chap5-b Graphical Method · 2020-02-23 · GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS Procedure for analysis Shear diagram • Since dV/dx = w, slope of the shear

Text

Mechanics of Materials

R.C. Hibbeler 8th Edition, Prentice Hall

CIVL222 STRENGTH OF MATERIALS

Chapter 5b

GRAPHICAL METHOD

Instructor: Assoc.Prof.Dr. Mürüde Çelikağ

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

CIVL222 Dr. Mürüde Çelikağ 2


RELATIONSHIP BETWEENLOAD AND SHEAR

Slope of shear diagram at each point

= distributed load intensity at each point

wdxdV

wdxdVdVVwdxV

Fy

0)(

0


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)


6Dr. Mürüde Çelikağ

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


Dr. Mürüde Çelikağ

dMdx

= V

nDegree

1nDegree

2nDegree

Area (A)

CIVL222 8



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




V = ∫ w(x) dx M = ∫ V(x) dx

Change in shear

Change in moment

= area under distributed loading

= area under shear diagram


summary

Shear force and load relationCIVL222 Dr. Mürüde Çelikağ 11

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 negative sign of M0





Procedure for analysis

Support 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




Shear 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 pointsCIVL222 Dr. Mürüde Çelikağ 18



Shear 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 pointCIVL222 Dr. Mürüde Çelikağ 19


Moment 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 points is equal to area under shear diagram

between the two points.




Moment 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 1Draw the

• SFD

• BMD


initialfinal VloadingofareaV

0



initialMSFDofareaM final

CIVL222 24

EXAMPLE 2

Draw the

• SFD

• BMD


0

initialfinal VloadingofareaV



0

initialMSFDofareaM final

27

EXAMPLE 3

(a)

0

(b)


0


EXAMPLE 4

Draw the

• SFD

• BMD


0



0

CIVL222 32

Given: A simply supported beam is loaded as shown.

Required :

a) Reactions at the supports

b) Shear Force Diagram. Use the graphical method

c) Bending Moment Diagram. Use the graphical method

Note: Label all key points on both the V and M diagrams with

both values and units.

18 kN/m

5m 5m 4.5m

A

B CD

100 kN 92 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 V and 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;

• 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


chap5-b Graphical Method · 2020-02-23 · GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS Procedure for analysis Shear diagram • Since dV/dx = w, slope of the shear

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