Moment Redistribution

Transcription

1 Moment Redistribution Objective To examine the application of plastic design to reinforced concrete structures. Introduction to Plastic Design What happens when an indeterminate beam is overloaded? Progressive development of hinges and consequent change in the bending moment diagram. Encastre beam example Continuous beam example Two Fundamental Theorems Lower-bound Theorem If a set of bending moments can be found that satisfies equilibrium throughout the structure and which does not violate the yield condition anywhere, then the corresponding load is either less that or equal to the correct collapse load. Upper-bound Theorem The external load that is obtained from any assumed collapse mechanism must be either greater than or equal to the correct collapse load. Advantages of plastic design Reduced peak moments Reduced sensitivity to support movements Application of plastic design to concrete Reinforced concrete can be designed using plastic analysis provided that the reinforced concrete is designed so that it can undergo plastic deformation, i.e. so long as rotation can be achieved at a plastic hinge.

2 Slab Design Objectives To understand the principles of slab design. To introduce the concept of moment redistribution in RC design. To introduce the principal methods of slab analysis design. To present the common forms of slab construction. Introduction Codes Reinforced Concrete Designer s Handbook Types of Slab Construction Solid slabs supported on beams or walls and spanning in one direction. Solid slabs simply supported on beams or walls and spanning in two directions. o Corners free to lift and no torsional reinforcement present in the corners. o Corners restrained in the vertical direction and torsional reinforcement provided Slabs supported by columns Ribbed slabs Slab analysis and design techniques Yield line method Hillerborg strip Elastic analyses

3 Classification of Slabs 1. One-way spanning 2. Two-way spanning, slab simply-supported on beams or walls and not held down 3. Two-way spanning, slab restrained in vertical direction and continuous or discontinuous at edges loading need not be uniform and all edges need not be supported. 4. Two-way spanning, slab supported on columns Slab Analysis Requirements Moments particularly peak moments Shears particularly peak shears Reactions necessary for the design of support members Elastic analysis Hillerborg Strip Yield-line analysis Slab Analysis Methods

4 One-way Spanning Slabs These slabs are designed as wide beams. The design codes give details of appropriate moment redistribution and details of suitable simplified curtailment. Note: If the length of the slab is more than twice its width the slab is generally designed as one-way spanning Simply-supported Unrestrained Slabs If the corners are not restrained from lifting and no torsional reinforcement is provided the moments in the two orthogonal directions can be calculated from the Grashof and Rankine formulae; M M dx dy = α = α x2 y2 nlx 8 nl y 8 Where n is the design load per unit m 2, α = α. Note l x is the shorter span. y2 1 x2 2 2 α x2 4 k =, and k 4 + 1

5 Restrained Slabs m m sx sy = β nl sx = β nl sy 2 x 2 y

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7 Ratio of spans k=l y /l x If the corners of the slab are prevented from lifting and torsional restraint is provided then a more exact elastic analysis is appropriate. Or, alternatively, Marcus s method may be used. Marcus s method involves multiplying the Grashof and Rankine coefficients by a further factor, which depends on the fixity at each slab edge.

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11 Flat Slabs The distinguishing feature of flat slabs is the absence of beams. The columns support the slabs directly. Advantages of Flat Slabs Ease of construction Uncluttered Soffit Potential reduction in storey height Potential disadvantages of Flat Slabs Concentrated shear stresses Reduction in structural depth hence greater forces and reduced stiffness

12 Column Heads In order to overcome problems of high shear stresses around column heads the column heads are sometimes modified to increase the shear perimeter.

13 Design of Flat Slabs Moments The design of flat slabs to resist bending moments is based on the concept of dividing the slab into a series of strips spanning in the two orthogonal directions. The strips are referred to a column strips and edge strips. Each continuous strip in each direction is designed to carry the total load. Therefore in some simple sense we can consider the slab to be twice as strong as required. The distribution of the forces within column and edge strips is not uniform. The central portion of the strip between the columns will carry most of the load. Thus the column and edge strips are subdivided into middle strips and column strips.

14 Middle Strip dimensions for columns with drop heads.

15 Bending Moments for Standard Flat Slabs Distribution of Moments Within Srips

16 Moment Transfer from slab to Column

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