5. Imperfections Josef Machacek
Czech Technical University y in Prague g
Objectives Types of imperfections
Objectives of the lecture
Introduction into analysis Global imperfections Imperfections for bracings
• This lecture describes various forms of imperfections of structures.
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
• It is shown, how these imperfections, whether at all, should be introduced into analysis.
Assessment 2 Examples
• Finally some basic examples are presented.
C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
Outline of the lecture
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
1. Types of imperfections 2. Introduction into analysis 3. Imperfections for global analysis 4. Imperfections p of structures for analysis y of bracings 5. Member imperfections p 6. Imperfections versus tolerances 7 Examples 7. 8. Conclusions
Lecture 5, V001, April 09
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Objectives Types of imperfections
1. Types of imperfections
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1
• Geometrical imperfections: – variance of dimensions of a structure or a member, e.g.: b±Δ
Member i imperfections f ti Imperfections vs. tolerances Assessment 2
– lack of verticality of a structure and straightness or flatness of a member member, e e.g.: g:
Examples C Conclusions l i Notes
h
φ
Lecture 5, V001, April 09
h
e0
φ
4
Objectives Types of imperfections
1. Types of imperfections
Introduction into analysis Global imperfections Imperfections for bracings
• Material imperfections: – variance of material properties, e.g.: σ
Assessment 1
±Δ
Member i imperfections f ti Imperfections vs. tolerances Assessment 2
ε
– residual stresses ((distribution in a cross section usually considered uniform along the member):
Examples
≈ fy
C Conclusions l i
σr
Notes
σr Lecture 5, V001, April 09
e. g. in I sections (both hot-rolled and welded)
5
Objectives Types of imperfections
1. Types of imperfections
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1
• Structural imperfections: – variance of boundary conditions, eccentricities in joints, e.g.:
Member i imperfections f ti
e0
Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
2. Introduction into analysis
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti
Introduction into analysis: • In strut ((frame)) analysis y all types yp of imperfections are usually introduced as equivalent geometrical imperfections (with i increased d value l off amplitude lit d e0d). )
Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i
• IIn plate l t (plated ( l t d structure) t t ) analysis l i geometrical ti l imperfections and residual stresses are introduced to derive buckling factors factors.
Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
2. Introduction into analysis
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
In frame analysis the following imperfections shall be introduced: • Global imperfections of frames or bracing systems (cover lack of verticality for frames or straightness of structure restrained by bracings)
Assessment 2 Examples C Conclusions l i Notes
• Local (member) imperfections of individual members (cover lack of straightness or flatness of a member and residual stresses of the member) Lecture 5, V001, April 09
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Objectives Types of imperfections
2. Introduction into analysis
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
• Other imperfections mentioned are covered by partial factors in the Limit State Design procedure. d In introduction of the equivalent geometrical imperfections (i.e. deflections) there is necessary to determine: 9 shape of the deflection (buckling mode); 9 amplitude of the deflection.
Lecture 5, V001, April 09
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Objectives Types of imperfections
3. Imperfections for global analysis
Introduction into analysis Global imperfections
1.
Imperfections for bracings Assessment 1
The amplitude of the shape (e0d) shall be d t determined i d ffrom E Eurocode d 3, 3 eq. 5.10, 5 10 securing required reliability in the most axially stressed cross section section.
Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
In general, the first critical buckling mode (ηcr) of the structure may be investigated and applied as imperfection shape for GNIA GNIA.
2.
Approximately, the global imperfection in sway mode (φ) and local geometrical imperfections (e0d) of individual members page: g mayy be introduced,, see next p
Lecture 5, V001, April 09
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Objectives Types of imperfections
3. Imperfections for global analysis
Introduction into analysis Global imperfections
• Global sway imperfections φ: V1
Imperfections for bracings
V2
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
φ V1 φ
≈
φ V2
V1 V2
For value of φ see Eurocode 3,, eq. q 5.5. In g general,, the sway imperfections are introduced into analysis as corresponding horizontal loadings Hi = φ Vi. Sway imperfections may be disregarded if the rate of horizontal/vertical loading is high (≥ 0,15), so that their contribution to internal forces is negligible.
