Chapter 9 Two-way slabs and yield-line analysis
Chapter 9 Two-way slabs and yield-line analysis
One-way slabs and two-way slabs ·One-way slabs which as the name implies span in one direction is principle analyzed and designed as beams and present no special problems ·Two-way slabs The design of two-way slabs presents varying degrees of difficulty depending on the boundary conditions For irregular cases,the yield-line theory provides a powerful design tool as we shall see in the following sections of this chapter
One-way slabs and two-way slabs • One-way slabs which as the name implies span in one direction is principle analyzed and designed as beams and present no special problems • Two-way slabs The design of two-way slabs presents varying degrees of difficulty depending on the boundary conditions For irregular cases, the yield-line theory provides a powerful design tool as we shall see in the following sections of this chapter
Yield-line analysis o Ultimate-load theory The yield-line theory pioneered by Johansen is an ultimate-load theory for slab design and is based on assumed collapse mechanisms and plastic properties of under-reinforced slabs ·Yield line The assumed collapse mechanism is defined by a pattern of yield line,along which the reinforcement has yielded and the location of which depends on the loading and boundary conditions
Yield-line analysis • Ultimate-load theory The yield-line theory pioneered by Johansen is an ultimate-load theory for slab design and is based on assumed collapse mechanisms and plastic properties of under-reinforced slabs • Yield line The assumed collapse mechanism is defined by a pattern of yield line, along which the reinforcement has yielded and the location of which depends on the loading and boundary conditions
Restriction of Yield-line analysis 1.For the yield-line theory to be valid,shear failures, bond failures.and primary compression failures in flexure must all be prevented; 2.The moment/curvature relationship must have a long horizontal portion when the yield capacity of the slab is reached: In practice,this restriction presents no difficulties because slabs are usually very much under-reinforced
Restriction of Yield-line analysis 1、For the yield-line theory to be valid, shear failures, bond failures. and primary compression failures in flexure must all be prevented; 2、The moment/curvature relationship must have a long horizontal portion when the yield capacity of the slab is reached; In practice, this restriction presents no dif iculties because slabs are usually very much under-reinforced
Typical yield-line patterns Simple supports Yield lines support Fre edge Simple Yield line SImple support (a) (b) Negative yield line Simple support a6pa paxid 77777777777 Simple support Yield ifne in different slab patterns
Typical yield-line patterns Yield line in different slab patterns
A full line represents a positive yield line caused by a sagging (yield moment,so that the concrete cracks in tension in the bottom face of the slab A broken line represents a negative yield line caused by a hogging (yield moment so that tensile cracking occurs in the top face. Negative yield line Simple support 77777777772 Simple support
• A full line represents a positive yield line caused by a sagging（下沉） yield moment, so that the concrete cracks in tension in the bottom face of the slab • A broken line represents a negative yield line caused by a hogging（翘曲） yield moment so that tensile cracking occurs in the top face
·The convention(惯例)for support conditions is as follows:. Single hatching represents a simply supported edge; Double hatching represents a built-in edge; A line by itself represents a free edge. Negative yield line Simple support 777777777777 Simple support
• The convention （惯例）for support conditions is as follows: Single hatching represents a simply supported edge; Double hatching represents a built-in edge; A line by itself represents a free edge
The following comments should be noted: 0 The yield lines divide the slab into several regions,called rigid regions,which are assumed to remain plane,so that all rotations take place in the yield lines. Yield lines are straight and they end at a slab boundary. A yield line between two rigid regions must pass through the intersection of the axes of rotation of the two regions (the supports form the axis of rotation). An axis of rotation usually lies along a line of support and passes over columns
The following comments should be noted: • The yield lines divide the slab into several regions, called rigid regions, which are assumed to remain plane, so that all rotations take place in the yield lines. • Yield lines are straight and they end at a slab boundary. • A yield line between two rigid regions must pass through the intersection of the axes of rotation of the two regions (the supports form the axis of rotation). • An axis of rotation usually lies along a line of support and passes over columns
A yield-line pattern indicates how a slab collapses,just as a plastic-hinge mechanism indicates how a framework collapses KW M 1) (2) M M - T7 L=6.1 Plastic-hinge mechanisms of a framework
A yield-line pattern indicates how a slab collapses, just as a plastic-hinge mechanism indicates how a framework collapses Plastic-hinge mechanisms of a framework
Energy dissipation in a yield line Figure 9.3(a)shows a positive yield line ab of length L and making angles a and a B respectively with the two axis eg and df,about which the rigid regions A and B rotate through the 88 angles日aand6B 日58 (a)
Energy dissipation in a yield line Figure 9.3(a) shows a positive yield line ab of length L and making angles αA and αB respectively with the two axis eg and df, about which the rigid regions A and B rotate through the angles θA and θB