Plastic hinge

Cellular materials can collapse by another mechanism. If the cell-wall material is plastic (as many polymers are) then the foam as a whole shows plastic behaviour. The stress-strain curve still looks like Fig. 25.9, but now the plateau is caused by plastic collapse. Plastic collapse occurs when the moment exerted on the cell walls exceeds its fully plastic moment, creating plastic hinges as shown in Fig. 25.12. Then the collapse stress (7 1 of the foam is related to the yield strength Gy of the wall by... 

References 7, 64, and 72 through 75 discuss expected consequences at various levels of stmctural response deformation ratios or plastic hinge... 

Damage Level Deformation Ratio Plastic Hinge Rotation... 

The first step in developing a resistance function is to determine the plastic section capacities, such as plastic moment, Mp, as shown in Figure 7.1. The next step is to determine the sequence of plastic hinge formation and the corresponding load and deformation values. This is done by incrementally applying loads until a collapse mechanism is formed as illustrated in Figure 7.2 for a fixed end beam with a uniform load. 

FIGURE 7.2 Resistance Function for Member With Sequential Plastic Hinges... 

Elastic Region - The deformation range from zero up to the formation of the first plastic hinge., ... 

White 1993, Plastic Hinge Based Methods for advanced Analysis and Design of Steel Frames, D. W. White and W. F. Chen (cds), Structural Stability Research Council, Lehigh University, Bethlehem, Pennsylvania, 1993... 

Physical vapor explosions 3-3 Plant operation 1-1 —1-2 Plastic hinges 5-3, 7-5... 

Structural dements resist blast loads by developing an internal resistance based on material stress and section properties. To design or analyze the response of an element it is necessary to determine the relationship between resistance and deflection. In flexural response, stress rises in direct proportion to strain in the member. Because resistance is also a function of material stress, it also rises in proportion to strain. After the stress in the outer fibers reaches the yield limit, (lie relationship between stress and strain, and thus resistance, becomes nonlinear. As the outer fibers of the member continue to yield, stress in the interior of the section also begins to yield and a plastic hinge is formed at the locations of maximum moment in the member. If premature buckling is prevented, deformation continues as llic member absorbs load until rupture strains arc achieved. 

The plastic hinge nonlinear material model is easier to use but usually can not consider axial load effects. Plastic hinge locations must usually be predetermined and are usually limited to the ends of the member. Analysis results which include displacements and plastic hinge rotations which are directly comparable against acceptance criteria. ... 

Connections must be sized to transfer computed reaction forces and to assure that plastic hinges can be maintained in the assumed locations. For reinforced concrete design, splices and development lengths are provided for the full yield capacities of reinforcing. For structural steel design, connections are designed for a capacity somewhat greater than that of its supported member. Further information is provided in later sections of this chapter. Typical connection details are provided in Chapter 8. 

The details discussed or illustrated in this chapter are some of those that have been found to be cost effective and easily constructed. Structural steel connections are designed to move plastic hinge formation away from the connection and into the member. Reinforced concrete connections must provide full development of reinforcing with ties to permit extended plastic deformations, The design details included are not intended to limit the use of alternate designs. 

Hinge Rotation - A measure of the energy absorbing capacity of a structural member. This is the angle of deformation at a plastic hinge. 

Ultimate Capacity - The load applied to a structural element as the final plastic hinge, or collapse mechanism, is formed. 

Materials must be varied to perform the many tasks required of them in today s society. Often they must perform them repeatedly and in a special manner. We get an ideal of what materials can do by looking at some of the behavior of giant molecules in our body. While a plastic hinge must be able to work thousands of times, the human heart, a complex muscle largely composed of protein polymers (Section 10.6), provides about 2.5 billion beats within a... 

Figure 14.13 Geometry of the plastic hinge that forms due to the shape discontinuity and produces overcurvature...