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Objectives Types of imperfections
3. Imperfections for global analysis
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1
• Local geometrical imperfections are given as values e0d/L in Eurocode 3 (Tab. 5.1), which may be b replaced l db by corresponding di ttransverse uniform if loadings giving the same bending moments.
Member i imperfections f ti
NEd
NEd
Imperfections vs. tolerances Assessment 2
e0d
Examples
4 NEde0d L
8 NEde0d L2
≈
C Conclusions l i Notes
NEd Lecture 5, V001, April 09
NEd
4 NEde0d L 12
Objectives Types of imperfections
3. Imperfections for global analysis
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
• Usually these local imperfections are ignored in global analysis and covered by reduction factors χ and χLT in member checks checks, unless the frame is sensitive to 2nd order effects, i.e.: - a member b h has att lleastt one momentt resistant end joint; - and has simultaneously high slenderness given in Eurocode 3, eq. 5.8.
C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
4. Imperfections of structures for analysis l i off bracings b i • Bracing systems may provide lateral stability of a strut in compression or a beam in bending. The strut/beam should be considered with a geometrical imperfection (initial bow) of amplitude e0 = L/500 or less, taking number of strut/beams into account according to Eurocode 3, eq. 5.12.
Assessment 2 Examples C Conclusions l i
• The bow with amplitude e0 may be replaced by t transverse uniform if loading l di qd.
Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
4. Imperfections of structures for analysis l i off bracings b i • The loading qd corresponds to impact of sum of the amplitude e0 and the in-plane deflection of the bracing system δq. Such analysis requires 2nd order calculation or iterative procedure, see Eurocode 3, eq. 5.13: member in compression ( compression (or i flange force of a beam)
NEd
e0
NEd
q d = NEd 8
e0 + δ q L2
C Conclusions l i Notes
bracing system
L Lecture 5, V001, April 09
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Objectives Types of imperfections Introduction into analysis Global imperfections Imperfections for bracings Assessment 1
4. Imperfections of structures for analysis l i off bracings b i • Members supporting a splice of compression members have to be verified for additional force NEd/100.
Member i imperfections f ti
NEd 100
Imperfections vs. tolerances Assessment 2
NEd
NEd
Examples C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
Formative Assessment Question 1
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i
• Describe types of imperfections. • How the imperfections p are introduced into design of a steel structure? • Describe form of global imperfections and their design model. • Explain form of imperfection for bracings (e (e.g. g rafter bracing in a roof of an industrial building) and how to encompass it for design. g
Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
5. Member Imperfections
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1
• In general, the influence of member (local) imperfections is covered by reduction factors (in columns and beams by χ, χLT, in plates by ρ, χw, χF).
Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i
• Instead, when using GNIA (geometrically nonlinear analysis with imperfections or approximate second order analysis), the imperfections of critical shape are taken with amplitudes as follows:
Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
5. Member Imperfections
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1
• For compression struts the equivalent initial bow may be used with form of flexural buckling and amplitude e0d in accordance with Eurocode 3 (Tab (Tab. 5.1, e0d/L given), e.g.:
Member i imperfections f ti
e0d
Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
L e0d
is system y length; g ; depends on buckling curve and type of analysis.
• For beams in bending only equivalent initial bow in the direction of weak axis of the beam may be used, with amplitude 0.5 e0d (where e0d is as above): 0 5 e0d 0.5 Lecture 5, V001, April 09
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Objectives Types of imperfections
5. Member Imperfections
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i
• For plates the geometric imperfections should have amplitude of 80 % of assembly tolerances and residual stresses according to fabrication (say - compression stresses from 0.10 fy to 0.25 fy) or equivalent initial plate deflections with amplitude b/200 only and equivalent initial stiffener bow with amplitude L/400. For unstiffened compression plating e.g.: e0 = b/200
•
Notes
welds
+ fy
+
+ -
σres = - 0.25 fy
b Lecture 5, V001, April 09
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Objectives Types of imperfections
6. Imperfections versus tolerances
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
• Tolerances of structures/members are given in product standards and Eurocode EN 1090-2. • Eurocode distinguishes essential tolerances (required for due resistance) and functional tolerances (class 1 and more rigorous class 2 for fit up and appearance requirements).
