Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

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

upper M Subscript upper A Baseline equals one-twelfth h squared upper E overTilde Subscript upper A Baseline ModifyingAbove epsilon With caret Subscript upper A Baseline plus one-twelfth h squared nu Subscript upper A Baseline upper E overTilde Subscript upper T Baseline ModifyingAbove epsilon With caret Subscript upper T Baseline comma (2.240)

upper M Subscript upper T Baseline equals one-twelfth h squared nu Subscript upper A Baseline upper E overTilde Subscript upper T Baseline ModifyingAbove epsilon With caret Subscript upper A Baseline plus one-twelfth h squared upper E overTilde Subscript upper T Baseline ModifyingAbove epsilon With caret Subscript upper T Baseline period (2.241)

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

(2.242)

nu Subscript upper A Baseline left-parenthesis epsilon overbar Subscript upper A Baseline plus one-half h ModifyingAbove epsilon With caret Subscript upper A Baseline right-parenthesis plus left-parenthesis epsilon overbar Subscript upper T Baseline plus one-half h ModifyingAbove epsilon With caret Subscript upper T Baseline right-parenthesis equals 0 comma (2.243)

upper M Subscript upper A Baseline equals one-twelfth h squared left-parenthesis upper E overTilde Subscript upper A Baseline ModifyingAbove epsilon With caret Subscript upper A Baseline plus nu Subscript upper A Baseline upper E overTilde Subscript upper T Baseline ModifyingAbove epsilon With caret Subscript upper T Baseline right-parenthesis comma (2.244)

nu Subscript upper A Baseline ModifyingAbove epsilon With caret Subscript upper A Baseline plus ModifyingAbove epsilon With caret Subscript upper T Baseline equals 0 period (2.245)

It is clear from (2.242), (2.243) and (2.245) that

(2.246)

From (2.215)₁, (2.244) and (2.246)

(2.247)

(2.248)

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

epsilon overbar Subscript normal upper A Baseline equals minus one-half ModifyingAbove epsilon With caret Subscript normal upper A Baseline h comma sigma Subscript normal upper T Baseline equals 0 comma (2.249)

upper M Subscript upper A Baseline equals one-twelfth h squared upper E overTilde Subscript upper A Baseline ModifyingAbove epsilon With caret Subscript upper A Baseline comma (2.250)

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