mm meshes are shown in Figure 3.2 with test 1 (left) to 6 (right). Larger elements led to too high forces in the force-displacement curves compared to smaller elements, and couldn’t capture the strain response properly due to the plates being very thick compared to the width.

Figure 3.2: Mesh with 10 mm elements of Broekhuijsen’s formability tests

3.2.3 Contact

The contact between the indenter and plate is modelled as general contact with a friction coefficient of 0.2 which is the same as both Broekhuijsen [12] and Storheim [13] used in their analysis and is normal for steel to steel contact. Figure 3.3 shows the setup for test 3 with the 20 mm elements.

The impactor is initially placed 0.001 mm above the plate to make sure the parts is not in contact before the test starts. The test is set to 0.05 seconds and is ramped up evenly the first 10% of the duration.

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Figure 3.3: Setup for formability test 3 with 20 mm elements

3.3 Results

This section shows the results of the formability tests. The following figure shows the result of the tests without a failure criterion with 5 mm mesh size. The solid lines are the different simulations of the tests and the dashed lines show the corresponding lab results.

Figure 3.4: Results of Broekhuijsen’s formability tests without BWH

The figure shows that we get a local neck at the end of the curves even without any fracture criterion for all the tests.

The following pages shows the results for the tests with BWH when assuming fracture at the first integration point through the thickness and the middle integration point, as originally intended for BWH, with and without mesh scaling. We can see that the first integration point gives more conservative results, especially for the first test. The mesh scaling is also pretty good for most of the tests.

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3.3.1 Test 1

(a) First integration point without mesh scaling (b) First integration point with mesh scaling

(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 3.5: Formability test 1 - Broekhuijsen

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3.3.2 Test 2

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3.3.3 Test 3

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3.3.4 Test 4

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3.3.5 Test 5

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3.3.6 Test 6

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4 Penetration resistance of stiffened steel panels

The experiments for these tests are from Alsos and Amdahl’s paper: ”On the resistance to penetration of stiffened plates” [14]. The tests are performed on plates welded to a strong box frame.

The tests are then carried out by forcing an indenter laterally into the plate from the center. Five different tests are performed: unstiffened plate (US), 1 and 2 flatbar stiffeners (1-FB and 2-FB) and 1 and 2 bulb stiffeners (1-HP and 2-HP).

4.1 Geometry

The geometries are shown in Figure 4.1. The plates are 5 mm thick, 1200 mm long and 720 mm wide. The stiffeners are 6 mm thick, 120 mm long and placed evenly apart, which implies that the tests with 1-FB and 1-HP stiffeners are placed in the center line, while 2-FB and 2-HP are placed 240 mm apart. The indenter has a cone shape with a spherical ”nose”, which has a 200 mm radius and a 90 degree spreading angle.

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(b)

Figure 4.1: Geometry of components - Alsos: (a) shows the transverse cross section of the component and (b) shows a longitudinal cross section with and without stiffeners [14]

4.2 Material

The bulb stiffeners and frame are made from high strength steel (S355NH EN10210), while the flat bar (FB) plates and flat bar stiffeners are made from mild steel (S235JR EN10025). The different parameters for the same material are due to the fact that the steel comes from different batches. The same hardening rule is used here as in Broekhuijsen’s tests and is shown in section 3.1 on page 10. The Young’s modulus is set to 210 000 MPa, poisson’s ratio to 0.3 and density to 7.85 tons/m³The material parameters are shown in the table below.

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Table 4.1: Material parameters for the different components [3]

Specimen Material type σY K n εplateau

Plate US,1-FB, 2-FB S235JR EN10025 285 740 0.24 0

Plate 1-HP S235JR EN10025 340 750 0.20 0

Plate 2-HP S235JR EN10025 260 640 0.22 0.003

Flat bar stiffeners S235JR EN10025 340 760 0.225 0.015 Bulb stiffeners S355NH EN10210 390 830 0.18 0.01

Frame S355NH EN10210 390 830 0.18 0.01

4.3 Numerical model

The frame is assumed stiff enough to assume fixed ends around the plate and is not modelled. The model is also modelled with one symmetry plane across the middle of the plate. The stiffeners and plate are connected with a tie constraint since they have different materials.

