第5章 超静定结构(I) new

§2.6 Element equivalent nodal load vector
Local: P = − F
e E Fe
Global: PEe = − F Fe
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§3 Global analysis
§3.1 Joint equil. eqs.
(as in S.D.) Note: • a joint disp. => a equil. eq. • constrained disp. => no equil. eq.
No. of global disp. = No. of joint equil. eqs E-Location vector (>0) = Global equil. eq. numbers
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§3.2 Joint disp. conforming
• Number global joint disp.; form global joint disp. vector ∆ • Form global equil. eqs. K∆ = P
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More examples:
real,target :: a,b real,pointer:: p,q a = 3.14 b = 2.0 p => a ! p = a = 3.14 q => b ! q = b = 2.0 q = p ! ( q pointing to b ) = ( p pointing to a ) ！i.e. all = 3.14
e 6×6
F 1 k11 k12 Δ1 F1 F + F = Blocked F 2 k21 k22 Δ2 F2
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e
e
e
e
Element stiffness matrix:
EA EA 0 0 − l l 12 EI 6 EI 0 0 3 2 l l 4 EI 0 6 EI 0 2 l l ke = EA l 0 对称 0 0 − 12 EI l3 6 EI − 2 l 0 12 EI l3 6 EI − 2 l 0 6 EI 2 l 2 EI l 0 6 EI − 2 l 4 EI l
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§1 Introduction
Features of S.I. structures:
• •
needs stiffness, material properties disp. conforming + static equil.
Prog. Approach — stiffness/disp. method • can be used for both S.D. and S.I. structures • disp. and internal forces are given same time
(1)
e FX = s eΔ + FXFe
e
(2)
Substitute (2) into (1)：
F =h
e
e
e e 6×3 3×6
e
s Δ +h F
e 6×3
Fe
e
Fe X
+F
e p
(3) F =k Δ +F {E-end forces}=[E-stiffness]{E-end disp.}+{Fixed end forces}
F =h F +F
e X e e e p
e
e
FN1 M1 M 2
e
0 − q l 2 0 + 0 q l − 2 0
e
(1)
{E-end forces} = [equil.]{E-internal} + {load}
i —— direction； j —— cause
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§2.5 Global coordinate transformation
( x, y)
e
α
( x, y )
e e T e
Local
T
TT F
Fe T
Global
Fe
F =T F ⇒ F =T F
=T F
Δ = T Δe ⇒ Δe = T TΔ
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5. 5. Statically Statically Indeterminate Indeterminate Structure Structure
§1 Introduction §2 Element analysis §3 Global analysis §4 Storage of variable band matrix §5 Decomposition of stiffness matrix §6 Illustrative examples
F = s Δ + FXFe
e X e e
(2)
{E-unknown forces} = [stiffness]{E-end disp.} + {Fixed end forces}
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§2.4 Element stiffness eq.
We know:
F = h FXe + F p
e e e
e
e
∴ F e = T T F e = T T k eΔe + T T F Fe
= T T k eTΔe + T T F Fe = k eΔe + F Fe
ke = T T k e T, F Fe = T T F Fe
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Properties of element stiffness matrix
F90 a feature a week
! pointer
Pointer is an attribute of a variable, and can be used to point to variables of the same data type; A pointed target must have target or pointer attribute a pointer is a nickname or a title real,target :: a,b,EDisp(6) !can be pointed real,pointer :: p1,p2 ! title: 班长、课代表 ！p1,p2 are pointers, can point real data real,pointer :: G(:) ! 先进集体 ！G is a pointer, can point to 1D real array
e
Fixed-end force vector:
0 ql − 22 − q l 12 = 0 ql − 2 ql 2 12
e
F Fe
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Physical meaning of stiffness coefficients
近4， 远2， 线负6
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Matrix form:
u1 EA − EA 0 0 0 0 υ 0 l 1 l F EI EI EI EI N1 θ 1 ql 2 0 − 6 2 2 + − M1 = 0 6 2 4 u2 12 l l l l 2 M 2 EI EI EI EI υ 0 6 2 2 0 − 6 2 4 2 ql l l l l 12 θ 2
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§2 Element analysis
§E-internal force equil. (review)
e.g. uniform load M2 F N1 M1 FN1 Equil.
q
Fx 2
qFx1l来自Fy 2M2M1
Fy1
M1 + M 2 ql − − l 2 M1 + M 2 ql − l 2
Expand the i-row (no loading):
F = k Δ + k Δ +!+ k Δ +!
If let: 0
e i
e i
e i1
e 1
e i2
e 2
e ij
e j
0
e i
1
1 Δ = 0
i= j i≠ j
F =k
e ij
k =
e ij
Force along the i-th direction due to the the j-th unit disp.
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• Pointer is a “title”, the above declaration established several “titles”, but not yet given to any variables, and hence are all pointing to null (no memory used). a = 3.0 p1 => a ! p1 pointing to a ! title p1 is given to a，the data of a has two names: fixed name a and a floating name p1; either one can be used (p1 — 班长, a — John) a = 4.0 ! a is 4.0, p1 is also 4.0 p1 => b ! 班长 is changed to s.b. else G => EDisp ! 先进集体 is issued ！EDisp(:) and G(:) have same length, same effect using G or using EDisp