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© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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CE 102 Statics
Chapter 5
Forces in Beams and Cables
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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Contents
IntroductionInternal Forces in MembersSample Problem 7.1Various Types of Beam Loading and
SupportShear and Bending Moment in a
BeamSample Problem 7.2Sample Problem 7.3Relations Among Load, Shear, and
Bending Moment
Sample Problem 7.4Sample Problem 7.5Cables With Concentrated LoadsCables With Distributed LoadsParabolic CableSample Problem 7.6
Catenary
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Introduction
• Preceding chapters dealt with:
a) determining external forces acting on a structure and
b) determining forces which hold together the various members of a structure.
• The current chapter is concerned with determining the internal forces (i.e., tension/compression, shear, and bending) which hold together the various parts of a given member.
• Focus is on two important types of engineering structures:
a) Beams - usually long, straight, prismatic members designed to support loads applied at various points along the member.
b) Cables - flexible members capable of withstanding only tension, designed to support concentrated or distributed loads.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Internal Forces in Members• Straight two-force member AB is in
equilibrium under application of F and -F.
• Internal forces equivalent to F and -F are required for equilibrium of free-bodies AC and CB.
• Multiforce member ABCD is in equil-ibrium under application of cable and member contact forces.
• Internal forces equivalent to a force-couple system are necessary for equil-ibrium of free-bodies JD and ABCJ.
• An internal force-couple system is required for equilibrium of two-force members which are not straight.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.1
Determine the internal forces (a) in member ACF at point J and (b) in member BCD at K.
SOLUTION:
• Compute reactions and forces at connections for each member.
• Cut member ACF at J. The internal forces at J are represented by equivalent force-couple system which is determined by considering equilibrium of either part.
• Cut member BCD at K. Determine force-couple system equivalent to internal forces at K by applying equilibrium conditions to either part.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.1
:0 yF
0N1800N2400 yE NEy 600
:0 xF 0xE
SOLUTION:
• Compute reactions and connection forces.
:0 EM
0m8.4m6.3N2400 F N1800F
Consider entire frame as a free-body:
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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Sample Problem 5.1Consider member BCD as free-body:
:0 BM
0m4.2m6.3N2400 yC N3600yC
:0 CM
0m4.2m2.1N2400 yB N1200yB
:0 xF 0 xx CB
Consider member ABE as free-body:
:0 AM 0m7.2 xB 0xB
:0xF 0 xx AB 0xA
:0yF 0N600 yy BA N1800yA
From member BCD,
:0 xF 0 xx CB 0xC
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.1• Cut member ACF at J. The internal forces at J are
represented by equivalent force-couple system.
Consider free-body AJ:
:0 JM
0m2.1N1800 M mN2160 M
:0 xF
07.41cosN1800 F N1344F
:0 yF
07.41sinN1800 V N1197V
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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Sample Problem 5.1
• Cut member BCD at K. Determine a force-couple system equivalent to internal forces at K .
Consider free-body BK:
:0 KM
0m5.1N1200 M mN1800 M
:0 xF 0F
:0 yF
0N1200 V N1200V
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Various Types of Beam Loading and Support
• Beam - structural member designed to support loads applied at various points along its length.
• Beam design is two-step process:
1) determine shearing forces and bending moments produced by applied loads
2) select cross-section best suited to resist shearing forces and bending moments
• Beam can be subjected to concentrated loads or distributed loads or combination of both.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Various Types of Beam Loading and Support
• Beams are classified according to way in which they are supported.
• Reactions at beam supports are determinate if they involve only three unknowns. Otherwise, they are statically indeterminate.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Shear and Bending Moment in a Beam• Wish to determine bending moment
and shearing force at any point in a beam subjected to concentrated and distributed loads.
• Determine reactions at supports by treating whole beam as free-body.
• Cut beam at C and draw free-body diagrams for AC and CB. By definition, positive sense for internal force-couple systems are as shown.
• From equilibrium considerations, determine M and V or M’ and V’.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Shear and Bending Moment Diagrams• Variation of shear and bending
moment along beam may be plotted.
• Determine reactions at supports.