4.3.1 Mesh

These tests are also modelled with S4R elements in Abaqus 2016. The mesh sizes that are used are: 5 mm (l_e/t_e= 1), 20 mm (l_e/t_e= 4) and 50 mm (l_e/t_e= 10). The 5 mm mesh for the test with two flat bar stiffeners is shown in the figure below.

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(b)

Figure 4.2: 5 mm mesh for 2 flat bar stiffeners

The welds are modelled in a similar way to what was done by alsos [3] and the method is also described by Wang et al. [15]. The welds vary between 5 and 7 mm in height and width, so 6 mm wide elements with increased thicknesses are used in the simulations. The increased thicknesses are 2 mm for the plate side, and 4 mm for the stiffener side. This is shown in Figure 4.3. The different thicknesses are modelled with a composite layup in Abaqus with a conventional shell element type.

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Figure 4.3: Cross section of the weld elements

4.3.2 Contact

The contact is modelled pretty similar to what is done for the Broekhuijsen tests described in section 3.2.3. The indenter is initially placed 0.001 mm above the plate and the velocity is ramped up the first 10% of the simulation which are also set to 0.05 seconds for these tests. The only difference is the friction coefficient that is set to 0.3 and is normal for coated steel to steel contact.

General contact is also used here. The setup for two bulb stiffeners is shown in the figure below with 20 mm elements.

Figure 4.4: Setup for 2-HP 20 mm elements

4.4 Results

The results for the different configurations are shown in the next pages. The circles indicate where the first element fails. As can be seen on the figures, the results without BWH (the dashed lines in the plots) also shows some necking behavior at for these tests.

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4.4.1 Unstiffened plate

(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 4.5: Results - Unstiffened plate

We see from the figure that it is not much difference between using the middle integration point and the first integration point. The first integration point is, as expected, more conservative, but the difference is barely noticeable. It is also observed that the mesh scaling for this case is a little too strong.

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4.4.2 One flat bar stiffener (1-FB)

(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 4.6: Results - 1-FB stiffener

The same observations are observed here, as in the unstiffened plate. Only a little more conservative using the first integration point and the mesh scaling is a bit too strong.

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4.4.3 Two flat bar stiffeners (2-FB)

(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 4.7: Results - 2-FB stiffeners

The same goes for these tests, as for the two previous ones, when looking at the difference between the integration points. However, here is the mesh scaling pretty good.

The evolution of the fracture is shown in Figure 4.8 for this configuration.

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(a) 2-FB right before fracture (b) 2-FB right after fracture

(c) 2-FB at the end of the simulation

Figure 4.8: Fracture evolution for two flat bar stiffeners

It is seen that the fracture occurs in the middle of the plate close to the weld, and develops outwards. This is pretty similar to what happened in the experiments. An image of the experiment after fracture is shown in Figure 4.9.

Figure 4.8c shows that a strain concentration occurs prior to fracture. The stiffener has large resistance and friction effects in the impact zone leads to large deformations in the plate section close to the stiffener. Also, the plate experience a change in geometry close to the weld, which implies larger strain concentration here. This leads to fracture occurring here first.

Figure 4.9: Experiment 2-FB after fracture [14]

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4.4.4 One bulb stiffener (1-HP)

(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 4.10: Results - 1-HP stiffener

The plots here show pretty similar results as for the 1-FB and unstiffened test, where the mesh scaling is a bit too strong, especially for the middle integration point here, and pretty similar results compared to using the first integration point. Figure 4.10a and 4.10b shows that fracture occurs a bit before we get a bigger drop in force. The reason for this is that the tip of the stiffener fractures first in that case. The bigger drop is when the test fractures in the plate. This can show that the stiffener is carrying small load compared to the plate.

