Chapter 6 Buckling of Plate Elements
1.
Differential Equation of Plate Buckling, a Linear Theory
Consider an isolated freebody of a plate element in the deformed configuration (necessary for
stability problems examining equilibrium in the deformed configu
AUBURN UNIVERSITY
Department of Civil Engineering
CIVL 7690 Analysis of Plate and Shell Systems
Final, Take Home, Due August 4, 2009
1.
A simply supported semi-infinite plate shown below is free at the cut-off edge. A vertical
line loading of a sinusoidal
Yield Line of Reinforced Concrete Slabs
1. Problem description
Figure 1 shows a group of slabs subjected to uniformly distributed loading. The cracking
pattern of these slabs is examined and compared to the fracture line (or yield line) estimated
from the
Principle of Minimum Total Potential Energy
In a linear elastic body, the strain energy stored in the body due to deformation is
U=
1
T dV
2 V
(1)
and the loss of the potential energy of the applied loads for a conservative system is the negative
of the
P Introduction to Calculus of Variations
The calculus of variations is a generalization of the minimum and maximum problem of
ordinary calculus. It seeks to determine a function, y = f ( x ) , that minimizes/maximizes a
definite integral
x2
I = F ( x, y,
Uniqueness theorem
Let us consider now whether our equations can have more than one correct solution
corresponding to given surface and body forces.
The equations of equilibrium expressed in terms of stresses have been established as
ij ,i + f j = 0
(a)
CIVL 7690 Analysis of Plate and Shell Systems/Yoo
Printed on September 24, 2011
AUBURN UNIVERSITY
Department of Civil Engineering
COPURSE:
CIVL 7690 Analysis of Plate and Shell Systems
CREDITS:
3 Semester Hours
PREREQUISITES:
Graduate Standing
DESCRIPTION
Plastic Analysis and Design of Beams
The current AISC specifications for structural steel buildings, both ASD and LRFD, allow the
plastic design of braced frame members (including beams) meeting the compact section criteria.
The mill specified yield stres
AUBURN UNIVERSITY
Department of Civil Engineering
CIVL 7690 Analysis of Plate and Shell Systems
Midterm Exam, Take Home, Due July 2, 2009
1.
Consider the plane stress problem shown in the sketch. Find the stresses at the
boundaries of the plate element, A
AUBURN UNIVERSITY
Department of Civil Engineering
CIVL 7690 Analysis of Plate and Shell Systems
Midterm Exam, Take Home, Due July 2, 2009
1.
Consider the plane stress problem shown in the sketch. Find the stresses at the
boundaries of the plate element, A
MECHANICS OF DIAGONAL
TENSION FIELD ACTION
TENSION
Chai H. Jay Yoo, Ph.D., P.E., F. ASCE
Professor Emeritus
Department of Civil Engineering
Auburn University
CIVL 7690
July 14, 2009
Yoo,
C.H., and Lee, S.C., Mechanics of
Web Panel Postbuckling Behavior i
AUBURN UNIVERSITY
Department of Civil Engineering
CIVL 7690 Analysis of Plate and Shell Systems
Final, Take Home, Due August 4, 2009
1.
A simply supported semi-infinite plate shown below is free at the cut-off edge. A vertical
line loading of a sinusoidal
Transformation of Double Integrals into Line Integrals - Greens theorem in the plane
(Reference, Advanced Engineering Mathematics, Erwin Kreyszig, 1967, Wiley, pg 313
Advanced Engineering Mathematics, Grossman/Derrick, 1988, Harper, pg 611
Mathematic of P
HW#10, Chajes, Problem 7-1
Using the energy method, investigate the behavior of the one-degree-of-freedom model of a
curved plate shown in Fig. P7-1. The model consists of four rigid bars pin connected to each
other and to the supports. At the center of t
HW#8
A rectangular plate of dimension a and b is clamped on all four edges. The plate is subjected to
a uniformly distributed load of q0 . Using the deflection equation of both ends clamped beam for
2
2
2
2
the assumed plate deflection equation, w ( x, y
HW#7
1.
a
q0
x
x
a
z
qo a4
wmax = , = 0.0020932, at x = 0.55a, y = 0.5a
D
y
2.
a
4
a
4
a
2
a
4
x
a
4
x
q0
x
x
a
2
x
R
y
(a) Find the expression for R.
