ANALYSIS REFERENCE Chapter 5. Algorithm

1) 기둥 좌굴

REFERENCE Gere et al1 ELEMENTS Beam elements, shell elements, solid elements MODEL FILENAME Buckling01.mpb

아래 그림은 기둥모델로 경계조건에 따른 좌굴모드 양상을 확인하기 위한 문제이다.

Material data Elastic modulus E = 10000 tonf/m2

Section property Rectangular cross-section No shear deform

0. 25 m x 1.0 m

Y

X

15

Units : mTop : RollerBottom : Pin

Top : FreeBottom : Fixed

Top : Laterally guidedBottom : Fixed

Top : RollerBottom : Fixed

1 tonf 1 tonf 1 tonf 1 tonf

1

0.25

2.4 좌굴해석 검증 예제

Figure 2.4.1.1 Column model

Section 2. 고유치 추출법 | 143

Chapter 5. Algorithm

ANALYSIS REFERENCE

Table 2.4.1.1 Critical loads in tonf obtained using beam elements

Table 2.4.1.2 Critical loads in tonf obtained using shell elements

Case 1 Case 2 Case 3 Case 4

Figure 2.4.1.2 Buckling mode shapes

Case 1 2 3 4

Reference 0.5712 0.1428 2.2846 1.1684

Element type Number of elements

Beam-2 15 0.5714 0.1428 2.2847 1.1685

Case 1 2 3 4

Reference 0.5712 0.1428 2.2846 1.1684

Element type Number of elements

TRIA-3 30 0.5741 0.1430 2.3323 1.1812

QUAD-4 15 0.5736 0.1429 2.3245 1.1790

144 | Section 2. 고유치 추출법

ANALYSIS REFERENCE Chapter 5. Algorithm

Table 2.4.1.3 Critical loads in tonf obtained using solid elements

Case 1 2 3 4

Reference 0.5712 0.1428 2.2846 1.1684

Element type Number of elements

HEXA-8 15 0.5747 0.1430 2.3421 1.1841

Section 2. 고유치 추출법 | 145

Chapter 5. Algorithm

ANALYSIS REFERENCE

2) 포틀 프레임

REFERENCE Timoshenko et al2 ELEMENTS Beam elements MODEL FILENAME Buckling02.mpb

아래 그림은 포틀 프레임 형태의 2 차원 라멘구조모델로 수직하중에 대해서 요소개수에 따라 좌굴하중이 어떻게 바뀌는지 확인하기 위한 문제이다.

Material data Elastic modulus E = 1 x 106 psi

Section property Area Moment of inertia

A = 1.0 in2 Ix = 1.0 in4

YX

100

Units : in

100

1 lbf 1 lbf

①

②

③ ④

⑤

⑥

Figure 2.4.2.1 Column model

146 | Section 2. 고유치 추출법

ANALYSIS REFERENCE Chapter 5. Algorithm

Table 2.4.2.1 Critical loads in tonf obtained using beam elements

Reference 737.9

Element type Number of elements

Beam-2

2 per member 739.8

4 per member 737.6

8 per member 737.5

Figure 2.4.2.2 Buckling mode shapes

Section 2. 고유치 추출법 | 147

Chapter 5. Algorithm

ANALYSIS REFERENCE

3) 고정구속된 정사각형 판

REFERENCE Chajes3 ELEMENTS Shell elements, solid elements MODEL FILENAME Buckling03.mpb

아래 그림은 면내 압축을 받는 정사각형 판모델을 나타낸다. 대칭조건을 이용하여 1/4 만 모델링하였다.

Material data Elastic modulus Poisson’s ratio

E = 11.164 x 106 psi ν = 0.3

Section property Thickness T = 0.01 in

Y

X

1

1.0

Units : in.

P = 1.0 lbf/in. P

t = 0.01

Figure 2.4.3.1 Clamped square plate model

148 | Section 2. 고유치 추출법

ANALYSIS REFERENCE Chapter 5. Algorithm

Table 2.4.3.1 Critical loads in lbf obtained using shell elements

Reference 100.7

Element type Number of elements

TRIA-3 32 111.7

QUAD-4 16 107.3

Table 2.4.3.2 Critical loads in lbf obtained using solid elements

Reference 100.7

Element type Number of elements

HEXA-8 16 98.0

Figure 2.4.3.2 Buckling mode shapes

Section 2. 고유치 추출법 | 149

Chapter 5. Algorithm

ANALYSIS REFERENCE

4) 직사각형 판의 면내좌굴

REFERENCE Timoshenko et al2 ELEMENTS Shell elements, solid elements MODEL FILENAME Buckling04.mpb

아래 그림은 면외 방향으로는 단순지지로 구속되어있고 면내로는 아래쪽 변이 Y 방향으로 구속되어있는 판모델을 나타낸다. 판 위쪽 중앙지점(E)에 집중하중이 가해졌을 때 면내좌굴 하중을 확인하기 위한 문제이다.

