chapter 2 結構力學總複習 review of structural mechanics
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Chapter 2
結構力學總複習Review of Structural Mechanics
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Contents
2.1. 結構分析問題的定義 Definition of Structural Analysis Problems
2.2. 控制方程式 Governing Equations
2.3. 解題方法:有限元素法 Solution Method: Finite Element Method
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第 2.1 節
結構分析問題的定義Definition of Structural Analysis Problems
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2.1.1 結構分析問題
• 結構分析是一個固態的實體( body )承受負載( loads )後,求解結構反應( responses )的過程
z
Q1 at S1
P2 at S2
D3 = 0 at S3
D4 at S4x
y Body
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2.1.2 Example (1/2)
• Body 樑• Loads 集中力 , 分佈力 , 及拘束• Responses 變形 , 應力 , 應變
P
Q
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2.1.2 Example (2/2)
• Body 樑• Loads 右端變位 D, 左端變位 0• Responses 變形 , 應力 , 應變
D
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2.1.3 loads
結構分析負載分類 SI單位 主要 ANSYS 命令
作用在物體界面
力表面分佈力 Pa SF
點集中力 N F
變位零變位 ( 固定 ) m D
非零變位 m D
作用在物體內部力
慣性力 ( 重力或離心力 ) N/m3 ACEL, OMEGA
其他體積分佈力 ( 靜電力或磁力等 ) N/m3 (none)
熱 溫度變化 K BF
Default boundary condition: zero surface force
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Comparison:Loads in Thermal Analysis
熱分析負載分類 SI單位 主要 ANSYS 命令
作用在物體界面熱流
Heat flux W/m2 SF
Heat flow W F
溫度固定溫度 K D
作用在物體內部 熱源Heat generation W/m3 BF
Default boundary condition: zero heat flux (adiabatic)
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2.1.4 Responses
結構反應 向量符號 分量數目
變位 (displacements) u 3
應變 (strains) 6
應力 (stresses) 6
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2.1.5 Displacements
變形前
(x, y, z)
u(x, y, z)
變形後
zyx uuuu
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2.1.6 Stresses (1/4)
• 應力是用來描述力的密度 (intensity) 及方向 (direction)
• 應力的大小以 SI 單位表示是 N/m2 (Pa)
• 應力需以兩個方向來描述• 數學上稱此種物理量為張量 (tensor)
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2.1.6 Stresses (2/4)
z
x
y
A
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2.1.6 Stresses (3/4)
x
y
x
y
x
y
z
xy
xy
yx
yx
zy
zx
zzyzx
yzyyx
xzxyx
σ
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2.1.6 Stresses (4/4)
zzyzx
yzyyx
xzxyx
σ
xzzxzyyzyxxy , ,
0,0,0 zyx MMM
zxyzxyzyx σ
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2.1.7 Strains (1/3)
• 應變是描述某一質點被拉伸或壓縮 , 及扭曲的程度
• 應變的 SI 單位是 m/m, 相當於無單位• 應變也是需要兩個方向來描述
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2.1.7 Strains (2/3)C’
AB
C
A’B’
B’’
D
B’’’
C’’
C’’’
x
y
AB
BD
AB
BDxyx
,
rad BBAAB
BDxy
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2.1.7 Strains (3/3)
xx
y
x
y
z
xy
xy
yx
yx
zy
zx
zzyzx
yzyyx
xzxyx
ε
y
xzzxzyyzyxxy , ,
zxyzxyzyx ε
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第 2.2 節
控制方程式Governing Equations
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2.2.1 控制方程式
控制方程式是描述此 15 個未知量之間的關係 , 包括 3 個力平衡方程式 , 6 個應變與變位關係 , 及 6 個應力與應變關係 ( 虎克定律 )
zyx uuuu
zxyzxyzyx σ
zxyzxyzyx ε
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2.2.2 力平衡方程式 0,0,0 zyx FFF
0
0
0
zzzyzx
yyzyyx
xxzxyx
fzyx
fzyx
fzyx
zzzyzyxzx
yzyzyyxyx
xzxzyxyxx
Tnnn
Tnnn
Tnnn
由力平衡條件
可以導出 , 在 body 的內部
或是在 body 的表面
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2.2.3 應變與變位關係
z
u
x
u
y
u
z
u
x
u
y
u
z
u
y
ux
u
xzzx
zyyz
yxxy
zz
yy
xx
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2.2.4 應力與應變關係
G
G
G
EEE
EEE
EEE
zxzx
yzyz
xyxy
zyxz
zyxy
zyxx
)1(2
EG
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第 2.3 節
解題方法 : 有限元素法Solution Method: Finite Element Method
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2.3.1 數值解法• 理論上 15 個方程式可以解 15 個未知量• 實務上只有很簡單的問題才有解析解• 大部份的問題必須以數值方式解答• 最普遍的數值方法是有限元素法
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2.3.2 有限元素法 : 基本構想• 將 body 切割成元素 , 元素之間假設僅以節點連接
• 先建立每個元素的方程式
• 再組成整體 body 的方程式
• 最後解出未知量
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2.3.3 自由度Degrees of Freedom
• 自由度是控制方程式的未知量
• 結構分析問題是指節點上的變位量
• 熱分析問題是指節點上的溫度
lz
ly
lx
kz
ky
kx
jz
jy
jx
iz
iy
ix uuuuuuuuuuuud
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2.3.4 形狀函數Shape Functions
形狀函數是用來連接變位場 u和節點變位d間的關係
x
y
未知函數 f(x)
近似函數
Ndu
lkji
lkji
lkji
NNNN
NNNN
NNNN
00000000
00000000
00000000
N
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2.3.5 Order of Element
• 一個元素的 order 是指形狀函數的次數• Order 越高 , 解答的精度越高 (p-method)• 提高精度的另一個方法是將元素切割得更細小 (h-method)
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2.3.6 Stiffness Matrices
元素的力平衡方程式
整體結構的力平衡方程式
fkd
FKD
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2.3.7 FEM Summary輸入
分析模型
建立元素方程式
建立結構方程式
解結構方程式
計算應變及應力