PLASTIC DESIGN OF CONTINUOUS BEAMS & PORTAL FRAMES
Through ductility, structural is able to absorb large deformations beyond elastic limit without the danger of fracture. It is this characteristics feature of steel that makes possible the application of plastic analysis to structural design.
Figure 1: Stress strain diagram (for mild steel)
Figure 2:Modified Plastic strain diagram
Stress-strain diagram for mild steel in tension is shown in figure-1. Let ‘ab’ represent the elastic range, ‘be’ the range where strain increases without load (plastic flow), ‘d’ represents the ultimate strength and ‘e’ the breaking load.
For applying the plastic theory to simple beams in bending the following assumptions are made:
- The plastic range is entered on reaching the yield point.
- Strain hardening is ignored.
- Stress-strain relation for tension is the same as that for compression.
- Plane sections remain plane, and
- The upper and lower points b and b’ merge into one.
Based on the above assumptions, the stress-strain diagram shown in figure 2 is used in plastic theory upto the point of failure; the limb ‘bc’ will represent the plastic strain.
Figure 3: Distribution of stress
Consider a simple beam under bending (figure 3(a)) subjected to a load W at mid point. Figure 3(b) represents the stress distribution due to both self weight (dead load) and live load W. as the load W increases, the stress in the extreme fibre reaches the yield point and the extreme fibre offer no further resistance but the inner fibres have not yet been stressed to the yield limit. In figure 3(c), some fibres are stressed to yield point and others still under-stressed, whereas in figure 3(d) all the fibres are stressed to the yield level. It can be assumed, at this point, that the section has become fully plastic. Any further increase of load is assumed to increase the deflection substantially and the fully plastic section may be treated as a plastic hinge. The deflection under the increased load will lead to collapse of the beam.
The ratio of the load producing collapse to the working load is called the load factor. As the working stress is dependent on the shape of the section, i.e. I and Z values, so also the collapse load is dependent on the shape of the section.
(Shown in figure 4)
Figure 4: Load factor
The moment of resistance under working load =
For collapsible load, the moment of resistance =
Load factor =
Assuming a factor of safety of 1.5, load factor = 1.5 x 1.5 = 2.25
The fully plastic stage in the section is said to have occurred when tensile as well as compressive zones both have at all the points.
The plastic moment will be given by
= Force x lever arm
This can be written as
Thus the bending strength of a rectangular member is given by , which is 1.5 times its yield strength . This ratio is called shape factor.
Shape factor f =
Shape factor can also be written as
It may be seen that the shape factor is a property dependent upon the geometry of the section only.
Table – 1: Shape factors for different sections
I – section
PLASTIC HINGE MECHANISM
Consider a simply supported beam of span L, carrying a concentrated load W at mid point. The beam will fail when the centre section becomes fully plastic. With simple supports at the ends and a plastic hinge at centre, the beam will transform into a mechanism consisting of two links. Figure 5 shows plastic zone shaded.
The length of the plastic zone depends upon the ratio to . Greater the ratio, larger will be the length of the plastic zone. The sections in this length will be at different stages of curve about the yield value. Figure 6 shows the curve for I-section.
Figure 5: Plastic Hinge
Figure 6: Idealised curve
Table – 2: Different Types of Mechanisms
The insertion of a real hinge or a pint joint, into a statically indeterminate frame reduces the number of indeterminate moments by one, so that if the number of indeterminacies is m, the addition of n hinges produces a simple statically determinate structure. The addition of one more hinge will allow the structure to movewith one degree of freedom, i.e. a mechanism is formed; then the number of hinges to form a mechanism is (n+1). This criterion must, of course be applied to each element of a structure as well as the structure as a whole, because collapse of one part represents practical failure. Typical collapse mechanisms are shown in figure 7.
Figure 7: Collapse Mechanism
Figure 8 shows a two span continuous beam, built at one end.
Figure 8 (b), (c) and (d) show possible failures in each span. The problem is then to determine the least load value to cause collapse, or conversely, the maximum plastic moment of resistance required for the section. Bending moment diagrams are given in (e), (f) and (g) of figure 8. The associated failure loads are
, , respectively.
Consider a pin-based frame shown in figure 9(a). The frame is indeterminate to the first degree; therefore, the number of plastic hinges at collapse should be 2. The final bending moment diagram is drawn in 2 parts for
- Statically determinate frame (shown in figure 9(b)), and
- Moment diagram due to indeterminate horizontal reaction, shown in figure 9(d).
These diagrams have been added to give two equal and opposite peaks of bending moment to satisfy the mechanism condition. figure (d) shows the combined bending moment diagram. One hinge forms at D and the length BC is fully plastic. Considering B, we get,