Assessment 2 Examples C Conclusions l i Notes
• Essential tolerances have to be confirmed by inspection and testing to determine quality of the structure.
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Objectives Types of imperfections
6. Imperfections versus tolerances
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1
• Comparison of some imperfections e0d for analysis and essential tolerances: – columns:
Member i imperfections f ti
e0d = L/100 ÷ L/350 tolerance: L/750
Imperfections vs. tolerances Assessment 2 Examples
– girders:
C Conclusions l i
0,5 e0d, i.e. L/200 ÷ L/700
Notes
tolerance: L/750
Lecture 5, V001, April 09
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Objectives Types of imperfections
6. Imperfections versus tolerances
Introduction into analysis Global imperfections
– unstiffened plates and stiffeners:
Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
b
plates:
stiffeners:
e0d = b/200
e0d = b/400
tolerance: b/100 (obviously incorrect)
tolerance: b/400
– frames, e.g. simple portal frame: Δ
C Conclusions l i
Δd = h/200 h
Notes
Lecture 5, V001, April 09
tolerance: h/500
23
Objectives Types of imperfections
Formative Assessment Question 2
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
• How member imperfections are commonly introduced into design? • How member imperfections are introduced into GNIA? • Explain differences between imperfections and tolerances.
Assessment 2 Examples C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
7. Examples
Introduction into analysis Global imperfections
Example 1: Two-bay braced frame of a building
Imperfections for bracings Assessment 1
Example 2: Portal frame
Member i imperfections f ti Imperfections vs. tolerances
Example 3: Rafter bracing
Assessment 2 Examples C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis
Assessment 1
3 600
Member i imperfections f ti
C Conclusions l i Notes
6 000
HE 160 B
Examples
HE 160 B
Assessment 2
HE 160 B
Imperfections vs. tolerances
11 400
3 600
Imperfections for bracings
Example E l 1: 1 Two-bay braced frame of a building The frames spaced at distance of 6 m, bracing each 12 m.
4200
Global imperfections
6 000
Geometry and cross sections: composite p floor beams: A= 9345 mm2, I = 127.4.106 mm4 Lecture 5, V001, April 09
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections
L di [kN] and Loading d reactions ti
Imperfections for bracings
153.0
306.0
153.0
56.0
137.5
275.0
137.5
62.0
137.5
275.0
137.5
30.8 imp 1
Assessment 1 Member i imperfections f ti
i imp 2
Imperfections vs. tolerances
imp 3
Assessment 2 Examples
HEd,1
C Conclusions l i Notes
VEd,1
HEd,2 VEd,2
HEd,3 VEd,3
Note: Wind (horizontal) loading is due to this bracing.
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
• For formulas see Eurocode 3: • Σ HEd = 148.8 kN i.e. < 0.15 ΣVEd = 256.8 kN (→ φ need to be considered) • Sway imperfection for global analysis: 2 2 2 but α h ,min = = αh = 3 h 11.4
Assessment 2 Examples C Conclusions l i Notes
⎛ ⎝
α m = 0. 5 ⎜ 1 +
1⎞ 1⎞ ⎛ 0 5 1 = + . ⎟ ⎜ ⎟ = 0.82 m⎠ ⎝ 3⎠
1 2 φ = φ0 α h α m = ⋅ ⋅ 0.82 = 0.0027 200 3 Lecture 5, V001, April 09
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti
• Global imperfections imp 1 = φ ∑V = 0.0027 ⋅ (153 + 306 + 153 ) = 1.6 kN imp 2 = imp 3 = = φ ∑ V = 0.0027 ⋅ (137 .5 + 275 + 137 .5 ) = 1.5 kN
Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
These imperfections Th i f ti b belongs l tto each h cross frame. For analysis of the bracing frame appropriate total values (as for wind loading) need to considered. Here they are doubled ((belonging g g to two cross frames): ) Lecture 5, V001, April 09
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings
• Internal forces due to loading + doubled imperfections:
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2
-0.46
-0.79
Examples
-0.46 -217
-918
-428
C Conclusions l i Notes
MEd [kNm] Lecture 5, V001, April 09
NEd [kN] 30
Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
• Local imperfections for global analysis only if simultaneously (bottom central column): - exists moment resistant end joint: no (MEd ≈ 0) - slenderness A fy 5425 ⋅ 235 λ > 0,5 = 0,5 = 0.60 NEd 918.0 ⋅ 103 true, because:
λy 4200 / 67.8 λ= = = 0.66 λ1 93.9
Assessment 2 Examples C Conclusions l i Notes
• Therefore, the local imperfections could be ignored for global analysis in this example.
