Unit II Shear and Bending in Beams of... · Syllabus: • Beams and Bending- Types of loads, supports - Shear Force and Bending Moment Diagrams for statically determinate beam with

Syllabus: • Beams and Bending- Types of loads, supports - Shear Force and

Bending Moment Diagrams for statically determinate beam with concentrated load, UDL, uniformly varying load.

• Theory of Simple Bending – Analysis of Beams for Stresses – Stress Distribution at a cross Section due to bending moment and shear force for Cantilever, simply supported and overhanging beams with different loading conditions

• Flitched Beams

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

1

Objective:

• To know the mechanism of load transfer in beams and the induced stress resultants.

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

2

• 1. Rajput.R.K. “Strength of Materials”, S.Chand and Co, New Delhi, 2007.

• Bhavikatti. S., "Solid Mechanics", Vikas publishing house Pvt. Ltd, New Delhi, 2010.

• Junnarkar.S.B. andShah.H.J, “Mechanics of Structures”, Vol I, Charotar Publishing House, New Delhi,1997.

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

3

References:

Types of Loads:

1. Point loads:

2. Uniformly distributed load (UDL):

The loads are uniformly applied over the entire length of the beam.

It can be shown as follows:

3. Uniformly varying load (UVL):

Triangular or trapezoidal loads fall under this category. The variation in intensities of such loads is constant.

It can be shown as follows:

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

4

2 kN

2 kN/m

2 kN/m 2 kN/m 2 kN/m 1 kN/m

Types of supports:

1. Roller support:

2. Hinged support:

3. Fixed support:

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

5

• 1. cantilever beams:

A cantilever beam which is fixed at one end and free at the other end.

• 2. Simply supported beams:

A simply supported beam rests freely on hinged support at one end and roller support at the other end.

• 3. Overhanging beams:

If a beam extends beyond its supports it is called an overhanging beam.

Over hanging portion could be either any one of the sides or both the sides.

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

6

Statically determinate beams:

• 1. Propped cantilever beam:

A cantilever beam with propped support at the free end.

• 2. Fixed beam:

A beam with both supports fixed.

• 3. Continuous beam:

A beam with more than two supports.

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

7

Statically indeterminate beams:

• Shear force at a section of a loaded beam may be defined as the algebraic sum of all vertical forces acting on any one side of the section.

• Sign Convention:

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

8

Shear force:

+ ve

Section line

- ve

Section line

• Bending moment at a section of a loaded beam may be defined as the algebraic sum of all moments of forces acting on any one side of the section.

• Sign Convention:

Unit II – Shear and Bending in Beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

9

Bending Moment:

Sagging moment

+ve

Sagging moment: Moments which bend the beam upwards and cause compression in the top fibre and tension in the bottom fibre are taken as positive.

Hogging Moment: Moments which bend the beam downwards and cause compression in the bottom and tension in the top fibre are taken as negative.

Hogging moment

-ve

1. Cantilever beam subjected to a point load at the free end:

• (i) Shear force (S.F.) Calculations:

Sign convention:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

10

W

‘L’ m

A B

+ ve

Section line

- ve

Section line

S.F. @ B= + W S.F. @ XX= +W S.F. @ A= + W

𝑥 X

X

W W W

S.F.D.

+ve

1. Cantilever beam subjected to a point load at the free end:

• (ii) Bending Moment(B.M.) Calculations:

Sign convention:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

11

W

‘L’ m

A B

B.M. @ B= 0 B.M. @ XX= -W𝑥 B.M. @ A= - WL

𝑥 X

X

B.M.D.

Sagging moment

+ve

Hogging moment

-ve

0

W𝑥

WL

-ve

1. Cantilever beam subjected to a point load at the free end:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

12

W

‘L’ m

A B

W W

S.F.D.

+ve

WL

-ve

B.M.D

(iii) S.F.D. & B.M.D. Diagrams:

2. Cantilever beam subjected to point loads as shown in Fig. Draw S.F.D. & B.M.D.:

• (i) Shear force (S.F.) Calculations:

Sign convention:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

13

+ ve

Section line

- ve

Section line

S.F. @ B= 0 S.F. @ C= +2 kN S.F. @ D (without Pt. Load at D)=+ 2 kN S.F. @ D (with Pt. Load at D)=+ 5 kN S.F. @ A= + 5 kN

0

2 kN

5 kN

S.F.D.

