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
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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