Chapter-2_bending-members.pdf

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

Types of restraining condition of beam 1. 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

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.

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

Load Distribution

One-way Spanning Slab Ly/Lx  2.0 Ly

Slab Beam

Baem

Lx

Beam

Beam

One-way Spanning Slab

Two-way Spanning Slab Ly/Lx  2.0 Ly

Slab

Beam

Baem

Lx

Beam

Beam

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 Ly Lx Lx Ly/Lx  2.0, one-way slab

Ly/Lx  2.0, one-way slab

Precast Concrete Slab

One way direction One way direction

One-way spanning slab

Cast In-situ Slab Ly/Lx  2.0, one-way slab Ly/Lx  2.0, two-way slab

Cast-insitu slab

Ly

Ly Lx

Lx Ly/Lx  2.0, one-way slab

Ly/Lx  2.0, two-way slab

Beam-to-column connection

SECONDARY BEAM FLOOR PLAN

MAIN BEAM

Main beam Column

Main beam

Secondary beam

Secondary beam

Main beam

Example 3.1: Load distribution

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.

I 4.0m

I A

I 1 Pre-cast panel 5.0m

I 2 B

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,

Shear check

VEd  1 .0 Vc ,Rd

In the absence of torsion, the shear resistance may be taken as the design plastic shear resistance, V pl, Rd The plastic shear resistance is basically defined as the yield strength in shear multiplied by a shear area Av (Clause 6.2.6(3). A

V pl , Rd 

Main beam

Av ( f y / 3 )

 M0

A Column

≈ 0.6 fy The yield strength in shear is taken as fy/√3 and this is used in a plastic shear resistance formulation.

Shear buckling The resistance of the web to shear buckling should also be checked, though this is unlikely to affect crosssections of standard hot-rolled proportions. Shear buckling need not be considered provided: hw   72 tw 

where

for unstiffene d webs

 

235 ;   1.0 ( from UK NA) fy

Example 3.2: Shear resistance

Assignment 2

Bending moment resistance, Clause 6.2.5

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-ρ)fy where ρ = [(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.

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

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.

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

Vertical deflection limits, NA.2.23 NA to BS EN 1993-1-1:2005 Design situation Cantilevers Beams carrying plaster or other brittle finish Other beams (except purlins and sheeting rails) Purlins and sheeting rails

Deflection limit Length/180 Span/360 Span/200 To suit cladding

Horizontal deflection limits NA.2.24 NA to BS EN 1993-1-1:2005 Design situation Tops of columns in single storey buildings, except portal frames Columns in portal frame buildings, not supporting crane runways In each storey of a building with more than one storey

Deflection limit Height/300 To suit cladding Height of storey/300

u is overall horizontal displacement over the building height H ui is horizontal displacement over a storey height Hi

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.

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.

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:

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

STEP: 1)Load, MEd, VEd

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)

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 and distribution of moment.

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.

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).

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 6.57 from Clause 6.3.2.2 (general case) and Clause 6.3.2.3 (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.

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)

αLT – refer Table 6.3 and 6.4

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 Mcr should 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:

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

Table 6.11: C1 values for end moment loading

Table 6.12: C1 values for transverse loading

Condition of restraints and Effective length

Design procedure for LTB check 1. Determine effective(buckling) length Lcr – depends on boundary conditions and load level 2. Calculate Mcr 3. Non-dimensional slenderness, λLT 4. Determine imperfection factor, α LT 5. Calculate buckling reduction factor, χLT 6. Design buckling resistance, Mb,Rd 7. Check for each unrestrained portion

Example 3.7: Lateral torsional buckling resistance A simply supported beam is required to span 10.8m and to support two secondary beams as shown in Figure 1. The secondary beams are connected through fin plates to the web of the primary beam and full lateral restraint may be assumed at these points. Select a suitable member for the primary beam assuming grade S275 steel.

Section properties for a 762 x 267 x 173 UB

CONCLUSION Restrained 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

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