+ve

2 kN 3 kN A

B 1 m 2 m 1 m

C D

2 kN

5 kN

2. Cantilever beam subjected to point loads as shown in Fig. Draw S.F.D. & B.M.D.:

• (ii) Bending Moment(B.M.) Calculations:

Sign convention:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

14

B.M. @ B= 0 B.M. @ C= 0

0

2 kN 3 kN

A B

1 m 2 m 1 m

C D

Sagging moment

+ve

Hogging moment

-ve

0

4 kNm B.M. @ D = −2 × 2 = −4 kNm

B.M. @ A= − 2 × 3 − 3 × 1 = −9 𝑘𝑁𝑚 9 kNm

-ve

B.M.D

2. Cantilever beam subjected to point loads as shown in Fig. Draw S.F.D. & B.M.D.:

• (i) S.F.D. & B.M.D.

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

15

2 kN 3 kN A

B 1 m 2 m 1 m

C D

0

2 kN

5 kN

S.F.D.

+ve 2 kN

5 kN

4 kNm

9 kNm

-ve

B.M.D.

3. Cantilever beam subjected to u.d.l as shown in Fig. Draw S.F.D. & B.M.D.:

• (i) Shear force (S.F.) Calculations:

Sign S.F. @ A= +

:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

16

+ ve

Section line

- ve

Section line

S.F. @ B= 0 S.F. @ XX= +𝑤𝑥

𝑤/𝑚

‘L’ m A B

𝑥 X

X

𝑤𝑥

S.F. @ A= +𝑤𝐿

𝑤𝐿

+ve

S.F.D.

3. Cantilever beam subjected to u.d.l as shown in Fig. Draw S.F.D. & B.M.D.:

• (i) Bending Moment(B.M.) Calculations:

Sign convention:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

17

B.M. @ B= 0

W /m

‘L’ m

Sagging moment

+ve

Hogging moment

-ve

A B 𝑥 X

X

B.M. @ A= −𝑤𝐿 ×𝐿

2= −𝑤𝐿2

2

𝑤𝑥2

2 B.M. @ XX= --𝑤𝑥 ×

𝑥

2=𝑤𝑥2

2

𝑤𝐿2

2 B.M.D.

-ve

3. Cantilever beam subjected to u.d.l as shown in Fig. Draw S.F.D. & B.M.D.:

• (iii) S.F.D & B.M.D:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

18

𝑥 X

X

A B W /m

‘L’ m

𝑤𝑥2

2

𝑤𝐿2

2 B.M.D.

-ve

𝑤𝑥

𝑤𝐿

+ve

S.F.D.

3. Cantilever beam subjected to u.d.l as shown in Fig. Draw S.F.D. & B.M.D.:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

19

A B 4 kN/m

1.5 m 1.5 m

3. Cantilever beam subjected to u.d.l as shown in Fig. Draw S.F.D. & B.M.D.:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

20

A B 4 kN/m

1.5 m

13.5 𝑘𝑁𝑚 B.M.D.

-ve

6 𝑘𝑁

+ve

S.F.D.

1.5 m

4.5 kNm

6 𝑘𝑁

4. A simply supported beam of span ‘L’ carries a central concentrated load ‘W’. Draw S.F.D. & B.M.D.:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

21

W

‘L’ m

4. A simply supported beam of span ‘L’ carries a central concentrated load ‘W’. Draw S.F.D. & B.M.D.:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

22

W

‘L’ m

W/2 W/2

+ve

W/2

W/2

-ve S.F.D.

𝑊𝐿/4

B.M.D.

+ve

4. A simply supported beam of span ‘L’ carries a eccentric concentrated load ‘W’ as shown in Fig. Draw S.F.D. & B.M.D.:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

23

W

‘L’ m 𝑎 𝑏

Solution:

(i) Calculation of reactions:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

24

W

‘L’ m 𝑎 𝑏

A B C

−𝑅𝐵 × 𝐿 +𝑊 × 𝑎 = 0

𝑀𝐴 = 0 +ve

∴ 𝑹𝑩 =𝑾𝒂

𝑳

𝐹𝑉 = 0 +ve

𝑅𝐴 𝑅𝐵

𝑅𝐴 + 𝑅𝐵 −𝑊 = 0

𝑅𝐴 +𝑊𝑎

𝐿−W = 0 ∴ 𝑹𝑨 =

𝑾𝒃

𝑳

Solution (contd…):

(ii) Shear force (S.F.) Calculations:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

25

W

‘L’ m 𝑎 𝑏

A B C

𝑊𝑏

𝐿

𝑊𝑎

𝐿

+ve

-ve

𝑊𝑏

𝐿

𝑊𝑎

𝐿

S.F.D.

