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
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