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Iu, Jerry & Bradford, M.(2015)Novel non-linear elastic structural analysis with generalised transverse el-ement loads using a refined finite element.Advanced Steel Construction, 11(2), pp. 223-249.
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https://doi.org/10.18057/IJASC.2015.11.2.6
1
y
z
xv
P2P1
Q qMz2
Mz1
Figure 1. Equilibrium of beam-column element about z-axis under element loadings
2
∆Q
∆Q
P P
∆Q
e ∆Q
e ∆Q
Figure 2. Numerical procedures using the conventional finite element method
a) No lateral movement at roller support
b) Vertical deflection by tangent stiffness
c) Unbalanced forced by secant stiffness
d) Axial deformation by tangent stiffness
e) Achieving equilibrium condition
3
∆Q
e ∆Q
e ∆Q
P P
Figure 3. Numerical procedures using the present approach
a) No lateral movement at roller support
b) Deformations by tangent stiffness
c) Achieving equilibrium condition
4
Qa=L/2
x=L/2∆
Figure 4. A propped cantilever subjected to a mid-span point load
5
Qa=L/2
x=L/2 ∆
Qa=L/2
x=L/2 ∆
Qa=L/3
∆x=L/2
Figure 5. Simply-supported beam subjected to a point load at different locations
a) Mid-span deflection of beam under a mid-span load
b) Mid-span deflection of beam under a third-point load
c) Third-point deflection of beam under a third-span load
6
q
x=L/2∆
a=L/3 b=L/3
x=L/2∆
qL/3 L/3 L/3
Figure 6. Simply-supported beam subjected to various trapezoidal loads
a) Mid-span deflection of beam under partial uniform load
b) Mid-span deflection of beam under trapezoidal load
7
Figure 7. Deflection of a beam under uniform distributed load at mid-span
EIqL
3844 4
EI
qLEI
qL3845
3845 44
4qLEI∆
q=P/L∆
P
q∆
8
Figure 8. Deflection of a beam under uniform distributed load at one-third of span
EIqL
9729 4
EI
qLEI
qL97211
97211 44
4qLEI∆
q=P/LP
∆
L/ 3
q∆
L/ 3
9
Figure 9. Deflection of a beam under a single point load at mid-span
EIQL64
3
EI
QLEI
QL4848
33
3PLEI∆
Q=PP
∆
Q
∆
10
Figure 10. Deflection of a beam under a single point load at one-third of span
Q=PP
∆L/ 3
EIQL
129618 3
EIQL
EIQL
129633.23
129623 33
3PLEI∆
Q
∆L/ 3
11
Figure11. Deflection of a beam under two point loads at mid-span
EIQL
3849 3
EI
QLEI
QL38411
38411 33
3PLEI∆
Q Q0.25L 0.25L0.5L
∆
Q=PQ=PP
0.25L 0.25L0.5L
∆
12
Figure12. Deflection of a beam under two point loads at a quarter of span
EIQL
307254 3
EI
QLEI
QL307263
307264 33
3PLEI∆
Q=PQ=PP
0.25L 0.25L0.5L
∆
Q Q0.25L 0.25L0.5L
∆
13
Figure 13. Load-deformation response of right-angle frame
4025.0 mI z =
23871.0 mA =
295.68 mmNE =
mL 4.25=me 254.0=
Qe
θ
14
∆
3.6575m
3.6575m
6.09m
q
q
αq
W 1
296
W 14 48
W 14 48
W 1
296
W 1
296
W 1
296
E = 200kN/m2
Figure 14. Geometry of two-storey building frame
15
Figure 15. Lateral drift ∆ and load factor relationship for two storey frame
101
=α
1001
=α
10001
=α
∆