Karger-Kocsis recorded the different fracture behaviors of non-nucleated and -modified PP (MFR 0.8 dg min 1) tested in a three-point bending configuration at 1 ms-1 at 23 °C, a-PP was semi-ductile and /3-PP ductile with a plastic hinge at - 40 °C a-PP was brittle, /i-PP ductile [72], The descriptors from the linear elastic fracture mechanics (LEFM), Kq, the stress intensity factor, and Gc, the energy release rate, used to quantify the toughness correlated well with the fracture picture. This conclusion is also valid for... 

The J-integral was originally defined by J. Rice [11] for elastic materials. In the case of an ideal plastic element with a plastic hinge the J-integral can be calculated as [9,13,14]... 

At the root section a plastic hinge circle is to be located so that each generator of the middle surface of the cylinder rotates around the hinge according to (12). 

The exact calculation of the J-integral and the tearing modulus (1) is quite complicated. It was shown by several authors that a good approximation for the J-integral in the case of rotation around a plastic hinge can be presented as [13,14]... 

When the force on the polymer material exceeds the fully plastic moment of the weakest element of the cell structure, plastic collapse occurs (characterized by region C in Figure 6.22). In effect, the weakest elements become a plastic hinge and deform to create a collapsing cell that responds to nearly constant stress (opl) with large deformations (see Figure 6.28). 

While elastic buckling of structural walls is fully recoverable, plastic collapse of the plastic hinge sections is not. The theory of plastic collapse corresponds well for materials with a relative density of 0.3 or less materials with greater relative densities do not follow theoretical predictions because their cell partitions are too thick to buckle or hinge readily. The data represented in Figure 6.18, showing SEM from 6.17a, estimates a relative density of 0.345. 

In general, it is not possible to measure CTOD, but rather the crack-opening displacement (COD). The latter quantity can be determined at the outer end of the notch with a suitable clip gauge. Thus, for a notched three-point-bend specimen, it was shown that a plastic hinge can form around the tip of the crack [Brostow and Comeliussen, 1986]. If the center of rotation is known, the CTOD can be calculated from the measured COD. A Standard has been published [BS 5762]. 

Necking will not occur on the tensile side of the heam, because of the support of the compressive side. For metals, the initial stages of yielding occur as in Fig. 8.7a, and the beam remains very slightly bent if the loads are removed. There is no evidence of permanent deformation in polymer beams before a plastic hinge forms. The non-linearity in the early part of the... 

Yielded zones in a three-point bend test when (a) The centre of the beam remains elastic and (b) a plastic hinge forms. The stress distributions, on the beam section under the load P, are shown for a material of infinite Young s modulus, and constant yield stress of 50 MPa. 

Euler buckling theory predicts collapse at a constant force. However, finite element analysis (FEA) shows that the onset of buckling causes the load bearing capacity to decrease (Fig. 8.8). At high axial deflections, plastic hinges develop at mid-length and the ends of these slender struts. 

When polystyrene foams are compressed, the 1-5 p,m thick, biaxially oriented, cell faces form permanent plastic hinges at intervals (Fig. 8.20), in directions perpendicular to the compression axis. The deformation mechanism is similar to that when thin sheet steel crumples in a car crash. This behaviour contrasts to the crazing and fracture that occurs when 2 mm thick polystyrene sheet is bent. Thirty-two micrometers thick, biaxially oriented polystyrene film, used in window envelopes, yields in tension rather than crazing and fracturing. In closed-cell polyethylene foams... 

Any pressure differential between neighbouring cells would cause bowing of the intervening face. The face deformation mode should cause equal pressure rises in neighbouring cells, for instance by having an even number of plastic hinges across buckled faces. If cell faces concertina as in a bellows, it allows the foam Poisson s ratio to be zero. 

Plots of compressive yield stress versus density on logarithmic scales, for polystyrene and HOPE foams, have slopes 1.5. If the lines are extrapolated to the density of the solid polymer, the yield stress is close to (tq, that measured for the solid. Consequently, the form of Eq. (8.20) is confirmed. Cell faces in HDPE foams behave in a non-linear viscoelastic manner when bent. The stress distribution, resembles that in a plastic hinge (Fig. 8.7b), so it is not surprising that the exponent in the yield stress-relative density relationship is the same as for polystyrene. We will return to these materials in the cycle helmet case study in Chapter 14. 

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