10/10/2013

1

CHAPTER 3:

BENDING MEMBERS

General- The usual requirement for a beam design is to provide

sufficient resistance to bending moment

- However in some cases it is also necessary to consider other criteria such as shear or lateral-torsional buckling

- In general, to design such members, the structure should be checked for the following at critical sections;

1. Combination of bending and shear force

2. Deflection

3. Lateral restraint

4. Local buckling

5. Web bearing and buckling

10/10/2013

2

Types of restraining

condition of beam1. Restrained beam

A beam where the compression flange is restrained against lateral deflection and rotation. Only vertical deflection exists.

2. Unrestrained beam

The compression flange is not restrained from deflect laterally and rotate about the plan of the section which is called lateral torsional buckling.

Three component of displacement i.e. vertical, horizontal and torsional displacement

10/10/2013

3

Laterally restrained beam

Cases where beams can be designed as fully restrained along the spans: 1. Beams carrying in-situ reinforced concrete slabs.

The friction of concrete floor to the compression flange of the beam can be assumed to provide full lateral restraint (Figure 3.1).

2. Beams with steel decking flooring system, with or without shear studs or by sufficient bracing member added.The shear studs function as a simple concrete anchor and can be employed to provide a permanent bond between steel and concrete; enabling the two materials to act compositely (i.e steel beam and concrete slab can act as one component) Figure 3.2.

10/10/2013

4

10/10/2013

5

As a result of full lateral restraint along the compression flange of the beam, bending will only take place about y-x plane.

In other words, the beam is prevented from moving sideways. Hence, the beam deforms in the vertical plane only.

Beam

Restrained beam Unrestrained beam

10/10/2013

6

Load Distribution

One-way Spanning Slab

Ly

Lx

Beam

Beam

Bae

m

Bea

m

Slab

Ly/Lx >>>> 2.0

10/10/2013

7

One-way Spanning Slab

Two-way Spanning Slab

Ly/Lx 2.0

Baem

Beam

Beam

Beam

Slab

Ly

Lx

10/10/2013

8

Two-way Spanning Slab

Precast Concrete Slab

Ly/Lx >>>> 2.0, one-way slab

Ly/Lx 2.0, one-way slab

Precast concrete hollow-core SLAB

Ly/Lx >>>> 2.0,

one-way slab

Ly/Lx 2.0,

one-way slab

Ly

Lx

Ly

Lx

10/10/2013

9

One-way spanning slab

One way direction

One way direction

Precast Concrete Slab

Cast In-situ Slab

Ly/Lx >>>> 2.0, one-way slab

Ly/Lx 2.0, two-way slab

Cast-insitu slab

Ly/Lx 2.0,

two-way slab

Ly/Lx >>>> 2.0,

one-way slab

Ly

Lx

Ly

Lx

10/10/2013

10

SECONDARY BEAM

MAIN BEAM

Beam-to-column connection

Main beam

FLOOR PLAN

Main beam

Main beam

ColumnSecondary beam

Secondary beam

Figure below shows a portion of plan view of a building. The slab system is precast slab with loading as below:

Permanent action, Gk

- self weight of precast slab, brick wall and furnishing = 5.0kN/m2

Variable action, Qk = 4.0kN/m2

Determine the shear force and moment maximum for beam 1/A-B.

Example 3.1: Load distribution

I

I

I

I

A B

1

2

4.0m

5.0m

Pre-cast panel

10/10/2013

11

Design checks for laterally restrained beam

Shear resistance, Clause 6.2.6

Bending moment resistance, Clause 6.2.5

Deflection

Shear resistance, Clause 6.2.6

The design shear resistance of a cross-section,

(Clause 6.2.6 EC3) , is denoted by Vc,Rd,

0.1V

V

Rd,c

Ed Shear check

10/10/2013

12

In the absence of torsion, the shear resistance may be taken as the design plastic shear resistance,

The plastic shear resistance is basically defined as the yield strength in shear multiplied by a shear area Av (Clause 6.2.6(3).

The yield strength in shear is taken as fy/3 and this is used in a plastic shear resistance formulation.

0

,

)3/(

M

yv

Rdpl

fAV

====

0.6 fy

A

A

Main beam

Column

RdplV ,

Shear buckling

The resistance of the web to shear buckling should also be checked, though this is unlikely to affect cross-sections of standard hot-rolled proportions.

Shear buckling need not be considered provided:

)(0.1;235

where

webs dunstiffenefor 72

NAUKfromf

t

h

y

w

w

========

10/10/2013

13

Example 3.2: Shear resistance

10/10/2013

14

10/10/2013

15

Assignment 2

Bending moment resistance,

Clause 6.2.5

10/10/2013

16

Bending and shear (Clause 6.2.8)

Bending moment and shear force acting in combination on structural members is commonplace.