• Cut beam at C and consider member AC,
22 PxMPV
• Cut beam at E and consider member EB,
22 xLPMPV
• For a beam subjected to concentrated loads, shear is constant between loading points and moment varies linearly.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.2
Draw the shear and bending moment diagrams for the beam and loading shown.
SOLUTION:
• Taking entire beam as a free-body, calculate reactions at B and D.
• Find equivalent internal force-couple systems for free-bodies formed by cutting beam on either side of load application points.
• Plot results.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.2SOLUTION:
• Taking entire beam as a free-body, calculate reactions at B and D.
• Find equivalent internal force-couple systems at sections on either side of load application points.
:0yF 0kN20 1 V kN201 V
:02 M 0m0kN20 1 M 01 M
mkN50kN26
mkN50kN26
mkN50kN26
mkN50kN26
66
55
44
33
MV
MV
MV
MV
Similarly,
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.2
• Plot results.
Note that shear is of constant value between concentrated loads and bending moment varies linearly.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.3
Draw the shear and bending moment diagrams for the beam AB. The distributed load of 40 lb/in. extends over 12 in. of the beam, from A to C, and the 400 lb load is applied at E.
SOLUTION:
• Taking entire beam as free-body, calculate reactions at A and B.
• Determine equivalent internal force-couple systems at sections cut within segments AC, CD, and DB.
• Plot results.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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Sample Problem 5.3SOLUTION:
• Taking entire beam as a free-body, calculate reactions at A and B.
:0 AM
0in.22lb400in.6lb480in.32 yB
lb365yB
:0 BM
0in.32in.10lb400in.26lb480 A
lb515A
:0 xF 0xB
• Note: The 400 lb load at E may be replaced by a 400 lb force and 1600 lb-in. couple at D.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.3
:01 M 04051521 Mxxx
220515 xxM
:02 M 06480515 Mxx
in.lb 352880 xM
From C to D:
:0yF 0480515 V
lb 35V
• Evaluate equivalent internal force-couple systems at sections cut within segments AC, CD, and DB.
From A to C:
:0yF 040515 Vx
xV 40515
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.3
:02 M
01840016006480515 Mxxx
in.lb 365680,11 xM
• Evaluate equivalent internal force-couple systems at sections cut within segments AC, CD, and DB.
From D to B:
:0yF 0400480515 V
lb 365V
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.3
• Plot results.
From A to C: xV 40515
220515 xxM
From C to D:lb 35V
in.lb 352880 xM
From D to B:lb 365V
in.lb 365680,11 xM
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Relations Among Load, Shear, and Bending Moment• Relations between load and shear:
wx
V
dx
dV
xwVVV
x
0lim
0
curve loadunder area D
C
x
xCD dxwVV
• Relations between shear and bending moment:
VxwVx
M
dx
dM
xxwxVMMM
xx
21
00limlim
02
curveshear under area D
C
x
xCD dxVMM
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Relations Among Load, Shear, and Bending Moment
• Reactions at supports, 2
wLRR BA
• Shear curve,
xL
wwxwL
wxVV
wxdxwVV
A
x
A
22
0
• Moment curve,
0at 8
22
2
max
2
0
0
Vdx
dMM
wLM
xxLw
dxxL
wM
VdxMM
x
x
A
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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Sample Problem 5.4
Draw the shear and bending-moment diagrams for the beam and loading shown.
SOLUTION:
• Taking entire beam as a free-body, determine reactions at supports.
• With uniform loading between D and E, the shear variation is linear.
• Between concentrated load application points, and shear is constant.
0 wdxdV
• Between concentrated load application points, The change in moment between load application points is equal to area under shear curve between points.
.constantVdxdM
• With a linear shear variation between D and E, the bending moment diagram is a parabola.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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Sample Problem 5.4
• Between concentrated load application points, and shear is constant.0 wdxdV
• With uniform loading between D and E, the shear variation is linear.
SOLUTION:
• Taking entire beam as a free-body, determine reactions at supports.
:0AM
0ft 82kips 12
ft 14kips 12ft 6kips 20ft 24
D
kips 26D
:0 yF
0kips 12kips 26kips 12kips 20 yA
kips 18yA
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.4• Between concentrated load application
points, The change in moment between load application points is equal to area under the shear curve between points.
