Design of Flexural Members 2

Restraints and design of restraints against flexural torsional bucklingAsst. Prof. Hang Thu Vu

hang@civil.uwa.edu.au

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

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.

Simply supported beam

Cantilever beam

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

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

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.

Restraints

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

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.

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*

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)

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

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

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:

Lateral restraint (LR)

LR: no lateral displacement of the critical flange. But twisting and rotation of the cross section is allowable

Example:

Lateral rotation restraint LR: significant restraint against lateral

rotation of the critical flange Configuration:

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

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

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

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

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

Member subjects to full lateral restraints

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

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.

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

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

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