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Minggu 7-2

Internal Forces

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Objectives

1. To show how to use the method of sections for

determining the internal loadings in a member

2. To generalize the procedure by formulating

equations that can be plotted so they describethe internal shear and moment throughout a

member

3. To analyze the forces and study the geometry

of cables supporting a load

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Shear Force and Bending

Moment DiagramsA BEAM is a long, slender, structural

member designed to support transverse

loadings.

A transverse loading is applied

perpendicular to the axis of the beam.

Beams classified by their supports.

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Need to know the internal shear

force and the internal bending

moment at all locations in thebeam. This information is usually

presented as a graph or diagram of

these values vs. position.

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Pw

a

L

b

x1

x2

x3

Simply Supported Beam

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Pw

a

L

b

x1

x2

x3

Cantilever Beam

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Sign Convention

1. Positive Shear Force will cause a

clockwise rotation of the member on

which it acts.

2. Positive Bending Moment will cause

compression on the top surface of the

beam.

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VV

VV

Positive Shear

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M M

MM

Positive Moment

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MM

Compression

Tension

Positive Moment

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Procedure

1. Find the support reactions.

2. Identify critical points.

3. At representative locations betweencritical points take a section cut.

4. Find V and M and representative locationsbetween critical points.

5. Use proper sign conventions for V and M.6. Plot results.

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Critical Points

1. End of Beam

2. Support

3. Point of application of concentrated

load4. Point of application of concentrated

couple moment.

5. Beginning of distributed load.6. End of distributed load.

7. Change in distributed load definition.

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3 m6 m

6 kN

Determine the internal forces just to the left and the right of the

external force

9 kN m

Critical Points

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4 ft 4 ft

50 lb/ft

A C

Critical Points

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Pw

a

L

b

x1

x2

x3

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Pw

a

L

b

x1x2

x3

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4 ft 4 ft

50 lb/ft

A C

Critical Points

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3 m6 m

6 kN

9 kN m

Shear Force and Bending

Moment Diagram

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FBD of Beam

3 m6 m

6 kN

9 kN m

AyDy

Dx

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x

x

y

y y

D

y

y

y

F 0

D 0

F 0

A 6 D 0

M 0

9 6(6) - A (9) 0

A 5 kN

D 1 kN

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FBD of Beam

3 m 6 m6 kN 9 kN m

5 kN 1 kN

x

x

9x3

3x0

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5 kN

x1

y

0 x 3

F 05 V 0

V 5 kN

M 0

M 5 kN x 0

M 5x kN m

V

M

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3 m 6 kN

5 kN

x

x

V

M

y

3 x 9

F 0

5 kN 6 kN V 0

V 1 kN

M 0

M 5 x 6 x 3 0

M 18 x kN m

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V (kN)

M (kN m)

V = -1 kN

V = 5 kN

M = 5x kN m M = (18-x) kN m

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4 ft 4 ft

50 lb/ft

A C

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4 ft 4 ft

50 lb/ft

Ax

Cx

Cy

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4 ft 4 ft

50 lb/ft

Ax

Cx

Cy

400 lb

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x

x

y

y y

A

y

y

y

F 0

C 0F 0

A C 50(8) 0

M 0

50(8)(4) - C (8) 0

A 200 lb

C 200 lb

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x

50 lb/ft

200 lb V

M

y

2

0 x 8

F 0V 200 lb 50x 0

V 50x 200 lb

M 0x

M 200x 50(x) 02

M 200x 25x lb ft

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x

50 lb/ft50 lb/ft

200 lbV

M

0Mlb200-V8,xat

0Mlb200V0,xat

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V (lb)

M (lb ft)

x50200V

2x25x200M

L

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Pw

a b

w 100 lb / ft

L 15 ft

a 5 ft

b 10 ft

P 1000 lb

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Pw

a

L

b

Ay

Ax

Cy

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x

x

y

y y

y y

A

y

y

y

F 0

A 0

F 0

A C 100(5) 1000 0

A C 1500

M 0

100(5)(2.5) -1000(10) C 15 0

A 750 lbC 750 lb

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x

x750 lb

100 lb/ft

V

M

y

2

0 x 5

F 0

V 750 lb 100x 0

V 750 100x lb

M 0x

M 750x 100(x) 02

M 750x 50x lb ft

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x

x750 lb

100 lb/ft

V

M

f tlb002Mlb250V5,xat

0Mlb750V0,xat

5

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x

750 lb

100 lb/ft

V

M

5 ft

y

5 x 10

F 0

V 750 lb 100(5) 0

V 250 lb

M 0

M 750x 100(5) x 2.5 0M 750x 500x 1250 lb ft

M 250x 1250 lb ft

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x

750 lb

100 lb/ft

V

M

5 ft

f tlb3750Mlb250V,10xat

f tlb2500Mlb250V5,xat

-

10 ft

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1000 lb

x

750 lb

100 lb/ft

V

M

10 ft

5 ft

y

10 x 15

F 0

V 750 lb 100(5) 1000 0

V 750 lb

M 0

M 750x 100(5) x 2.5 1000 (x 10) 0

M 750x 11250 lb ft

10 ft

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1000 lb

x

750 lb

100 lb/ft

V

M

10 ft

5 ft

f tlb0Mlb750-V,15xat

flb3750Mlb750-V,10xat

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2

0 x 5

V 750 lb 100 x

M 750 x 50 x lb ft

5 x 10

V 250 lbM 250 x 1250 lb ft

10 x 15

V 750 lb

M 750 x 11250 lb ft

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V (lb)

M (lb ft)

x100lb750V

lb750V

ftlbx50x750M2

lb250V

ftlb1250x250M

ftlb11250x750M

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V

-1000

-500

0

500

1000

0.0 5.0 10.0 15.0

V

M

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 5.0 10.0 15.0

M

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