S.F. at A=𝑊𝑏

𝐿

S.F. at C (between A and C)=𝑊𝑏

𝐿

S.F. at C(between C and B)=𝑊𝑎

𝐿

S.F. at B=𝑊𝑎

𝐿

Sign Convention:

+ ve

Section line

- ve

Section line

Solution:

(ii) Bending Moment(B.M.) Calculations:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

26

W

‘L’ m 𝑎 𝑏

A B C

𝑊𝑏

𝐿

𝑊𝑎

𝐿

B.M.D.

𝑀𝐴=0

𝑀𝐶=𝑊𝑏

𝐿× 𝑎 =

𝑊𝑎𝑏

𝐿

𝑀𝐵 = 0

𝑊𝑎𝑏

𝐿

+ve

Sagging

+ve

Sign Convention:

Hogging

-ve

5. A simply supported beam of span ‘L’ carries two point loads as shown in Fig. Draw S.F.D. & B.M.D.:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

27

3 kN 4 kN

1.5 𝑚 3.5 𝑚 1 m

Solution:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

28

3 kN 4 kN

1.5 𝑚 3.5 𝑚 1 m 𝑅𝐴 𝑅𝐵

(i) Calculation of reactions:

𝑀𝐴 = 0 +ve

−(𝑅𝐵 × 6) + (3 × 5) + (4 × 1.5) = 0

A B

∴ 𝑹𝑩 = 𝟑. 𝟓 𝒌𝑵

𝐹𝑉 = 0 +ve

𝑅𝐴 + 𝑅𝐵 − 4 − 3 = 0

𝑅𝐴 + 3.5 − 4 − 3 = 0 ∴ 𝑹𝑨 = 𝟑. 𝟓 𝒌𝑵

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

29

3 kN 4 kN

1.5 𝑚 3.5 𝑚 1 m 3.5 kN

3.5 kN

+ -

S.F.D.

(ii) Shear force (S.F.) Calculations:

S.F. at A=3.5 𝑘𝑁

S.F. at C (between A and C)=3.5 𝑘𝑁 S.F. at C(between C and D)=−0.5 𝑘𝑁 S.F. at D (between D and B)=-3.5 kN

A C D B

3.5 kN

0.5 kN

3.5 kN

Sign Convention:

+ ve

Section line

- ve

Section line

S.F. at B=−3.5 𝑘𝑁

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

30

3 kN 4 kN

1.5 𝑚 3.5 𝑚 1 m

3.5 kN 3.5 kN

B.M.D.

(iii) B.M.Calculations:

A C D B

Sagging

+ve

Sign Convention:

Hogging

-ve

𝑀𝐴=0 𝑀𝐶=3.5 × 1.5 = 5.5 𝑘𝑁𝑚 𝑀𝐷 = 3.5 × 1 = 3.5 𝑘𝑁𝑚 𝑀𝐵 = 0

5.5 kNm

3.5 kNm

+

5. A simply supported beam of span ‘L’ carries u.d.l throughout the . Draw S.F.D. & B.M.D.:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

31

′𝐿′ 𝑚

𝑤 𝑘𝑁/𝑚

5. A simply supported beam of span ‘L’ carries u.d.l throughout the . Draw S.F.D. & B.M.D.:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

32

′𝐿′ 𝑚

𝑤 𝑘𝑁/𝑚

wL/2 wL/2

wL/2

wL/2

+

- S.F.D.

A B

S.F. @ XX=0

X

X 𝑥

𝑤𝐿

2− 𝑤𝑥 = 0

∴ 𝑥 = 𝐿/2

M @ L/2 =𝑤𝐿2

8

𝑤𝐿2

8

+

B.M.D.

Over hanging beams:

Problem:

An over hanging beam of length 10 m is loaded as shown in Fig. Draw the S.F.D. and B.M.D. Mark the values at salient points.

