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Course:CIVL222 Strength of Materials
Chapter 2 (continued)
TextMechanics of Materials
R.C. Hibbeler 8th Edition, Prentice Hall
GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS
• A simpler method to construct shear and moment diagram, one that is based on two differential equations that exist among distributed load, shear and moment
6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS
RELATIONSHIP BETWEENLOAD AND SHEAR
wdxdV
wdxdVdVVwdxV
Fy
0)(
0
Slope of shear diagram at each point
= distributed load intensity at each point
B
AAB
B
A
wdxVVdV
Change in shear between points A and B
Area under the distributed load diagram between points A and B
wdxdV using
Regions of distributed load
= w(x)dVdx
nDegree
1nDegree
Area (A)
Area (A)
RELATIONSHIP BETWEENSHEAR AND BENDING MOMENT
VdxdM
VdxdM
dMMdxVdxdxwM
M
0)()()
2)((
00
Slope of the BMD at a point = shear at that point
B
AA
B
AB VdxMMdM
Change in moment between points A and B
Area under SFD between points A and B
VdxdM using
dMdx
= V
nDegree
1nDegree
2nDegree
Area (A)
6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT
DIAGRAMSRegions of distributed load
dVdx
= w(x)dMdx
= V
Slope of shear diagram at each point
Slope of moment diagram at each point
= distributed load intensity at each point
= shear at each point
6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS
Regions of distributed load
V = ∫ w(x) dx M = ∫ V(x) dxChange in shear
Change in moment
= area under distributed loading
= area under shear diagram
summary
Shear force and load relation
summary
Shear force and bending moment relation
RELATIONSHIP BETWEENSHEAR AND CONCENTRATED LOAD
0
0)(
0
PdVdVVPV
Fy
Change in shearat the point of application of aconcentrated load
Step change having the same sign as P
RELATIONSHIP BETWEENBENDING MOMENT AND APPLIED COUPLE
0
0
0
0)(0
MdMdMMMM
M
The change in bending moment =Step change having a negativesign of M0
6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT
DIAGRAMS
Procedure for analysisSupport reactions• Determine support reactions and resolve forces
acting on the beam into components that are perpendicular and parallel to beam’s axis
Shear diagram• Establish V and x axes• Plot known values of shear at two ends of the
beam
6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT
DIAGRAMSProcedure for analysisShear diagram• Since dV/dx = w, slope of the shear diagram at
any point is equal to the intensity of the distributed loading at that point
• To find numerical value of shear at a point, use method of sections and equation of equilibrium or by using V = ∫ w(x) dx, i.e., change in the shear between any two points is equal to area under the load diagram between the two points
6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT
DIAGRAMSProcedure for analysisShear diagram• Since w(x) must be integrated to obtain V, then if
w(x) is a curve of degree n, V(x) will be a curve of degree n+1
Moment diagram• Establish M and x axes and plot known values of
the moment at the ends of the beam• Since dM/dx = V, slope of the moment diagram at
any point is equal to the shear at the point
6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT
DIAGRAMSProcedure for analysisMoment diagram• At point where shear is zero, dM/dx = 0 and
therefore this will be a point of maximum or minimum moment
• If numerical value of moment is to be determined at the point, use method of sections and equation of equilibrium, or by using M = ∫ V(x) dx, i.e., change in moment between any two pts is equal to area under shear diagram between the two pts
6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT
DIAGRAMS
Procedure for analysisMoment diagram• Since V(x) must be integrated to obtain M, then
if V(x) is a curve of degree n, M(x) will be a curve of degree n+1
EXAMPLE 1
Draw the
• SFD
• BMD
initialfinal VloadingofareaV
0
initialMSFDofareaM final
EXAMPLE 2
Draw the
• SFD
• BMD
0
initialfinal VloadingofareaV
0
initialMSFDofareaM final
EXAMPLE 3
(a)
0
(b)
0
EXAMPLE 4
Draw the
• SFD
• BMD
0
0
18 kN/m
5m 5m 4.5m
AB C D
Given:A simply supported beam is loaded as shown. Required :a) Reactions at the supportsb) Shear Force Diagram. Use the graphical methodc) Bending Moment Diagram. Use the graphical method
Note: Label all key points on both the V and M diagrams with both values and units.
100.8158 kN 92.6417 kN
CHAPTER REVIEW
• Shear and moment diagrams are graphical representations of internal shear and moment within a beam.
• They can be constructed by sectioning the beam an arbitrary distance x from the left end, finding Vand M as functions of x, then plotting the results
• Another method to plot the diagrams is to realize that at each point, the slope of the shear diagram is, w = dV/dx;
• and slope of moment diagram is the shear,V = dM/dx.
• Also, the area under the loading diagram represents the change in shear, V = ∫ w dx.
• The area under the shear diagram represents the change in moment, M = ∫ V dx. Note that values of shear and moment at any point can be obtained using the method of sections
CHAPTER REVIEW
CHAPTER REVIEW
Shear force and load relation
CHAPTER REVIEW
Shear force and bending moment relation
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