Download - Three Moment Theorem2
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Lecture outline
The purpose of carrying out structural analysis is to calculate the maximum values of actions (moment, shear force and axial force) for design member or check member capacity
Continuous beam with n spans which subjects to various loading configurations is one of most popular structural member for building
Continuous beam is indeterminate structure with the number of unknown reactions is more than the number of static equilibrium equations (3 equations).
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Lecture outline A simple technique to solve continuous beam
problem is to use the “Theorem of Three bending moments”. First, the bending moment can be constructed straightforward by applying the theorem. Then, other unknowns can be found from the beam equations. For example, the shear force can be found by differentiating the bending moment
dx
dMV
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Lecture outline
This lecture will detail the theorem and its applications Theorem of Three bending moments Applications
Example 1: 4 spans, patch load Example 2: 4 spans, 2-span loaded Example 3: 4 spans, loads on alternate spans Example 4: Fixed ends
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Theorem of Three bending moments
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Revision: Structural analysis
Free body diagram
Vertical reaction Ay, equivalent load P. At the cut of distance x from left end: shear force V, bending moment M
w (kN/m)
L (m)
M
V
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Revision: Structural analysis The reaction at A is The equivalent point load at a distance x/2 from A is Take moment about the cut,
Force equilibrium for y direction,
Maximum shear force happens at end Maximum bending moment happens at middle
wLAy 2
1
wxP
22
1
022
1
2wxwLxM
xwxwLxM
wxwLV
wxwLV
2
1
02
1
wLV2
1max
8
2
max
wLM
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Revision: Structural analysis
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The area underneath the bending moment curve Integral
The distance from either support to centroid of A: x1=x2=L/2
1246
223222
333
0
23
0
2
0
wLwLwL
xwLxwdx
wxLwxMdxA
LLL
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Point load In a similar manner, the bending moment M
and bending moment area A can be obtained for span subjected to point load
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Point load
Distance from left end to centroid of bending moment area A is
bxaPaPL
Pbx
axL
Pbx
M
0
2
PabA
31
Lax
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Three bending moment theorem The theorem applies to any two adjacent
spans in a continuous beam. For constant section, the theorem states
Where MA, MB, MC are the bending moment values at three subsequent supports A, B, C
22
22
11
11
2
2
2
2
1
1
1
1 62IL
xA
IL
xA
I
LM
I
L
I
LM
I
LM CB
A
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For any load system
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Solution procedure
For each patch formed by 2 adjacent spans of the continuous beam, write the theorem equation. Establish the system of n linear independent equations for n unknown bending moments at supports
Solve the system for support bending moments The BM of the continuous beam is constructed
individually for each span by summing the basic bending moment MLoad (due to applied load on the single span) and the linear function fitted through computed bending moments at 2 corresponding supports, M3BMtheorem
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Applications
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Example 1: UDL on 4 spans
To generate the bending moment diagrams, nominate values for UDL w and span length L. Assume w = -10 kN/m (negative sign to indicate downward force) and L = 10 m.
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Example 1: UDL on 4 spans Apply the “Three bending moment theorem”
equation to spans (AB, BC), (BC, CD), (CD, DE). Substitute L=10, A1 = A2 = -833.33, MA = ME = 0 into 3 equations, we obtain
For each span , fit a linear function through the bending moment at its 2 ends.
14.107
43.71
14.107
500
500
500
410
141
014
D
C
B
D
C
B
M
M
M
M
M
M
Lx 0
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Example 1: UDL on 4 spans
Revise mathematics: linear function y(x) fitted through point 1 (x1, y1) and point 2 (x2, y2) satisfies
12
1
12
1
xx
xx
yy
yy
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Example 1: UDL on 4 spans Sum the bending moment due to applied load
MLoad and the fitted function M3BMtheorem to obtain the bending moment M on each span
Plot the bending moment for the continuous beam from constructed bending moment M in the above table
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Example 1: UDL on 4 spans
UDL over 4 equal spans
-100
-50
0
50
100
150
0 5 10 15 20 25 30 35 40
Distance (m)
Be
nd
ing
mo
me
nt
M (
kN
.m)
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Example 2: UDL on 2 spans
Having L=10; spans (AB, BC): A1 = A2 = -833.33, spans (BC, CD): A1 = -833.33, A2 =0, spans (CD, DE): A1 = A2 = 0; MA = ME = 0.
