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6

Plate and Shell Analyses

6.1 Introduction
Many structure members can be categorized as plates and shells, which are extensions of
the 1-D straight beams and curved beams to 2-D cases, respectively. Examples of struc-
tures that can be modeled as plates are shear walls, floor panels, and shelves, while those
that can be modeled as shells include shell structures in nature (such as sea shells and egg
shells), various containers, pipes, tanks, roofs of buildings (such as the superdome), and
bodies of cars, boats, and aircrafts. Figure 6.1 shows an airplane (Boeing 787) that is con-
structed mainly using plate and shell structure members.
The advantages of using plate and shell structures are their light weight, superior load-
carrying capabilities, and sometimes, simply their artistic appeals.

6.2 Review of Plate Theory
Plates are flat surfaces applied with lateral loading, with bending behaviors dominating
the structural response. Shells are structures which span over curved surfaces; they carry
both membrane and bending forces under lateral loading. The thickness t of a plate or a
shell is much smaller than the other dimensions of the structure. Theories related to plates
and shells are briefly reviewed below.

6.2.1 Force and Stress Relations in Plates
Consider an infinitesimally small element in a plate under lateral loading. The internal
shear forces and bending moments acting on the element, and the induced stresses are
shown below in Figure 6.2 and Figure 6.3, respectively.
The induced stresses are related to the internal moments and forces in the following
manner:

Bending moments (per unit length):

⋅
M x = ∫ t/2 x σ zdz,( Nm m) (6.1)
/
t − /2

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