Lesson 31 Nodal Forces and Two-way Slabs Version 2 CE IIT, Kharagpur
Instructional Objectives: At the end of this lesson, the student should be able to: •derive the expression for determining the work done by bending and twisting moments when the yield lines are at angles with the directions of reinforcing bars, •state the need for considering the nodal forces and to estimate their values when one yield line meets another yield line or a free edge, •to select the possible yield pattern of a two-way slab supported at four sides either by simple supports or fixed supports, •to finalise the yield patterns and to evaluate the collapse loads of two-way slabs, either simply supported or clamped at four sides, •apply the theory in solving numerical problems of slabs to finalise the yield pattern and to determine the collapse load employing (i) the method of segmental equilibrium and (ii) the method of virtual work. 12.31.1 Introduction Lesson 30 introduces the yield line analysis, which is an upper bound method of analysis for slabs. The different rules for predicting the yield lines are stated. The two methods i.e., (i) method of segmental equilibrium and (ii) method of virtual work are explained. Applications of both the methods are illustrated through numerical problems of one-way slabs – either simply supported or continuous. This lesson presents the derivations of the expressions for determining bending and torsional moments when yield lines are at angles with the directions of reinforcement. The need for the nodal forces and their determinations are explained when yield line meets another yield line or the free edge. Thereafter, different possible yield patterns of two-way slabs are explained. Numerical illustrative problems of two-way slabs with or without nodal forces are illustrated. Version 2 CE IIT, Kharagpur
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12.31.2 Work Done by Yield Line Moments Normally, the reinforcing bars are placed in two mutually perpendicular directions parallel to the sides of rectangular and square slabs. However, the yield lines may be at an angle with the direction of reinforcing bars as shown in Fig. 12.31.1, in which the yield line AB of length Lhas bending moment Mband twisting moment Mtper unit length of the yield line. The slab segment is undergoing rigid body rotation whose components are θxand θy. The horizontal and vertical projections of the yield line are having moment capacities of Mxand Myper unit length, respectively. All moments and rotations are shown using the right hand thumb rule. The following expression is derived for obtaining the absolute values of the work done by Mband Mton the yield line AB. With reference to Fig. 12.31.1, the total work done by the bending and twisting moments Mband Mtis, W = MbL(θxcos θ+ θysin θ) + MtL(-θxsin θ+ θycos θ) (12.19) Version 2 CE IIT, Kharagpur