Properties for Design of Composite Structures. Neil McCartney
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upper T Baseline plus one-third h ModifyingAbove epsilon With caret Subscript normal upper T Baseline right-parenthesis plus one-half nu overTilde Subscript normal t Baseline sigma Subscript normal t Baseline minus one-half upper E overTilde Subscript normal upper T Baseline alpha overTilde Subscript normal upper T Baseline upper Delta upper T right-bracket period"/>(2.239)
On using (2.235) and (2.236) it follows that
2.18.3 Some Special Cases
It is useful now to consider some important special cases that arise very often when considering the bending deformation of materials.
2.18.3.1 Four-point Bending Tests
The previous analysis can be used to determine the stress and strain state in beams subject to four-point bending. The analysis will apply near the mid-plane between the planes normal to the beam axis that contain the contact points of the inner rollers used in the experiments. For this case σA=σT=σt=MT=ΔT=0 and the relations (2.235), (2.236), (2.240) and (2.241) reduce to the form
It is clear from (2.242), (2.243) and (2.245) that
From (2.215)1, (2.244) and (2.246)
It is noted from (2.211) that the quantities ε¯A+12ε^Ah and ε¯T+12ε^Th appearing in (2.242), (2.243) and (2.246) are the in-plane strains on the mid-plane x3=12h (i.e. the neutral plane) which are zero for the loading case under consideration.
2.18.3.2 Plane Strain Bending
Plane strain bending conditions are characterised by a zero transverse strain everywhere in the beam so that ε¯T=ε^T=0. It is also assumed that σA=σt=ΔT=0. It then follows that the relations (2.235), (2.236), (2.240) and (2.241) reduce to