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections
• Imperfections for local analysis
Imperfections for bracings Assessment 1 Member i imperfections f ti
• In LA (linear analysis) the local imperfections are covered byy reduction factors (χ and χLT )).
Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti
• In GNIA analysis (see lecture 6 for details) •
The following imperfections should be used: 1. Generally either together with sway imperfections also approximate pp local ((sinusoidal)) bows with amplitudes p in accordance with Eurocode 3, Tab. 5.1:
Imperfections vs. tolerances
Buckling curve
Assessment 2 Examples C Conclusions l i Notes
Lecture 5, V001, April 09
Elastic analysis Plastic analysis e0 /L
a0
1/350
1/300
a
1/300
1/250
b
1/250
1/200
c
1/200
1/150
d
1/150
1/100
33
Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections
- i.e. at columns (HE profile): e0 = L/250 = 4 200/250 = 16.8 mm
Imperfections for bracings Assessment 1
(b kli curve b) (buckling - at composite beam approx.:
Member i imperfections f ti Imperfections vs. tolerances
e0 = L/200 = 4 200/200 = 21.0 21 0 mm (buckling curve c) - at bracing diagonals (L profile):
Assessment 2
e0 = L/250 = 3 662/250 = 15.0 mm (buckling curve b)
Examples C Conclusions l i Notes
Note: For this example however, the local imperfections can be ignored as shown above.
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti
2. Or unique global and local imperfection in the shape of the critical buckling mode (received from LBA) corresponding p g to buckling g of respective p member with amplitude e0. The first buckling mode in the present frame corresponds to buckling of the central column (non (non-sway sway mode):
Imperfections vs. tolerances Assessment 2
The first critical buckling mode:
Examples
e0
C Conclusions l i
αc,1 c 1 = 5.51 (Note: The first sway mode is the 15th, where αcr,15 = 144.08)
Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
Warning: In some cases the first critical mode corresponds to other (less important) members, e.g. hinged diagonals. The critical mode corresponding to required member (e (e.g. g column) may be of higher level. level This higher mode shall be taken for column imperfections (otherwise the solution is conservative). Example: If in our frame the bottom diagonals are 2L 70x6 and overhead diagonals 2L 60x6, the 4th critical mode should be taken for column design:
Assessment 2 Examples C Conclusions l i Notes
1st mode d αcr,1 = 1.66 Lecture 5, V001, April 09
2nd mode d αcr,2 = 2.16
3rd mode d αcr,3 = 4.80
4th mode d αcr,4 = 5.30 36
Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections
According to Eurocode 3, eq. 5.10:
Imperfections for bracings Assessment 1
2
(
e0 = α λ − 0.2
Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
Rk
χλ γ M1
Rk
1− χ λ
) MN
1−
2
- where for central bottom column: Nc,Rk = A fy = 5425 ⋅ 235 = 1275 ⋅ 10 3 N
α ult,k =
