Statics of Building Structures I., ERASMUSfast10.vsb.cz/koubova/SoBSI_theme3_frames.pdfVierendeel...
Transcript of Statics of Building Structures I., ERASMUSfast10.vsb.cz/koubova/SoBSI_theme3_frames.pdfVierendeel...
Department of Structural Mechanics
Faculty of Civil Engineering, VŠB-Technical University of Ostrava
Statics of Building Structures I., ERASMUS
Frame structure
• Basic properties of plane frame structure
• Simple open frame structure
• Simple closed frame structure
2 / 52Basic properties of plane frame structure
Examples of simple open plane frame
Types of plane frames
Frames: a) right-angled
b) oblique
c) branched
d) open (a), (b), (c)
3 / 52Basic properties of plane frame structure
Types of plane frames
Frames: a) right-angled
b) oblique
c) branched
d) closed (a), (b), (c)
Examples of simple closed plane frame
4 / 52Basic properties of plane frame structure
Branched frame
Types of plane frames
5 / 52
Examples of right-angled and oblique coupled frames
Types of plane frames, coupled frames
Basic properties of plane frame structure
Coupled frames – originates by combining several simple
open frames
6 / 52
Vierendeel truss and Storey frame
Types of plane frames
Basic properties of plane frame structure
Vierendeel truss – originates by combining several closed
frames side by side
Storey frame - originates by combining several closed frames
above each other
7 / 52Simple open frame structure
The first step of the Force method
Force method, simple open frame
8 / 52
Different ways to create a basic statically determinate structure within the second
step of the Force method
Force method, simple open frame
Simple open frame structure
9 / 52
Replacement of removed links by reactions or interactions in the third step of
the Force method
Force method, simple open frame
Simple open frame structure
10 / 52
Equation in pictures illustrating decomposition into loading states
Force method, simple open frame
0
0
0
:loading thermaland forcefor
conditions nalDeformatio
30333232131
20323222121
10313212111
XXX
XXX
XXX
Simple open frame structure
11 / 52
Force method, simple open frame
j
m
j
l
j
j
itj
m
j
l
jit
j
m
j
l
j
i
j
m
j
l
j
i
ikki
j
m
j
l
j
ki
j
m
j
l
j
ki
iki
dxh
tMdxtN
dxAE
NNdx
IE
MM
mdxAE
NNdx
IE
MM
jj
jj
jj
1 0
,1
1 0
,0i,0
1 0
0
1 0
0
i,0
,,
1 01 0
ki,
s0,
n
1k
k,
s
:loading thermal todue tscoefficien naldeformatio ofn Calculatio
:loading force todue tscoefficien naldeformatio ofn Calculatio
frame theof bars ofnumber is
:tscoefficien naldeformatio ofn Calculatio
)1,.....n i(for X
:formin structure ateindetermin statically timesnfor
written becan equations) (canonical conditions nalDeformatio
s
12 / 52
Force method, simple open frame, support shifting
Deformational conditions for support shifting:
s0,
1
,
330333232131
220323222121
110313212111
n1,...,ifor ii
n
k
kki dX
dXXX
dXXX
dXXX
s
13 / 52
Calculation of deformational coefficients due to support shifting
Force method, simple open frame, support shifting
vuMuHu
lwMwRw
M
udwdd
ww
aaaaaab
aaaaaab
aaa
bb
a
ba
b
b
)(
)(
)(
)(clockwise),(clockwise
),(u ),(u ),( ),( :shifts of directions and shiftingSupport
33
*
30
22
*
20
1
*
10
321
b
ba
Simple open frame structure
14 / 52
Force method, simple open frame, support shifting
330333232131
220323222121
110313212111
dXXX
dXXX
dXXX
vuuXXX
lwwXXX
XXX
vulw
udwdd
aab
aab
ab
aaaa
abbb
333232131
323222121
313212111
3020
10321
is
,
, , , ,
15 / 52
Two constraints in the axis of the same bar
Warning
Simple open frame structure
Bar “c-d” is supported against movement in the direction of
the axis of the bar.
It is necessary to take into account the influence of normal
forces on the deflection of the bar “c-d”.
Otherwise, the system of canonical equations singular.
