ulrich häussler-combe computational methods for reinforced concrete...
TRANSCRIPT
Ulrich Häussler-Combe
Computational Methods for Reinforced Concrete Structures
Es As
F2 = As Es ε2 = −As Es α α γ
sc ssc/s = α n = z/sc = z/(s α)
Ftie =z
sc/ αF2 = −z
As2
sEs
2 α α γ
V = −F2 α
Vtie = z as2 Es3 α α γ
as = As/s
∂Vtie
∂γ= z as2 Es
3 α α
∂V
∂γ=
∂V
∂γ+
∂Vtie
∂γ= z
(bEc
2 θ 2 θ + as2 Es3 α α
)α = π/2 θ = π/4
∂V
∂γ=
1
4zbEc =
1
2zbGc
Gc
3 α zb = αc A, αc = z/hαc 3
Vθ = π/4
20 ≤ θ ≤ 45
σε
σ = · ε
σ = · ε+ σp
σp =
(Np
Mp
)= −F p
(αp
−zp αp
)F p zp
αp = zp/ x
e =
∫Le
T · σ x =
∫Le
T · · ε x+
∫Le
T · σp x = εe +
pe
− pe
•ε•p
−N ε′ −Np′ = px−M ε′′ −Mp′′ = pz
−N ε′ = px − (F p αp)′
−M ε′′ = pz + (zpFp αp)
′′
F p ≈ const., αp ≈ 1
−M ε′′ = pz + z′′pFp
z′′pFp z z′′p
pz
σpp
zp, αp x
F p0
F p0
x
F p0 Fp
zp =[
r3
4 − 3r4 + 1
2Ler
3
8 − Ler2
8 − Ler8 + Le
8 − r3
4 + 3r4 + 1
2Ler
3
8 + Ler2
8 − Ler8 − Le
8
]·
⎛⎜⎜⎝zpIαpI
zpJαpJ
⎞⎟⎟⎠Le
αP =∂zp∂x
=∂zp∂r
∂r
∂x= z′p
zpI , αpI zpJ , αpJ −1 ≤ r ≤ 1zp(−1) = zpI , z
′p(−1) = αpI zp(1) = zpJ , z
′p(1) = αpJ
e
LPe =
Le
2
∫ 1
−1
√(x′p)2 + (z′p)2 r
xp
x′p = 1⎛⎜⎜⎝zpIαpI
zpJαpJ
⎞⎟⎟⎠ =
⎛⎜⎜⎝zp0Iαp0I
zp0Jαp0J
⎞⎟⎟⎠zp0I , αp0I , zp0J , αp0J
x′p = 1 + ε⎛⎜⎜⎝zpIαpI
zpJαpJ
⎞⎟⎟⎠ =
⎛⎜⎜⎝zp0I + wI
αp0I + φI
zp0J + wJ
αp0J + φJ
⎞⎟⎟⎠ε
wI , φI , wJ , φJ LP
LP0 LP
F p =LP
LP0
F p0
Δεp(x) = Δε(x)− zp Δκ(x)
Δε, Δκ
F p(x) = F p0 + EpAp Δεp(x)
Ep Ap
F p(x)
L = 10
b = 0.2, h = 0.4 fcd = 38 /As1 = As2 = 12.57 , d1 = d2 = 5
Mu ≈ 0.20 N = 0qu = 8Mu/L
2 = 15.2 /
σc0 = −10 /F p0 = 0.8
hp
zp = 4hp
(x2
L2− x
L
)hp = 0.15
Ap = 6fp0,1 = 1600 / fp = 1800 /
Ep = 200 000 / σp0 = 1333 /
εp0 = 6.67
q = 5 /
qp = 25 /
F p/F p0 = 1.002 hp/L = 1/67
≈−2.2 −3.5
Mq = 0.03 × 102/8 =
0.375 Mc = 0.255Mp = 0.120
0.12Mc = 0.220 Mp = 0.155
F p
L ∫ L
0
δεT · σ s =
∫ L
0
δ T · ¯ s+ δ T · ¯
sσ ε
αe
υe
υe = (αe) · υe
˜(r) = (r) · υe, ε(r) = (r) · υe
δεT = δυTe · T (αe) · T (r)
e =T (αe) · Le
2
∫ 1
−1
T (r) · σ(r) r
Le αe
Te =∂ e
∂υe
Te = TMe + TGe
TMe = T · Le
2
∫ 1
−1
T · ∂σ∂ε
· ∂ε
∂υe· ∂υe
∂υer
= T · Le
2
∫ 1
−1
T · T · r ·= T · ˜ Te ·