Material data Elastic modulus Poisson’s ratio

E = 200 GPa ν = 0.3

Section property Thickness t = 0.01 mm

Y

XZ A

B C

D

E

P = 1 kN

2

1

Units : mm

t = 0.01

Figure 2.4.4.1 Clamped square plate model

150 | Section 2. 고유치 추출법

ANALYSIS REFERENCE Chapter 5. Algorithm

Table 2.4.4.1 Critical loads in kN obtained using shell elements

Reference 330

Element type Number of elements

TRIA-3 144 342

QUAD-4 72 327

Table 2.4.4.2 Critical loads in kN obtained using solid elements

Reference 330

Element type Number of elements

HEXA-8 72 298

Section 2. 고유치 추출법 | 151

Chapter 5. Algorithm

ANALYSIS REFERENCE

5) 축압축력을 받는 실린더

REFERENCE Simo et al.4 ELEMENTS Shell elements, solid elements MODEL FILENAME Buckling05.mpb

아래 그림은 축압력을 받는 실린더 모델로 양단의 횡방향으로 고정구속되어있다. 대칭성을 이용하여, 실린더 축방향으로 1/2 과 평면의 원형에 대해서는 1/4 만 모델링하였다.

Material data Elastic modulus Poisson’s ratio

E = 567 Pa ν = 0.3

Section property Thickness t = 0.247 m

P = 0.209 N/m

ClampedR = 100

L = 35.95

Units : m

Figure 2.4.5.1 Axially compressed cylinder model

152 | Section 2. 고유치 추출법

ANALYSIS REFERENCE Chapter 5. Algorithm

Table 2.4.5.1 Critical load factor obtained using shell elements

Reference 1.0833

Element type Number of elements

TRIA-3 840 1.2016*

QUAD-4 420 1.0734

* obtained from 10th buckling mode Table 2.4.5.2 Critical load factor obtained using solid elements

Reference 1.0833

Element type Number of elements

HEXA-8 420 1.0631

Figure 2.4.5.2 Buckling mode shapes

Section 2. 고유치 추출법 | 153

Chapter 5. Algorithm

ANALYSIS REFERENCE

6) L 형 브라켓

REFERENCE Simo et al.4 and Argyris et al.5 ELEMENTS Shell elements, solid elements MODEL FILENAME Buckling06.mpb

아래 그림은 L 형 브라켓 판 모델로 면내방향 하중을 받는 모델이다. 면내방향 모멘트에 대해서 횡좌굴이 발생하는 현상을 확인하기위한 문제이다.

Material data Elastic modulus Poisson’s ratio

E = 71.24 GPa ν = 0.3

Section property Thickness t = 0.6 mm

240

240

30

Units : mm

t = 0.6XY

P = 1 N

Figure 2.4.6.1 L-bracket plate model

Figure 2.4.6.2 Buckling mode shapes

154 | Section 2. 고유치 추출법

ANALYSIS REFERENCE Chapter 5. Algorithm

Table 2.4.6.1 Critical loads in N obtained using shell elements

Reference 1.137 (Simo et al.), 1.155 (Argyris et al.)

Element type Number of elements

TRIA-3 136 1.187

QUAD-4 68 1.199

Table 2.4.6.2 Critical loads in N obtained using solid elements

Reference 1.137 (Simo et al.), 1.155 (Argyris et al.)

Element type Number of elements

HEXA-8 68 1.198

Section 2. 고유치 추출법 | 155

Chapter 5. Algorithm

ANALYSIS REFERENCE

References

1 J.M. Gere and S.P. Timoshenko, “Mechanics of Materials”, 2nd Edition, Thomson Brooks/Cole, California, New York, 1984

2 S.P. Timoshenko and J.M. Gere, “Theory of Elastic Stability”, 2nd Edition, McGraw-Hill, New York, 1961

3 A. Chajes, “Principles of Structural Stability Theory”, Prentice-Hall, Englewood Cliffs, N.J, 1974

4 J.C. Simo, D.D. Fox and M.S. Rifai, “On a Stress Resultant Geometrically Exact Shell Model. Part III: The Computational Aspects of the Nonlinear Theory”, Computer Methods in Applied Mechanics and Engineering, Vol. 79, pp.21-70, 1990

5 J.H. Argyris, H.Balmer, J.St. Doltsinis, P.C. Dunne, M. Haase, M. Kleiber, G.A. Malejannakis, H.P. Malejenek, M. Muller and D.W. Scharpf, “Finite Element Method – Ther natural approach”, Computer Methods in Applied Mechanics and Engineering, VOl. 17/18, pp. 1-106, 1979

156 | Section 2. 고유치 추출법