Comparison between different finite element method

Comparison between different finite element method and different finite element software to the stability of cold-formed steel structure.

Contents 1.

Introduction ........................................................................................................................1

2.

Finite element method .......................................................................................................1

3.

Comparison between each method under different softwares ........................................4 3.1

3.2.

3.3.

Use CUFSM4 to simulate C-section ............................................................................4 3.1.1

Simulate C-section without round corner ........................................................6

3.1.2

Simulate C-section with round corner ..........................................................11

Use Abaqus to simulate ............................................................................................17 3.2.1

Simulate C-section without consider geometric imperfection ......................17

3.2.2

Consider the geometric imperfection ............................................................21

3.2.3

Simulate C-section which has round corner...................................................27

Use solidworks to simulate.......................................................................................31 3.2.1

Simulate C-section without round corner ......................................................31

3.2.1

Simulate C-section with round corner............................................................35

Reference .................................................................................................................................39

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1. Introduction: Development of an efficient FE model to represent local buckling as well as combined local/distortional buckling under different boundary conditions of cold-formed steel column is the main objective in this paper. This paper use different softwares to simulate one model to see the difference between different softwares and analyze reason and compare to each other which is better. Elastic buckling analysis is the most commonly used technique to define the shape and distribution of initial imperfections. The lowest Eigenmode is often selected as the shape of the initial geometry of the structure [1]. However, this is not always the case and selecting an appropriate Eigenmode is sometimes very challenging in some boundary conditions since it should represent the buckling modes which are triggered in the member. Kwon and Hancock [2] found that the mode of imperfection had a more significant effect on the behaviour of their sections than the amplitude of initial imperfections. Yu and Schafer [3] used the finite strip software CUFSM to obtain the shape of initial imperfections. In the present study, it was found to be difficult to obtain an appropriate Eigenmode representative of pure local buckling along the constant axial force. It was, therefore, decided to generate the initial geometric imperfections manually with the aid of finite strip software CUFSM [4] and sinusoidal functions rather than using the conventional method of feeding the Eigenmodes from the ABAQUS elastic buckling analysis into the ABAQUS nonlinear analysis. The sensitivity of FE models to the magnitude of initial imperfections has also been addressed in this paper using different CDF values proposed by Schafer and Peköz [5] for imperfection amplitudes.

2. Finite element method: This method is actually part of the approximate methods, but it is the most versatile tool when solving plate exercises. The solving of these problems is based in the hypothesis in the turning of the normal lines to the middle plain and two theories exist. The thin plate theory of Kirchhoff establishes that the normal lines to middle plain stay orthogonal to the deformed form of the middle plain, which allows us to despise the shear strain deformation. The Reissner-Mindlin theory keeps the condition of the straight deformation of the normal line but does not demand the ortogonality after the deformation [6]. The FEM requires discretization in every dimension of the exercise, and so requires many more variables that other approximate methods. If the geometrical and mechanical properties are constant in a direction, like in plates or prismatic structures, in which the transversal section does not vary in the longitudinal direction, the analysis can be simplified. Combining the FEM and Fourier series we can solve this kind of structures. This procedure is known as the Finite Strip Method (FSM). The comparative table between FSM and FEM is shown in the following: Table 1: Comparison between the Finite Element Method and the Finite Strip Method

FEM

FSM

Applicable to any type of geometry, boundary conditions or material variation. Extremely powerful and usable in nearly every case.

In static analysis it is used for structures with two opposite simply supported ends. In dynamic analysis it is used with all boundary conditions and with discrete supports.

Implies a great number of equations and extremely big matrix. Can be very expensive

Usually has a much smaller quantity of equations and matrix are also smaller. This 2

and even impossible to use sometimes because of the demanding computing facilities.

leads to a much shorter computing time to find a solution with nearly the same accuracy.

Large quantity of input data which can lead to mistakes.

Very small amount of input data due to the smaller number of meshing.

Large quantity of output. Normally displacements of all the nodes are listed.

Easier to specify only those nodes which displacements and stresses are required.

Difficult to program and a very big computational requirement.