TGe =∂ T
∂αe· e · ∂αe
∂υe
T
e =Le
2
∫ 1
−1
T · σ r
∂ T
∂αe=
⎡⎢⎢⎢⎢⎢⎢⎣− αe − αe 0 0 0 0
αe − αe 0 0 0 00 0 0 0 0 00 0 0 − αe − αe 00 0 0 αe − αe 00 0 0 0 0 0
⎤⎥⎥⎥⎥⎥⎥⎦
e =
⎛⎜⎜⎜⎜⎜⎜⎝NI
VI
MI
NJ
VJ
MJ
⎞⎟⎟⎟⎟⎟⎟⎠ ,∂αe
∂υe=
1
Le
⎛⎜⎜⎜⎜⎜⎜⎝αe
− αe
0− αe
αe
0
⎞⎟⎟⎟⎟⎟⎟⎠
TGe =
⎡⎢⎢⎢⎢⎢⎢⎣−AI αe AI αe 0 AI αe −AI αe 0BI αe −BI αe 0 −BI αe BI αe 0
0 0 0 0 0 0−AJ αe AJ αe 0 AJ αe −AJ αe 0BJ αe −BJ αe 0 −BJ αe BJ αe 0
0 0 0 0 0 0
⎤⎥⎥⎥⎥⎥⎥⎦
Ai = αeNi + αeVi, Bi = αeNi − αeVi, i = I, J
¯e =Le
2
∫ 1
−1
T (αe) · T (r) · (αe) · ¯(r) r
¯e =
(NI VI MI NJ VJ MJ
)T¯e ¯
υe, ε, σ
(υ) =
= ¯ + ¯
u,w, φαe = π/2
T = αe α
TM =
⎡⎣ 12EJL3 0 6EJ
L2
0 EAL 0
6EJL2 0 4EJ
L
⎤⎦
TG = NJ0TG,
0TG =
⎡⎣ 1L 0 00 0 00 0 0
⎤⎦0TG
( TM + TG) · υ =
υ
TM · υ = −NJb0TG · υ
NJb
NJb =3EJ
L2
9wJ +6Lφ = 0 π2 EJ/4L2 ≈ 2.47EJ/L2
NJb
NJb
αe Le
ε σ
T
αe, Le
L = 5.0 h = 0.4b = 0.2
nE = 10
Ec = 33 000 / fc = 38 /εc1 = −0.0023, εcu1 = −0.0035
fyk =
500 /2
ft = 525 /2
εy0 = 2.5 εu = 25
� As2 = As1 = 12.57d2 = d1 = 5
Pb =π2
4
EJ
L2=
π2
4
33 000 · 0.0010675.02
= 3.47
P = 2 e =0.032
uu = 0.071
P = 2 e > 0.032
Mz=4.94 = 0.067e P → 0.064
Mz=0.06 = 0.206(e + uu)P → 0.206
E = 33 000 /Mx=0.06 = 0.172
Mu = 0.261
εcu1 = −0.0035
zc1, zc2z0
m Θ
m = �A, Θ = � J
� AJ Θ
Θ
�
· υ(t) + (t) = (t)
t
e
υe
e
e = ¯e + ¯e
(t) tt
t = 0
(t) = · υ(t)
e
· υ(t) + · υ(t) = (t)
,(t)
· υ(t) + · υ(t) = 0
t
υ = ξ ωt
ξ ω
T =2π
ω
· ξ = ω2 · ξn ξi, ωi, i = 1, . . . , n n
ξi
ξi Ξυ = Ξ ·υ ˜ = ΞT · ·Ξ
˜ = ΞT · · ΞΞT n
ω1
ω1 =
(√υT · · υυT · · υ
)=
2π
T1
υ T1
m k
ω =
√k
m, T = 2π
√m
k
n > 1ω1 T1 υ
mw + EJ w′′′′ = 0
EJ
w(x, t) =π x
Lωt
L
w =∂2w
∂t2= −ω2 π x
Lωt, w′′′′ =
∂4w
∂x4=
π4
L4
π x
Lωt
w(0, t) = w(L, t) = 0 EJ w′′(0, t) = EJ w′′(L, t) = 0
ω =π2
L2
√EJ
m
T =2L2
π
√m
EJ
ν = 1/T
E = 33 000 /25 / g ≈ 10 / � =0.025/10 = 2.5× 10−3 / A = 0.2 · 0.4
m = 0.2× 10−3 /
T = 2L2
π
√mEJ =
0.038 ν = 26
P0
P (t) = P0 f(t)
f(t) =
{1 t ≤ td0 t > td
P0, td P0 = −0.07 td = 0.1
nE = 20
Δt = 0.001 0.06
0.01060.0053 P0
td P0 = −0.07
tdtd = 0.001
P0 = −0.07 td = 0.1
As2 = 12.57 , d2 = 5
εc ≈ −1 εs ≈ 2
f(t)
· υ(t) + · υ(t) + · υ(t) = (t)