a study on the development of equivalent beam analysis

Journal of Mechanical Science and Technology 25 (9) (2011) 2401~2411

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0608-4

A study on the development of equivalent beam analysis model on pedestrian

protection bumper impact† Dong-Kyou Park1,* and Chang-Doo Jang2

1Department of Mechanical and Automotive Engineering, Daeduk University, Daejeon, 305-715, Korea 2RIMSE, Department of Naval Architecture & Ocean Engineering, Seoul National University, Seoul, 151-742, Korea

(Manuscript Received January 6, 2011; Revised May 27, 2011; Accepted June 3, 2011)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract This paper presents a dynamically equivalent beam analysis model on pedestrian protection bumper impact instead of a non-linear fi-

nite element impact analysis method. Equivalent beam analysis model was developed by substituting the femur and tibia for dynamically equivalent Euler beam. Dynamically equivalent forces of bumper beam, upper stiffener and lower stiffener are found by a finite element analysis results and applied to the Euler beam model of lower legform impactor. This equivalent beam analysis model was used to obtain a bending angle of lower legform impactor by using finite element beam theory. Peak acceleration of the tibia was obtained by develop-ing an approximate acceleration equation. A linear interpolation of non-linear finite element analysis results considering the dimension variation of bumper beam factors affecting the acceleration was used to get an approximate acceleration equation. The accuracy of this simple analysis model was tested by comparing its results with those of the non-linear finite element analysis. Tested bumper beam types were press type beam and roll forming beam used widely in the current car bumpers. The differences of maximum acceleration of the tibia between the two models did not exceed 10% and the bending angle did not exceed 20%. This accuracy is enough to be used in the early stage of bumper beam design to check the bumper pedestrian performance quickly. Use of equivalent beam analysis model is ex-pected to reduce the analysis time with respect to the non-linear finite element analysis significantly.

Keywords: Pedestrian protection bumper impact; Dynamically equivalent Euler beam; Simple analysis model; Press type bumper beam; Roll forming

bumper beam; Approximate acceleration equation ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Currently, cars produced for European export must satisfy the first stage of pedestrian protection regulation, and from 2012 they will have to satisfy the second stage of this regula-tion. The criteria of the second stage [1] are as follows:

Impact velocity : 40 kph 1) Peak acceleration of the tibia < 170G 2) Bending angle of the legform < 19° 3) Lateral shear displacement at the knee < 6 mm In the domestic market, cars must meet the pedestrian pro-

tection performance of bumper impact from 2013. The criteria of domestic regulation are the same as the second stage of the European regulation. To analyze pedestrian protection bumper impact performance, a non-linear finite element impact analy-sis is widely used. But finite element analysis takes much time

to get a result. To this end, many studies have been performed to reduce

the bumper impact analysis time. Park et al. [2] and Yim et al. [3] proposed a bumper impact analysis technique by using intermediate response surface modeling (IRSM) and com-pared its results with non-linear finite element analysis. Kim et al. [4] developed a mathematical expression satisfying re-quirements for both bumper barrier and pole impact. Kim et al. [5-7] developed a simple analysis model for pendulum bumper impact by using the energy method and compared its results with non-linear finite element analysis.

In this paper, we propose a dynamically equivalent beam analysis model of pedestrian protection bumper impactor. Dynamically equivalent forces of bumper beam and bending resisting stiffener are found by ANSYS [8] analysis results and applied to the Euler beam model of lower legform impac-tor. This equivalent beam analysis model was used to obtain a bending angle of lower legform impactor by using finite ele-ment beam theory. The peak acceleration of the tibia was ob-tained by developing an approximate acceleration equation. A linear interpolation of LS-DYNA [9] analysis results consider-ing the dimension variation of bumper beam factors affecting

† This paper was recommended for publication in revised form by Associate EditorJeong Sam Han

*Corresponding author. Tel.: +82 42 866 0236, Fax.: +82 42 866 0389 E-mail address: [email protected]

© KSME & Springer 2011

2402 D.-K. Park and C.-D. Jang / Journal of Mechanical Science and Technology 25 (9) (2011) 2401~2411

affecting the acceleration was used to get an approximate ac-celeration equation.

The accuracy of this simple analysis model was tested by comparing its results with those of non-linear finite element analysis. Tested bumper beam types were press type beam and roll forming beam used widely in current car bumpers. The relative error of acceleration between the simple analysis model and LS-DYNA did not exceed 10% and for the bending angle did not exceed 20%.