.constantVdxdM
• With a linear shear variation between D and E, the bending moment diagram is a parabola.
048
ftkip 48140
ftkip 9216
ftkip 108108
EDE
DCD
CBC
BAB
MMM
MMM
MMM
MMM
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.5
Sketch the shear and bending-moment diagrams for the cantilever beam and loading shown.
SOLUTION:
• The change in shear between A and B is equal to the negative of area under load curve between points. The linear load curve results in a parabolic shear curve.
• With zero load, change in shear between B and C is zero.
• The change in moment between A and B is equal to area under shear curve between points. The parabolic shear curve results in a cubic moment curve.
• The change in moment between B and C is equal to area under shear curve between points. The constant shear curve results in a linear moment curve.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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Sample Problem 5.5
• With zero load, change in shear between B and C is zero.
SOLUTION:
• The change in shear between A and B is equal to negative of area under load curve between points. The linear load curve results in a parabolic shear curve.
awVV AB 021 awVB 02
1
0,at wdx
dVB
0,0,at wwdx
dVVA A
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Statics
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Sample Problem 5.5• The change in moment between A and B is equal
to area under shear curve between the points. The parabolic shear curve results in a cubic moment curve.
• The change in moment between B and C is equal to area under shear curve between points. The constant shear curve results in a linear moment curve.
aLawMaLawMM
awMawMM
CBC
BAB
3061
021
203
1203
1
0,0,at Vdx
dMMA A
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Cables With Concentrated Loads
• Cables are applied as structural elements in suspension bridges, transmission lines, aerial tramways, guy wires for high towers, etc.
• For analysis, assume:a) concentrated vertical loads on given
vertical lines,b) weight of cable is negligible,c) cable is flexible, i.e., resistance to
bending is small, d) portions of cable between successive
loads may be treated as two force members
• Wish to determine shape of cable, i.e., vertical distance from support A to each load point.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Cables With Concentrated Loads• Consider entire cable as free-body. Slopes of
cable at A and B are not known - two reaction components required at each support.
• Four unknowns are involved and three equations of equilibrium are not sufficient to determine the reactions.
• For other points on cable,
2 yields02
yMC yxyx TTFF , yield 0,0
• constantcos xx ATT
• Additional equation is obtained by considering equilibrium of portion of cable AD and assuming that coordinates of point D on the cable are known. The additional equation is .0 DM
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Cables With Distributed Loads• For cable carrying a distributed load:
a) cable hangs in shape of a curveb) internal force is a tension force directed along
tangent to curve.
• Consider free-body for portion of cable extending from lowest point C to given point D. Forces are horizontal force T0 at C and tangential force T at D.
• From force triangle:
0
220
0
tan
sincos
T
WWTT
WTTT
• Horizontal component of T is uniform over cable.
• Vertical component of T is equal to magnitude of W measured from lowest point.
• Tension is minimum at lowest point and maximum at A and B.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Parabolic Cable
• Consider a cable supporting a uniform, horizontally distributed load, e.g., support cables for a suspension bridge.
• With loading on cable from lowest point C to a point D given by internal tension force magnitude and direction are
,wxW
0
2220 tan
T
wxxwTT
• Summing moments about D,
02
:0 0 yTx
wxM D
0
2
2T
wxy
or
The cable forms a parabolic curve.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.6
The cable AE supports three vertical loads from the points indicated. If point C is 5 ft below the left support, determine (a) the elevation of points B and D, and (b) the maximum slope and maximum tension in the cable.
SOLUTION:
• Determine reaction force components at A from solution of two equations formed from taking entire cable as free-body and summing moments about E, and from taking cable portion ABC as a free-body and summing moments about C.
• Calculate elevation of B by considering AB as a free-body and summing moments B. Similarly, calculate elevation of D using ABCD as a free-body.
• Evaluate maximum slope and maximum tension which occur in DE.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.6SOLUTION:
• Determine two reaction force components at A from solution of two equations formed from taking entire cable as a free-body and summing moments about E,
06606020
041512306406020
:0
yx
yx
E
AA
AA
M
and from taking cable portion ABC as a free-body and summing moments about C.