Due to the reduction in the number of degrees of freedom, the computational requirements are smaller.

For the finite element modeling: Modeling Details and Loading/Boundary Conditions: The cold-formed steel columns (same as beams when only subject to compression) are modeled using 4-node, quadrilateral, shell elements (S4R in ABAQUS). One end of column is constrained and the other ends are: simply-supported, clamped and free. And we will see the effects of different B.C. to the column. Geometric Imperfections: I considered the geometric imperfection in Abaqus. To define an imperfection based on the deformed geometry of a previous static analysis (“Unstable collapse and postbuckling analysis”), specify the result file and step (and, optionally, the increment number) from a previous static analysis. The response of some structures depends strongly on the imperfections in the original geometry, particularly if the buckling modes interact after buckling occurs. Hence, imperfections based on single buckling mode tend to yield nonconservative results. By adjusting the magnitude of scaling factors of various buckling modes, the imperfection sensitivity of the structure can be assessed. Normally, a number of analyses should be conducted to investigate the sensitivity of a structure to imperfections. Structures with many closely spaced eigenmodes tend to be imperfection sensitivity, and imperfections with shapes corresponding to the eigenmode for the lowest eigenvalue may not give the worst case. The imperfect structure will be easier to analyze if the imperfection is large. If the imperfection is small, the deformation will be quite small (relative to the imperfection) below the critical load. The response will grow quickly near the critical load, introducing a rapid change in behavior. On the other hand, if the imperfection is large, the postbuckling response will grow steadily before the critical load is reached. In this case the transition into postbuckled behavior will be smooth and relatively easy to analyze. Material Modeling: Material nonlinearity in the cold-formed steel beams was modeled with von Mises yield criteria and isotropic hardening. Measured stress-strain relations taken from tensile coupons from the beams were employed. All other components were modeled as elastic, with E = 29500 ksi (203 GPa) and ν = 0.3, except for the hot-rolled steel tubes and the loading beam which used an artificially elevated modulus (10E) so that they would effectively act as rigid bodies. Residual stresses were ignored. [7]

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3. Comparison between each method under different simulation software: In this paper, I use the member’s cross-section 600S162-54, boundary condition are: one end is fixed, the other end constrain the displacement in 3 different ways, C-C, C-F, C-S. Uniform pressure applied in the cross-section other than constraint one. On the other hand, set the yield stress is 50ksi as well. 3.1. Use cufsm4 to simulate C-section. In the finite strip method, the structure is only discretized in the cross-section. The other dimension is usually represented using a shape trigonometrical function. In the following figure we can see the axis taken for the exercise. Local coordinates are named with small letters (x-y-z) and will always be associated with the strip element. [8] Displacements are represented with the translation U-V-W and the rotation θ for global displacements and u-v-w and ϕ for local displacements. The subscript p refers to the half-wave number (number of longitudinal terms).

Figure 1: Strip situation

Figure 2: Strip degrees of freedom definition

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Figure 2: Strip stress distribution

The shape functions for the transverse direction are assumed to be the same polynomial function for every boundary condition. On the other hand, in the longitudinal direction, trigonometrical functions are taken. These functions have to satisfy the pre-set boundary conditions. The out of plane displacement will use a shape cubic polynomial function for all boundary conditions. Therefore, the expressions for general displacements are as follows:

Where 𝜇𝑝=𝑝·𝜋 and 𝑝 is the half-wave number. 𝑌𝑚 is the function for the longitudinal direction, which varies depending on the boundary conditions. Boundary conditon Simply-Simply Clamped-Clamped

Shape function

p y a p   y  y Yp  sin  sin a a Yp  sin

Simply-Clamped

Yp  sin Clamped-Free

Clamped-Guided

( p  1)    y p 1 p   y ( ) sin a p a

1 ( p  )   y 2 Yp  1  cos a

1 ( p  )   y p   y 2 Yp  sin  sin a 2a

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A typical signature curve after analysis are shown in the following:

Figure 3: Signature curve

3.1.1 Use cufsm4 to simulate C-section without round corner. (a). Geometry and material property The web depth: 6’’, style is: stud or joist sections, flange width: 1.625’’, mil thickness: 0.054’’, design thickness: 0.0566’’, and assume the lips are both 0.54’’. I build the model with accuracy 10 nodes. Material property: Elastic modulus is: 29500ksi. Poisson’s ratio is 0.3. Shear modulus is 11346.15ksi. Building model in CUFSM 4:

Figure 4: Section details and material property 6

Figure 5: Calculated section properties

After set the material and dimension of cross section, then use the general boundary condition solution type for the half-wave length, because traditional BC only fit for S-S. ①. Clamped-Simply supported B.C. Set 20 eigenvalues and make m=2 means only two term of triangular series will be used for the specified half- wave length in the finite strip method for the C-S boundary condition.

Figure 6: Boundary condition figures for m=2 (C-S)

We just set the yield stress is 50 ksi, then Calculate P,M and B, but just check P part, uncheck the other options because I just assume the cross-section only subject to compression:

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Figure 7: Load condition figures

Open all the base vectors to analyze:

Figure 8: CUFSM4 analysis

Then, we get the results as shown in the following, means the local buckling controls and when applied load is 0.28682*50ksi=14.341ksi, the elastic critical local buckling will occurs.

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Figure 9: CUFSM results for C-S boundary condition, mode 1

②. Clamped-Clamped B.C. This time I use m=4 in the C-C boundary condition. Because after analysis, the result with m=4 can be closer to the Abaqus analysis result without consider geometric imperfection

Figure 10: Boundary condition figures for m=4 (C-C) 9

After analysis, we get the results as shown in the following, means the local buckling controls and when applied load is 0.28682*50ksi=14.341ksi, the elastic critical local buckling will occurs.

Figure 11: CUFSM results for C-C boundary condition, mode 1

③. Clamped-Free B.C. This time I use m=1 in the C-F boundary condition. Because under this boundary condition, it seems like the simply supported B.C., relative weak than other two B.C., so use minimum m value.

Figure 12: Boundary condition figures for m=1 (C-F) 10

After analysis, we get the results as shown in the following, means this time global buckling controls and when applied load is 0.063387*50ksi=3.17ksi, the elastic critical local buckling will occurs.

Figure 13: CUFSM results for C-F boundary condition, mode 1

3.1.2 Use cufsm4 to simulate C-section with round corner. This time use web depth 6’’ which the same as without round corner in section 3.1.1, style is: stud or joist sections, flange width the same as well: 1.625’’, mil thickness: 0.054’’, design thickness: 0.0566’’, and assume the lips are both 0.54’’. But, the connection between 1.Flange and web. 2. Flange and lips becomes with a 0.25’’ radius round corner. Material property: Elastic modulus is: 29500ksi. Poisson’s ratio is 0.3. Shear modulus is 11346.15ksi. I building model in CUFSM 4 and the cross-section details and property are as shown in the following:

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Figure 14: Section details and material property

Figure 15: Calculated section properties

After set the material and dimension of cross section, then use the general boundary condition solution type for the half-wave length, because traditional BC only fit for S-S and C-F (m=1).

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①. Clamped-Simply supported B.C. Set 20 eigenvalues and make m=2 means only two term of triangular series will be used for the specified half- wave length in the finite strip method for the C-S boundary condition.

Figure 16: Boundary condition figures for m=2 (C-S)

We just set the yield stress is 50 ksi, then Calculate P,M and B, but just check P part, uncheck the other options because I just assume the cross-section only subject to compression:

Figure 17: Load condition figures

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Open all the base vectors to analyze:

Figure 18: CUFSM4 analysis

Then, we get the results as shown in the following, means the local buckling controls and when applied load is 0.30151*50ksi=15.0755ksi, the elastic critical local buckling will occurs.

Figure 19: CUFSM results for C-S boundary condition, mode 1 14

②. Clamped-Clamped B.C. This time I use m=4 in the C-C boundary condition. Because after analyze and comparison, it is more close to Abaqus analysis result without consider geometric imperfection.

Figure 20: Boundary condition figures for m=4 (C-C)

After analysis, we get the results as shown in the following, means the local buckling controls and when applied load is 0.30556*50ksi=15.278ksi, the elastic critical local buckling will occurs.