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4.4.5 Two bulb stiffeners (2-HP)

(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 4.11: Results - 2-HP stiffeners

This is the only test that shows less conservative results for using the first integration point compared to the middle. It is observed from Figure 4.11a that the 50 mm mesh does not predict fracture and the 12 mm mesh predict instability too late. The weld is getting large strain concentration (see Figure 4.12 and is not working as originally intended by easing the transition between the stiffener and the plate. However, comparing the results to what Alsos got [3], and to the experiments, the results for the middle integration point is pretty good. For the larger mesh sizes, the weld elements get pretty thin and long, and therefore takes up the plastic equivalent strain in the model. So, the reason for the first integration point with 30 mm mesh does not capture the fracture, could be because of this large stress concentration happening in the weld.

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Figure 4.12: Plastic equivalent strain concentration in weld elements 2-HP 30 mm mesh size - middle integration point

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5 Low-velocity impact behaviour and failure of stiffened steel plates

These tests are the quasi-static tests from Sindre Hellem Sølvernes’ master thesis in 2015 [16].

The test is also studied in the paper by Gruben et al. in 2017 [17]. Two types of indenters are used, and denoted indenter A and indenter B. The numerical model in this chapter was recieved from David Morin and is the model Sindre made during his work. The model is still described here.

5.1 Geometry

5.1.1 Test specimen

The test specimen consists of a 1250 mm wide and 1375 long plate with 3 mm thickness and six stringers are welded to it with intermittent welds. The throat size is 3 mm and length of the weld is 15 mm. These are placed with a center-to-center distance of 45 mm. The stringers have a height of 65 mm and a thickness of 3 mm. The stringers are an L-shape and the width of the bottom is 15 mm (thickness not included). The geometry of the test specimen is shown in the figure below.

Figure 5.1: (a) Stiffened steel plate seen from the bottom, (b) cross section of test specimen, (c) stiffener cross-section, and (d) length and center-to-center distance of the fillet welds. [17]

5.1.2 Test rig frame

This specimen is set between two frame parts that are bolted together by using 8 M16 bolts.

Teflon sheets are used between the frame and test specimen to reduce friction. Shim plates with a thickness of 8 mm are also used to give extra support for the test specimen. It is also 50 mm cutouts for the stiffeners in the bottom frame. The frame with the test specimen installed, is shown in Figure 5.2

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Figure 5.2: (a) Specimen clamped between bottom and top frame, (b) cross-section in the longitudinal direction, (c) lower support frame side, (d) profile in the width direction, (e) cross section in longitudinal direction, and (f) cross section in width direction. [17]

5.1.3 Indenters

The geometry for indenter A and B, are shown in Figure 5.3. Indenter A has a cylindrical shape with rounded corners with 25 mm radius, and has a length of 350 mm. Indenter B has a hemispherical shape with a radius of 50 mm.

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(a) Indenter A

(b) Indenter B

Figure 5.3: Geometry of the indenters [16]

5.2 Material

The material used for the test specimen is steel grade DOMEX 355 MC E. The material model for the test specimen is the same as used in the previous tests, and described in section 3.1 on page 10, but the material parameters are of course different. These are shown in the following table.

Table 5.1: Material parameters for the stiffened steel plates σ₀[MPa] E [GPa] K[MPa] n ε_plateau ν ρ[tons/m³]

404 210 772 0.173 0.024 0.33 7.85

The material used in the frame is elastic-perfectly plastic with a yield limit of 355 MPa. The material used for the bolts is just elastic with the same Young’s modulus.

5.3 Numerical model

5.3.1 Mesh

This test is run with three different mesh sizes for the test specimen: 3 mm (le/te= 1), 12 mm (le/te = 4) and 30 mm (le/te = 10). The mesh with 3 mm elements is shown in Figure 5.4 and has just an area around the indenter that are set to 3 mm to reduce computational time. S4R elements in Abaqus are used in the plate. The indenters are modelled as discrete rigid with R3D elements with a mesh size of 6 mm. The frame is also modelled using S4R elements, and has a 15 mm mesh. The bolts has a mesh size of 10 mm. A symmetry plane across the middle of the plate is also utilized in the simulations.