Use of this simple analysis model reduces the analysis time significantly with respect to the non-linear finite element anal-ysis.

2. Equivalent simple analysis model of pedestrian

bumper impact

2.1 Overview of pedestrian protection impact

Pedestrian protection impact includes child headform im-pact, adult headform impact, upper legform impact and lower legform impact. The diameter of the child headform is 165 mm and its weight is 3.5 kg. The diameter of the adult head-form is 165 mm and its weight is 4.8 kg. Impact velocity of child and adult headform is 35 kph. Headform impact criterion is HIC (head injury criteria) value. This HIC value should be below 1000 for child headform and adult headform. Among these impact conditions, lower legform impact is related to the car bumper system. The human leg is modeled by lower leg-form impactor, and this impactor is impacted to the bumper system by 40 kph. Fig. 1 shows the configuration of pedes-trian protection bumper impact.

Lower legform impactor for pedestrian bumper impact is composed of femur and tibia as shown in Euro-NCAP pedes-trian testing protocol [1]. A joint ligament between femur and tibia is used to get the bending angle of the impactor. Acceler-ometer is attached to tibia. The criteria of lower impactor are for peak acceleration, bending angle and shear displacement at knee. Usually, shear displacement at knee is within the criteria. Peak acceleration and bending angle are considered in this paper. Its weight is 13.4 kg. The overview of lower legform impactor is shown in Fig. 2.

2.2 Simple analysis model of lower legform impactor

Femur and tibia of lower legform impactor is substituted by beam element, and linear spring elements are used to support the beam at the joint and the center of femur and tibia. The properties of femur and tibia beam element are listed in Table 1.

Bumper beam impact force is substituted by equivalent dis-tributed force, and the impact forces of upper and lower stiff-ener are substituted by equivalent concentrated forces. Con-figuration of equivalent forces is shown in Fig. 3.

A simple impactor model for calculating equivalent forces is shown in Fig. 4. The distance between the center of the tibia

Fig. 1. Configuration of pedestrian protection impact.

Table 1. Properties of femur and tibia beam element.

Young’s Modulus (E) 207000 N/mm2

Area (A) 3847 mm2

Moment of inertia (IX) 1.178×106 mm4

Moment of inertia (IY) 1.178×106 mm4

Fig. 2. Lower legform impactor.

Fig. 3. Configuration of equivalent forces.

D.-K. Park and C.-D. Jang / Journal of Mechanical Science and Technology 25 (9) (2011) 2401~2411 2403

0, =+ jijib σ

and impactor joint is 235 mm and it is a dimension of finite element lower legform impactor. The distance between the center of the femur and impactor joint is 216 mm and it is also a dimension of finite element impactor. Tibia length is 494 mm and femur length is 432 mm. The distance between upper stiffener and impactor joint is 144 mm and the distance be-tween lower stiffener and impactor joint is 194 mm. Springs are used to support the joint and the centroids of femur and tibia. Spring type is a linear elastic spring and its stiffness is determined by adjusting with nonlinear finite element analysis result in ANSYS program. Spring stiffness at femur and tibia is 1000 N/mm and spring stiffness at joint is 200 N/mm.

Pedestrian protection bumper impact is composed of six cases by the combination of bumper beam, bumper foam and upper and lower stiffener. Six cases of pedestrian protection bumper impact are listed in Table 2 and corresponding items are shown in Fig. 5. Equivalent forces are found by changing these forces for six cases until getting the same LS-DYNA results in ANSYS model. These equivalent forces are applied

to the simple beam model of lower legform impactor. These forces are calculated for a press type bumper beam and a roll forming bumper beam. By these equivalent forces, the bent configuration of lower legform is shown in Fig. 6.

2.3 Simple analysis model for impactor bending angle

Finite element beam model incorporating the dynamically equivalent forces is used to calculate the bending angle of the lower legform.

Governing equation is expressed in Eq. (1) as follows:

(1)

where bi is body fore andσij,j is stress tensor. Displacement condition is expressed in Eq. (2) as follows:

( , ) ( ),

0,( , ) ( )

u x z z xvw x z w x

β=−=

= (2)

where u is the deformation in the x-axis direction, v is the deformation in the y-axis direction, w is the deformation in the z-axis direction and βis the rotation angle of beam cross section.

Strain-displacement relation is expressed in Eq. (3) as fol-lows:

x

u z zx x

βε κ∂ ∂= = − ≡ −∂ ∂

, (3)

xz

w u wx z x

γ β∂ ∂ ∂= + = −∂ ∂ ∂

where ε is the axial strain, γ is the shear strain and κ is the curvature of beam cross section.