0610305
:0
yx
C
AA
M
Solving simultaneously,kips 5kips 18 yx AA
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.6• Calculate elevation of B by considering AB
as a free-body and summing moments B.
020518:0 BB yM
ft 56.5By
Similarly, calculate elevation of D using ABCD as a free-body.
0121562554518
:0
Dy
M
ft83.5Dy
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 5.6
• Evaluate maximum slope and maximum tension which occur in DE.
15
7.14tan 4.43
cos
kips 18max T kips 8.24max T
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Catenary
• Consider a cable uniformly loaded along the cable itself, e.g., cables hanging under their own weight.
• With loading on the cable from lowest point C to a point D given by the internal tension force magnitude is
wTcscwswTT 022222
0
,wsW
• To relate horizontal distance x to cable length s,
c
xcs
c
sc
csq
dsx
csq
ds
T
Tdsdx
ssinhandsinh
coscos
1
022
220
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Catenary• To relate x and y cable coordinates,
c
xcy
cc
xcdx
c
xcy
dxc
xdx
c
sdx
T
Wdxdy
x
cosh
coshsinh
sinhtan
0
0
which is the equation of a catenary.
40
Problem 5.7
A
BC
D
E
2 ft 3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G
A 200-lb load is applied at point G of beam EFGH, which is attached to cable ABCD by vertical hangers BF and CH. Determine (a) the tension in each hanger, (b) the maximum tension in the cable, (c) the bending moment at F and G.
41
A
BC
D
E
2 ft 3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G
A 200-lb load is applied at point G of beam EFGH, which is attached to cable ABCD by vertical hangers BF and CH. Determine (a) the tension in each hanger, (b)
Solving Problems on Your Own
1. Identify points of the cable where useful information (position, slope,etc.) exists. Cut the cable at these points and draw a free-body diagram of one of the two portions of the cable.2. Use M = 0 if you know the position, or Fx = 0 and Fy = 0 if you know the slope, to generate needed equations of equilibrium.
Problem 7.158
the maximum tension in the cable, (c) the bending moment at F and G.
42
4. For a cable supporting vertical loads only, the horizontal component of the tension force is the same at any point. For such a cable, the maximum tension occurs in the steepest portion of the cable.
3. The tension in each section can be determined from the equations of equilibrium.
A
BC
D
E
2 ft 3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G
A 200-lb load is applied at point G of beam EFGH, which is attached to cable ABCD by vertical hangers BF and CH. Determine (a) the tension in each hanger, (b)
Solving Problems on Your Own
Problem 7.158
the maximum tension in the cable, (c) the bending moment at F and G.
43
+
A
BC
D
E
2 ft 3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G
Problem 7.158 Solution
Identify points of the cable where useful information exists. Draw a free-body diagram of one of the two portions of the cable. Use M = 0 to generate the equations of equilibrium.
A
B
2 ft
5 ft
FBF
TBC
T0
Ay
Free Body: Portion AB
MB = 0: T0(2 ft) - Ay(5 ft) = 0
Ay = 0.4 T0
44
Problem 7.158 Solution
+
A
BC
D
E
2 ft 3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G
Identify points of the cable where useful information exists. Draw a free-body diagram of one of the two portions of the cable. Use M = 0 to generate the equations of equilibrium.
A
B3 ft
5 ft
FBF
TCDT0
Free Body: Portion ABC
MC = 0: T0(3 ft) - 0.4 T0 (10 ft) + FBF (5 ft) = 0
0.4T0
C
5 ft
FCH
FBF = 0.2 T0
45
+
A
BC
D
E
2 ft 3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G
Identify points of the cable where useful information exists. Draw a free-body diagram of one of the two portions of the cable. Use M = 0 to generate the equations of equilibrium.
3 ft TBCT0
Free Body: Portion CD
MC = 0: Dy(5 ft) - T0 (3 ft) = 0
C
5 ft
FCH
FBF = 0.2 T0
Dy
Dy= 0.6 T0D
Problem 7.158 Solution
46
+
A
BC
D
E
2 ft 3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G
Identify points of the cable where useful information exists. Draw a free-body diagram of one of the two portions of the cable. Use M = 0 to generate the equations of equilibrium.