Lecture 5, V001, April 09
Nc,Rk NEd
1275 ⋅ 10 3 = = 1.39 3 918 ⋅ 10
37
Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings
- for diagonal 2 x L110/10 (other members not relevant): Nc, 0 ⋅ 235 35 = 996.4 ⋅ 10 03 N c Rk = A f y = 4240
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2
α ult,k =
N c,Rk NEd
996 .4 ⋅ 10 3 = = 4.4 3 228 .3 ⋅ 10
Lower, i.e. column decides. λ=
Examples
α ult,k 1.39 = = 0.50 5.51 α cr
C Conclusions l i Notes
For buckling curve b (α = 0.34): χ = 0.88
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.1 Example 1
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2
Mpl,Rk = Wplfy = 354 ⋅ 10 3 ⋅ 235 = 83.2 ⋅ 10 6 kNm
Resulting amplitude of imperfections in the shape of the first critical buckling mode: 2
(
e0 = α λ − 0.2
Notes
Rk
Rk
Examples C Conclusions l i
) MN
χλ 1− γ M1 1− χ λ
2
=
0.88 ⋅ 0.50 2 1− 83.2 ⋅ 10 6 1.00 = 8.5 mm = 0.34 ⋅ (0.50 − 0.2) ⋅ ⋅ 3 2 996.4 ⋅ 10 1 − 0.88 ⋅ 0.50
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections Imperfections for bracings
Example 2: Portal frame • Imperfections for global linear analysis
Assessment 1
12 kN/m'
Member i imperfections f ti Imperfections vs. tolerances
IPE 550
HE 340 B
Assessment 2 Examples
40 kN
imp 1 10000
HEd,1
HEd,2
24000
VEd,2 Ed 2
VEd,1
C Conclusions l i Notes
40 kN
geometry and cross sections Lecture 5, V001, April 09
loading and reactions
40
Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
• For formulas see Eurocode 3: • Σ HEd = 0 i.e. < 0.15 VEd ((consider φ) • Sway imperfection for global analysis (imp 1): 2 2 2 α = αh = = b t but h ,min 3 h 10 ⎛
Assessment 2 Examples C Conclusions l i Notes
1⎞ 1⎞ ⎛ ⎟ = 0.5⎜1 + ⎟ = 0.87 m⎠ 2⎠ ⎝ ⎝ 1 2 φ = φ0 α h α m = ⋅ ⋅ 0.87 = 0.0029 200 3
α m = 0. 5 ⎜ 1 +
• •
imp 1 = φ ∑ V = 0.0029 ⋅ (12 ⋅ 24 + 80 ) = 1.07 kN
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections
• Internal forces (loading + imperfections):
Imperfections for bracings Assessment 1
-374,6
387,1
144,5 -38,7
Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
143,5
-483,2 -183,5 MEd [kNm] [kN ]
MEd [kNm]
-184,5 NEd [kN]
NEd [kN]
-37,5
38,7 VEd [kN]
VEd [kN]
C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1
• Local imperfections for global analysis only if simultaneously (column concerned): - exists i t momentt resistant i t t end d joint: j i t - slenderness
Member i imperfections f ti
λ > 0,5
Imperfections vs. tolerances
Examples
Notes
NEd
= 0,5
17090 ⋅ 235 = 2,33 3 184.5 ⋅ 10
not true, true because λ 10000 / 146.5 λ= y = = 0.73 93.9 λ1
Assessment 2
C Conclusions l i
A fy
OK
• There, the local imperfections can be ignored in global analysis. Lecture 5, V001, April 09
43
Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections
• Imperfections for local analysis
Imperfections for bracings Assessment 1 Member i imperfections f ti
• In LA (linear analysis) the local imperfections are covered byy reduction factors (χ and χLT )).
