L5-2 Beams - External and Internal Effects.pdf

Statics

1. 2.Beams – External
and
Internal Effects
DISTRIBUTED FORCES

2. Learning objectives
•Define beam

•Determine beam external and internal
effects

3. 5.0 Beams & Trusses
•Beams are structural elements designed to
transmit loads over long spans.

5.1 Introduction

4. 5.0 Beams & Trusses
•A beam is structural members which offer
resistance to bending due to applied loads.

5.2 Beams

5. 5.0 Beams & Trusses
There are three common types of beams;
1.Simply supported beam (both ends supported)
2.Over hang beam (one end or both ends suspended in air)
3.Cantilever beam (one end is fixed and the other is
suspended)
5.2.1 Types of Beams

6. 5.0 Beams & Trusses
There are mainly three types of loading that act
on beams;
1.Concentrated load or point load (acts at a point)
2.Uniformly distributed load (acts uniformly over a length)
3.Uniformly varying load (varies with respect to change in length)

5.2.2 Beam Loading
L/3
L/2
L
q (N/m)
L
P (N/m)
PL qL/2 W
1 W
2
R
B
R
A
R
B
R
B
R
A
R
A

7. Sample problem 1
Determine the reactions on the beam loaded as shown in fig 3. Neglect
the thickness and mass of the beam.
Solution
Fig. 3 – Over hang Beam 0;AM 
   B10 2 20sin60 6 (30sin45 13) (40sin80 17) F 17 0           0
x
F

Fig. 3 (b) – FBD
2.The system has three unknowns, F
Ax, F
Ay and F
B
as shown. Applying moment equilibrium
equation at A eliminates two unknowns at A
making it easy to solve for reaction F
B. 62.9BF kN 0
y
F
 F 20cos60 30cos45 40cos80 0Ax     4.3AxF kN F 10 20sin60 30sin45 40sin80 0Ay B F       F 25.0Ay kN
1.Create FBD
5.0 Beams & Trusses
F
B F
Ay
F
Ax

8. Sample problem 2
What force and moment is transmitted to the supporting wall at A in the
given cantilever beam shown in fig.4 (a). (Beam weight & thickness are neglected)
Solution
Fig. 4 (a) – Cantilever Beam 0;AM 
 7.5 0.75 15 2.0 0AM kN m kN m     0
x
F

Fig. 4 (b) – FBD
2.The system has three unknowns, F
Ax, F
Ay and M
A
as shown. Applying moment equilibrium
equation at A eliminates two unknowns at A
making it easy to solve for reaction M
A. 24.4AM kN   0
y
F
 F0Ax F 7.5 15 0Ay kN kN    F 7.5Ay kN
1.Create FBD and convert varying load to
point load by finding area under the graph.
15kN

F
Ay
F
Ax
M
A
7.5kN

0.75m

2m
0.5m

5.0 Beams & Trusses

9. Sample problem 3
Determine the reactions at A for the cantilever beam shown in fig.5 (a).
(Beam weight & thickness are neglected)
Solution
Fig. 5 (a) – Cantilever Beam 0;AM 
 6 2 2 4.5 0AM kN m kN m     0
x
F

Fig. 5 (b) – FBD
2.The system has three unknowns, F
Ax, F
Ay and M
A
as shown. Applying moment equilibrium
equation at A eliminates two unknowns at A
making it easy to solve for reaction M
A. 21.0AM kN 0
y
F
 F0Ax F 6 2 0AykN kN    F 8.0Ay kN
2kN

F
Ay
F
Ax
M
A
6kN

2m

4.5m
1.5m
intPo load Area of triangle 1
int 6
2
Po load b h kN     1
int 1
3
Po load acts at h m from base of triangle
1.Create FBD and convert varying load to
point load by finding area under the graph.
5.0 Beams & Trusses

10. •Due to external applied loads and couples,
beams develops internal effects which includes;
5.2.2 Beams and their Internal Effects
5.0 Beams & Trusses
1)Axial force or Normal force (N)
2)Shear force (SF)
3)Bending moments (BM)
•The effects of the normal force is in most cases
insignificant, but the magnitudes of the other
two effects must be known in order to predict
the behavior of the structures

