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notes_23_plate_bendin

# notes_23_plate_bendin - Plate Bending not so long plate...

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y ε x ν ε y + collect σ x , E , ν ε y x y z x z y Plate Bending not so long plate previously have shown: M 2 x w d d 2 := . this was for single axis bending. D this relationship holds for the partial derivative in the respective direction fo both x and y; assumptions: plane cross section remains plane small deflections w max < 3/4 t stress < yield w w(x , y) = 1 d 2 1 d 2 R x = 2 w(x,y) R y = dy 2 w(x,y) dx we are making no statement with respect to ε y (or any other ε ) as we did in long plate. d 2 ε x := R 1 x ε x := z d d 2 2 w(x,y) ε y := R 1 y ε y := z d 2 w(x,y) σ x ν σ σ ε x := σ ε y := σ y ν σ E E E E ε ) + ν ε => σ x := E ( ε x + ν ε 1 − ν 2 1 notes_23_plate_bending.mcd dz z r x r y dy dx t

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γ y x y ν t t y x y x y y substituting into ε y := z d 2 2 w x , y) ε x := z d 2 2 w x , y) σ x := E ( ε x + ν ε ( ( ε ) d d 1 − ν 2 σ x E z d d d w x , y) ν⋅ z dy y w x , y) or ... σ x z ( ( 1 − ν 2 dx x ( d d d ( ( ) := z 1 E ν 2 d d 2 2 w x , y) + ν⋅ d d y 2 2 w x , y) ( ( similarly: σ y := z E d 2 2 w x , y) + ν⋅ d 2 2 w x , y) 1 − ν 2 d dx as in bending (applied in each of x and y direction): see figure Hughes 9.3 (below) the lower case m denotes moment per unit length
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notes_23_plate_bendin - Plate Bending not so long plate...

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