BY: NTDEGUMA
THEORY OF STRUCTURES (MODULE 2)
IV. VIRTUAL WORK EQUATION
BY: NTDEGUMA
THEORY OF STRUCTURES (MODULE 2)
INDETERMINATE BEAMS
DEFLECTION OF TRUSS
II. CASTIGLIANO'S THEOREM
DEFLECTION:
VIRTUAL WORK METHOD
y = deflection
DEFLECTION:
I. THREE-MOMENT EQUATION
y = deflection
L
y=
0
Mm
dx
EI
M = actual moment at element
under consideration
m = moment due to application
of a unit load at the section
under consideration
M1L1 + 2M2 L 1 + L 2 + M3 L 2 + 6A1a1 + 6A2 b2 = 0
L1
L2
SuL
AE
=
where:
S = stress in member due to
actual loads
L = length of the member
u = stress in member due to
virtual unit load
II. MOMENT DISTRIBUTION METHOD
SLOPE :
A = cross-sectional area
of the member
E = modulus of elasticity
STEPS TO BE FOLLOWED:
L
0
M = actual moment at element
under consideration
m = moment due to application
of a unit couple at the
section under consideration
Mm
EI
L
VIRTUAL WORK EQUATION
y=
0
DEFLECTION OF FRAMES
M
dx
EI
SLOPE:
L
1. Solve for k = I / L
2. Solve for Distribution Factor (DF)
DF = K / S K
DF = 1 for external hinge
DF = 0 for fixed end support
M
P
M = actual moment at the
element under
consideration
m = moment due to
application of a P
at the section under
consideration
0
M M
m EI
M = actual moment at the
element under
consideration
m = moment due to
application of moment
"m" at the section
under consideration
I. VIRTUAL WORK METHOD
DEFLECTION:
3. Solve for the fixed end moments
APPROXIMATE ANALYSIS OF STRUCTURES
4. Balance the moments
V. CASTIGLIANO'S THEOREM
COM = 1/2 moment of the other end
DEFLECTION:
y = deflection
L
y=
0
M
P
M
dx
EI
M = actual moment at the
element under
consideration
m = moment due to
application of a P
at the section under
consideration
CECC-3
0
For any loading condition, the moment at the fixed end
can be solved by integration.
MA
L-x
b
y
M M
m EI
M = actual moment at the
element under
consideration
m = moment due to
application of moment
"m" at the section
under consideration
DESIGN AND CONSTRUCTION
B
L
CANTILEVER METHOD
ASSUMPTIONS:
1. A point of inflection occurs at the midspan of each girder.
2. A point of inflection occurs at the midheight of each column.
3. The axial force in each column is directly proportional to its
distance from the center of gravity of all columns on that level.
SLOPE:
MB
x1
Mm
dx
EI
where:
y = deflection
M = actual moment at element
under consideration
m = moment due to application
of a unit load at the section
under consideration
P = ydx
xa
A
L
y=
III. FIXED-END MOMENTS
SLOPE:
0
L
5. Carry over the moments (COM)
PORTAL METHOD
L
ASSUMPTIONS:
x1
MA = -
1
2
L
Pab
x2
x1
2
MB = -
1
2
L
0
Pba
2
where:
y = deflection
M = actual moment at element
under consideration
m = moment due to application
of a unit load at the section
x2
3/12
Mm
EI
CECC-3
DESIGN AND CONSTRUCTION
1. The building frame is divided into independent portals.
2. A point of inflection occurs at the midspan of each girder.
3. A point of inflection occurs at the midheight of each column.
4. The horizontal shear at a given storey is distributed among
the columns such that each interior columns resists twice
as much as each exterior column.
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