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(a) close-up of the mesh close to the indenter (b) Seen from below

(c) Whole model with 3 mm elements Figure 5.4: 3 mm mesh of the stiffened steel plates

5.3.2 Contact

A general contact formulation is also used in these tests. The friction coefficient is also here set to 0.3 in the tangential direction. Small stabilizing forces of 0.01 kN is added for the shim plates to push the plates towards the frame in order to avoid that the parts are set in motion by propagating stress waves. A friction coefficient of 0.04 is adopted between the teflon sheets and the plate. This coefficient between teflon and steel is taken from Engineers Handbook [18].

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5.4 Results

5.4.1 Indenter A

(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 5.5: Results - Indenter A

The results shown for the middle integration point is of course similar to what Sindre got in his simulations using the BWH criterion [16]. Using the first integration point instead, leads to more conservative results and could be good for design where high safety factors are used.

When using the mesh scaling technique, Figure 5.5d (middle integration point) shows a bit too strong scaling, while Figure 5.5b (first integration point) actually shows a small, but a bit too weak scaling, however, it is still more conservative compared to using the middle integration point.

The plate does not fracture for the 30mm mesh without the mesh scaling but when using it, the test specimen fractures first in tip of the stringers below the indenter. The force continues to increase due to this, and fails in the plate where the curve gets a rapid drop in force. All the elements below the indenter fails fast when reaching this point, and drops to zero. This happens much sooner when using the first integration points, as seen on the figure.

For the 3mm mesh, the curve continues in a pretty straight line after the first element fails.

When looking at how the fracture develops (see Figure 5.6), we can see that the fracture develops in a straight line which allows the membrane stresses to continue to be carried in the transverse direction.

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(a) Right before fracture

(b) At 200 mm indentation

Figure 5.6: Plastic equivalent strain before fracture and for 200 mm indentation for 3 mm mesh with BWH using first integration point

The fracture for the 12 mm mesh seems to develop in arbitrary directions after the stringer fails first. The plastic equivalent strain for the 12 mm mesh with the first integration point and mesh scaling right before the fast drop in force is shown in the following figure:

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Figure 5.7: Plastic equivalent stress at 150 mm indentation for 12 mm mesh first integration point with mesh scaling

After this, the elements above the indenter fails fast and the drop in force is observed.

5.4.2 Indenter B

The results for indenter B are shown in Figure 5.8. It is also observed here that when using the middle integration point, the mesh scaling is too strong. For the first integration point, the biggest mesh size (30 mm) with mesh scaling is too weak, but instability occurs at the same spot as the middle integration point. The first integration point also gives more conservative results here, especially for the 3 mm and 12 mm mesh. The sudden drop in force is because of the shape of the indenter, which penetrates the plate and goes through pretty fast after first fracture occurs.

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(c) Middle integration point without mesh scaling (d) Middle integration point with mesh scaling Figure 5.8: Results - Indenter B

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6 Conclusion

The results show more conservative or pretty much the same results for all cases studied, except for the 2-HP case explained in section 4.4.5, when using the first integration point instead of the middle integration point that is intended for BWH. This could be interesting for design where large safety factors are used, but more tests and simulations are needed on this topic.

The mesh scaling technique used in this study shows overall pretty good results for most cases.

However, it seems to give too conservative results for for some cases. The reason for this is prob- ably that the scaling is used for the whole test specimen in all the cases instead of only at places where it is needed, as explained in 2.2.3.

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References

[1] M. Storheim. Structural response in ship-platform and ship-ice collisions [doctoral thesis].

Norwegian University of Science and Technology, 2016.

[2] S. Ehlers et al. Simulating the collision response of ship side structures: A failure criteria benchmark study. International Shipbuilding Progress, 55:127–144, 2008.

[3] H.S. Alsos et al. On the resistance to penetration of stiffened plates, part ii: Numerical analysis. International Journal of Impact Engineering, 36:875–887, 2009.

[4] Abaqus 2016. Dassault syst`emes, 2015.