Imperfections vs. tolerances Assessment 2 Examples
C Conclusions l i Notes
Lecture 5, V001, April 09
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Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections
•
Alternative GNIA analysis
Imperfections for bracings
•
In GNIA the following imperfections should be used:
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances
1. Generally either together with sway imperfections also approximate pp local ((sinusoidal)) bows with amplitudes p in accordance with Eurocode 3, Tab. 5.1: at columns: e0 = L/250 = 10000/250 = 40 mm
Assessment 2 Examples
(buckling curve b) at beam:
C Conclusions l i Notes
e0 = L/300 = 24000/300 = 80 mm (buckling curve a)
Note: For this example however, the local imperfections can be ignored as shown above. Lecture 5, V001, April 09
45
Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections Imperfections for bracings
2. Or unique global and local imperfection in the shape of the first critical buckling mode received from LBA, with amplitude p e0:
Assessment 1
e0
Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
The first critical buckling mode: αcr,1 = 6.93 6 93
C Conclusions l i Notes
Lecture 5, V001, April 09
46
Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections
According to Eurocode 3, eq. 5.10: 2
Imperfections for bracings Assessment 1
(
e0 = α λ − 0.2
Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
Rk
1− χ λ
) MN
2
- where for columns: Nc,Rk = A fy = 17090 ⋅ 235 = 4016 ⋅ 10 3 N
C Conclusions l i Notes
Rk
χλ 1− γ M1
α ult,k =
Lecture 5, V001, April 09
N c,Rk NEd
4016 ⋅ 10 3 = = 21,8 3 184 .5 ⋅ 10
47
Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
C Conclusions l i Notes
- for beam: Nc,,Rk = A fy = 13440 ⋅ 235 = 3158 ⋅ 103 N
α ult,k =
N c,Rk NEd
3158 ⋅ 10 3 = 81,6 = 3 384 .7 ⋅ 10
Lower, i.e. column decides. α ult,k 21,8 λ= = = 1.77 α cr 6.93
For buckling curve b (α = 0.34): χ = 0.26 Lecture 5, V001, April 09
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Objectives Types of imperfections
7.2 Example 2
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti
Imperfections vs. tolerances Assessment 2
M pl,Rk = Wplfy = 2408 ⋅ 10 3 ⋅ 235 = 565.9 ⋅ 10 6 kNm
Resulting amplitude of imperfections in the shape of the first critical buckling mode: 2
(
e0 = α λ − 0.2
Notes
Rk
Rk
Examples
C Conclusions l i
) MN
χλ 1− γ M1 1− χ λ
2
=
0.26 ⋅ 1.77 2 1− 565.9 ⋅ 10 6 1.00 = 0.34 ⋅ (1.77 − 0.2) ⋅ ⋅ = 75.2 mm 3 2 4016 ⋅ 10 1 − 0.26 ⋅ 1.77
Lecture 5, V001, April 09
49
Objectives Types of imperfections
7.3 Example 3
Introduction into analysis
Imperfections for bracings
Example 3: Rafter bracing purlin
Assessment 1
rafter IPE 550
Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
L=8x3=2 24 m
Global imperfections
qd = 1 kNm 6m 24 m
δq = 4.5 mm
10 x 6 = 60 m
C Conclusions l i Notes
Plan of the roof
Lecture 5, V001, April 09
Deflection of bracing
50
Objectives Types of imperfections
7.3 Example 3
Introduction into analysis Global imperfections Imperfections for bracings
•
Initial deflections with amplitude e0 of the bracing system will be replaced by equivalent stabilizing force qd: qd
Assessment 1
≈
Member i imperfections f ti
e0 = αmL/500
Imperfections vs. tolerances Assessment 2 Examples
C Conclusions l i Notes
D t from Data f former f calculations: l l ti - max. moment in the rafter: MEd = 362.0 kNm - max. force in the compression flange of the rafter: NEd = MEd/h = 362/0.5328 362/0 5328 = 679 679.4 4 kN - external loading per one bracing system: qd,ext = 3.70 kN/m Number of braced flanges per one bracing system: m = 11/3 = 3.67
Lecture 5, V001, April 09
51
Objectives Types of imperfections
7.3 Example 3
Introduction into analysis Global imperfections
•
Amplitude e0: ⎛ ⎝
Imperfections for bracings
α = 0.5⎜1 +
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples
C Conclusions l i Notes
1⎞ 1 ⎞ ⎛ ⎟ = 0 .5 ⋅ ⎜ 1 + ⎟ = 0.80 m⎠ ⎝ 3.67 ⎠
e0 = α mL / 500 = 0.80 ⋅ 24000 / 500 = 38.4 mm
•
Equivalent stabilizing loading qd requires iterative procedure. To avoid the iteration, suitable guess of the total deflection δq,(0) from stabilizing loading qd and all g qd,ext y Say: y external loading d ext is necessary. δ q(0 ) ≈
Lecture 5, V001, April 09
L = 48.0 mm 500
52
Objectives Types of imperfections
7.3 Example 3
