CE2033 (5 ch) Structural Analysis
Positive Moments
imply point lies below tangent line
Examples
Determine the beam rotation at
A and the deflections at F and C.
Since the problem is symmetric the rotation at the centreline
must be zero. Therefore we
will
CE2033 (5 ch) Structural Analysis
Determine the vertical deflection under the
load P. The load is applied gradually from zero
to a value of P. Use the method of real work.
V(x)! = -P
M(x)! = -Px
We!= P
Wi! = 0 L M/2EI dx + k 0 L V/2GA dx
! = (1/2EI) 0 L (
CE2033 (5 ch) Structural Analysis
Determine the member end moments for the
beam and loading indicated using moment distribution.
Prodecure
1. Evaluate stiffnesses, distribution factors,
carry over factors and fixed end moments
Element end stiffness
KAB
CE2033 (5 ch) Structural Analysis
Virtual Work
From basic physics we know that the work done done by a force or moment is
equal to that force or moment multiplied by the distance (for forces) or rotation
(for moments) through which they act.
Also, if a ri
CE2033 (5 ch) Structural Analysis
!
!
= -0.00433 rad
DC!
! !
! !
= [(2)(-50)(2)]/EI
= -66.7/[(200x106 kN/m)(50x10-6 mm4)]
= -0.0067 m ! below the tangent
D! !
! !
= 2C + DC = 2(0.004333) + 0.0067
= 0.0153 m !
MAX! between B & C = 0
we need the area to th
CE2033 (5 ch) Structural Analysis
Determine the deflection at "c" using virtual work.
Determine reactions
Ma = 0 [+ ]
6Dy - (3.6)(4)(2 + 4/2) = 0
Dy = +9.6 = 9.6 k
Md = 0 [+ ]
-3.6Ay + (3.6)(4)(4/2) = 0
Ay = +4.8 = 4.8 k
Develop M/EI diagrams based on c
CE2033 (5 ch) Structural Analysis
Shear & Bending Moment Diagrams
In general, an engineer would like to know the variation in shear and bending moment
over the length of a beam. This information would allow the determination of maximum
values, the locatin
CE2033 (5 ch) Structural Analysis
Member BC is a high strength, strand
wire, steel cable for which EA = 7337 K
and the maximum recommended design
load is 39 K; member AB is a steel
beam for which EI = 422,917 Kft.
Choose the tension in the cable as the
re
CE2033 (5 ch) Structural Analysis
Using virtual work to calculate the end rotations due the real loading on the statically determinate base structure.
A! = (1/EI)[L(0.281wL)() + (L)(-0.281wL)(0.8125)]
! = 0.03659wL/EI
fAA!= (1/EI)[(1)(L)()
! = 0.333L/EI
B
CE2033 (5 ch) Structural Analysis
In both examples above, a unit rotation is applied at the point where we want our maximum bending moment and then sketch out how all the beams and columns would deflect assuming all the beams and columns are connected rig
CE2033 (5 ch) Structural Analysis
Determine the displacement
at C and the slope at B. EI
is constant. Use the conjugate beam method.
Develop M(x) using normal
shear and bending moment
calculations.
Fortunately for this example, when we convert from
the re
CE2033 (5 ch) Structural Analysis
Approximate Beam/Frame Analysis
Vertical loading
Lateral loading
Portal method
Cantilever method
Beams/Frames under Vertical
Loading
Assumptions
points of inflection at 0.10L from
any fixed or moment resisting
joints
CE2033 (5 ch) Structural Analysis
Influence Coecients
help to explain the relationship between member displacements and member forces
influence coefficients could be described as the response at one point in a structure
to an action taken at another poi
CE2033 (5 ch) Structural Analysis
The virtual moment curve is always either a straight line or a series of straight lines. As
a result any line segment for the m diagram from point j to k can be represented as
m = ax + b.
Wi!
! !
but
! !
! !
= Wi = xj xk
CE2033 (5 ch) Structural Analysis
Deflections
"It is probably obvious by this time that these deflections, be they large or small, generate the forces of resistance which make a solid hard and stiff and resistant to external
loads. In other words, a solid
CE2033 (5 ch) Structural Analysis
In addition, free body diagrams can be drawn for
each of the individual pieces that make up the frame.
If you consider just the free bodies from the individual
pieces that make up the frame, you will see that
there are a
CE2033 (5 ch) Structural Analysis
4. Force frame to sway.
Consider the case where the problem frame is given a
lateral deflection or sway. In this case the fixed end moments that are set up in AB and CD [6EI/L] times the lateral deflection. However, we do
CE2033 (5 ch) Structural Analysis
Conjugate (analogous) Beam
Introduction
method was developed and presented in a paper in 1868 by Otto Mohr
Description
If a conjugate (analogous) beam is loaded with the M/EI curve of the original beam, the
slope and def
CE2033 (5 ch) Structural Analysis
Determine expressions for shear force and bending moment across the footing.
It can be seen that there is on discontinuity in loading when moving from the left hand
end to the right; the point load and moment at the centr
CE2033 (5 ch) Structural Analysis
Determine the deflection of the hinge
at C and the joint rotation at B for the
braced frame. EI is constant in all
members.
NOTE:
Since member BCD contains a hinge
its elastic curve is not continuous.
Therefore it is not
CE2033 (5 ch) Structural Analysis
Displace the top bar to the
left.
Both sections of the beam now
make and angle of with respect to the horizontal.
Therefore the two sections of
the beam are parallel.
There is one time when this
particular analogy breaks
CE2033 (5 ch) Structural Analysis
Slope Deflection Method
Introduction
Displacement or stiffness method of analysis of statically indeterminate structures
composed of moment resisting elements.
In the previous force (or flexibility) analysis internal fo
CE2033 (5 ch) Structural Analysis
5. Combine no-sway and sway conditions to get final results.
At this point we know three things:
under the real load and no sway, an extra force of 18.3 k to the right is required
assuming sway and fixed end moments of
CE2033 (5 ch) Structural Analysis
Elastic Strain Energy (internal work) - Bending Moment
From Strength of Materials
d/dx = M/EI
d = (M/EI)dx
The internal work associated with dx is
dWi = Md
dWi = M[(M/EI)dx] = (M/EI)dx
Total strain energy stored in the be
CE2033 (5 ch) Structural Analysis
Examples
Determine the mid span deflection of the simply
supported beam shown using virtual work and
visual integration.
Mmax = wL/8 = 25(30)/8 = 2813 kft
This leads to the M(x) diagram shown.
Now apply a virtual load at
CE2033 (5 ch) Structural Analysis
Really should check other locations for all cases above.
Distributed Loads and Influence Lines
Consider the loading on a differential length of beam
dP = w(x)dx
The contribution of this differential
load to the response q
CE2033 (5 ch) Structural Analysis
2. Carry out actual moment distribution and plot results.
AB
BA BC CB CD DC
Carry over factor
0.0
<=0.5 0.5=> <=0.5 0.0=> <=0.5
=>
Distribution factor
0.000 0.667 0.333 0.308 0.692 1.000
Fixed end moment
0
0
-150 +150 0
0
CE2033 (5 ch) Structural Analysis
Determine the end moments for each
of the three elements that make up
this frame. Use slope deflection.
Consider this typical solution
Note that there are zero rotations at A
and D due to the fixed supports there.
Also, i