11. 1.Shear Force (S.F)
Shear force is the algebraic sum of all the vertical
forces at any section of the beam to the right or left of
the section.
5.2.2 Beams and their Internal Effects
Fig. (a) – Simply Supported Beam
Fig. (b) – Internal forces at section
(Sign Conventions)
5.0 Beams & Trusses

12. 2.Bending Moment (B.M)
•Bending moment is the algebraic sum of all the
moment of all the forces and couples acting to the
right or left of the beam section.
•The bending moment at a point is positive when
the external forces acting on the beam tend to
bend the beam as shown.
Fig. (c) – Effects of External Forces
(positive bending moment)

Fig. (d) – Effects of External Forces
(positive Shear Force)

5.0 Beams & Trusses

13. Sample problem 4
Determine the shear force and bending moment produced in the simply
supported beam shown in fig.6 (a). (Beam weight & thickness are neglected)
Solution
Fig. 6 (a) – Simple Beam 0;AM 
 4 6 10 0BykN m F m     0
x
F

Fig. 6 (b) – FBD
2.Solve for the reactions from the entire beam’s FBD. 2.4ByF kN 0
y
F
 F0Ax F 4 0Ay ByF kN  
F
Ay
F
Ax
4 kN

6 m

4 m

1.Create FBD.
F
By F 1.6Ay kN
5.0 Beams & Trusses

14. Sample problem 4
Fig. 6 (c) – Simple Beam 110;M 
 110;AyF - V 0
x
F

Fig. 6 (d) – FBD
3.To determine the shear force and bending
moment the beam is divided into two AC and CB.
AC is cut at distance x and analyze the left of
section 1-1 as shown in Fig.6 (d). 11V 1.6kN 0
y
F
 F0Ax
F
ay
F
Ax
V

x

1

1

M
1-1
S.F.
1-1
B.M.
1-1 11M F 0 ;Ayx   1.6AyF kN 11M 1.6 .x kN m
Note: Values of S.F. and B.M. apply throughout the
section of the beam to the left of 4kN. x6At m 11M 9.6 .kN m x0At m 11M 0 .kN m
5.0 Beams & Trusses

15. Sample problem 4
Fig. 6 (e) – Simple Beam 220;M 
 220;AyF -4- V 0
x
F

Fig. 6 (f) – FBD
4.To determine the shear force and bending moment
between CB, cut the beam with section 2-2 at
distance x
1 as shown in Fig.6 (f). 22V 2.4 kN   0
y
F
 F0Ax
F
ay
F
Ax
V
2-2 x
1
2

2

M
2-2
S.F.
2-2
B.M.
2-2 2 2 1 1M F 4 (x 6) 0 ;Ayx      1.6AyF kN 2 2 1M (24 2.4 ) .x kN m
4 kN
6 m
1x6At m 22M 9.6 .kN m 1x 10At m 22M 0 .kN m
5.0 Beams & Trusses

16. Q1
In each case, calculate the reaction at A and then draw the free-
body diagram of segment AB of the beam in order to determine the
internal loading at B. (Weight and thickness are neglected)
Problems
(a)
(b)

17. 1.The beam is loaded as shown.
a)Determine the internal normal force, shear force, and
moment at point C.
Problems
Q2

18. 1.Determine the reactions for the simply
supported beam loaded as shown.
Additional Problems
Fig.3

19. 3.Determine the shear force
and bending moment produced
in a simply supported beam
loaded as shown in figure 1.
Additional Problems
Fig.1
Fig.2
4.Determine the shear force
and bending moment produced
in the simply supported beam
loaded as shown in figure 2.
Fig.3
5.Determine the shear force and
bending moment produced in a
simply supported beam loaded
as shown in figure 3.

20. 1. Meriam J. L & Kraige L. G (2002) Engineering
Mechanics: Statics, John Wiley & Sons, Inc: New York
Acknowledgements
2.Malhotra M.M &Subramanian R (1994) Textbook of
Applied Mechanics, New Age International: New Delhi
3.Beer F.P, Johnson E.R, Elsenberg E.R & Mazuke D.F
(2010) Vector Mechanics for Engineers, 9th edition,
McGraw hill education: New York

21. THE
END