The thin rectangular plate (Fig. 1, not to scale) with sides 6 m and 4 m of thickness 112 mm is subjected tothe stresses ox, Oy and txy. An Airy stress function is proposed to be:$(x, y) = Ax° + Bx'y+Cxy, with A=B=C =1(a) Determine the values of stresses (ox, Oy and txy) at point (x_y) which satisfy the stress function(b) Draw a stress element showing the stresses on the four faces of the element (Use the face bounded bythe X- and Y-axes with a line representing a face, that is, a 2-D stress element).(c) Neglecting body forces, determine the resultant normal and shearing forces on the right face of the platethat are consistent with o (Use the face bounded by the X- and Y-axes with a line representing a face, thatis, a 2-D stress element).112 mm4 m6 mFig. 1Biharmonic Equationô*o(x, y)ôxo*o(x.y) ô*¢(X.y)=v*o(x,y) =0+2where o(x.y) is defined as:ô*o(x,y)o o(x, y)ôxcy

Given ϕ(x,y)=Ax3+Bx2y+Cxy3 with A=B=C=1 Thus ϕ(x,y)=x3+x2y+xy3

The thin rectangular plate (Fig. 1, not to scale) with sides 6 m and 4 m of thickness 112 mm 1s subjected the stresses Ox, Oy and Txy. An Airy stress function is proposed to be: $(x, y) = Ax° + Bx y+Cxy², with A= B =C =1 (a) Determine the values of stresses (Or, Gy and Ty) at point (x.y) which satisfy the stress function (b) Draw a stress element showing the stresses on the four faces of the element (Use the face bounded b the X- and Y-axes with a line representing a face, that is, a 2-D stress element). (c) Neglecting body forces, determine the resultant normal and shearing forces on the right face of the plat that are consistent with o (Use the face bounded hy the Yand V aves with a line anting a fce the

The thin rectangular plate (Fig. 1, not to scale) with sides 6 m and 4 m of thickness 112 mm 1s subjected the stresses Ox, Oy and Txy. An Airy stress function is proposed to be: $(x, y) = Ax° + Bx y+Cxy², with A= B =C =1 (a) Determine the values of stresses (Or, Gy and Ty) at point (x.y) which satisfy the stress function (b) Draw a stress element showing the stresses on the four faces of the element (Use the face bounded b the X- and Y-axes with a line representing a face, that is, a 2-D stress element). (c) Neglecting body forces, determine the resultant normal and shearing forces on the right face of the plat that are consistent with o (Use the face bounded hy the Yand V aves with a line anting a fce the

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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Concept explainers

Stress transformation

The term stress indicates the measure of internal forces that develop due to an external force's application on an object. When a specific external force is applied to an object, internal stress generates in the material. The value of stress can obtain by taking the ratio of the externally applied force and cross-sectional area of the object. There are different types of stress depending on the type of external force.

Question

The thin rectangular plate (Fig. 1, not to scale) with sides 6 m and 4 m of thickness 112 mm is subjected to
the stresses ox, Oy and txy. An Airy stress function is proposed to be:
$(x, y) = Ax° + Bx'y+Cxy, with A=B=C =1
(a) Determine the values of stresses (ox, Oy and txy) at point (x_y) which satisfy the stress function
(b) Draw a stress element showing the stresses on the four faces of the element (Use the face bounded by
the X- and Y-axes with a line representing a face, that is, a 2-D stress element).
(c) Neglecting body forces, determine the resultant normal and shearing forces on the right face of the plate
that are consistent with o (Use the face bounded by the X- and Y-axes with a line representing a face, that
is, a 2-D stress element).
112 mm
4 m
6 m
Fig. 1
Biharmonic Equation
ô*o(x, y)
ôx
o*o(x.y) ô*¢(X.y)=v*o(x,y) =0
+2
where o(x.y) is defined as:
ô*o(x,y)
o o(x, y)
ôxcy

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