However, in the majority of cases (particularly when standard rolled section are adopted), the effect of shear force on moment resistance is negligible and may be ignored.

Clause 6.2.8(2) states that if the applied shear force is less than half the plastic shear resistance, its effect on the moment resistance may be neglected

For cases where the applied shear force is greater than half the plastic shear resistance of the cross section, the moment resistance should be calculated using a reduced design strength for the shear area, given by the equation;

fyr = (1-)fywhere = [(2VEd/Vpl,Rd)-1)2

for VEd > 0.5Vpl,Rd

Vpl,Rd may be obtained from Clause 6.2.6 and when torsion is present, it should be replaced by Vpl,T,Rd obtained from Clause 6.2.7.

10/10/2013

17

For I-cross section with equal flanges and bending about major axis, the reduced design plastic resistance moment allowing for the shear force may be alternatively be obtained from;

where, Aw = hw tw

10/10/2013

18

Example 3.3: Cross-section resistance under combined bending and shear

A short-span (1.4m), simply supported, laterally restrained beam is to be designed to carry a central point load of 1050kN as shown in Fig.1. The arrangement resulted in a maximum design shear force VEd of 525kN and a maximum design bending moment MEd of 367.5kNm. In this example a 406x178x74 UB in grade S275 steel is assessed for its suitability for this application.

10/10/2013

19

Deflection

Excessive deflections may impair the function of a structure, for example, leading to cracking of plaster, misalignments of crane rails, causing difficulty in opening doors, etc.

From the UK National Annex, NA 2.23 & 2.24, deflection checks should be made under unfactored variable actions Qk.

Table A1.4 (EN 1990): Design value of actions for use in the combination of actions

10/10/2013

20

Vertical deflection limits, NA.2.23

Design situation Deflection limit

Cantilevers Length/180

Beams carrying plaster or other brittle finish Span/360

Other beams (except purlins and sheeting rails) Span/200

Purlins and sheeting rails To suit cladding

NA to BS EN 1993-1-1:2005

Design situation Deflection limitTops of columns in single storey buildings, except portal frames

Height/300

Columns in portal frame buildings, not supporting crane runways

To suit cladding

In each storey of a building with more than one storey Height of storey/300

Horizontal deflection limits NA.2.24

NA to BS EN 1993-1-1:2005

u is overall horizontal displacement over the building height H

ui is horizontal displacement over a storey height Hi

10/10/2013

21

10/10/2013

22

Example 3.4 Deflection

A simply supported roof beam of span 5.6m is subjected to the following (unfactored) loading:

- Dead load: 8.6kN/m

- Imposed roof load: 20.5kN/m

- Snow load: 1.8kN/m

Choose a suitable UB such that the vertical deflection limits are not exceeded.

Example 3.5: Restrained Beam Design

The simply supported 610 x 229 x 125 UB of S275 steel shown below has a span of 6m. Check moment resistance, shear and deflection of the beam.

10/10/2013

23

Resistance of the web to

transverse force

-Refer to BS EN 1993-1-5 Clause 6

Design calculations are required for concentrated transverse forces applied to girders from supports, cross beams, columns, etc.

The concentrated loads are dispersed through plates, angles and flanges to the web of the supporting girder.

10/10/2013

24

The deformation that occur to the supporting beam due to transverse concentrated load: yielding of flange and local buckling of the web

The design resistance is expressed as:

10/10/2013

25

Example 3.6

The beam shown below is fully laterally restrained along its length and has bearing length of 50mm at the unstiffened supports and 75mm under the point load. Design the beam in S275 steel for the loading shown below.

Given:

Actions (loadings),

Permanent actions:

Uniformly distributed load (including self weight) g1 = 15kN/m

Concentrate load G1 = 40kN

Variable actions:

Uniformly distributed load q1 = 30kN/m

Concentrate load Q1 = 50kN

The variable actions are not due to storage and are not independent of each other

10/10/2013

26

STEP:

1)Load, MEd, VEd

10/10/2013

27

2)Cross-section classification

3)Shear resistance (also shear buckling) (6.2.6)

4)Bending moment resistance (6.2.5) and also check bending & shear (6.2.8)

5)Resistance of the web to transverse forces

- only required when there is bearing on the beam (refer to BS EN 1993-1-5 Clause 6 Resistance to transverse force)

10/10/2013

28

6)Deflection

Laterally unrestrained beam Lateral torsional buckling is the member buckling

mode associated with slender beams loaded about their major axis, without continuous lateral restraint.

