6/11/2021

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Tran Kim Bang

Department of Engineering Mechanics (DEM),
Faculty of Applied Sciences,
Ho Chi Minh City University of Technology,
Email: tkbang@hcmut.edu.vn

Structural Analysis Chapter 4. Plate and Shell Analyses 1

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Chapter 4. Plate and Shell

Analyses

Structural Analysis Chapter 4. Plate and Shell Analyses 2

 6/11/2021

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

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 structures 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.

Structural Analysis Chapter 4. Plate and Shell Analyses 3

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

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.

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Forces and moments acting on an Stresses acting on the infinitesimally small

infinitesimally small element in a plate element in the plate

Structural Analysis Chapter 4. Plate and Shell Analyses 5

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Thin Plate Theory (Kirchhoff Plate Theory)

The thin plate theory is based on assumptions that a straight line normal to the
midsurface remains straight and normal to the deflected mid-surface after loading

(Negligible transverse shear deformations)

Deflection and rotation after loading of a

plate according to Kirchhoff plate theory

Structural Analysis Chapter 4. Plate and Shell Analyses 6

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Thick Plate Theory (Mindlin Plate Theory)

If the thickness t of a plate is not small, for example, when t/L ≥ 1/10 (L = a
characteristic dimension of the plate main surface), then the thick plate theory by
Mindlin should be applied. The theory accounts for the angle changes within a
cross section, that is

(transverse shear deformations)

Displacement and rotation based on the

Mindlin thick plate theory

Structural Analysis Chapter 4. Plate and Shell Analyses 7

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Shell Theory
Unlike the plate models, where only bending forces exist, there are two types of
forces in shells, that is, membrane forces (in plane forces) and bending forces (out
of plane forces)
Similar to the plate theories, there are two types of theories for modeling shells,
namely thin shell theory and thick shell theory. Shell theories are the most
complicated ones to formulate and analyze in mechanics.

Internal forces acting in a cylinder under internal pressure load

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Case Study with ANSYS Workbench

Vases are decorative pieces that can be of any artistic shapes. The figure below
gives the dimensions of a flower vase made of glass. Assume that the vase has a
uniform thickness of 4 mm. The water level reaches 100 mm below the opening
of the vase. Determine the maximum deformation and von Mises stress in the
vase under the hydrostatic pressure.

Structural Analysis Chapter 4. Plate and Shell Analyses 9

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 1: Start an ANSYS Workbench Project Launch ANSYS Workbench and save
the blank project as “Vase.wbpj.”
Step 2: Create a Static Structural (ANSYS) Analysis System Drag the Static
Structural (ANSYS) icon from the Analysis Systems Toolbox window and drop it
inside the highlighted green rectangle in the Project Schematic window to create a
standalone static structural analysis system.

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 3: Add a New Material

Double-click on the Engineering Data cell to add a new material. In the following
Engineering Data interface which replaces the Project Schematic, type “Glass” as
the name for the new material, and double-click Isotropic Elasticity under Linear
Elastic in the leftmost Toolbox window. Enter “70E9” for Young’s Modulus and “0.17”
for Poisson’s Ratio in the Properties window. Click the Return to Project button to go
back to Project Schematic.

Structural Analysis Chapter 4. Plate and Shell Analyses 11

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 4: Launch the DesignModeler Program

Ensure Surface Bodies is checked in the Properties of Schematic A3: Geometry
window (select Properties from the View drop-down menu to enable display of this
window). Choose 3D as the Analysis Type in this Properties window.

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Doubleclick the Geometry cell to launch DesignModeler, and select “Millimeter” in the
Units pop-up window.

Structural Analysis Chapter 4. Plate and Shell Analyses 13

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 5: Create a Profile Sketch

Click on the Sketching tab. Select the Draw toolbox and then Construction Point. Draw
five construction points A, B, C, D, and E as shown below.

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Draw a horizontal line from the origin to point A. Next, draw a spline passing through
points A, B, C, D, and E; right-click at the last construction point and choose Open
End from the context menu to finish the spline creation.

Structural Analysis Chapter 4. Plate and Shell Analyses 15

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 6: Create a Surface Body

Switch to the Modeling tab and click on the Revolve feature. The default Base Object is
set as Sketch1 in the Details of Revolve1. Click on the y-axis in the Graphics window
and apply it to the Axis selection. Click Generate.

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

A Surface Body will be created from revolving the profile sketch. Exit the
DesignModeler program.

Structural Analysis Chapter 4. Plate and Shell Analyses 17

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 7: Launch the Static Structural Program

Double-click on the Model cell to launch the Static Structural program. Click on Surface
Body under Geometry in the Outline tree. In the Details of “Surface Body,” set
Thickness as 4e-3 m. Click to the right of the Material Assignment field and select
Glass from the drop-down menu.

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 8: Generate Mesh

Right click on Mesh in the Project Outline. Select Insert and then Sizing from the
context menu. In the Details of “Face Sizing,” enter “2e-3 m” for the Element Size.
Click on the side wall and the bottom surface of the vase in the Graphics window and
apply the two faces to the Geometry selection.

Structural Analysis Chapter 4. Plate and Shell Analyses 19

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Right-click on Mesh and select Generate Mesh.

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 9: Apply Boundary Conditions

Right-click on Static Structural (A5). Choose Insert and then Fixed Support from the
context menu. Apply this support to the bottom surface of the vase.

Structural Analysis Chapter 4. Plate and Shell Analyses 21

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 10: Apply Loads

Right-click on Static Structural (A5). Choose Insert and then Hydrostatic Pressure. In
the Details of “Hydrostatic Pressure,”

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Ctrl-click the side wall and the bottom surface of the vase, and apply the two faces to
the Geometry selection.

Structural Analysis Chapter 4. Plate and Shell Analyses 23

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Change the Shell Face to Bottom, because the hydrostatic pressure is applied to the
inside of the shell surface.

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Enter 1000 kg/m3 for Fluid Density. Change the Define By selection to Components,
and enter 9.8 m/s2 for the Y Component of Hydrostatic Acceleration. Enter 180e-3 m in
the field of Y Coordinate of the Free Surface Location.

Structural Analysis Chapter 4. Plate and Shell Analyses 25

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Step 11: Retrieve Solution

Insert a Total Deformation item by right-clicking on Solution (A6) in the Outline tree.
Insert an Equivalent Stress item by right-clicking on Solution (A6) in the Outline tree.
Right-click on Solution (A6) in the Outline tree and select Solve. Click on Total
Deformation in the Outline to review the deformation results.

Structural Analysis Chapter 4. Plate and Shell Analyses 26

 6/11/2021

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Structural Analysis Chapter 4. Plate and Shell Analyses 27

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Problem 1

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Problem 2

Structural Analysis Chapter 4. Plate and Shell Analyses 29

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Problem 3

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Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics
Homework 1

Structural Analysis Chapter 4. Plate and Shell Analyses 31

Vietnam National University Ho Chi Minh City Faculty of Applied Science

University of Technology Department of Engineering Mechanics

Homework 2

Structural Analysis Chapter 4. Plate and Shell Analyses 32