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What is Yield Line Theory?The yield line theory of analysis is a factored or ultimate load method of analysis. The yield line theory is conducted based on the bending moment of the structural element at its collapse state. The yield analysis was proposed by Ingerslev in 1923. The importance of the theory is more reflected in the analysis of slabs.
Yield Lines in Concrete SlabsAs mentioned before, the analysis is conducted based on the collapse load of the slab under consideration. Under this load, cracks are formed. This crack formation is seen in under reinforced concrete slabs and the yield line theory considers only under reinforced concrete slabs. During the crack formation in the slabs, the reinforcement starts to yield. This yielding happens at the point of maximum bending moments. As the cracks propagates, the yield lines started to develop. Finally, under uncontrollable load, the slab will collapse leaving maximum yield line representing the achievement of maximum bending moments. The main principle of yield line theory is to determine the location of appropriate yield lines. In the case of two-way slab systems, which are statically indeterminate, detailed inelastic analysis (yield line analysis) is conducted. The moment coefficients given in the Code IS 456:200(Table 26) for two-way rectangular slabs with various possible edge conditions are based on Yield Line Analysis. ‘Yield line analysis’ is the equivalent for a two-dimensional flexural member (plate or slab) of the limit analysis of a one-dimensional member (continuous beam).
Characteristics of Yield Lines in R.C SlabsThe following are the characteristics of yield lines formed in reinforced concrete slabs under ultimate loads: i) Yield lines are straight ii) Yield lines end at supporting edges of slab iii) Yield lines passes through intersection of axis of rotation of adjacent slab elements iv) Axis of rotation lies along lines of supports and passes over columns
Assumptions of Yield Line TheoryThe following are the assumptions of the yield line analysis of reinforced concrete slabs:
- At the collapse stage, the steel reinforcement will be fully yielded along the yield lines.
- During the collapse, the slab is deformed plastically and they gets separated into segments. Elastic behaviour is followed by the individual segments.
- In yield theory, only plastic deformations are taken into consideration. The elastic deformations are neglected. The so called deformations are happening along the yield lines. Even in the collapse state, the elements will remain plane.
- Along the yield lines, there is uniform distribution of bending and twisting moments. The amount of reinforcement in the section will reflect the capacity that will give the maximum moment capacity of the section.
- The yield lines are lines of intersection of two planes, hence they will remain straight.
Rules of yield linesThe positive and negative yield lines, are used to designate the yield lines for positive bending moments having tension at the bottom and negative bending moments having tension at the top of the slab, respectively. The guidelines for predicting the yield lines and axes of rotation are mentioned below:
- Straight yield line occure between two intersecting planes.
- In one way slabs, the positive yield line occur at the mide span.
- At the supports, the yield lines are negative in addition to the mid positive yield lines for one way continuous slabs.
- For slabs with a point load, the yield lines will be projecting out of the point of application.
- For two planes intersecting, the yield line will pass through the intersection of axes of rotation of two planes.
- The end of yield lines are either at the boundary of the slab or at another yield line.
- The axes of rotation is represented by yield lines
- The edges of support too act as axes of rotation. If the support is fixed, the support will have negative yield lines. This negative yield line will have resistance to rotation. In the case of simply supported edges, the axes of rotation at the support will not have resistance to rotation.
- For a column support, the axis of rotation pass over it. The orientation will depend on other considerations
Methods of Analysis in Yield Line TheoryBased on the rules and the assumptions, a general load pattern and the axis of rotation is located. The below mentioned methods can be used to determine the final pattern, axes of rotation and the collapse load. The methods used are:
- Method of Segmental Equilibrium
- Method of Virtual Work