Unformatted text preview: a are free of shearing stress and no normal stress acts on edge
x = a.
3.3. In bending of a rectangular plate (Fig. P3.3), the state of stress is expressed
by
x = c1y + c2xy xy = c31b2  y22 (a) What conditions among the constants (the c’s) make the preceding expressions possible? Body forces may be neglected. (b) Draw a sketch
showing the boundary stresses on the plate.
3.4. Given the following stress field within a structural member,
x
y
z = a[y2 + b1x2  y22] xy 2 2 = ab1x + y 2
2 =  2abxy yz = a[x + b1y  x 2]
2 = xz =0 2 where a and b are constants. Determine whether this stress distribution
represents a solution for a plane strain problem. The body forces are
omitted.
y
b
x
b
a FIGURE P3.1.
136 FIGURE P3.3.
Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 137 3.5. Determine whether the following stress functions satisfy the conditions of
compatibility for a twodimensional problem:
£ 1 = ax2 + bxy + cy 2 (a) £ 2 = ax3 + bx2y + cxy 2 + dy3 (b) Here a, b, c, and d are constants. Also obtain the stress fields that arise from
£ 1 and £ 2.
3.6. Figure P3.6 shows a long, thin steel plate of thickness t, width 2h, and
length 2a. The plate is subjected to loads that produce the uniform stresses
o at the ends. The edges at y = ; h are placed between the two rigid walls.
Show that, by using an inverse method, the displacements are expressed by
u= 1E 2
o x, v = 0, w= 11 + 2
E oz 3.7. Determine whether the following stress distribution is a valid solution for a
twodimensional problem:
x =  ax2y y 1
=  ay3
3 xy = axy 2 where a is a constant. Body forces may be neglected.
3.8. The strain distribution in a thin plate has the form B ax3
axy 2 axy2
R
ax2y in which a is a small constant. Show whether this strain field is a valid solution of an elasticity problem. Body forces may be disregarded.
3.9. The components of the displacement of a thin plate (Fig. P3.9) are given by
u =  c1y2 + x2 v = 2cxy Here c is a constant and v represents Poisson’s ratio. Determine the stresses
x, y, and xy. Draw a sketch showing the boundary stresses on the plate.
3.10. Consider a rectangular plate with sides a and b of thickness t (Fig. P3.10).
(a) Determine the stresses x, y, and xy for the stress function £ = px3y,
where p is a constant. (b) Draw a sketch showing the boundary stresses on
the plate. (c) Find the resultant normal and shearing boundary forces
( Px, Py, Vx, and V ) along all edges of the plate.
y FIGURE P3.6.
Problems 137 ch03.qxd 12/20/02 7:21 AM Page 138 y b
x
b
a a FIGURE P3.9. FIGURE P3.10. 3.11. Redo Prob. 3.10 for the case of a square plate of side dimensions a and
p
£= 2a2 1x2y2 + 1xy32
3 where p is a constant.
3.12. Resolve Prob. 3.10 a and b for the stress function of the form
p
£= b3 xy 213b  2y2 where p represents a constant.
3.13. A vertical force P per unit thickness is applied on the horizontal boundary
of a semiinfinite solid plate of unit thickness (Fig. 3.11a). Show that the
stress function £ =  1P/ 2y tan11y/x2 results in the following stress field
within the plate:
2P
x = x3
,
1x2 + y222 2P
y = xy2 ,
1x2 + y222 2P
xy = yx2 1x2 + y222 Also plot the resulting stress distribution for x and xy at a constant depth
L below the boundary.
3.14. The thin cantilever shown in Fig. P3.14 is subjected to uniform shearing
stress o along its upper surface 1y = + h2 while surfaces y =  h and
x = L are free of stress. Determine whether the Airy stress function FIGURE P3.14. 138 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 139 £= 1
4o ¢ xy  xy3
Ly 3
xy2
Ly 2
 2+
+ 2≤
h
h
h
h satisfies the required conditions for this problem.
3.15. Figure P3.15 shows a thin cantilever beam of unit thickness carrying a uniform load of intensity p per unit length. Assume that the stress function is
expressed by
£ = ax2 + bx2y + cy3 + dy 5 + ex2y3
in which a, Á , e are constants. Determine (a) the requirements on
a, Á , e so that £ is biharmonic; (b) the stresses x, y, and xy.
3.16. Consider a thin square plate with sides a. For a stress function
£ = 1p/a2211x2y2  1y42, determine the stress field and sketch it along the
2
6
boundaries of the plate. Here p represents a uniformly distributed loading
per unit length. Note that the origin of the x, y coordinate system is located
at the lowerleft corner of the plate.
3.17. Consider a thin cantilever loaded as shown in Fig. P3.17. Assume that the
bending stress is given by
Mzy
x = = I p2
xy
2I (P3.17) and z = xz = yz = 0. Determine the stress components y and xy as
functions of x and y.
3.18. Show that for the case of plane stress, in the absence of body forces, the
equations of equilibrium may be expressed in terms of displacements u and
v as follows:
1+
02u
02u
+ 2+
2
10x
0y
1+
02v
02v
+ 2+
2
10y
0x 0 0u
0v
¢ + ≤=0
0x 0x
0y
0 0v
0v
¢ + ≤=0
0y 0y
0x (P3.18) [Hint: Substitute Eqs. (3.10) together with (2.3) into (3.6).]
3.19. Determine whether the following compatible stress field is possible wi...
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 Force, Stress, Eqs., plane stress

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