2 ft T0
Free Body: Portion BCD
MB = 0: 0.6T0 (10 ft) - T0 (2 ft) - FCH (5ft) = 0
C
5 ft
FCH
FBF = 0.2 T0
0.6 T0
5 ft
TAB
0.2 T0
BD
FCH = 0.8 T0
Problem 7.158 Solution
47
A
BC
D
E
3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G FCH = 0.8 T0
2 ftThe tension in each section can be determined from the equations of equilibrium.
Free Body: Beam EFH
E7.5 ft 200 lb
F
H
G
Ey
Ex
0.8 T00.2 T0
5 ft 5 ft
FBF = 0.2T0
+
0.2T0 (5 ft) + 0.8 T0 (10 ft) - (200 lb)(7.5 ft) = 0
ME = 0:
T0 = 166.67 lb
FCH = 0.8(166.67) FCH = 133.33 lb T
FBF = 0.2(166.67) FBF = 33.33 lb T
Problem 7.158 Solution
48
A
BC
D
E
3 ft
5 ft 5 ft 5 ft
7.5 ft 200 lb
F H
G
2 ftThe maximum tension occurs in the steepest portion of the cable.
T0
TmDy= 0.6 T0
T0 = 166.67 lb
Tm = T02 + (0.6T0)2 = 1.1662T0
= 1.1662(166.67)
Tm = 194.4 lb
D
Problem 7.158 Solution
49
+
E7.5 ft
200 lb
F H
G
Ey
Ex
5 ft 5 ft
FCH = 133.33 lbFBF = 33.33 lb
Beam EFH
133.33 lb
2.5 ft
VMG
GMG = 0: (133.33 lb)(2.5 ft) -MG = 0 MG = + 333 lb-ft
133.33 lb
2.5 ft
VMF
G
2.5 ft
200 lb
+ MF = 0: (133.33 lb)(5 ft) - (200 lb)(2.5 ft) -MF = 0 MF = +166. 7 lb-ft
The moments at points G and F are determined by using free-body diagrams for two sections of the beam.
Problem 7.158 Solution
50
Problem 5.8
For the beam and loading shown, (a) write the equations of the shear and bending-moment curves, (b) determine the magnitude and location of the maximum bending moment.x
y
L
AB
w = w0 cos x2L
51
Solving Problems on Your Own
For beams supporting a distributed load expressed as a function w(x), the shear V can be obtained by integrating the function -w(x) , and the moment M can be obtained by integrating V (x).
For the beam and loading shown, (a) write the equations of the shear and bending-moment curves, (b) determine the magnitude and location of the maximum bending moment.
x
y
L
AB
w = w0 cos x2L
Problem 7.159
52
Problem 7.159 Solution
x
y
L
AB
w = w0 cos x2L
The shear V can be obtained by integrating the function -w(x)and the moment M can be obtained by integrating V (x).x
= -w = -w0 cosdVdx
x2L
V = - wdx = - w0 sin + C1( )2L
x2L
dMdx
= V = - w0 sin + C1( )2L
x2L
M = Vdx = w0 cos + C1x + C2( )2L
2 x2L
53
x
y
L
AB
w = w0 cos x2L
The shear V can be obtained by integrating the function -w(x)and the moment M can be obtained by integrating V (x).x
V = - w0 sin + C1( )2L
x2L
M = w0 cos + C1x + C2( )2L
2 x2L
Boundary conditions
At x = 0 : V = C1 = 0 C1 = 0
At x = 0 : M = w0 (2L/)2 cos (0) + C2 = 0
C2 = -w0 (2L/)2
Problem 7.159 Solution
54
x
y
L
AB
w = w0 cos x2L
The shear V can be obtained by integrating the function -w(x)and the moment M can be obtained by integrating V (x).x
V = - w0 sin( )2L
x2L
M = w0 cos - w0 ( )2L
2 x2L ( )2L
2
M = w0 ( -1 + cos )( )2L
2 x2L
Mmax at x = L: Mmax = w0 [-1 + 0]( )2L
2
42
Problem 7.159 Solution
Mmax = w0 L2
55
Problem 5.9
It has been experimentally determined that the bending moment at point K of the frame shown is 300 N-m. Determine (a) the tension in rods AE and FD, (b) the corresponding internal forces at point J.