Figure 21: CUFSM results for C-C boundary condition, mode 1 15

③. Clamped-Free B.C. This time I use m=1 in the C-F boundary condition. Because under this boundary condition, it seems like the simply supported B.C., relative weak than other two B.C., so use minimum m value.

Figure 22: Boundary condition figures for m=1 (C-F)

After analysis, we get the results as shown in the following, means this time global buckling controls and when applied load is 0.059504*50ksi=2.9752ksi, the elastic critical local buckling will occurs.

Figure 23: CUFSM results for C-F boundary condition, mode 1 16

Then I summary the result in following table, with make all the other conditions are same and only change the cross-section. Load factor

C-S B.C.

C-C B.C.

C-F B.C.

C-section without round corner

0.28682

0.28946

0.063387

C-section with round corner

0.30151

0.30151

0.059504

Difference (%)

5.12%

4.16%

-6.13%

From the table, I can make a conclusion that section with round corner can improve load factor a little under C-S and C-C boundary conditions and reduce a little under C-F B.C.

3.2. Use ABAQUS to simulate. 3.2.1. Without consider the geometric imperfection: Materials: E=29500ksi, v=0.3. Dimension are same as part2, thickness is 0.0566’’.Length is 96’’. Boundary condition: Set BC for stress perturbation and buckling mode calculation. Property: Set the elastic and plastic material’s property as shown below (Use elastic perfectly plastic model, do not consider strain-hardening now):

Figure 24: material property setting in ABAQUS

Step: set the analysis step as shown in the following

Figure 25: Buckling analysis step in ABAQUS 17

Load and boundary condition: Apply load equal to the yield stress, means apply 50ksi to the cross-section. For the thin-wall structures, I use an equivalent load application method, treat the uniform load as a concentrate load which apply on the center of gravity of cross section which is not encased, as shown in the following. So P=-50ksi*0.554=-27.7k, negative value because the force is pressure. For the fixed end, I use ENCASTRE (U1=U2=U3=UR1=UR2=UR3=0) in ABAQUS.

Figure 26: Load and boundary condition settings in ABAQUS

Mesh: Use the mesh kind as shown in following, the shell element S4R is employed throughout.

Figure 27: Mesh of the member

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①. Clamped-Simply supported B.C. For the simply-supported end, apply the pressure and constrain displacement in x and y direction

Figure 28: B.C. setting for C-S condition

Then we get the mode 1 outcome in the following figure. The color bar means resultant displacement of this member. Eigenvalue means the critical buckling force are eigenvalue*applied force in Abaqus. Mode shape 1: eigenvalue=0.28998, buckling stress=50*0.28998=14.5ksi.

Figure 29: Abaqus result for C-S boundary conditions (mode 1)

After the buckling analysis, we consider geometric imperfection in this time, because one end of column is fixed and the other end of column is simply-supported.

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②. Clamped-Clamped B.C. All the conditions are same as C-S situation except for the B.C. of where load applies. This time, constrain x,y direction of displacement and 3 direction of rotations. But only accept displacement in z-direction.

Figure 30: B.C. setting for C-C condition

Figure 31: Abaqus result for C-C boundary conditions (mode 1)

Mode shape 1: eigenvalue=0.29038, buckling stress=50*0.29038=14.52ksi.

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③. Clamped-Free B.C. This time delete the boundary condition at where load applies. After that, we can get the outcomes as shown in the following

Figure 32: Abaqus result for C-F boundary conditions (mode 1)

Mode shape 1: eigenvalue=0.0607, buckling stress=50*0.0607=3.035ksi. The outcomes comparison:

C-C

C-S

C-F

CUFSM4

0.28946

0.28682

0.063387

Abaqus

0.29038

0.28998

0.0607

Difference (%)

+0.32%

+1.1%

-4.4%

3.2.2. Consider the geometric imperfection: ①. Clamped-Simply supported B.C. I consider the geometric imperfection with apply a concentrate load in horizontal direction with a very small value (0.0001kip) in each conditions to see what happens. I create a new step with consider geometric imperfection instead of the older one and set the same incrementation as shown in the following in the all three B.C..

