US 12,129,610 B2 · Combined plate-beam unit analysis method considering residual stress effect of orthotropic plate

Abstract

A combined plate-beam unit analysis method considering a residual stress effect of an orthotropic plate, which is used for analyzing an orthotropic steel bridge deck welded by a top plate of a bridge deck and a trapezoidal rib, the top plate of the bridge deck is analyzed by a flat shell unit, while each sub-plate forming a trapezoidal rib is analyzed by a plate-beam unit. The welding residual stress of a top plate and a trapezoidal rib is obtained by a residual stress self-balancing condition, and the initial deformation of a top plate and left and right web plates of a trapezoidal rib is obtained by the stress-strain relationship. The combined plate-beam unit analysis method has the advantages of less freedom and high calculation accuracy, so it is especially suitable for structural analysis of the trapezoidal rib orthotropic plates.

A combined plate-beam unit analysis method considering a residual stress effect of an orthotropic plate, comprising the steps of:

S1, an analysis object serving as an orthotropic steel bridge deck welded by a top plate of a bridge deck and a trapezoidal rib, wherein the trapezoidal rib orthotropic plate is discretized into a combined plate-beam unit, the top plate is analyzed by a flat shell unit, and each sub-plate forming the trapezoidal rib is analyzed by a plate-beam unit;
S2:
the top plate having four nodes 1, 2, 3 and 4, wherein the analysis of the top plate is superimposed by the plane stress problem and the thin plate bending problem with small deflection, and each node has five degrees of freedom, including the linear displacement degrees of freedom u and v corresponding to the plane stress problem, and the linear displacement and the rotational angle degrees of freedom w, θx and θy corresponding to the thin plate bending problem with small deflection,
establishing a displacement array of top plate nodes of the combined plate-beam unit;
δ=[δ1δ2δ3δ4]T (1),
where δi=[ui vi wi θxi θyi] (i=1,2,3,4);
S3:
each plate-beam sub-unit forming the trapezoidal rib in mn section having the axial displacement adopting the first-order polynomial in the longitudinal direction and the vertical displacement adopting the third-order polynomial in the longitudinal direction, wherein the interpolation functions are as follows:
m
=
[
m
1
⁢
⁢
m
2
]
m
1
=
1
-
ζ
m
2
=
ζ
}
,
(
2
)
n
=
[
n
1
⁢
⁢
n
2
⁢
⁢
n
3
⁢
⁢
n
4
]
n
1
=
1
-
3
⁢
ζ
2
+
2
⁢
ζ
3
n
2
=
(
ζ
-
2
⁢
ζ
2
+
ζ
3
)
⁢
d
n
3
=
3
⁢
ζ
2
-
2
⁢
ζ
3
n
4
=
(
-
ζ
2
+
3
⁢
ζ
3
)
⁢
d
}
,
(
3
)
where
ζ
=
x
d
,
d is the length of the trapezoidal rib in mn section;
S4:
obtaining the displacement of each node of the trapezoidal rib according to the deformation coordination condition between the top plate and the trapezoidal rib, and obtaining the longitudinal displacement of the trapezoidal rib nodes 7 and 8 by combining the cross-sectional size and displacement parameters of the combined plate-beam unit and the displacement field of the plane stress unit:
u
7
=
1
2
⁢
(
1
-
a
k
)
⁢
u
1
+
1
2
⁢
(
1
+
a
k
)
⁢
u
4
,
(
4
)
u
8
=
1
2
⁢
(
1
+
a
k
)
⁢
u
1
+
1
2
⁢
(
1
-
a
k
)
⁢
u
4
,
(
5
)
where a is the distance between the upper ends of a trapezoidal rib web plate;
k is the width of the top plate of the combined plate-beam unit;
the rotation angles of nodes 7 and 8 around y axis are:
θ
y
⁢
⁢
7
=
1
2
⁢
(
1
-
a
k
)
⁢
θ
y
⁢
⁢
1
+
1
2
⁢
(
1
+
a
k
)
⁢
θ
y
⁢
⁢
4
,
(
6
)
θ
y
⁢
⁢
8
=
1
2
⁢
(
1
+
a
k
)
⁢
θ
y
⁢
⁢
1
+
1
2
⁢
(
1
-
a
k
)
⁢
θ
y
⁢
⁢
4
,
(
7
)
obtaining the vertical displacement of nodes 7 and 8 in combination with the displacement field of thin plate bending with small deflection:
w
7
=
1
4
⁢
(
1
-
a
k
)
⁡
[
2
-
a
k
-
(
a
k
)
2
]
⁢
w
1
+
k
8
⁢
(
1
-
a
k
)
⁡
[
1
-
(
a
k
)
2
]
⁢
θ
x
⁢
⁢
1
+
1
4
⁢
(
1
+
a
k
)
⁡
[
2
+
a
k
-
(
a
k
)
2
]
⁢
w
4
-
k
8
⁢
(
1
+
a
k
)
⁡
[
1
-
(
a
k
)
2
]
⁢
θ
x
⁢
⁢
4
,
(
8
)
w
8
=
1
4
⁢
(
1
+
a
k
)
⁡
[
2
+
a
k
-
(
a
k
)
2
]
⁢
w
1
+
k
8
⁢
(
1
+
a
k
)
⁡
[
1
-
(
a
k
)
2
]
⁢
θ
x
⁢
⁢
1
+
1
4
⁢
(
1
-
a
k
)
⁡
[
2
-
a
k
-
(
a
k
)
2
]
⁢
w
4
-
k
8
⁢
(
1
-
a
k
)
⁡
[
1
-
(
a
k
)
2
]
⁢
θ
x
⁢
⁢
4
,
(
9
)
S5:
considering the same rotation angle of each section, without considering the extrusion of longitudinal fibers of a plate-beam sub-unit, wherein the rotation angles of left and right web plates at m end around y axis are the same as those of nodes 8 and 7 around y axis, respectively, and the rotation angle of the bottom plate center around y axis is linearly interpolated between nodes 5 and 6, namely:
θ
lcm
=
1
2
⁢