AB
D
EJ
k
F
100 mm
100 mm
100 mm
120 mm
280 mm
C
56
Solving Problems on Your Own
1. Cut the member at a point, and draw the free-body diagram of each of the two portions.
2. Select one of the two free-body diagrams and use it to write the equations of equilibrium.
It has been experimentally determined that the bending moment at point K of the frame shown is 300 N-m. Determine (a) the tension in rods AE and FD, (b) the corresponding internal forces at point J.
AB
D
EJ
k
F
100 mm
100 mm
100 mm
120 mm
280 mm
C
Problem 7.161
57
Problem 7.161 Solution
AB
D
Tx
J T100 mm
100 mm
120 mm
Cut the member at a point, and draw the free-body diagram of each of the two portions.
D100 mm
280 mmC
TykV
FMK = 300 N-m
k
MK = 300 N-m
V
F
Tx
T Ty
300
280 - 120 = 160
3002 + 1602 = 340
A
D
A
D
8
1517
Ty = T1517
Tx = T817
58
+
AB
D
J200 mm
120 mm
kV
FMK = 300 N-m
Ty = T1517
Tx = T8
17 Select one of the two free-body diagrams and use it to writethe equations of equilibrium.
Mk = 0: 300 N-m - T(0.2 m) - T(0.12 m) = 0 1517
817
T = 1500 N
Free Body: ABK
Problem 7.161 Solution
59
AB
D
J100 mm
120 mm
Ty = (1500) = 1323.53 N1517
Tx = (1500) = 705.88 N8
17
Free Body: ABJ
VF
MJ
+ MJ = 0: MJ - (705.88 N)(0.1 m) - (1323.53 N)(0.12 m) = 0
MJ = 229 N-m
Fx = 0: 705.88 N - V = 0 V = 706 N +
Fy = 0: -F - 1323.53 N = 0 F = 1324 N+
Problem 7.161 Solution
60
Problem 5.10
Cable ACB supports a load uniformly distributed along the horizontal as shown. The lowest point C is located 9 m to the right of A. Determine (a) the vertical
B2.25 m
C
A
a
9 m 6 m
distance a, (b) the length of the cable, (c) the components of the reaction at A.
60 kg/m
61
Solving Problems on Your Own
Cable ACB supports a load uniformly distributed along the horizontal as shown. The lowest point C is located 9 m to the right of A.
B2.25 m
C
A
a
9 m 6 m
Determine (a) the vertical distance a, (b) the length of the cable, (c) the components of the reaction at A.
1. Identify points of the cable where useful information (position, slope,etc.) exists. Cut the cable at these points and draw a free-body diagram of the two portions of the cable.
2. Use M = 0 if you know the position, or Fx = 0 and Fy = 0 if you know the slope, to generate needed equations of equilibrium.
3. The length of the cable can be determined from (7.10).
60 kg/m
Problem 7.162
62
+
Problem 7.162 Solution
C
Aa
9 m
Ay
T0
T0
Identify points of the cable where useful information exists. Cut the cable at these points and draw a free-body diagram. Use M = 0 if you know the position, or Fx = 0 and Fy = 0 if you know the slope to generate needed equations ofequilibrium.
Free Body : Portion AC
Fy = 0: Ay - 9w =0, Ay = 9w
4.5 m9w
MA = 0: To a - (9w)(4.5 m) = 0, To a = 40.5w (1)
+
63
BC
6 mIdentify points of the cable where useful information exists. Cut the cable at these points and draw a free-body diagram. Use M = 0 if you know the position, or Fx = 0 and Fy = 0 if you know the slope to generate needed equations ofequilibrium.