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Figure 33: Geometric imperfection consideration

In C-S boundary condition, I apply the horizontal load very near the simply-supported side. Because it is simply-supported, so I can’t apply load in the exactly cross-section because the x-direction’s displacement has been restrained. I apply load as shown in the following:

Figure 34: Geometric imperfection application method for C-S.

After apply the geometric imperfection, we get the results as shown in the following, we can see it converges when increment=47, so the load factor is 0.3598 at this condition, means buckling stress is 50ksi*0.3598=17.99ksi

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Figure 35: Abaqus result with consideration of Geometric imperfection for C-S

②. Clamped-Clamped supported B.C. In C-C boundary condition, I apply the horizontal load very near the clamped side which load applies. Because it is clamped, so I can’t apply load in the exactly cross-section because the xdirection’s displacement has been restrained. I apply load as shown in the following:

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Figure 36: Geometric imperfection application method for C-C.


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Figure 37: Abaqus result with consideration of Geometric imperfection for C-C

③. Clamped-Free supported B.C. In C-F boundary condition, I apply the horizontal load in the exactly free end as shown in the following:

Figure 38: Geometric imperfection application method for C-F.

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Figure 39: Abaqus result with consideration of Geometric imperfection for C-F

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I summary results in following table. C-C

C-S

C-F

Abaqus without consider geometric imperfection

0.29038

0.28998

0.0607

Abaqus with consider geometric imperfection

0.5863

0.3598

0.0596

Difference (%)

+102%

+24%

-1.8%

From the table above, I can make a conclusion that the C-C B.C. has the most significant effects with consider geometric imperfection. 3.2.3. Simulate C-section which has round corner: In this situation, make both property, load, step, interaction, load and mesh same, only change the cross section as shown in the following to see what will happen? Materials: E=29500ksi, v=0.3. Dimension are same as part2, thickness is 0.0566’’.Length is 96’’. Boundary condition: Set BC for stress perturbation and buckling mode calculation. Property: Set the elastic and plastic material’s property as shown below (Use elastic perfectly plastic model, do not consider strain-hardening now):

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Figure 40: C-section with round corner

For the simply-supported end, apply the pressure and constrain displacement in x and y direction

Figure 41: B.C. setting for C-S condition

Then we get the mode 1 outcome in the following figure, because for buckling, we only consider the minimum buckling mode. The color bar means resultant displacement of this member. And I change the deformation scale factor to 3 just in order to see the deformation better. Eigenvalue means the critical buckling force are eigenvalue*applied force in Abaqus. For mode shape 1: 28

eigenvalue=0.30919, buckling stress=50*0.30919=15.46ksi.

Figure 42: Abaqus result for C-S boundary conditions (mode 1)

②. Clamped-Clamped B.C. All the conditions are same as C-S situation except for the B.C. of where load applies. This time, constrain x,y direction of displacement and 3 direction of rotations. But only accept displacement in z-direction.

Figure 43: B.C. setting for C-C condition

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Figure 44: Abaqus result for C-C boundary conditions (mode 1)

For mode shape 1: eigenvalue=0.30905, buckling stress=50*0.30905=14.453ksi. ③. Clamped-Free B.C. This time delete the boundary condition at where load applies. After that, we can get the outcomes as shown in the following

Figure 45: Abaqus result for C-F boundary conditions (mode 1)

For mode shape 1: eigenvalue=0.05077, buckling stress=50*0.05077=2.5385ksi.

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The outcomes comparison: Between CUFSM4 result and Abaqus result: C-C

C-S

C-F

CUFSM4

0.30556

0.30151

0.059504

Abaqus

0.30905

0.30919

0.05077

Difference (%)

+1.13%

+2.48%

-17.2%

Abaqus result between C-section with round corner and without round corner: C-C

C-S

C-F

Without round corner

0.29038

0.28998

0.0607

With round corner

0.30905

0.30919

0.05077

Difference (%)

+6.43%

+6.624%

-16.4%

From the table, I can make a conclusion that section with round corner can improve load factor a little under C-S and C-C boundary conditions and reduce a little under C-F B.C. It is the same outcome as analysis by the CUFSM4.