Stress-strain relation is expressed in Eq. (4) as follows:

,x x

xz xz

EkG

σ ετ γ

=

= (4)

Table 2. Cases of pedestrian protection bumper impact.

Case 1 With foam + bumper beam + upper/lower stiffener

Case 2 With foam + bumper beam + lower stiffener

Case 3 With foam + bumper beam

Case 4 No foam + bumper beam + upper/lower stiffener

Case 5 No foam + bumper beam + lower stiffener

Case 6 No foam + bumper beam

Fig. 4. Beam model of lower legform impactor (initial impact).

Fig. 5. Configuration of pedestrian bumper system.

Fig. 6. Beam model of lower legform impactor (after impact).


where E is Young’s modulus, G is the shear modulus, τ is the shear stress and k is the shear coefficient (for a rectangular section k=5/6).

Total internal virtual work is expressed in Eq. (5) as fol-lows:

( )0

12

L

x x xz xzU A dxε σ γ τ= +∫ .

(5)

Incorporation Eqs. (1)-(5), variation of total internal virtual

work is expressed in Eq. (6) as follows:

( )10 0

L L w wU EI dx kGA dxx x x xβ βδ β β∂ ∂ ∂ ∂⎛ ⎞⎛ ⎞= + − −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

∫ ∫ .

(6)

Total external virtual work and its variation are expressed in

Eq. (7) as follows:

( )0

LW wp x dx= ∫ (7) ( ) ( )1

0

LW wp x dxδ = ∫ . By the principle of virtual work, total potential variation is

expressed in Eq. (8) as follows:

( ) ( ) ( )

( )

1 1

0 0 0

0

L L L

U W

w wEI dx kGA dx wp x dxx x x x

δ π δ

β β β β

= −

∂ ∂ ∂ ∂⎛ ⎞⎛ ⎞= + − − −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠=

∫ ∫ ∫

(8) The impactor beam model can be modeled by Euler-

Bernoulli beam without distributed forces. The displacement equation is expressed in Eq. (9) as follows:

4

40d w

dx= .

(9)

The general expression of displacement is shown in Eq.

(10) as follows: ( ) 2 3

1 2 3 4w x x x xα α α α= + + + . (10) The nodal displacement value is expressed in Eq. (11) as

follows:

' '1 2 3 4(0) , (0) , ( ) , ( ) .w a w a w L a w L a= = = = (11)

By using Eqs. (10) and (11), the general solution of dis-

placement equation is expressed in Eq. (12) as follows:

( )1

2 3 2

3

4

1w x x x x

αααα

⎧ ⎫⎪ ⎪⎪ ⎪=⎡ ⎤⎨ ⎬⎣ ⎦⎪ ⎪⎪ ⎪⎩ ⎭

( )( )( )( )

1

2 32 3

2

01 0 0 000 1 0 0

110 1 2 3

ww

x x xw LL L Lw LL L

− ⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥ ′⎪ ⎪⎢ ⎥=⎡ ⎤ ⎨ ⎬⎣ ⎦⎢ ⎥ ⎪ ⎪

⎢ ⎥ ′⎪ ⎪⎣ ⎦ ⎩ ⎭

(12)

2

21 3x

L= −

( )( )( )( )

3 2 3 2 3 2 3

3 2 2 3 2

00

2 2 3 2

wwx x x x x x xxw LL L L L L L Lw L

⎧ ⎫⎪ ⎪′⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪+ − + − − + ⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎪ ⎪′⎪ ⎪⎩ ⎭

.

By substituting Eq. (12) into Eq. (8), finite element equation

is expressed in Eq. (13) as follows:

0 0

L LT TEI dx pdx=∫ ∫B B d N (13)

where

2 3 2 3 2 3 2 3

2 3 2 2 3 21 3 2 2 3 2x x x x x x x xx

L L L L L L L L⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= − + − + − − +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

N

[ ]2

2 2 3 2 2 3 2

6 4 6 6 12 2 612d x x x xdx L L L L L L L L

⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − = − − − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦B N

. Therefore, element stiffness matrix is given as Eq. (14) as

follows:

2 2

3

2 2

12 6 12 66 4 6 212 6 12 6

6 2 6 4

e

L LL L L LEI

L LLL L L L

−⎡ ⎤⎢ ⎥−⎢ ⎥=− − −⎢ ⎥⎢ ⎥−⎣ ⎦

K

.