(
1
+
a
k
)
⁢
θ
y
⁢
⁢
1
+
1
2
⁢
(
1
-
a
k
)
⁢
θ
y
⁢
⁢
4
,
(
10
)
θ
rcm
=
1
2
⁢
(
1
-
a
k
)
⁢
θ
y
⁢
⁢
1
+
1
2
⁢
(
1
+
a
k
)
⁢
θ
y
⁢
⁢
4
,
(
11
)
θ
bcm
=
1
2
⁢
θ
lcm
+
1
2
⁢
θ
rcm
,
(
12
)
S6:
obtaining the displacement mode of the m end,
wherein the longitudinal displacement at the centroid of the left and right web plates at the m end is:
u
lcm
=
u
8
-
h
2
⁢
θ
lcm
,
(
13
)
u
rcm
=
u
7
-
h
2
⁢
θ
rcm
,
(
14
)
where h is the height of a trapezoidal rib web plate;
the longitudinal displacement of nodes 5 and 6 is:
u5=u8−hθlcm (15),
u6=u7−hθrcm (16),
linear interpolation is performed between nodes 5 and 6 to obtain the longitudinal displacement at the centroid of the bottom plate at the m end:
ubcm=½u5+½u6 (17),
the vertical displacement of the left and right web plates at the m end at the centroid is expressed as:
wlcm=w8 (18),
wrcm=w7 (19),
linear interpolation is performed between nodes 5 and 6 to obtain the vertical displacement at the centroid of the bottom plate at the m end:
wbcm=½wlcm+½wrcm (20);
S7:
obtaining the displacement mode of the n end according to the method of obtaining the displacement mode of the m end in S6;
S8:
obtaining the longitudinal and vertical displacements of the left web plate and the bottom plate,
wherein the node displacement parameters of longitudinal and vertical displacements at the centroid of the left web plate-beam sub-unit are expressed by ul* and wl*, respectively:
ul*=[ulcmulcn]T (21),
wl*=[wlcmθlcmwlcnθlcn]T (22),
the longitudinal and vertical displacements are:
ulc=nul*=nAδ (23),
wlc=mwl*=nBδ (24),
where A is the transformation matrix of ul* and δ;
B is the transformation matrix of wl* and δ;
S9:
obtaining the longitudinal and vertical displacements of the right web plate and the bottom plate according to the method of obtaining the longitudinal and vertical displacements of the left web plate and the bottom plate in S8;
S10:
obtaining the stiffness matrix of the trapezoidal rib by using the potential energy variational method according to the displacement modes of each plate-beam sub-unit of the trapezoidal rib obtained in S4 to S9:
Π=Πq+Πl+Πr+Πb−FeTδ (25),
where:
Πq is the strain energy of a top plate unit;
Πl, Πr, and Πb are the strain energy of the left and right web plates and the bottom plate unit of the trapezoidal rib, respectively;
FeT is an external force load array;
the strain energy of the left and right web plates and the bottom plate unit of the trapezoidal rib is expressed as:
Πl=½∫0dEAl(n′ul*)2dx+½∫0dEIyl(m′wl*)2dx=½δTKleδ (26),
Πr=½∫0dEAr(n′ur*)2dx+½∫0dEIyr(m′wr*)2dx=½δTKreδ (27),
Πb=½∫0dEAb(n′ub*)2dx+½∫0dEIyb(m′wb*)2dx=½δTKbeδ (28),
where:
Kle, EAl and EIyl are the stiffness matrix, the axial stiffness and the vertical bending stiffness of the left web plate of the trapezoidal rib, respectively;
Kre, EAr and EIyr are the stiffness matrix, the axial stiffness and the vertical bending stiffness of the right web plate of the trapezoidal rib respectively;
Kbe, EAb and EIyb are the stiffness matrix, the axial stiffness and the vertical bending stiffness of the bottom plate of the trapezoidal rib, respectively;
S11:
assuming that the residual stress in the top plate along the x direction is a constant σpx0, and σsx0 the longitudinal residual stress σsx(z) of the trapezoidal rib gradually changes from σpx0 to σsx0 along the z direction,
according to the self-balance condition of the residual stress, obtaining:
1
2
⁢
tk
⁢
⁢
σ
px
⁢
⁢
0
+
t
1
⁢
∫
0
g
1
⁢
l
g
1
⁢
σ
px
⁢
⁢
0
⁢
dl
-
t
1
⁢
∫
0
g
2
⁢
l
g
2
⁢
σ
sx
⁢
⁢
0
⁢
dl
=
0
,
(
29
)
according to formula (29), g1 and g2 is obtained from the residual stresses σpx0 and σsx0, and g1 and g2 represent the distribution width of the residual stresses in two directions, respectively,
for the top plate, the initial deformation of the top plate is obtained by substituting the residual stress σpx0 into the stress matrix of the plane strain unit,
for the trapezoidal rib, the initial deformation of the left and right web plates is obtained by combining the stiffness matrix and the residual stress distribution of the left and right web plates, wherein the stiffness matrix of the left web plate is Kle, the stiffness matrix of the right web plate is Kre, and the residual stress is σsx(z).

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Combined plate-beam unit analysis method considering residual stress effect of orthotropic plate

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Combined plate-beam unit analysis method considering residual stress effect of orthotropic plate

Life of this patent

Abstract

Claims1