design of flexural members 2
TRANSCRIPT
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Restraints and design of restraints against flexural torsional bucklingAsst. Prof. Hang Thu Vu
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What is ‘lateral’ or ‘flexural-torsional’ buckling of beams? Slender beams buckle sideways even under
perfectly vertical loads. In this context, slenderness is defined by ratio
For example, if the beam bends about major axis xx. “Slender” means ratio is more than the nominal limit
yrl /
axisminor about gyration of radius :
lengthsegment beam :
where
r
lr
l
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Member behaviours Look at movement of a cross section
The beam curves in the horizontal plane (lateral displacement) and the vertical plane.
The beam rotates about the minor axis yy (lateral rotation) and twists about its longitudinal axis zz.
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Simply supported beam
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Cantilever beam
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Effect on member moment capacity If the beam segment is ‘full lateral restraint’,
the moment capacity is the section capacity
that we derived in last lecture If the beam segment is partially restrained or
not restrained, the moment capacity reduces to
yes fZM
factor reduction sslendernesa :
factor onmodificatimoment a : where
s
m
sssmb MMM
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What do we focus in this lecture? Restraints:
Restraint overview Beam segments and sub-segments
consideration Definitions and usages for different types of
restraints Full restraint Partial restraint Continuous lateral restraint Lateral restraint
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What do we focus in this lecture?
Lateral rotation restraint Examples: practise to distinguish restraint types Design of restraints Examples: practise to compute design loads acting on
different restraints Member subjects to full lateral restraints
Study ways to generate full lateral restraint condition for beam segments
Examples: practise to compute different length limits for different cross section shape. Use the derived length limit to restrain segment and sub-segment length.
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Restraints
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Overview Restraint: An element or connection detail used to
prevent a beam cross-section from being affected by flexural torsional buckling
Restraints at beam supports are often supplemented by additional restraints along the span
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Segment and sub-segments Segment—a portion of a beam between fully
(F) or partially (P) or nil (U) restrained cross-sections. Restraint combinations (left and right) can be FF, PP or FP, FU, PU
Segment length, l—length of the segment Subsegment—a segment can be further
subdivided into portions by intermediate lateral (L) restraints to the critical flange. Restraint combinations can be FL, PL or LL.
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Critical flange – AS4100 Clause 5.5 Flange which displaces laterally and rotates
further than other flange(s) It is compression flange of a simple beam
and tension flange of a cantilever (refer to figures in slides 4, 5)
Critical section: the cross section that governs the design, i.e. where M = M*
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Full restraint (F)
Strongest configuration: no lateral displacement of critical flange, no section twisting (1)
Soften criteria: no lateral displacement of critical flange, partially prevent section twisting (2)
Soften criteria: no lateral displacement of a point within cross section, no section twisting (3)
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Example
Is it F? Type (1), (2) or (3)?
Answer: Left section: no lateral displacement for C, partially
prevent section twisting. Type (2) Right: no lateral displacement of a point within cross
section. Full twisting. Hence not F
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Partial restraint (P) P: No lateral displacement of some point within
cross section, partially prevent section twisting Example: Is it P?
Answer: Left: bottom flange cannot displace. It partly
prevents whole section from twisting. P Right: part of web can not displace. It partly
prevents whole section from twisting. P
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Continuous lateral restraint C: continuous restraint applied to the critical
flange by concrete slab, chequer plate, timber floor. Segment ends must be fully or partly restrained.
Example:
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Lateral restraint (LR)
LR: no lateral displacement of the critical flange. But twisting and rotation of the cross section is allowable
Example:
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Lateral rotation restraint LR: significant restraint against lateral
rotation of the critical flange Configuration:
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Design restraints Capacity of restraint element for
preventing lateral deflection for F, P, LR
When restraints are spaced more closely than necessary, the design action for the restraint element is reduced. Refer 5.4.3.1
supportselement at thesegment th theof action design the:
restraint theof action design the: where
025.0
*
*
**
f
r
fr
N
N
NN
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Example: Design force in lateral restraint element
The restraint element is connected to 2 adjacent segments. Segment 1: L1,UDL design load w. Segment 2: L2, design load P.
Solution: Design action for the lateral restraint element is
4025.0
8025.0
max 2
21
*
PL
wL
N r
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Design of restraints (cont.) Lateral restraints for parallel beams Where a restraining element continues over
several parallel beams into a reaction point, it is necessary to use factor 0.025 only for the most critical beam segment and 0.0125 for each of the remaining beams.
*__
*__
* 0125.0025.0 segotherfcritmostfr NNN
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Example: Design force for lateral restraint element for parallel beams
Design loads on the 3 parallel beams are uniform distributed load w/2, w and w/2. Hence the design load for the restraint element is:
8
2/0125.02
8025.0
22* LwwL
N r
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Design restraints
Capacity of restraint element for preventing section twisting for F, P
Similar as for lateral restraint. Clause 5.4.3.2
supportselement at thesegment th theof action design the:
restraint theof action design the: where
025.0
*
*
**
f
r
fr
N
N
NN
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Member subjects to full lateral restraints
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Moment capacity If the beam segment is considered ‘full
lateral restraint’, flexural-torsional buckling is effectively prevented and
The moment capacity is the section capacity
There are three different cases for a segment to be considered full lateral restrained
yes fZM
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Segment with Continuous lateral restraint (C) is considered to be fully lateral restraint. No further requirement. Clause 5.3.2.2.
Segment with intermediate lateral restraints (LR). Clause 5.3.2.3. (*)
Segment with full (F) or partial (P) restraint at segment ends. Clause 5.3.2.4. (**)
Sub-segment length in (*) and segment length in (**) must satisfy the length limit stated in Clause 5.3.2.4. Different limit formulas for different cross section shapes.
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Example: Compute length limit for universal beam The beam is OneSteel 300Plus 310 UB 20.4.
It is restrained laterally by intermediate lateral restraints (LR). Compute maximum spacing for restraints so that no flexural-torsional buckling occurs
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Example (cont.) Having:
fy = 320 MPa, ry = 38.3 OneSteel 300Plus 5th ed.
For the universal beam, the length limit is
Clause 5.3.2.4
The sub-segments subject to load Q, hence
Clause 5.3.2.4 Hence maximum sub-segment length is
y
my frl
2505080
8.0m
mm 1354320
2508.050803.38max l
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Next lecture
We will study how to compute Mb for member with segment/sub-segment without full lateral restraint
Reading: AS 4100 - 1998.
Section 5 “Member subject to bending”. Clause 5.6 to 5.8
AS 4100 Suppl – 1999. Section C5 “Steel designers’ handbook”, Chapter 5