The prime factors that influence the buckling strength of beams are un-braced span, cross sectional shape, type of end restraint anddistribution of moment.

10/10/2013

29

10/10/2013

30

Cross-sectional and member bending resistance must be verified

Lateral Torsional Buckling (LTB)

It exhibits vertical movement (bending about y-y axis), lateral displacement (bending about z-z axis) and rotation (about x-x axis).

It occurs when the buckling resistance about z-z axis and torsional resistance about the x-x axis are low.

10/10/2013

31

LTB is considered to be prevented if the compression flange is prevented from moving laterally.

Thus, intersection member or frictional restrained from floor units can prevent lateral movement of the compression flange.

For this beam failure will occur in another mode, generally in-plane bending (and/or shear).

Characteristics of LTB

Initially the beam bends about the major axis.

As the load increases the sideway displacement occurs.

Twisting of cross section

The sideway displacement bends about the minor axis.

The way to prevent LTB is to have adequate lateral bracing at the compression flange at adequate intervals along the beam.

Check should be carried out on all unrestrained segments of beams (between the points where lateral restraint exists).

10/10/2013

32

Design Buckling Resistance, Mb,Rd(Clause 6.3.2.1)

The design buckling resistance of an unrestrained beam (or unrestrained segment of beam) should be taken as

3 Methods to Check LTB

1. The primary method adopts the lateral torsional buckling curves given by equations 6.56 and 6.57from Clause 6.3.2.2 (general case) and Clause6.3.2.3 (for rolled sections and equivalent welded sections).

2. A simplified assessment method for beams with restraints in buildings, Clause 6.3.2.4

3. The third is a general method for lateral and lateral torsional buckling of structural components, given in Clause 6.3.4.

10/10/2013

33

Method 1: Lateral torsional buckling

curves (6.3.2.2 &6.3.2.3)

For the general case (6.3.2.2)

For rolled or equivalent welded sections case

(6.3.2.3)

(6.3.2.2)

(6.3.2.3)

LT refer Table 6.3 and 6.4

10/10/2013

34

Elastic critical moment for lateral

torsional buckling, Mcr

EC3 offers no formulations and gives no guidance on how Mcr should be calculated

It only mentioned in Clause 6.3.2.2(2) that Mcrshould be based on gross cross sectional properties and should take into account the loading conditions, the real moment distribution and the lateral restraints

The Mcr of a beam of uniform symmetrical cross-section with equal flanges, under standard conditions of restraint at each end loaded through the shear centre and subject to uniform moment is given by equation:

For uniform doubly-symmetric cross-sections, loaded through the shear centre at the level of the centroidal axis and with the standard conditions of restraint, Mcr may be calculated by:

10/10/2013

35

Standard condition of restraint at each end of the beam: restrained against lateral movement, restrained against rotation about the longitudinal axis and free to rotate on plan.

C1 factor: used to modify Mcr,0 (Mcr = Mcr.,0) to take account of the shape of bending moment diagram.

C1 factor for end moment may be approximated by equation:

where is the ratio of end moment from Table 6.11 and 6.12

10/10/2013

36

Table 6.11: C1 values for end moment loading

Table 6.12: C1 values for transverse loading

10/10/2013

37

Condition of restraints and Effective length

10/10/2013

38

Design procedure for LTB

check1. Determine effective(buckling) length Lcr depends on

boundary conditions and load level

2. Calculate Mcr3. Non-dimensional slenderness, LT4. Determine imperfection factor, LT5. Calculate buckling reduction factor, LT6. Design buckling resistance, Mb,Rd7. Check for each unrestrained portion

10/10/2013

39

Example 3.7: Lateral torsional

buckling resistance

A simply supported beam is required to span 10.8m and tosupport two secondary beams as shown in Figure 1. Thesecondary beams are connected through fin plates to the webof the primary beam and full lateral restraint may be assumedat these points. Select a suitable member for the primarybeam assuming grade S275 steel.

10/10/2013

40

Section properties for a 762 x 267 x 173 UB

CONCLUSIONRestrained beam

1. Design load, Design shear force, VEd, Design bending moment, MEd

2. Cross-section classification

3. Bending moment resistance Cl. 6.2.5

4. Shear resistance Cl. 6.2.6

- check also shear buckling

5. Combined bending and shear Cl. 6.2.8

6. Deflection Actual deflection < Deflection limit

7. Resistance to transverse force EC3-1-5 Cl. 6.

- only applied for beam with bearing

10/10/2013

41

Unrestrained beam

1. Same as restrained beam

2. Same as restrained beam

3. Same as restrained beam

4. Same as restrained beam

5. Same as restrained beam

6. Same as restrained beam

7. Buckling resistance in bending Cl. 6.3.2