2
(b) If R = qo a , find . Take a minimum of 4 terms in the summation.
B0.1868, for 9 terms
B0.18538 ,
H.W.#6
y
a
x
q ( x)
The above simply supported plate at x = 0 and x = a and of infinite length in the y-direction is
subjected to the following loadings which vary in the x direction only:
q0
( a)
x
q0
( b)
c
b
q ( x ) = q0 sin
( c)
x
x
a
x
Develop the ge
Home Work #5
Evaluate P ( x ) as a Fourier series expression for the followings:
c
( a)
R=P
ac
c
c
P
x
a
x
a
P
( b)
ca
P
cc a
a
1
R=P
ac
ac
c
Solution, Fourier series
(a) The loading is an even function Bn = 0, A0 = 0 ( from def., Eq. 3-21) , l = a
2l
n x
Home Work # 4
y
10"
h=1
o
x
45o
10" A
B
12"
C
(a) Consider the stress function, = xy 3 .
(1) Compute the corresponding stresses.
(2) Show on the sketch the sense and magnitude of the boundary forces at point A and
along the edge BC.
(3) Compute the displa
Home Work #3
Using the generalized Hookes law, prove that the maximum value of Poissons ratio, , cannot
exceed 0.5.
Solution
The strain-stress relationship in a three-dimensional stressed body is given by
1
x ( y + z )
E
1
y = y ( x + z )
E
1
z = z (
Home Work #1
An equilateral triangle ABC is to be installed at a corner of a tunnel complex at a municipal
wastewater treatment plant to facilitate a passage of a utility vehicle. The equivalent uniformly
distributed load intensity of the vehicle may be a
Galerkins Method
The requirement that the total potential energy of a column has a stationary value is
shown in the following equation:
( EIy
l
0
iv
+ Py ) ydx + ( EIy ) y l ( EIy ) y 0 = 0
0
l
(1)
where y is virtual displacement.
Assume that it is poss
AUBURN UNIVERSITY
Department of Civil Engineering
CIVL 7690 Analysis of Plate and Shell Systems
Final, Take Home, Due August 4, 2009
1.
A simply supported semi-infinite plate shown below is free at the cut-off edge. A vertical
line loading of a sinusoidal
Development of Structural Mechanics
Structural Mechanics is a branch of structural engineering concerned with applying Newtonian
mechanics to the analysis of deformations, internal forces and stresses within framed and/or
continuum structural elements and
Chapter 7 Behavior of Thin Shells
1.
Deformation of a Shell Element
Let ABCD shown in Fig. 7-1 represents an infinitesimal element of thin shell cut out by two pairs
of adjacent planes normal to the middle surface. The thickness of the shell is h and is a
Chapter 5 Approximate Methods of Analysis of Plates
1.
Validity of Classical Plate Theory
As in the case of the classical beam theory, there are obvious discrepancies in the classical theory
of plates that has been presented with regard to transforming a
Chapter 4 Plates Subjected to Transverse Loads
1.
Equations of Plates
g
Assumptions
(1)
At the boundary, the plates are assumed to move freely in the plane of the plate; thus the
reactive forces at the edges are normal to the plate.
(2)
The deflections ar
Chapter 3 Plates Subjected to In-plane Forces
1.
Derivation of Airys Stress Function, 2-D
From equations of equilibrium, we have (by crossing out the third row and column)
x xy
+
= 0
x
y
y yx
+
= 0
y
x
( no body force )
(3-1)
From the 2-D compatibility
Chapter 2. Review of Elasticity
1.
Stress
h
Plane-Strain - The plane-strain distribution is based on the assumption that
= uz = 0
z
(2-1)
where z represents the lengthwise direction of an elastic elongated body of constant cross section
subjected to unifo