(14)

By using this element stiffness matrix, the nodal displace-

ment of impactor beam element is expressed in Eq. (15) as follows:

1[ ] [ ] [ ]e

eqx K F−= ⋅ (15)

where Feq is the equivalent forces of bumper beam, upper stiffener and lower stiffener.

From these nodal displacements, bending angle of impactor can be calculated by trigonometric function. 2.4 Simple analysis model for impactor acceleration

For a simple analysis model of impactor acceleration, a sen-sitivity analysis has been done to find the main factors affect-ing the impactor acceleration. Two types of bumper beam are considered: press type beam and roll forming beam. Dimen-sional factors affecting acceleration are shown in Fig. 7.

For six cases of pedestrian bumper impact, sensitivity analysis has been done for dimensional factors by using LS-DYNA analysis results. Sensitivity analysis model is made for press type bumper beam and roll forming bumper beam as shown in Figs. 8 and 9.

For case 1 (with foam + bumper beam + upper/lower stiff-

.


ener), sensitivity results for D and H values are shown in Ta-ble 3. Pedestrian analysis has been done according to the bumper beam section types with different D and H values. As H value increases, acceleration value also increases. It is found that H value is strongly related to the acceleration. For the sensitivity of the upper and lower stiffener position, the accel-eration value is almost the same according to the change of lower stiffener position.

But acceleration value is changed according to the variation of upper stiffener position. Sensitivity results for the upper and lower stiffener position are shown in Table 4 and related di-

mensions are shown in Fig. 10. For case 1 (with foam + bumper beam + upper/lower stiffener), H value and upper stiffener position are main factors affecting on acceleration.

For case 2 (with foam + bumper beam + lower stiffener), sensitivity results for D and H values are shown in Table 5. As in case 1, the acceleration value increases as long as H value increases. Lower stiffener position does not affect the accel-eration as in Table 4. It is found that the H value is only re-lated to the acceleration in case 2.

For case 3 (with foam + bumper beam), sensitivity results for D and H values are shown in Table 6. Acceleration value increases as long as the H value increases. It is found that the H value is related to the acceleration value.

For case 4 (without foam + bumper beam + upper/lower stiffener), sensitivity results are shown in Table 7. In case 4, acceleration is independent on D and H values. For the sensi-tivity of the upper and lower stiffener position, the accelera-tion value changes according to the variation of lower and

Table 3. Sensitivity results for D and H values for case1.

Factor Press type beam Roll forming beam

D 54 mm 52 mm

H 60 mm 110 mm

Acceleration 320G 403G

Fig. 7. Factors affecting on acceleration of pedestrian impactor.

Fig. 8. LS-DYNA sensitivity analysis model of press type beam.

Fig. 9. LS-DYNA sensitivity analysis model of roll forming beam.

Table 4. Sensitivity results for upper and lower stiffener position for case1.

No. Factor Press type beam

Roll forming beam

① Upper stiffener (144 mm) Lower stiffener (192 mm) 320G 403G

② Upper stiffener (144 mm) Lower stiffener (100 mm) 321G 405G

③ Upper stiffener (100 mm) Lower stiffener (150 mm) 350G 433G

④ Upper stiffener (144 mm) Lower stiffener (150 mm) 320G 403G



D 54 mm 52 mm

H 60 mm 110 mm




D 54 mm 52 mm

H 60 mm 110 mm


Fig. 10. Dimensions of upper stiffener and lower stiffener.


upper stiffener position as shown in Table 8. As the lower stiffener position is only changed, acceleration is varied. Ac-celeration is also varied when upper stiffener position is changed, excluding the acceleration change by the variation of lower stiffener position. It is found that upper and lower stiff-ener positions are main factors in case 4.

For case 5 (without foam + bumper beam + lower stiffener), sensitivity results are shown in Table 9. Acceleration is in-creased as long as the H value is increased. As in case 2, lower stiffener position does not affect the acceleration. It is found that the H value is the main factor in case 5.

For case 6 (without foam + bumper beam), sensitivity re-sults are shown in Table 10. Acceleration value is increased as long as the H value is increased. It is found that the H value is the main factor in case 6.

By sensitivity analysis results from case 1 to case 6, main factors affecting pedestrian impact acceleration can be sum-marized as in Table 11.

Acceleration equation is obtained by using linear interpola-tion of sensitivity analysis results. The expanded rate of bum-per foam is also considered by using the foam factor for 30 times and 40 times foams.