3.3 Use Solidworks to simulate. 3.3.1 Simulate C-section without round corner In Solidworks, I design the member’s cross-section as shown in the following, and the length of member is 96 inches:

Figure 46: cross-section of member

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Figure 47: material property of member Mesh: I use standard mesh in Solidworks

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Figure 48: mesh in Solidworks

①. Clamped-Simply supported B.C. Mode shape 1: Load factor is 0.28106, so the critical buckling stress is: 50ksi*0.28106=14.05ksi

Figure 49: Solidworks result for C-S B.C. (mode 1)

②. Clamped-clamped B.C. Mode shape 1: Load factor is 0.51532, so the critical buckling stress is: 50ksi*0.51532=25.766ksi

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Figure 50: Solidworks result for C-C B.C. (mode 1)

③. Clamped-free B.C. Mode shape 1: Load factor is 0.034889, so the critical buckling stress is: 50ksi*0.034889=1.7445ksi

Figure 51: Solidworks result for C-F B.C. (mode 1)

I summary results between Solidworks results and Abaqus results with consideration of geometric imperfection. C-S

C-C

C-F

Solidworks result

0.28106

0.51532

0.0349

Abaqus with consider geometric imperfection

0.3598

0.5863

0.0596

Difference (%)

+21.9%

+13.77%

-70.8%

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3.3.2 Simulate C-section without round corner I design the member’s cross-section as shown in the following, and the length of member is 96 inches, all the parameters are the same as before only change the shape of cross-section:

Figure 52: cross-section of member Mesh: This time, I use curvature-based mesh in Solidworks

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Figure 53: mesh in Solidworks

①. Clamped-Simply supported B.C. Mode shape 1: Load factor is 0.46571, so the critical buckling stress is: 50ksi*0.46571=23.29ksi

Figure 54: Solidworks result for C-S B.C. (mode 1)

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②. Clamped-clamped B.C. Mode shape 1: Load factor is 0.7393, so the critical buckling stress is: 50ksi*0.7393=36.97ksi

Figure 55: Solidworks result for C-C B.C. (mode 1)

③. Clamped-free B.C. Mode shape 1: Load factor is 0.059586, so the critical buckling stress is: 50ksi*0.059586=2.98ksi

Figure 56: Solidworks result for C-F B.C. (mode 1) 37

I summary results between Solidworks results between with round corner and without round corner: C-S

C-C

C-F

Solidworks without round corner

0.28106

0.51532

0.0349

Solidworks result with round corner

0.46571

0.7393

0.0596

Difference (%)

+65.7%

+43.5%

70.8%

Base on the analysis result from solidworks, I can make a conclusion that the round corner will increase the load capacity of member in each 3 B.C.s.

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Reference [1]. Moreyra ME, Peköz T, Finite element studies on lipped channel flexural members. In: Proceedings of the twelfth international specialty conference on cold-formed steel structures. University of Missouri-Rolla, St Louis, MO; 1994.p. 57-74 [2]. Y.B. Kwon, G.J. Hancock, Post-buckling analysis of thin-walled channel sections undergoing local and distortional buckling [3]. C. Yu, B.W. Schafer, Simulation of cold-formed steel beams in local and distortional buckling with applications to the direct strength method [4]. CUFSM4 Version 4.03 by Ben Schafer. Department of Civil Engineering, Johns Hopkins University; 2006. 〈http://www.ce.jhu.edu/bschafer/cufsm/〉 [5]. B.W. Schafer, T. Peköz, Computational modelling of cold-formed steel: characterizing geometric imperfections and residual stresses [6]. JOVICEVIC, J and OÑATE, E: “Analysis of Beams and Shells Using a Rotation” [7]. C. Yu, B.W. Schafer, Finite Element Modeling of Cold-formed Steel Beams Validation and Application. [8] Alejandro Lifante Mira, Non-linear geometric elastic analysis of thin-walled beam by the finite strip element method.

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