Introduction into analysis Global imperfections Imperfections for bracings
and therefore the equivalent stabilizing loading : qd = ∑ NEd 8
e0 + δ q (0)
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2
L2
(
)
= 3.67 ⋅ 679.4 ⋅ 103 ⋅ 8 ⋅
38.4 + 48.0 = 2.99 N/mm N/ 2 24000
Check of the guess of δq(0): δ q( 1) = (qd + qd,ext )δ q (q=1) = (3.70 + 2.99 ) ⋅ 4.5 = 30.1 mm
Examples
C Conclusions l i Notes
The guess was OK, OK because conservative: δq(0) = 48.0 mm > δq(1) = 30.1 mm
Lecture 5, V001, April 09
53
Objectives Types of imperfections
8. Conclusions
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i
Notes
1) Imperfections significantly influence strength of structures. 2) In frame structures equivalent geometrical imperfections (initial deflections) may substitute all kinds of imperfections. 3) In plated structures preferably both initial deflections and residual stresses should be introduced into design. y, global g and local imperfections p have to 4)) Generally, be considered.
Lecture 5, V001, April 09
54
Objectives Types of imperfections
8. Conclusions
Introduction into analysis Global imperfections Imperfections for bracings Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i
Notes
5) Shape of the initial deflections is generally given by the first critical mode, approximately in the form of initial sway imperfection and individual bow imperfections of members. 6) Amplitude of the initial imperfections shall correspond to values given in Eurocode 3 (chapter 5.3.2) to secure required reliability of design. 7) In common design, the influence of imperfections is usually covered by global geometrical i imperfections f ti and d reduction d ti ffactors t ffor members. b 8) Compression residual stresses shall correspond to expected mean values values. Lecture 5, V001, April 09
55
Objectives Types of imperfections
Notes to Users of the Lecture
Introduction into analysis Global imperfections
•
This session Thi i iis ffor iimperfections f ti off structures t t and d requires i about 60 minutes lecturing and 60 minutes for tutorial session.
Imperfections for bracings
•
Within the lecturing, three types of imperfections necessary to account for in analysis of a structure are described described. In particular, introduction of global imperfections, imperfections for bracing systems and imperfections of individual members in compression and bending are shown. Attention is also paid to tolerances required by Eurocode EN 1090 for execution execution.
•
Further readings on the relevant documents from website of www.access-steel.com and relevant standards of national standard institutions are strongly recommended. recommended
•
Formative questions should be well answered before the summative questions completed within the tutorial session.
•
Keywords for the lecture:
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
initial deflections, residual stresses, global imperfections, imperfections for bracing systems, member imperfections, buckling mode mode, equivalent horizontal force force, tolerances tolerances. Lecture 5, V001, April 09
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Objectives Types of imperfections
Notes for lecturers
Introduction into analysis Global imperfections Imperfections for bracings
•
S bj t IImperfections Subject: f ti off structures. t t
•
Lecture duration: 60 minutes plus 60 minutes tutorial.
•
Keywords: initial deflections deflections, residual stresses stresses, global imperfections, imperfections for bracing systems, member imperfections, buckling mode, equivalent horizontal force, tolerances.
•
Aspects to be discussed: types of imperfections, necessity of their introduction into analysis.
•
Within the lecturing, lecturing the introduction of global and member imperfection should be practised and imperfections for bracing system in a roof of an industrial building as well.
•
Further F th reading: di relevant l td documents t www.access-steel.com t l and relevant standards of national standard institutions are strongly recommended.
•
Preparation P ti for f tutorial t t i l exercise: i see examples l within ithi th the lecture.
Assessment 1 Member i imperfections f ti Imperfections vs. tolerances Assessment 2 Examples C Conclusions l i Notes
Lecture 5, V001, April 09
57