For case 1, a simple expression of acceleration is obtained by using the changing rate of acceleration depending on main factors. As in Table 3, acceleration increment is 83 as long as the increment of the H value is 50. Therefore, the acceleration rate for the H value is 1.667. As in Table 4, acceleration in-crement is 30 as long as the decrement of upper stiffener posi-tion is 44. Therefore, acceleration rate for upper stiffener posi-tion is 0.682. By using these two rates, acceleration is ex-pressed as in Eq. (16) as follows:

320 + (H - 60)×1.667 + (144 - UP)×0.682. (16) Eq. (16) is based on 30 times foam. In pedestrian protection

impact, 30 and 40 times bumper foam are widely used, and a factor for 40 times foam is calculated as in Table 12.

Average foam rate factor is 0.755. The bumper foam thick-ness factor is also considered as in Table 13. Average bumper foam thickness factor is expressed as like Eq. (17).

1.0 - (F_thk - 50)×0.001 - 0.01 (17)

where F_thk is the bumper foam thickness.

Average bumper foam thickness factor is expressed as in Eq. (18) as follows:

1.0 - (F_thk - 50)×0.001 - 0.01 (18)



D 54 mm 52 mm

H 60 mm 110 mm


Table 8. Sensitivity results for upper and lower stiffener position for case4.

Factor Press type beam

Roll forming beam

Upper stiffener (144 mm) Lower stiffener (192 mm) 221G 289G






D 54 mm 52 mm

H 60 mm 110 mm




D 54 mm 52 mm

H 60 mm 110 mm


Table 11. Factor analysis result on effectiveness for deceleration.

No Combinations Main factors

Case 1 With foam + bumper beam + upper/lower stiffener

H value & Upper stiffener pos.

Case 2 With foam + bumper beam + lower stiffener H value

Case 3 With foam + bumper beam H value

Case 4 No foam + bumper beam + upper/lower stiffener

Upper stiffener pos. & Lower stiffener pos.

Case 5 No foam + bumper beam + lower stiffener H value

Case 6 No foam + bumper beam H value

Table 12. Bumper foam rate factor for case 1.

Expanded value Press type beam Roll forming beam

30 times 320G 378G

40 times 251G 275G

Foam rate factor 0.785 0.726

Table 13. Bumper foam thickness factor for case 1.

Press type Roll forming Foam thickness Accel. Factor Accel. Factor

Average factor

50 mm 251G 1.0 275G 1.0 0.99

60 mm 251G 1.0 269G 0.979 0.98

70 mm 251G 1.0 265G 0.964 0.97

80 mm 251G 1.0 263G 0.958 0.96


where F_thk is the bumper foam thickness. Therefore, a simple expression of acceleration for case 1 is

finally expressed in Eq. (19) as follows: [{320 + (H - 60)×1.667 + (144 - UP)×0.682}×F_times] ×{1.0 - (F_thk - 50)×0.001 - 0.01} (19)

where F_times is 1.0 for 30 times foam, 0.755 for 40 times foam and F_thk is the bumper foam thickness.

For case 2, as in Table 5, acceleration increment is 83 as long as the increment of the H value is 50. Therefore, the ac-celeration rate for the H value is 1.667. Factor for 40 times foam is calculated as in Table 14.

Average foam rate factor is 0.667. Factor for bumper foam thickness is shown in Table 15. Average bumper foam thick-ness factor is expressed in Eq. (20) as follows:

1.0 - (F_thk - 50)×0.004 - 0.01. (20) Therefore, a simple expression of acceleration for case 2 is

finally expressed in Eq. (21) as follows: [{270 + (H - 60)×1.667×F_times]×{1.0 - (F_thk - 50)×

0.004 - 0.01} (21)

where F_times is 1.0 for 30 times foam, 0.667 for 40 times foam and F_thk is the bumper foam thickness.

For case 3, as in Table 6, acceleration increment is 56 as long as the increment of the H value is 50. Therefore, accel-eration rate for the H value is 1.12. Factor for 40 times foam is calculated as in Table 16. Average foam rate factor is 1.715. A bumper foam thickness factor is shown in Table 17. Factor difference between press type beam and roll forming beam is large, and two types of bumper foam thickness factor are used for each beam type as Eqs. (22) and (23).

1.0 - (F_thk - 50)×0.009 (22)

where F_thk is the bumper foam thickness of press type beam.

1.0 - (F_thk - 50)×0.02 (23)

where F_thk is the bumper foam thickness of roll forming beam.

