Axial force, shear force and bending moment : Method of Sections



Determine support reactions
for the beam loaded as shown
in Figure.
Also determine the shear force
and bending moment at a
distance 3.5 m from left
RAy
support.
Determine shear force and
bending moment at a distance
3.3 m from right support.
REx
RCy
ΣFx = 0,
REy
REx = 0
ΣFy = 0, RAY + RCY +REY – 80*(2.4+0.6)-160 = 0
RAY + RCY +REY = 400
ΣMA = 0, RCY* 3+REY* 6– 80*(2.4+0.6)*3/2-160*(3+1.5) = 0
3 RCY + 6 REY = 1080
CE 211
Dr. Mohammad Al Amin Siddique
RCY + 2REY = 360
BUET
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Axial force, shear force and bending moment : Method of Sections
ΣFx = 0,
RBx
RBx = 0
RAy
RBy
ΣMB = 0, RAY* 2.4 - 80*(2.4) * 2.4/2 = 0
RAY = 96 kN ( )
RAY + RCY +REY = 400
96 + RCY +REY = 400
RCY + 2REY = 360
RCY +REY = 304
REY = 56 kN ( )
RCY = 248 kN ( )
CE 211
Dr. Mohammad Al Amin Siddique
BUET
2
Axial force, shear force and bending moment : Method of Sections
At a section F-F:
Shear force is VF
Bending moment is MF
Axial force = 0
MF
VF
56 kN
ΣFy = 0,
VF - 160+ 56 = 0
VF = 104 kN ( )
ΣME = 0,
VF* 2.5 + M - 160*1.5 = 0
MF = -20 kN.m
V
ΣFy = 0,
96 - 80*2.7 - V = 0
V = -120 kN ( )
96 kN
ΣMA = 0,
V* 2.7 – M +80*2.7*2.7/2= 0
-120* 2.7 – M + 291.6 = 0
M = -32.4 kN.m
CE 211
Dr. Mohammad Al Amin Siddique
BUET
3
Axial force, shear force and bending moment diagrams
 Because of the applied loadings, beams develop an internal axial force, shear
force and bending moment that, in general, vary from point to point along the
axis of the beam.
 One way to do this is to express N, V and M as functions of their arbitrary
position x along the beam’s axis, and then plot these functions. They represent
the axial, shear and moment diagrams, respectively.
CE 211
Dr. Mohammad Al Amin Siddique
BUET
4
Axial force, shear force and bending moment diagrams
 In order to properly design a beam it therefore becomes important to
determine the maximum shear and moment in the beam.
 The maximum values of V and M can then be obtained directly from these
graphs.
 Also, since the shear and moment diagrams provide detailed information
about the variation of the shear and moment along the beam’s axis, they
are often used by engineers to decide where to place reinforcement
materials within the beam or how to proportion the size of the beam at
various points along its length.
CE 211
Dr. Mohammad Al Amin Siddique
BUET
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Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
6
Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
7
Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
8
Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
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Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
10
Axial force, shear force and bending moment diagrams
 Draw the shear and moment diagrams for the beam shown in Fig. below.
RC
RA
ΣMA = 0, RC* 10 - 80 - 15 * 5 - 5 *5* (5+5/2) = 0
RC = 34.25 kN
RA = 5.75 kN
Shear and Moment Functions:
Since there is a discontinuity of distributed load and also a concentrated
load at the beam’s center, two regions of x must be considered in order
to describe the shear and moment functions for the entire beam.
CE 211
Dr. Mohammad Al Amin Siddique
BUET
11
Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
12
Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
13
Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
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Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
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Axial force, shear force and bending moment diagrams
A plane frame with an overhang is supported at points A and D shown in fig. A linearly
varying distributed load of peak intensity q0 =160 N/m acts on span AB. Concentrated
moment M0 = 380 N.m is applied at A, and an inclined concentrated load P = 200 N
acts at C. Force P also acts at mid-height of column BD. The lengths of segments AB
and BD are L = 4 m, and the length of the overhang BC is 2 m.
 Draw the axial-force, shear-force, and bending-moment diagrams for this frame.
CE 211
Dr. Mohammad Al Amin Siddique
BUET
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Axial force, shear force and bending moment diagrams
Member AB:
CE 211
Dr. Mohammad Al Amin Siddique
BUET
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Axial force, shear force and bending moment diagrams
Member BC:
Member DB:
CE 211
Dr. Mohammad Al Amin Siddique
BUET
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Axial force, shear force and bending moment diagrams
CE 211
Dr. Mohammad Al Amin Siddique
BUET
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