Reinforced Concrete Structures : Design according to CSA A23.3-04

8. Approximate Frame Analysis Method for Continuous RC Beams and One-Way Slabs

Chapter 8 Approximate Frame Analysis Method for Continuous RC Beams and One-Way Slabs 8.1 Notations Used in This Chapter CM Dimensionless moment coefficient CV Dimensionless shear force coefficient D Dead load L Live load L1 Length of span 1 L2 Length of span 2 Ln Clear span M Bending moment Mf Moment due to factored load Vf Factored shear force 248 Chapter 8 ldh Development length of standard hook in tension, measured from critical section to outside end of hook wf Factored uniformly distributed load αD Dead load factor αL Live load factor 8.2 Justification for Using the Approximate Frame Analysis Method RC building structures (beams, columns, floors, and slabs) are generally cast in place and are therefore considered as monolithic. These structures behave therefore as tridimensional statically indeterminate systems (Figure 8.1a). Their analysis is often long, complex, and tedious, especially when considering the various loading conditions and their combinations. The use of the approximate frame analysis method, which is recommended by CSA A23.3-04 for the analysis of simple and regular structures such as continuous RC beams and one-way slabs, to determine the moment and shear envelopes can be a good alternative . The method uses approximate coefficients and thereby lets the designer avoid resorting to complex analysis using software that is often time-consuming and difficult to implement. Figure 8.1 – Modeling: from a 3D Structure to a Simplified Model as per CSA A23.3-04 Approximate Frame Analysis Method for Continuous RC Beams and One-Way Slabs 249 8.3 Description of the Approximate Frame Analysis Method ■ ■ Principle Each floor is considered separately (usually the slab floor and the roof floor) in each direction, including the columns above and below the floor under consideration (see Figure 8.1c). ■ ■ Assumptions ➟ The vertical loads are taken by the columns. ➟ The lateral loads (wind and seismic loads), if any, are resisted by a dedicated bracing system. As a result, the floor supports only the vertical loads and acts as a diaphragm for transmitting the lateral forces to the bracing system. ■ ■ Conditions for Using the Approximate Frame Analysis Method The approximate frame analysis method can be used if the following conditions are met: Table 8.1 – Conditions for Using the Approximate Frame Analysis Method (CSA A23.3-04 Cl. 9.3.3) (a) Number of spans (b) Lengths L1 and L2 of adjacent spans (L1 > L2) (c) Type of applied loads (d) Ratio of factored live load to factored dead load (e) Geometry of members ≥ 2 L L L 1 2 1 20 − ≤ % Uniformly distributed α α L D L D ≤ 2 0 . Prismatic members Constant crosssectional dimensions along the span 8.4 Approximate Bending Moments and Shear Forces at Characteristic Points The factored bending moments, Mf, and the factored shear forces, Vf, at characteristic points of frames (end supports and mid-spans) can be determined based on the following equations, where the moment and shear coefficients CM and CV are as presented in Figure 8.2 for the various cases. 250 Chapter 8 M C w L f M f n = 2 (8.1) V C w L f V f n =       2 (8.2) Moments Shear forces a) Two spans b) More than two spans Figure 8.2 – Moments and Shear Forces at End Supports and at Mid-Span End span End span Exterior column support or spandrel beam Mid-span ; End span Exterior column support or spandrel beam ; End span Interior span Mid-span Mid-span (b3) (b2) (b1) Ln1/2 Ln2/2 Ln3/2 Exterior column support Spandrel beam or girder Spandrel beam or girder Exterior column support Approximate Frame Analysis Method for Continuous RC Beams and One-Way Slabs 251 8.5 Moment Envelope Diagram and Bar Cutoffs The moment envelope diagram and the locations of inflection points, obtained from the approximate frame analysis method, are required to determine the cutoffs of longitudinal bars. However, the standard does not provide any method for bar cutoffs. Two approaches can be used for this purpose: a) static analysis of spans; b) an approximate approach giving the bar-cutoff points. a) Two spans b) More than two spans Figure 8.3 – Moment Envelope Diagrams and Inflection Points for Common Cases Exterior column support Spandrel beam or girder 0.108Ln1 0.224Ln1 0.24Ln2 0.2Ln3 0.098Ln1 0.145Ln1 0.146Ln2 0.147Ln3 0.146Ln2 0.24Ln2 0.164Ln1 0.24Ln1 0.124Ln2 0...