Therefore, a simple expression of acceleration for case 3 is finally expressed in Eq. (24) as follows:

[{313 + (H - 60)×1.12×F_times]×{1.0 - (F_thk - 50)×

Beam_type} (24)

where F_times is 1.0 for 30 times foam, 1.715 for 40 times foam and Beam_type is 0.009 for press type beam, 0.02 for roll forming beam.

For case 4, acceleration of the press type beam is similar to the roll forming beam for the same space between bumper beam and bumper cover as shown in Table 18. For the press type beam, acceleration change is analyzed according to the main factor variation of an upper stiffener position and lower stiffener position. Acceleration change according to lower stiffener position is shown in Table 19. Acceleration incre-ment is 15 as long as the decrement of lower stiffener position is 92. Therefore, acceleration increment rate for lower stiff-ener position is 0.163. Acceleration change according to upper stiffener position is shown in Table 20. Acceleration incre-ment is 35 as long as the decrement of upper stiffener position is 44. Therefore, acceleration increment rate for upper stiff-ener position is 0.796. Acceleration change according to the

Table 16. Bumper foam rate factor for case 3.


30 times 257G 313G

40 times 514G 448G

Foam rate factor 2.0 1.431 Table 17. Bumper foam thickness factor for case 3.

Press type Roll forming Foam thickness Accel. Factor Accel. Factor

50 mm 514G 1.0 448G 1.0

60 mm 470G 0.91 343G 0.8

70 mm 423G 0.82 265G 0.6

80 mm 371G 0.73 210G 0.4 Table 18. Acceleration analysis result for case 4.

Type Press type beam Roll forming beam

Acceleration 221G 223G Table 19. Acceleration analysis result according to lower stiffener position variation.

Item Press type

Upper stiffener position 144 mm

Lower stiffener position 192 mm 100 mm


Table 14. Bumper foam rate factor for a case 2.


30 times 279G 330G

40 times 193G 212G

Foam rate factor 0.692 0.642

Table 15. Bumper foam thickness factor for case 2.

Press type Roll forming Foam thick-ness Accel. Factor Accel. Factor

Average factor

5 0mm 193G 1.0 212G 1.0 0.99

60 mm 183G 0.947 200G 0.947 0.95

70 mm 177G 0.918 192G 0.907 0.91

80 mm 169G 0.876 186G 0.881 0.87


space variation between bumper cover and bumper beam is shown in Table 21. The space variation between bumper cover and bumper beam does not affect the acceleration, and a simple expression of acceleration for case 4 is finally ex-pressed in Eq. (25) as follows:

220 + (192 - DWN)×0.163 + (144 - UP)×0.796. (25) For case 5, as in Table 9, acceleration increment is 22 as

long as the increment of the H value is 50. Therefore, accel-eration rate for the H value is 0.44. Acceleration change ac-cording to the space variation between bumper cover and bumper beam is shown in Table 22. Average factor for the space variation between bumper cover and bumper beam is expressed as in Eq. (26) as follows:

1.0 - (Space - 50)×0.01 (26)

where Space is the distance the space between bumper cover and bumper beam.

Therefore, a simple expression of acceleration for case 5 is expressed in Eq. (27) as follows:

{166 + (H - 60)×0.44}×{1.0 - (Space - 50)×0.01}. (27) For case 6, as in Table 10, acceleration increment is 32 as

long as the increment of the H value is 50. Therefore, accel-

eration rate for the H value is 0.64. Therefore, a simple ex-pression of acceleration for case 6 is expressed in Eq. (28) as follows:

674 + (H - 60)×0.64. (28) Simple expressions for acceleration from case1 to case 6

can be summarized as in Table 23. A flow-chart of equivalent simple analysis program of pedestrian bumper impact is shown in Fig. 11.

3. Verification of a simple analysis program of pedes-

trian bumper impact

To verify the accuracy of an equivalent simple analysis pro-gram, a non-linear finite element program, LS-DYNA, is used for a press type bumper beam and a roll forming bumper beam, which are widely used these days. For the two kinds of bumper beam types, it is verified for pedestrian combination cases. Press type bumper beam is shown in Fig. 12.

Result comparison of bending angle between a simple analysis program and LS-DYNA result is shown in Table 24. Bending angle is the sum of tibia rotation angle and femur

Table 20. Acceleration analysis result according to upper stiffener position variation.

Item Press type

Upper stiffener position 144 mm 100 mm

Lower stiffener position 150 mm


Table 21. Acceleration change according to the space variation be-tween bumper cover and bumper beam for case 4.