Hence,
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Example 2: UDL on 2 spans
Fitting the linear function through the bending moment data at ends for each span
93.8
714.35
07.116
0
250
500
410
141
014
D
C
B
D
C
B
M
M
M
M
M
M
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Example 2: UDL on 2 spans
M = MLoad + M3BMtheorem
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Example 2: UDL on 2 spansUDL on 2 adjacent spans
-100
-50
0
50
100
150
0 5 10 15 20 25 30 35 40
Distance (m)
Be
nd
ing
mo
me
nt
M (
kN
.m)
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Example 3: Load on alternate spans
Having L=10; spans (AB, BC): A1 = -833.33, A2 =0; spans (BC, CD): A1 =0, A2 = -833.33; spans (CD, DE): A1 = -833.33, A2 =0; MA = ME = 0. Hence,
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Example 3: Load on alternate spans
Fitting the linear function through the bending moment data at ends for each span
571.53
714.35
571.53
250
250
250
410
141
014
D
C
B
D
C
B
M
M
M
M
M
M
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Example 3: Load on alternate spans M = MLoad + M3BMtheorem
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Example 3: Load on alternate spans
UDL loaded on alternate spans
-120
-100
-80
-60
-40
-20
0
20
40
60
80
0 5 10 15 20 25 30 35 40
Distance (m)
Be
nd
ing
mo
me
nt
M (
kN
.m)
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Example 4: Continuous beam with fixed ends For a continuous beam of (n) spans, (n-1) equations
can be generated from the three moment theorem. The total of supports on the beam are (n+1).
If 2 end supports are pins, the number of unknown BMs at supports is (n-1). We can solve the system of (n-1) equations for these (n-1) BMs as detailed in previous examples
For continuous beam with one fixed end, the number of unknown BMs is (n). For continuous beam with two fixed ends, the number of unknown BMs is (n+1). We can not solve for these unknown BMs as the number of equations is smaller than the number of unknowns.
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Example 4: Continuous beam with fixed ends Consider this example of 2-span continuous
beam ABC of const. cross section, built in fixed ends at A and C, supported at B
One equation for 3 unknowns MA, MB, MC
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Example 4: Continuous beam with fixed ends Effect of a fixed end is similar to placing a
mirroring span at the wall end
B1 is mirror of B over A, B2 is mirror of B over C. Have MB1= MB, MB2= MB
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Example 4: Continuous beam with fixed ends With , from the left hand side,
having AB
1A = -157.5, AAB = -157.5,
ABC = -125, ACB2 = -125
Consider spans (B1A, AB)
12
3wLA
)1.(5.1572
3
5.15.157
3
5.15.157633323
1
1
EqMM
MM
MMM
BA
BB
BAB
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Example 4: Continuous beam with fixed ends Consider spans (AB, BC)
Consider spans (BC, CB2)
)2.(5.28217.133.4
5.3
75.1125
3
5.15.157633323
1
EqMMM
MMM
CBA
BAB
)3.(1.1072
5.3
75.1125
5.3
75.112565.35.35.325.3
2
2
EqMM
MM
MMM
CB
BB
BCB
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Example 4: Continuous beam with fixed ends From Eqs. (1), (2), (3), the bending moments at A,
B, C are found
After the bending moments at all supports are known, the bending moment is recovered for each span in the usual manner. The BMD for the continuous beam can then be plotted.
84.31
47.43
01.57
C
B
A
M
M
M
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Next lecture
Design of tension member Reading:
AS4100:1998, Section 7 and related parts in Section 6 and Section 9