Chapter 3 BENDING MEMBERS

general This chapter is concerned with the design of members which are predominantly in bending. The rolled I section as shown in previous chapters is used widely for the bending structure. The usual requirement for a beam design is to provide sufficient resistance to bending moment.

Laterally restrained beam Cases where beams can be designed as fully restrained along the spans: 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). Insitu RC slab Figure 3.1: In-situ concrete slab

Beams with steel decking flooring system, with or without shear studs. 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. Compression flange of the beam is restrained from moving sideways by the cast in-situ concrete and studs. Full lateral restraint beam Shear stud Welded connection Steel plate Figure 3.2

Load distribution Beam is a bending member that it is usually subjected to lateral load (from concrete floor). Loads from slab are normally defined in ‘q’ kN/m2 and these loads are then transferred to supporting beams in either ‘w’ kN/m or ‘W’ kN.

Loads from reinforced concrete solid slab may be distributed to beams by the following method; One way slabs carry load in one direction Ly/Lx > 2.0

Precast Slab Loads supported by precast concrete slabs (Figure 3.4) are distributed to beams in one direction only. Hence the precast slab is one way spanning slab.

Two way spanning slab , where Ly/Lx ≤ 2.0

Complete this example and submit it within a week!!!

Design for laterally restrained beam Design check for restrained beam; Shear resistance, Clause 6.2.6 Bending moment resistance, Clause 6.2.5 Deflection

Shear resistance, Clause 6.2.6

Shear Area, Av

Shear Buckling

Example: Shear resistance

Exercise Determine the shear resistance for a 533x 210 x 92UB in bending assuming grade S 275steel.

Bending moment resistance, Clause 6.2.5

Bending and shear

Example 3.3

Deflection

Example 3.4 :Deflection

Resistance of the web to transverse forces Web Bearing and Buckling are modes of failure that arise from concentrated forces being transversely applied onto the flanges of beams or columns Web bearing failure means that the web yields at its most vulnerable location, close to the root radius adjacent to the flange where the force is applied, as illustrated in Figure 1.a. Figure 1.a: Web bearing failure

Resistance of the web to transverse forces Buckling of the web happens when the web is too slender to carry the transverse force being transferred from the flange. In this mode the web has to work as a strut in compression and it buckles as shown in Figure 1.b. It is assumed that the flange is adequately restrained in the lateral direction and therefore it can neither rotate nor move laterally. Figure 1.b: Web buckling failure

Resistance of the web to transverse forces Eurocode presents a single check to deal with these two failure modes. This single check accounts for the bearing and buckling of the web when the member is subject to a transverse force . BS EN 1993-1-1 does not give design verifications for the resistance of webs thus, designer are referred to BS EN 1993-1-5 Clause 6. Design calculations is required for concentrate transverse forces applied to girders from supports, cross beam, columns, ect. The concentrated loads are dispersed through plates, angles and flanges to the web of the supporting girder.

To determine, kF

To determine, ly

From eq. 6.13

Unrestrained Beam Eurocode 3 states, as with BS 5950, that both cross-sectional and member bending resistance must be verified:

Lateral torsional buckling, LTB It is important to recognize the characteristics of 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 the buckling mode associated with slender beams loaded about their major axis, without continuous lateral restraint. LTB is considered to be prevented if the compression flange is prevented from moving laterally.

Unrestrained Beam Cross section twisting and moving laterally

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

Unrestrained Beam There are three methods to check LTB in EC3; 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). A simplified assessment method for beams with restraints in buildings, Clause 6.3.2.4 The third is a general method for lateral and lateral torsional buckling of structural components, given in clause 6.3.4.

Unrestrained Beam

Unrestrained Beam Cross section twisting and moving laterally