16 / 52
Example 5.1
I1=0,002m2, I2=I3=0,004m2
6,03,5
2,1cos ,8,0
5,3
8,2sin
033,59)33333,1(1,2
8,2
5,38,21,2
0
22
1,
arctgarctg
mll ca
Simple open frame structure
17 / 52
Loading states and bending moment diagrams for loading states of Example 5.1
Example 5.1, solution
)(1H ),(7,5
8,2 ),(
7,5
8,2
0H ),(7,5
1 ),(
7,5
1
0H ,0 ),(30
a222
a111
a000
kNRkNR
kNRkNR
RkNR
ba
ba
ba
Simple open frame structure
18 / 52
Example 5.1, solution continuation
EE
EE
EEE
EEE
EEE
4,64989
3
5,376842.163
.002,0
1
6,41585)
3
36842,063158,0(
2
5,363
.002,0
1
5,2762
3
6,376842,1
.004,0
1
3
5,376842,1
.002,0
1
4,15026,3
004,03
63158,076842,1)
3
36842,063158,0(
2
5,376842,1
.002,0
1
1,1304
004,03
6,363158,0))36842,0
3
263158,0(
2
5,3036842
2
63158,15,363158,0(
.002,0
1
20
10
22
22
2112
2
11
0
0
20222121
10212111
XX
XX
Deformational conditions:
Simple open frame structure
19 / 52
Example 5.1, solution of linear equations
kNX
kNX
XX
XX
EEEEE
906,165,1345370
2274471
4,15024,15025,27621,1304
4,1502)6,41585()4,64989(1,1304
814,125,1345370
17240145
4,15024,15025,27621,1304
4,649894,1502)4,64989(1,1304
04,649895,27624,1502
06,415854,15021,1304
4,64989 ,
6,41585 ,
5,2762 ,
4,1502 ,
1,1304
2
1
21
21
201022211211
0
0
20222121
10212111
XX
XX
Deformational conditions:
Simple open frame structure
20 / 52
Example 5.1, completion, reactions and internal forces diagrams
)(557,16
)(381,10)557,16(7,5
8,2)814,12(
7,5
10
clockwise)(counter 814,12
)(557,16)557,16(100
)(381,40)557,16(7,5
8,2)814,12(
7,5
130
2
22110
1
22110
22110
kNXH
kNXRXRRR
kNmXM
kNXHXHHH
kNXRXRRR
b
bbbb
a
aaaa
aaaa
Simple open frame structure
21 / 52
Removal of internal links and its replacement by interactions
Simple closed frame
Simple open frame structure
22 / 52
First three steps of the Force method
Simple closed frame
.
3acy indetermin statical of Degree
constIE
ns
Simple open frame structure
23 / 52
Loading states and corresponding bending
moments diagrams for Example 5.2
Example 5.2, loading states
)(8
)(33320
)(6669
czech)in stav" "0. (i.e. "0" state loading
in theonly reactions Nonzero
0
0
0
kNRR
kN,RR
kN,RR
axax
bzbz
azaz
Simple open frame structure
Notice to bending moment diagram (see pic.):
“Bottom side” of horizontal bars is down.
“Bottom side” of vertical bars is considered at
the right side of bar.
24 / 52
Example 5.2, canonical equations
IEIE
IEIE
IEIE
IEIE
XXX
XXX
XXX
732,78))7,2(6,3)7,2(22
3
7,2)7,2(2
3
7,2)7,2((
1
088,101)6,34,56,3
3
6,36,3
3
6,3)6,3((
1
0 0
40,32))6,3(14,5
2
6,316,3)
2
6,3(16,3(
1
18)116,3)1()1(6,3114,5)1()1(4,5(
1
0
0
0
22
33
22
22
32233113
2112
11
30333232131
20323222121
10313212111
Calculation of deformational coefficients:
25 / 52
Example 5.2, canonical equations
kNVVXkNNNXkNmMMX
XXX
XXX
XX,X
IEIE
IEIE
IEIE
edecedecedec 667,2 ,602,8 ,209,2
:
952,20932,7870 0
012,7980 088,1014,32
95,238 0 432 18
952,209))7,2(
2
6,3)8,28(
6
9,547,2)7,2()
3
8,289,548,28(
2
7,27,2(
1
12,798))6,3(6,3
2
)8,28(6,37,2
2
9,546,37,2
2
9,548,28(
1
95,238)16,3
2
)8,28()1(7,2
2
9,54)1(7,2
2
9,548,28(
1
321
321
321
321
20
20
10
equations canonical ofSolution
:equations canonicalin on Substituti
:tscoefficien naldeformatio ofn Calculatio
26 / 52
Example 5.2, completion
Internal forces can be determined:
a) From the equilibrium conditions with
knowledge of reactions and statically
indeterminate forces
b) By superposition of loading states,
taking into account real value of
statically indeterminate forces (see
bellow)
Ad b):
xxxxx
xxxxx
xxxxx
NXNXNXN
VXVXVXVV
MXMXMXMM
3322110
3322110
3322110
N
Simple open frame structure
27 / 52
Example 5.2, completion
X1=-2,209kNm, X2=-8,602kNm, X3=-2,667kN
26,141kNmM
9,409kNmM 9,409kNmM
4,991kNmM 4,991kNmM
21,5573kNmM 21,5573kNmM
7,159kNmM (column) 7,159kNmM
m ax
dbdc
cdca
babd
abac
)0(667,2)6,3(602,8)1()209,2(9,54
)7,2(667,2)0(602,8)1()209,2(0
