structural design ii - eth z · 2017-04-27 · structural design ii ... bending-flection relation...
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Structural Design II
1http://www.block.arch.ethz.ch/eq/
Philippe Block ∙ Joseph Schwartz
Structural design I+II
1. Introduction
2. Equilibrium and graphic static
3.+4. Cables
13. Cable-net and membrane structures
5.+7. Arches
14.+15. Vaults, domes and shells
16. Spatial arch-cable-structures
6.+8. Arch-cable-structures
12. Materials and dimensioning
9. Trusses
17. Spacial trusses
10.+11. Beams and frames
16. Shear walls and plates
20. Columns
19. Bending
Structural design I
Structural design II
Course overview
Bending
Bending and flectionBending stiffnessDeformation of bending beamsResistance of bending beamsResistance of framesStructures efficiency
3
Nc cN
tNNt
A B
F F
Bending and flection
Bending beam with flection
4
2
∆l
∆l
∆l
∆l∆l
1
5 5
2
∆l
∆l
44
∆l
1
Nc5
Nc4
c5N
c4N
Nt1 Nt1
Nt2 Nt2
Bending and flection
Distribution of the deformation along the girders heigth
5
rc
φ
r
th r z
cN
Nt
Nc
Nt
x = 1/r
𝑥 = 1 r
Bending and flection
Distortion of a bent beam segment
6
Bending
Bending and flectionBending stiffnessDeformation of bending beamsResistance of bending beamsResistance of framesStructures efficiency
7
r
M
r2
* z = * z =
z
1
1
φ φ
2 * z = * z = 2
z
1 t2
c1
t1
c1
t1
c2
t2
c1
c2
t1 NN N
N
N
N
N
N
NN
N
N
c2
t2
r
M
r2
* z = * z =
z
1
1
φ φ
2 * z = * z = 2
z
1 t2
c1
t1
c1
t1
c2
t2
c1
c2
t1 NN N
N
N
N
N
N
NN
N
N
c2
t2
Bending stiffness
Bending at different beams heights
8
∆l
∆l [mm]120
10
20
30
240 360
1∆l [mm]N [kN]S =
N
∆l
N [kN]
N = S *
Bending stiffness
Force-deformation diagram
9
φ
zr
* z = * z = M [Nmm]NN c
χ = 1/r [1/mm]
tN
cN
tN
Nc
t
ΧZ * zB =
1 φ
zr
* z = * z = M [Nmm]NN c
χ = 1/r [1/mm]
tN
cN
tN
Nc
t
ΧZ * zB =
1
χ = 1 r
χ
Bending stiffness
Bending moment-flection diagram
10
σ
z
σ
h
φ
r
t
Nc
tN
cN
Nt
z = 2/3 * h
Bending stiffness
Internal forces in a bent beam with rectangualr section at linear-elastic material behaviour
11
φ/2
2*h
h
t
φ/8
φ
h2*h
t
φ/2
2*t
φ φ/8
2*t
NcN
Nc
tN Nt Nt Nt
Nt Nt
cN cNcN
c
Bending stiffness
Influence of width and height of the cross-section to the bending strenght
12
Flansch
Flansch
Steg
A1 = A2
Steg
A2A1
Bending stiffness
Rectangular section and I-section with same height and same amount of material
Web
Flange
Flange
13
φ
r
b
z
t
h
Nt
Nc
cN
tN
Bending stiffness
Internal forces in a bent beam with I-section at linear-elastic material behaviour
14
z b
t
t
φ
r
cN /2
tN /2N /2c
N /2c
N /2c
tN /2
tN /2tN /2
z = 2/3 * b
Bending stiffness
Internal forces in a 90o bent beam with I-section at linear-elastic material behaviour
15
Bending
Bending and flectionBending stiffnessDeformation of bending beamsResistance of bending beamsResistance of framesStructures efficiency
16
w
h
l/4
t
l/4
l
A B
A B
F F
F F
Deformation of bending beams
Deflection of a beam with two single loads and the influence of the span l on the bending
17
2*l/4 2*l/4
8*w
2*l
BA
A B
F
FF
F
Deformation of bending beams
Deflection of a beam with two single loads and the influence of the span l on the bending
18
wz
l
cN
Nt
F
B
AF
F
Bv
Deformation of bending beams
Deflection of a cantilever with point load
19
w`
l
w
0.5*l
tN
cN
q
BA
q
Bv
w > w`
Deformation of bending beams
Deflection of a beam and a cantilever with distributed load
20
f
l
l`/l = 0.5
l`
l`/l = 0.4
0.84f
w = 38%
f
l`/l = 0.35
w = 23%
0.16fl
l`
0.5f
w
l`
l
w = 80%
l`
0.33f
l`
l`/l = 0.2
l` = 0
w = 100%
w = -20%
l`
0.5f
0.67f
l
l` l l`
Deformation of bending beams
Arch-cable structures and the deflection of a overhanging beam
21
Zug: Verlängerung
Zug: Verlängerung
Zug: Verlängerung
Druck: Verkürzung
Druck: Verkürzung
Druck: Verkürzung
Deformation of bending beams
Deflection of continuos beam
