P.7.1. A plate 10 𝑚𝑚 thick is subjected to bending moments 𝑀𝑥 equal to 10 𝑁𝑚/𝑚𝑚
and 𝑀𝑦 equal to 5 𝑁𝑚/𝑚𝑚. Calculate the maximum direct stresses in the plate.
A.7.1
P.7.2. For the plate and loading of problem P.7.1, find the maximum twisting moment
per unit length in the plate and the direction of the planes on which this occurs.
A.7.2
P.7.3. The plate of the previous two problems is subjected to a twisting moment of
5 𝑁𝑚/𝑚𝑚 along each edge in addition to the bending moments of 𝑀𝑥 = 10 𝑁𝑚/𝑚𝑚
and 𝑀𝑦 = 5 𝑁 𝑚/𝑚𝑚. Determine the principal moments in the plate, the planes on
which they act, and the corresponding principal stresses.
A.7.3
Solution 1
Solution 2
P.7.4. A thin rectangular plate of length 𝑎 and width 2𝑎 is simply supported along the
edges 𝑥 = 0, 𝑥 = 𝑎, 𝑦 = −𝑎, and 𝑦 = +𝑎. The plate has a flexural rigidity 𝐷, a
𝜋𝑥
Poisson’s ratio of 0.3, and carries a load distribution given by 𝑞(𝑥, 𝑦) = 𝑞0 𝑠𝑖𝑛( ). If
𝑎
the deflection of the plate is represented by the expression
𝑞𝑎4
𝜋𝑦
𝜋𝑦
𝜋𝑦
𝜋𝑥
𝑤=
+
𝐴
cosh
+
𝐵
sinh
sin
(1
)
𝐷𝜋 4
𝑎
𝑎
𝑎
𝑎
determine the values of the constants 𝐴 and 𝐵.
A.7.4.
P.7.5. A thin, elastic square plate of side a is simply supported on all four sides and
supports a uniformly distributed load 𝑞. If the origin of axes coincides with the center
of the plate show that the deflection of the plate can be represented by the expression
where 𝐷 is the flexural rigidity, 𝑣 is Poisson’s ratio and 𝐴 is a constant. Calculate the
value of 𝐴 and hence the central deflection of the plate.
A.7.5
P.7.6. The deflection of a square plate of side 𝑎 which supports a lateral load
represented by the function 𝑞(𝑥, 𝑦) is given by
where 𝑥 and 𝑦 are referred to axes whose origin coincides with the center of the plate
and 𝑤0 is the deflection at the center. If the flexural rigidity of the plate is 𝐷 and
Poisson’s ratio is 𝑛, determine the loading function 𝑞, the support conditions of the
plate, the reactions at the plate corners, and the bending moments at the center of the
plate.
A.7.6.
Answer is Book is different for Mx and My, rest all is correct….. the cosine terms in
step 6 box for Mx and My is not there in Book Answer
P.7.7. A simply supported square plate 𝑎 × 𝑎 carries a distributed load according to the
formula
𝑥
𝑞 (𝑥, 𝑦) = 𝑞0
𝑎
where 𝑞0 is its intensity at the edge 𝑥 = 𝑎. Determine the deflected shape of the plate.
A.7.7.
P.7.8. An elliptic plate of major and minor axes 2𝑎 and 2𝑏 and of small thickness t is
clamped along its boundary and is subjected to a uniform pressure difference 𝑝 between
the two faces. Show that the usual differential equation for normal displacements of a
thin flat plate subject to lateral loading is satisfied by the solution
2
𝑥 2 𝑦2
𝑤 = 𝑤0 (1 − 2 − 2 )
𝑎
𝑏
where 𝑤0 is the deflection at the center, which is taken as the origin. Determine 𝑤0 in
terms of 𝑝 and the relevant material properties of the plate and hence expressions for
the greatest stresses due to bending at the center and at the ends of the minor axis.
A.7.8
A.7.9
P.7.10. If, in addition to the point load 𝑊, the plate of problem P.7.9 supports an inplane compressive load of 𝑁𝑥 per unit length on the edges 𝑥 = 0 and 𝑥 = 𝑎, calculate
the resulting deflected shape.
A.7.10.
A.7.11.
A.7.12.