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

33

A

3 m

15 kN/m 25 kN

1 m 4 m 2 m

C D B E

20 kN/m

(i) Support Reactions:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

34

𝑀𝐴 = 0 +ve

𝑅𝐴 𝑅𝐵

− 𝑅𝐵 × 8 + 20 × 2 × 9 + 25 × 4 +1

2× 3 × 15 ×

2

3× 3 = 0

A

3 m

15 kN/m 25 kN

1 m 4 m 2 m

C D B E 20 kN/m

∴ 𝑹𝑩 = 𝟔𝟑. 𝟏𝟐𝟓 𝒌𝑵 𝐹𝑉 = 0 +ve

𝑅𝐴 + 63.125 −1

2× 3 × 15 − 25 − 20 = 0

∴ 𝑹𝑨 = 𝟐𝟒. 𝟑𝟕𝟓 𝐤𝐍

(ii) S.F. calculations:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

35

A

3 m

15 kN/m 25 kN

1 m 4 m 2 m

C D B E 20 kN/m

𝑆. 𝐹.@ 𝐴 = +24.37 𝑘𝑁

𝑆. 𝐹.@ 𝑋𝑋= +24.37

−1

2× 𝑥 × 5𝑥

= 24.37 − 2.5 𝒙𝟐

S.F. @ C=

24.37−1

2× 3 × 15

=1.88 kN

24.37

1.88

S.F. between C&D=1.88 kN

Parabola

24.37 kN 63.13 kN

X

X

(ii) S.F. calculations:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

36

𝑅𝐴=24.37 𝑅𝐵 = 63.13

A

3 m

15 kN/m 25 kN

1 m 4 m 2 m

C D B E 20 kN/m

S.F. between D&B=

24.37−1

2× 3 × 15 −25

=−23.13 𝑘𝑁

24.37

1.88

Parabola S.F. @B (including reaction at B)= -23.13+63.13=+40 kN

S.F. @ E=40 − 20 × 2 = 0

+ +

-

S.F.D.

40

23.13 23.13

(iii) B.M. calculations:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

37

𝑅𝐴=24.37 𝑅𝐵 = 63.13

A

3 m

15 kN/m 25 kN

1 m 4 m 2 m

C D B E 20 kN/m

𝑀𝐴 = 0

𝑀𝐶 =+ 24.37 × 3 −1

2× 3 × 15 ×

1

3× 3

=50.68 kNm

𝑀𝐷=24.37× 4 −1

2× 3 × 15 × 1 +

1

33

=52.5 kNm

50.68 52.5 kNm

(iii) B.M. calculations:

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

38

𝑅𝐴=24.37 𝑅𝐵 = 63.13

A

3 m

15 kN/m 25 kN

1 m 4 m 2 m

C D B E 20 kN/m

𝑀𝐵 = −(20 × 2 × 1) = −40 kNm

𝑀𝐸=0

+

- B.M.D.

X

X 𝑀𝑋𝑋 = 0

− 20 × 2 × 𝑥 − 1 + 63.13 × (𝑥 − 2) =0

𝑥

∴ 𝒙 = 𝟑. 𝟕𝟑 𝒎

Point of contraflexure

40 kNm

52.5 kNm 50.68 Cubic parabola

Parabola

Simply supported beam:

Problem:

A simply supported beam is loaded as shown in Fig. Draw the S.F.D. and B.M.D. Mark the values at salient points.

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

39

A

1.5 m

2 kN/m 2 kN

0.5 m 1m 1 m

D B E 3 kNm

C

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

40

A

1.5 m

2 kN/m 2 kN

0.5 m 1m 1 m

D B E 3 kNm

C

2.18 kN 2.82 kN

2.18

0.82

2.82 2.82

0.82

+

-

S.F.D.

S.F.Diagram:

S.F. @ XX=0

X

X

𝑥 2.18 − 2𝑥 = 0

∴ 𝑥 = 1.09 𝑚

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

41

A

1.5 m

2 kN/m 2 kN

0.5 m 1m 1 m

D B E 3 kNm

C

2.18 kN 2.82 kN

1.18

3.62 kNm 2.82 kNm

0.62

B.M.D.

B.M.Diagram:

1.05

M @ 1.09 m from A =1.18 kNm

X

X

1.09 𝑚

+

Over hanging beam:

Problem:

Draw the shear force and bending moment diagram for the overhanging beam shown in Fig. Indicate the salient values on them.

Shear force and Bending moment diagrams for Statically determinate beams

Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE

42

B

2 m

5 kN/m

20 kN

3m 2 m

D E 2 kNm

C

5 kN

A

1 m