Press type Roll forming Space

Accel. Factor Accel. Factor

50 mm 221G 1.0 224G 1.0

60 mm 225G 1.017 225G 1.004

70 mm 222G 1.004 223G 0.996

80 mm 221G 1.001 225G 1.005

Table 22. Acceleration change according to the space variation be-tween bumper cover and bumper beam for case5.

Press type Roll forming Space

Accel. factor Accel. factor Average

factor

50 mm 166G 1.0 188G 1.0 1.0

60 mm 145G 0.875 156G 0.831 0.9

70 mm 130G 0.779 128G 0.678 0.8

80 mm 125G 0.750 130G 0.691 0.7

Table 23. Summary of simple expression for acceleration.

No Combinations Acceleration expression

Case 1With foam + bumper beam + upper/lower

stiffener

[{320+ (H-60)×1.667+ (144-UP)×0.682}×F_times]×{1.0-

(F_thk- 50)×0.001 - 0.01}

Case 2 With foam + bumper beam + lower stiffener

[{270+ (H-60)×1.667× F_times ]×{1.0- (F_thk-50)×

0.004 - 0.01}

Case 3 With foam + bumper beam

[{313+(H- 60)×1.12× F_times]×{1.0-(F_thk - 50)×

Beam_type}

Case 4No foam + bumper

beam + upper/lower stiffener

220 + (192 - DWN)×0.163 + (144 - UP)×0.796

Case 5 No foam + bumper beam + lower stiffener

{166+ (H-60)×0.44}×{1.0- (Space-50)×0.01}

Case 6 No foam + bumper beam 674 + (H - 60)×0.64

Fig. 11. Flow-chart of pedestrian bumper impact analysis.


rotation angle. It is calculated from beam nodal displacements by using trigonometric function.

In case of no stiffener, relative error is less than other cases. The reason is that the equivalent forces of stiffener and bumper beam may contain some difference with LS-DYNA result. Maximum relative error of bending angle between two results does not exceed 16%.

Result comparison of acceleration between a simple analy-sis program and LS-DYNA result is shown in Table 25. Sim-ple acceleration expression formulated by main factors shows the similar results with LS-DYNA. Maximum relative error of acceleration between two results does not exceed 10%.

LS-DYNA finite element model for press type bumper beam is shown in Fig. 13, and its pedestrian bumper impact deformation is shown in Fig. 14.

The roll forming bumper beam is shown in Fig. 15 and re-sult comparison of bending angle between a simple analysis program and LS-DYNA result is shown in Table 26. Maxi-mum relative error of bending angle for roll forming bumper beam does not exceed 20%. Relative error is increased com-pared to press type bumper beam in Table 24. The reason is that equivalent forces of bumper beam and stiffener are ob-tained by using press type bumper beam and these forces are

Table 24. Bending angle comparison for press type bumper beam.

LS-DYNA Simple program Combinations

Bending angle [ °]

Relativeerror (%)

With foam + beam + upper/lower stiffener 29.1 26.0 10.6

With foam + beam + lower stiffener 27.2 27.9 2.9

With foam + beam 39.1 39 0.3 No foam + beam +

upper/lower stiffener 16.7 18.8 12.8

No foam + beam + lower stiffener 8.5 9.8 15.4

No foam + beam 37.3 37.5 0.5

Table 25. Acceleration comparison for press type bumper beam.


Acceleration [G]

Relative error (%)



With foam + beam 257.1 282.0 9.7 No foam + beam +



No foam + beam 674.2 639.4 5.2

Fig. 12. Configuration of press type bumper beam.

Table 26. Bending angle comparison for roll forming bumper beam.


Bending angle [ °]

Relative error (%)






No foam + beam 37.5 37.5 0.0

Fig. 13. LS-DYNA FE model for press type bumper beam.

Fig. 14. LS-DYNA analysis result for press type bumper beam.

Fig. 15. Configuration of roll forming bumper beam.


used for roll forming bumper beam. LS-DYNA finite element model for roll forming bumper beam is shown in Fig. 16, and its pedestrian bumper impact deformation is shown in Fig. 17.

Result comparison of acceleration between a simple analy-sis program and LS-DYNA result is shown in Table 27. Sim-ple acceleration expression formulated by main factors shows similar results with LS-DYNA. Maximum relative error of acceleration between two results does not exceed 10%.

4. Conclusions

A simple analysis program for pedestrian bumper impact was presented, using equivalent beam analysis model. This simple analysis program was validated by a non-linear finite element program, LS-DYNA.

