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