)7,2(667,2)0(602,8)1()209,2(0
)7,2(667,2)6,3(602,8)1()209,2(0
)7,2(667,2)6,3(602,81)209,2(8,28
max
3322110
M
M
M
M
M
MXMXMXMM
dc
ca
bd
ac
xxxxx
Simple open frame structure
28 / 52
Example 5.2, bending moments calculation
X1=-2,209kNm, X2=-8,602kNm, X3=-2,667kN
26,141kNmM
9,409kNmM 9,409kNmM
4,991kNmM 4,991kNmM
21,5573kNmM 21,5573kNmM
7,159kNmM (column) 7,159kNmM
m ax
dbdc
cdca
babd
abac
)0(667,2)6,3(602,8)1()209,2(9,54
)7,2(667,2)0(602,8)1()209,2(0
)7,2(667,2)0(602,8)1()209,2(0
)7,2(667,2)6,3(602,8)1()209,2(0
)7,2(667,2)6,3(602,81)209,2(8,28
max
3322110
M
M
M
M
M
MXMXMXMM
dc
ca
bd
ac
xxxxx
Simple open frame structure
29 / 52
Example 5.2, another way to calculate bending moments
X1=-2,209kNm, X2=-8,602kNm, X3=-2,667kN
kNmM
VMM
kNmMM
kNmM
NMM
kNM
NVMM
kNmMM
kNmMM
ba
ababba
acab
ac
eccaac
ac
ececceac
cdca
cecd
559,21
7,2304,5333,12159,77,2304,5
159,7
159,7
6,3602,86,38991,46,36,38
159,76,3602,87,2667,26,38209,2
6,37,26,38
991,4
991,47,2667,2209,2
:better
It is necessary to know some components
of internal forces in this shortened
calculation (e.g. shear force Vab)
Simple open frame structure
30 / 52
Resulting reactions, interactions and diagrams of internal forces for Example 5.2.
Example 5.2, completion
Simple open frame structure
31 / 52
Steel frame structure of industrial hall
Span 20,5 m
Examples of frame structures
32 / 52
Hall for manufacturing components for nuclear power plants,
Vitkovice
• Ground130 x 320 m
• Cranes with capacity 80 and 200 t
• Undermined area
Examples of frame structures
33 / 52
Steel frame structure of coupled hall, Vítkovice
• Span 30 a 24 m
• Cranes with capacity 80 a 50 t
• Undermined area
Examples of frame structures
34 / 52
Sport hall Slavia, Prague
Examples of frame structures
35 / 52
Administration Building, Glasgow, UK
Examples of frame structures
Space steel frame with bracing
36 / 52
Administration Building, Glasgow, UK
Examples of frame structures
Space steel frame with bracing
37 / 52
Administration Building, Glasgow, UK
Examples of frame structures
Space steel frame with bracing - detail
38 / 52
San Sebastian, Auditorium, Spain
Examples of frame structures
Space frame
39 / 52
San Sebastian, Auditorium, Spain
Examples of frame structures
40 / 52
Congress Centre, Brno Exhibition Centre
Visible space frame supporting structure
Examples of frame structures
41 / 52
University Children's Hospital, Brno
Supporting space frame structure with overhanging ends
Examples of frame structures
42 / 52
Elementary School, Brumov – Bylnice
Frame structure with bracing
Examples of frame structures
43 / 52
Aula, VŠB-TU Ostrava
Examples of frame structures
Space frame of reinforced concrete
44 / 52
Aula, VŠB-TU Ostrava
Examples of frame structures
Space frame of reinforced concrete - detail
45 / 52
Radio Free Europe, Prague
Vierendeel truss of
1968:
• Ground 59x83 m
• 6 pillars
Examples of frame structures
46 / 52
Radio Free Europe, Prague
Vierendeel truss of
1968:
• Ground 59x83 m
• 6 pillars
Examples of frame structures
47 / 52
Radio Free Europe, Prague
Vierendeel truss of
1968:
• Ground 59x83 m
• 6 pillars
Examples of frame structures
48 / 52
Road bridge, Karviná – Darkov Spa
RC arched bridge of 1925:
• Vierendeel truss
• Unique cross-bracing
• Height 6,25 m
• Deck length 55,8 m
• Width 6,25 m
Examples of frame structures
Photo: Ing. Renata Zdařilová
49 / 52
Road bridge, Karviná – Darkov Spa
Reinforced Concrete
arched bridge of
1925:
Examples of frame structures
Photo: Ing. Renata Zdařilová
50 / 52
Road bridge, Karviná – Darkov Spa
Examples of frame structures
Photo: Ing. Renata Zdařilová
Reinforced Concrete
arched bridge of
1925:
51 / 52
Road bridge, Karviná – Darkov Spa
Examples of frame structures
52 / 52
Road bridge, Karviná – Darkov Spa
Photo: Ing. Renata Zdařilová
Examples of frame structures