Tension: extension
Tension: extension
Tension: extension
Compression: shortening
Compression: shortening
Compression: shortening
22
Bending
Bending and flectionBending stiffnessDeformation of bending beamsResistance of bending beamsResistance of framesStructures efficiency
23
Bruch
Materialverhalten
Χ = 1/r [1/mm]
M
linear elastischesMaterialverhalten
1
z h
* z = * z = M [Nmm]
BB
1
plastisches
Nt Nc
Nt
Nc
χ = 1 r
Resistance of bending beams
Bending-flection relation at plastic material behaviour
linear elastic material behaviour
plasticmaterial behaviour
24
l
h/2
h/2
h
t
h/2
a
f * (t*h)/2
a
h/2
R,dQQR,d
QR,dQR,d
R,dQ
R,dQ
R,dQ R,dQ
R,dQ QR,d
QR,d
Nt
cN
d
Resistance of bending beams
Mechanical resistance of a beam at plastic material behaviour
25
a
h
t
h
a
h
a
2h
f * (t*h)/2
f * (t*h)
h
a
t
l
l
h
R,dQQR,d
R,dQR,d
QR,dQR,d
R,dQ QR,d
QR,d
QR,d
Q
d
d
Resistance of bending beams
Collapse load comparison between a beam with double height and two beams on top of the other
26
h
b
f * t *b
z = h
- t
t
a a
t
l
R,dQ
QR,d QR,d
QR,d QR,d
d f
f
f
Resistance of bending beams
Collapse load comparison of a beam with I-section
27
Bending
Bending and flectionBending stiffnessDeformation of bending beamsResistance of bending beamsResistance of framesStructures efficiency
28
ξ = 1/3
ξ = 0
ξ = 1
ξ = 3
ξ = inf
1
5QQ
1.143=
1
3
Q
1
2
Q
2Q =Q
Q =Q
1Q
Q3
Q
1 31/3
Q4
ξ Q
1.455=
3Q
4
4
5
52
3
Q
Q
3QQ
2
2QQ
1QQ
Q
1
5
1
1
4
4
Widerstandder Flansche
ξ * h
h
Maximum capacity load (collapse load) analysis
Resistance of frames 29
ξ = 3
ξ = inf
ξ = 1
ξ = 1/3
ξ = 0
Q =Q
Q
1Q
3
1
Q
Q =QQQ
1
2
Q
2
=Q
A
Q
1Q
=
4
5
Q
3Q
A
Q3
1.143
5
4
1
5
4
1
4
4
QQQ
1
Q
3
Q
2
4
1Q
2
Q1
Q
4
Q
1
3
A
1.455
1
Q
2
5
5
A
A
A
3
3
3
Q
A
2
5
A
4
2
2
A
5
A
Widerstandder Flansche
Resistance of frames
Maximum capacity load (collapse load) analysis
30
ξ = 1/3
ξ = 0
ξ = 1
ξ = 3
ξ = inf
1
5QQ
1.143=
1
3
Q
1
2
Q
2Q =Q
Q =Q
1Q
Q3
Q
1 31/3
Q4
ξ Q
1.455=
3Q
4
4
5
52
3
Q
Q
3QQ
2
2QQ
1QQ
Q
1
5
1
1
4
4
Widerstandder Flansche
ξ * h
h
Maximum capacity load (collapse load) analysis
Resistance of frames 31
eQ
Maximum capacity load (collapse load) analysis
Resistance of frames 32
Bending
Bending and flectionBending stiffnessDeformation of bending beamsResistance of bending beamsResistance of framesStructures efficiency
33
Structures efficiency
Comparison of cross-sections (under vertical loads) with increasing efficiency under bending stress from left to right
efficiency under bending stress
34
F
F
F B
B
B
A
A
A
A
F
A
F
B
B
Structures efficiency
Possible cross-sections for a beam loaded in the middle
35
12
8
f
2
205
14[m3]
Volumen (Q *l)
0
10
4
15 25
variable Querschnitte
konstante Querschnitte
6
l/h10
d
d
Structures efficiency
Required amount of material as a function of the slenderness l/h
constant section
variable section
36
l/h2010
0.004
0.012
0
variable Querschnitte
0.010
0.014
konstante Querschnitte
15 255
0.002
0.006
w/l
hw
l0.008
F
BA
Structures efficiency
Deflections at the midspan as a result of a centered load as a function of the slenderness l/h (εmax = 0.001)
constant section
variable section
37
prof. schwartz Tragwerksentwurf III 2Allgemein
Links: Erforderliche Materialmengen in Funktion der Schlankheit l/h Rechts: Richtwerte für die Schlankheit von Tragwerken
* Fachwerk mit konstanten Diagonalen * Bei Konsolen entspricht l der zweifachen Spannweite der Konsole, bei Balken
mit Konsolen bzw. Durchlaufträgern entspricht l dem Abstand der beiden
zwischen den Auflagern liegenden Gelenken.
l
l
l
l/h
Structures efficiency
Useful slenderness rate of structures
Material Structure
Vierendeel girderTruss girderSpace trussBeamBeamPlate
BeamPlateBeam
economic slenderness
Steel
Wood
Prestressedconcrete
Reinforcedconcrete
38
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