[5] O.S. Hopperstad, T. Børvik. Materials Mechanics - Part I. 2015.

[6] O.S. Hopperstad, T. Børvik. Impact Mechanics – Part 1: Modelling of plasticity and failure with explicit finite element methods. 2017.

[7] Engineers edge. Von Mises criterion. https://www.engineersedge.com/material_

science/von_mises.htm, Accessed april 2018.

[8] Z. Marciniak et al. Mechanics of sheet metal forming. Butterworth-Heinemann, 2002.

[9] R. Hill. On discontinous plastic states, with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids, 1:19–30, 1952.

[10] J.D. Bressan, J.A. Williams. The use of a shear instability criterion to predict local necking in sheet metal deformation. International Journal of Mechanical Sciences, 25:155–168, 1983.

[11] H.S. Alsos et al. Analytical and numerical analysis of sheet metal instability using a stress based criterion. International Journal of Solids and Structures, 45:2042–2055, 2008.

[12] J. Broekhuijsen. Ductile failure and energy absorption of y-shape test section [master thesis].

The Netherlands: Delft University of Technology, 2003.

[13] M. Storheim et al. A damage-based failure model for coarsely meshed shell structures. In- ternational Journal of Impact Engineering, 83:59–75, 2015.

[14] H.S. Alsos et al. On the resistance to penetration of stiffened plates, part i – experiments.

International Journal of Impact Engineering, 36:799–807, 2009.

[15] T. Wang et al. Finite element analysis of welded beam-to-column joints in aluminium alloy enaw 6082 t6. Finite Elements in Analysis and Design, 44:1–16, 2007.

[16] S. Sølvernes. Impact behaviour of stiffened steel plates [master thesis]. Norwegian University of Science and Technology, 2015.

[17] G. Gruben et al. Low-velocity impact behaviour and failure of stiffened steel plates. Marine Structures, 54:73–91, 2017.

[18] Engineers Handbook. Coefficient of friction.http://www.engineershandbook.com/Tables/

frictioncoefficients.htm, Accessed june 2018.

Finite element modelling of steel bridge structures exposed to ship collisions

MASTER’S THESIS 2018

for

Kevin Mandt Ofstad

Finite element modelling of steel bridge structures exposed to ship collisions

In this project, numerical methods for simulating collisions between large steel structures, e.g. ship against bridge, will be tested and evaluated.

The main tasks of the research project are as follows:

1. To perform a literature review on experimental and numerical studies of collisions between large steel structures, e.g. ships and bridges.

2. To perform a literature review on failure models for large-scale simulations of steel structures 3. To formulate mathematically, implement numerically and verify computationally (for simple

cases) the modified BWH model and the regularized Cockcroft-Latham model.

4. To perform a numerical study of a generic plane-stress problem, namely a sheet under biaxial stretching, in order to compare the accuracy and the mesh sensitivity of the two failure models.

A solid element model of the problem with the Cockcroft-Latham criterion will be used as the baseline case.

5. Perform simulations of existing experiments from the literature of collisions between large steel structures to assess the credibility and robustness of the two failure models.

6. Report the research work.

Odd Sture Hopperstad, Torodd Berstad, David Morin (NTNU)

The report should be written in the style of a scientific article and submitted to the Department of Structural Engineering, NTNU, no later than June ??, 2018.

NTNU, January 15

, 2018

Odd Sture Hopperstad

Contents

List of Figures

List of Tables

1 Introduction

1.1 Background

1.2 Literature review

1.3 Objective and scope

2 Modelling of plasticity and ductile failure

2.1 Plasticity theory

2.2 Bressan-Williams-Hill failure criterion

3 Formability tests of ship construction steel

3.1 Material

3.2 Numerical model

3.3 Results

4 Penetration resistance of stiffened steel panels

4.1 Geometry

4.2 Material

4.3 Numerical model

4.4 Results

5 Low-velocity impact behaviour and failure of stiffened steel plates

5.1 Geometry

5.2 Material

5.3 Numerical model

5.4 Results

6 Conclusion

References