The expression for deflection surface of plate whichsatisfies the boundary conditions Eq.. 27-388b and solving for C Equation 27-390 for w becomes The expression for Mx, Myand Mxy The maxi
Trang 1The expression for deflection surface of plate which
satisfies the boundary conditions Eq (27-389) and
Eq (27-388b)
Substituting Eq (27-390) in Eq (27-388b) and solving
for C
Equation (27-390) for w becomes
The expression for Mx, Myand Mxy
The maximum deflection and bending moments,
which occur at midpoint of plate
The maximum deflection and bending moments for a
square plate
The shearing forces from Eqs (27-368)
w ¼ C sin x
a sin
y
4 D
1
a2þ 1
b2
4 D
1
a2þ 1
b2
sin x
a sin
y
Mx¼ q0 2
1
a2þ 1
b2
2
1
a2þ v
b2
sin x
a sin
y b ð27-391aÞ
My¼ q0 2
1
a2þ 1
b2
2
v
a2þ 1
b2
sin x
a sin
y b ð27-391bÞ
Mxy¼ q0ð1 vÞ 2
1
a2þ 1
b2
2 ab
sin x
a sin
y
b ð27-391cÞ
wmax¼ q0 4 D
1
a2þ 1
b2
Mx max¼ q0
4 D
1
a2þ 1
b2
2
1
a2þ v
b2
ð27-392bÞ
My max¼ q0
2
1
a2þ 1
b2
2
v
a2þ 1
b2
ð27-392cÞ
wmax¼ q0a4
4 4D; Mx max¼ My max¼ð1 þ vÞq0a2
4 2 ð27-393Þ
Qx¼ q0 a
1
a2þ 1
b2
cos x
a sin
y
Qy¼ q0 b
1
a2þ 1
b2
sin x
a cos
y
27.82 CHAPTER TWENTY-SEVEN
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies All rights reserved.
Any use is subject to the Terms of Use as given at the website.
APPLIED ELASTICITY
Trang 2The reactive forces at the support edges at x ¼ a and
y ¼ b respectively
The resultant reaction concentrated at the corners of
the plate
The total pressure on all four edges of plate
The four corners reactions, which are equal due to
symmetry
The maximum bending stress if a > b is due to My
which is greater than Mx
Using Eq (27-395d), the expression for maximum
shear stress which is at the middle of the longer side
of the plate
Vx¼
Qx@Mxy
@y
x ¼ a
ð27-395aÞ
Vx¼ q0 a
1
a2þ 1
b2
2
1
a2þ2 v
b2
sin y b ð27-395bÞ
Vy¼
Qy@Mxy
@x
Vy¼ q0 b
1
a2þ 1
b2
2
1
b2þ2 v
a2
sin x a ð27-395dÞ
R ¼ 2ðMxyÞx ¼ a¼ 2q0ð1 vÞ
2 ab
1
a2þ 1
b2
2 ð27-396Þ
2
ðb
0 vxdy þ 2
ða
0vydx
¼4q0ab
4 þ 8q0ð1 vÞ 2
ab
1
a2þ 1
b2
R ¼ 8q0ð1 vÞ 2
ab
1
a2þ 1
b2
2
which is the second term on the right hand side of Eq (27-397)
y max¼6My max
h2
2
h2
1
a2þ 1
b2
2
v
a2þ 1
b2
ð27-398Þ
ðyzÞmax¼ 3q0
2 bh
1
a2þ 1
b2
2
1
b2þ2 v
a2
ð27-399Þ
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies All rights reserved.
Any use is subject to the Terms of Use as given at the website.
APPLIED ELASTICITY
... My maxẳ1 ỵ vịq0a24 2 27 -39 3ị
Qxẳ q0 a
a2ỵ 1...
a2ỵ 1
b2
27 -39 8ị
yzịmaxẳ 3q0
2 bh
a2ỵ...
b2
2 ab
sin x
a sin
y
b 27 -39 1cị
wmaxẳ q0 4 D
a2ỵ