St. Venant Torsion Multi-cell
applied torque
T
this is a combination of
nomenclature from Hughes
section 6.1 Multiple Cell
Sections and Mixed Sections
page 227 ff and Kollbrunner
sectin 2.3 Multicelluar Box
Section Members
cross section (partial)
qi-1
qi
Grillage Overview/Introduction
degrees of freedom:
flex only beam
1
3
4
2
we will divert to smartsketch to develop the structure for application of matrix analysis
y
19
22
11
12 c
Mx, x
=2
4
Mz, z
=3
b
15
a
5
26
24
14
13
25
23
21
10
fy, v
=1
x
20
7
27
17
General Method for Deriving an Element Stiffness Matrix
step I: select suitable displacement function
beam likely to be polynomial with one unknown coefficient for each (of four) degrees of freedom
v1
v'
1
dof = = in matrix notation:
v2
v'2
say .
2
Matrix Analysis Example Hughes figure 5.12 page 191 ff
6
C
4
4
3
5
FYB
A
3
c
b
1
2
2
3
B
1
a
4
2
1
input data
n_elements := 3
f, element; F, structure; m = element
n_nodes := 3
ie := 1 . n_elements
1
3
a
n_free := 2 number of degrees of freedom per node
n
Matrix Analysis, Grillage, intro to Finite
Element Modeling
suppose we were to analyze this pin-jointed structure
Py
what are some of the analysis tools we would use?
is this statically determinant?
when we write down the model, what equations result
sing
Intro to Matrix Analysis
ORIGIN := 1
consider a 2-D structure consisting of four elements, linked at pinned joints with six nodes:
Fx and Fy are the external forces applied at node 4
p a distributed pressure on element #4
Fy4
Y
3
2
1
2
1
3
Fx4
4
p
4
we at
Section 14.2 Ultimate Strength of Stiffened Panels
three failure types
compression in flange of stiffener (negative bending moment) Mode I
compression in plate (positive bending moment) Mode II
tension in flange of stiffener (high positive moment) Mode II
Buckling of Stiffened Panels 1
overall buckling vs plate buckling
PCCB Panel Collapse Combined Buckling
Various estimates have been developed to determine the minimum size
stiffener to insure the plate buckles while the stiffener remains straight. this
is
Plate Buckling
ref. Hughes Chapter 12
Buckling of a plate simply supported on loaded edges treated as a wide column results in
similar Euler stress, with EI replaced byD*(b):
dividing by area (b*t):
2
2
D b
Pe :=
e :=
2
a
D
E t
D :=
3
(
12 1
2
a t
)
2
Buckling general Up to this point, the stress and deflections have been proportional to an applied load: y bending stress proportional to moment e.g. x = M I maximum deflection of a simply supported beam subject to uniform load per unit length q => y max
Solution of Plate Bending Equation
Uniform Load Simply Supported Free to pull in
via sinusoidal loading
y
x
p( x , y ) := po sin sin
a
b
loading
pxy
w=0
for
mx = my = 0
2
mx = my = 0
=>
d
2
dx
w=
x=0
2
d
dy
2
1
w=0
y=0
x=0
x=b
y=0
y=a
x=b
y=a
w( x , y )
Plate Bending
not so long plate
previously have shown: M := D
2
d
w . this was for single axis bending.
2
dx
this relationship holds for the partial derivative in the respective direction fo both x and y;
assumptions:
plane cross section remains plane
ry
Plate Bending Introduction
see: bending with z load sheet for
derivations
review general beam, simply supported, clamped long plate
long plate, boundary conditions (end restrained) not so long plate
simply supported beam:
w
q b
Q( x) :=
2
M
x q x
b
2
x
Yield Criteria
Ref: Shames section 7.5 and 9.2
or Crandall and Dahl section 5.11 page 312 ff
general state of stress => expressing maximum shear stress on octahedral plane closing in on a point = oct
oct :=
1
3
where 1, 2, and 3 are principal stresses
(
Overall Design Approach (top level overview)
Ship Structure Design is a stochastic and time dependent process.
What parameters are uncertain?
Loads
waves, sea state, speed, direction, etc
Load effects
Assumptions in analysis
Variation in application
M
Dissimilar material such as a composite
structure:
what if E and I are not constant?
y
assuming bending only; Mz applied; determine Iz
In this cross section, the upper region has a
modulus = E2 where the remainder has modulus
E1
E2
as with Euler bending,
Shear stress due to Shear load (pure bending) multi-cell closed cross-section
Q
1
2
i-1
i
i+1
n
i th W ith resulting distribution of shear flow q(x,y) or q(s)
=
open section with shear flow q*(s)
1
q1
q 1
2
i-1 i-1
i
i+1
n
q 2 q1 q1
q2 q2 q2 q3
qi-1
qi
Shear Stress from Shear Load in closed non symmetric section
Q
t1
t2
Shear load is applied such that (pure) bending occurs
can't use symmetry to determine where to start s = 0 arc length parameter
approach: divide into two problems and superpose:
Q
t1
=
t
Lecture 6 - 2003
Torsion Properties for Line Segments and Computational Scheme
for Piecewise Straight Section Calculations
this consists of four parts (and how we will treat each)
A - derivation of geometric algorithms for section properties (cover quickl
Differential equation and solution
aka Pure and Warping Torsion
aka Free and Restrained Warping
ref: Hughes 6.1 (eqn 6.1.18)
the development of warping torsion up to this point was assumed to be "pure" or "free" i.e. it was the
only effect on a beam and i
Lecture 5 - 2003
Twist closed sections
As this development would be almost identical to that of the open section, some of the
development is simply repeated (copied) from the open section development.
pure twist around center of rotation D => neither axia
Lecture 4 - 2003
Pure Twist
pure twist around center of rotation D => neither axial () nor bending forces (Mx, My) act on
section; as previously, D is fixed, but (for now) arbitrary point.
as before:
a) equilibrium of wall element:
b) compatibility (she
Lecture 3 - 2003
Kollbruner Section 5.2 Characteristics of Thin Walled Sections and .
Kollbruner Section 5.3 Bending without Twist
thin walled => (cross section shape arbitrary and thickness can vary)
axial stresses and shear stress along center of wall g
13.122 Lecture 2
Shear force and bending moment in floating platform
Ship or freely floating offshore structure is a beam in equilibrium
Overall summation and forces and moments = 0
But shear force and bending moments can and do exist
Net force along leng
Introduction to course: Design process Structural design process
13.122 Lecture 1
Primary Load: Bending Moment
and Shear Force
General course content: 13.122 Ship Structural Design A. Loads on ship/offshore platforms Calculation of loads buoyancy, shea
Shear lag
ref: Hughes 3.8
we have assumed "plane sections remain plane" during our bending analysis
? how valid is this assumption
? are there situations where this is not the case
consider a wide flange "T" beam in bending due to two load senarios
load i
Torsion Properties for Line Segments and Computational Scheme
for Piecewise Straight Section Calculations
Closed Thin walled Sections
the new material consists of the "corrections for and Q
A = enclosed area
definition of
s
dc = hc
J
1
ds = dc
2 A t
b