Free Body : Portion CB
T0T0
By
yB
3 m6w
+
Fy = 0: By - 6w = 0, By = 6w
MB = 0: (6w)(3 m) - To yB = 0, To yB = 18w
+
Problem 7.162 Solution
64
B2.25 m
C
A
yB = a - 2.25
9 m 6 m
Identify points of the cable where useful information exists. Cut the cable at these points and draw a free-body diagram. Use M = 0 if you know the position, or Fx = 0 and Fy = 0 if you know the slope, to generate needed equations of equilibrium.
T0
6w
T0
9w
15w7.5 m
+ MA = 0: 6w (15 m) - 15w(7.5 m) + To (2.25 m) = 0
To a = 40.5w (1)
Free Body : Entire Cable
To = 10w
Using (1) (10w) a = 40.5w a = 4.05 m
Problem 7.162 Solution
65
B2.25 m
C
A
4.05
9 m 6 m
6w9w
15w7.5 m
(b) Length of AC & CB
The length of the cable can be determined from (7.10).
10w
10w
Portion AC xA = 9 m, yA = a = 4.05 m; = = 0.45yAxA
4.059
SAC = xA 1 + - + ...yAxA
23 ( )
4[ ]
yAxA
( )2
25
SAC = 9 m [ 1 + (2/3)(0.45)2 - (2/5)(0.45)2 + ... ]= 10.067 m
Portion CB xB = 6 m, yB = 4.05 - 2.25 = 1.8 m; = 0.3yBxB
SCB = 6 m [ 1 + (2/3)(0.3)2 - (2/5)(0.3)2 + ... ]= 6.341 m
SABC = SAC + SCB = 10.067 + 6.341 SABC = 16.41 m
Problem 7.162 Solution
66
B2.25 m
C
A
4.05
9 m 6 m
6w9w
15w7.5 m
(c) Components of Reaction at A
10w
10w
Ay = 9w = 9(60 kg/m)(9.81 m/s2) = 5297.4 N
Ax = 10w = 10(60 kg/m)(9.81 m/s2) = 5886 N
Ay = 5300 NAx = 5890 N
Problem 7.162 Solution
67
Problem 5.13
For the beam and loading shown, (a) draw the shear and bending-moment diagrams, (b) determine the magnitude and location of the maximum absolute value of the bending moment.
4 ft 6 ft
6000 lb
1500 lb/ft
A BC
68
Solving Problems on Your Own
1. Draw a free-body diagram for the entire beam, and use it to determine the reactions at the beam supports.
2. Draw the shear diagram.
3. Draw the bending-moment diagram by computing the area under each portion of the shear curve.
For the beam and loading shown, (a) draw the shear and bending-moment diagrams, (b) determine the magnitude and location of the maximum absolute value of the bending moment.
4 ft 6 ft
6000 lb
1500 lb/ft
A BC
Problem 7.163
69
+
Problem 7.163 Solution
4 ft 6 ft 6 kips
9 kips
A BC
Draw a free-body diagram for the entire beam, and use it to determine the reactions at the beam supports.
3 ft
MA = 0 : (6 kips)(10 ft) - (9 kips)(7 ft) + B (4 ft) = 0
B = 0.75 kips
Fy = 0 : A + 0.75 kips + 6 kips - 9 kips = 0
A = 2.25 kips
+
70
6 kips
1.5 kips/ft
A B C
2.25 kips 0.75 kips
4 ft 6 ft
Draw the shear diagram.
x+ 2.25 kip
+ 3 kip
- 6 kip
A B C
D
2 ft 4 ft
V (kips)
BD = =VB
w3 kips
1.5 kips/ft
Problem 7.163 Solution
71
6 kips
1.5 kips/ft
A B C
2.25 kips 0.75 kips
4 ft 6 ft
Draw the bending-moment diagram.
x+ 2.25 kip
+ 3 kip
- 6 kip
A B C
D
2 ft 4 ft
V (kips)
BD = =VB
w3 kips
1.5 kips/ft
xA B CD
M (kip-ft)
(+9 kip-ft)
(3 kips)(2 ft) = (+3 kip-ft)12
(- 12 kip-ft)
(+9 kip-ft)(+12 kip-ft) Mmax = 12 kip-ft
Problem 7.163 Solution
6 ft from A