A simple analysis program for the pedestrian bumper im-pact was developed by substituting the femur and tibia for dynamically equivalent Euler beam. And dynamically equiva-

lent forces of bumper beam and stiffeners were found by a finite element analysis and applied to this beam model.

The accuracy of the simple analysis program for the pedes-trian bumper impact was tested by comparing their results with those of a non-linear finite element code LS-DYNA. The relative error of bending angle between two models did not exceed 20%.

The relative error of acceleration between the simple analy-sis program and LS-DYNA analysis result did not exceed 10%.

Using this simple analysis program for pedestrian impact will reduce the analysis time, and it will also easily secure the pedestrian impact performance in the bumper design process.

Nomenclature

A : Beam sectional area B : Strain-displacement matrix bi : Body force D : Bumper beam depth DWN : Lower stiffener position from the impactor joint E : Young’s modulus Feq : Dynamically equivalent force F_thk : Bumper foam thickness F_times : Bumper foam factor G : Shear modulus H : Bumper beam height I : Moment of inertia Ke : Element stiffness matrix k : Shear coefficient N : Displacement interpolation matrix p : External force U : Internal virtual work UP : Upper stiffener position from the impactor joint U : Beam displacement in x-direction V : Beam displacement in y-direction W : External virtual work W : Beam displacement in z-direction w’ : Slope of beam displacement in z-direction w : Virtual beam displacement in z-direction β : Rotation angle of beam cross section β : Virtual rotation angle of beam cross section γxz : Shear strain at the x-z plane εx : Axial strain κ : Curvature of beam cross section π : Total potential energy σij,j : Stress tensor σx : Axial stress τxz : Shear stress at the x-z plane

References

[1] European new car assessment program, Pedestrian Testing Protocol Version 4.2 (2008).

[2] D. K. Park, C. D. Jang, S. B. Lee, S. J. Heo, H. J. Yim and

Table 27. Acceleration comparison for roll forming bumper beam.


Acceleration [G]

Relative error (%)






No foam + beam 705.8 700.0 0.8

Fig. 16. LS-DYNA FE model for roll forming bumper beam.

Fig. 17. LS-DYNA analysis result for roll forming bumper beam.


M. S. Kim, Optimizing the shape of a bumper beam section considering pedestrian protection, International Journal of Automotive Technology, 11 (4) (2010) 489-494.

[3] H. J. Yim, M. S. Park, J. Park, S. J. Heo and D. K. Park Shape optimization of bumper beam cross section for low speed crash, Proceedings of the World Congress, Society of Automotive Engineers, 2005-01-0880 (2005).

[4] H. Kim and S. G. Hong, Optimization of bumper system under various requirements, Proceedings of the World Con-gress, Society of Automotive Engineers, 2001-01-0354 (2001).

[5] M. H. Kim, S. H. Kim and S. K. Ha, Structural analysis and optimal design of front bumper for automobiles by using the beam theory, Transactions of the Korean Society of Me-chanical Engineers, 23 (12) (1999) 2309-2319.

[6] M. H. Kim, S. S. Jo and S. K. Ha, Design and structural analysis of aluminum bumper for automobiles, Transactions of the Korean Society of Automotive Engineers, 7 (4) (1999) 217-227.

[7] M. H. Kim, S. H. Kim and S. K. Ha, Design and Structural Analysis of bumper for automobiles, Proceedings of the An-nual Meeting, The Korean Society of Automotive Engineers, 2 (1997) 1019-1024.

[8] ANSYS, User’s Manual Version11.0 (2008). [9] LS-DYNA, User’s Keyword Manual Version 971 (2007).

Dong Kyou Park received his B.S. and M.S. degrees in the Department of Naval Architecture in Seoul National University in 1988 and 1990. He received his Ph.D. in the Department of Industrial Engineering and Naval Architecture in Seoul National University in 2011. He is a professor in the Department of

Automotive Engineering at Daeduk University, Daejeon, Ko-rea. His research interests include impact analysis, crash analy-sis, structure strength analysis and optimum structural design.

Chang Doo Jang received his B.S. de-gree in the Department of Naval Archi-tecture in Seoul National University in 1973. He received his M.S. degree in the Department of Naval Architecture in Kyushu University in 1977. He received his Ph.D. degree in the Department of Marine Engineering in University of

Tokyo in 1980. He is a professor in the Department of Indus-trial Engineering and Naval Architecture at Seoul National University, Seoul, Korea. His research interests include opti-mum structural design, welding distorsion analysis, impact analysis and fatigue analysis.

a study on the development of equivalent beam analysis

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