2. Unit 2- Stresses in Beams
Topics Covered
Lecture -1 – Review of shear force and bending
moment diagram
Lecture -2 – Bending stresses in beams
Lecture -3 – Shear stresses in beams
Lecture -4- Deflection in beams
Lecture -5 – Torsion in solid and hollow shafts.
4. What are beams
A structural member which is long when compared
with its lateral dimensions, subjected to transverse
forces so applied as to induce bending of the
member in an axial plane, is called a beam.
5. Objective
When a beam is loaded by forces or couples,
stresses and strains are created throughout the
interior of the beam.
To determine these stresses and strains, the
internal forces and internal couples that act on the
cross sections of the beam must be found.
7. Load Types on Beams
Types of loads on beam
Concentrated or point load
Uniformly distributed load
Uniformly varying load
Concentrated Moment
8. Sign Convention for
forces and moments
P M M
Q Q
“Happy” Beam is +VE
+VE (POSITIVE)
9. Sign Convention for
forces and moments
M M
P
Q Q
“Sad” Beam is -VE
-VE (POSITIVE)
10. Sign Convention for
forces and moments
Positive directions are denoted by an internal shear
force that causes clockwise rotation of the member
on which it acts, and an internal moment that
causes compression, or pushing on the upper arm
of the member.
Loads that are opposite to these are considered
negative.
11. SHEAR FORCES AND BENDING
MOMENTS
The resultant of the stresses must be such as to
maintain the equilibrium of the free body.
The resultant of the stresses acting on the cross
section can be reduced to a shear force and a
bending moment.
The stress resultants in statically determinate
beams can be calculated from equations of
equilibrium.
13. Shear Force and Bending
Moment
Shear Force: is the algebraic sum of the vertical
forces acting to the left or right of the cut section
Bending Moment: is the algebraic sum of the
moment of the forces to the left or to the right of the
section taken about the section
14. SF and BM formulas
Cantilever with point load
W
x Fx= Shear force at X
A Mx= Bending Moment at X
B
L
W SF Fx=+W
BM Mx=-Wx
WxL
at x=0=> Mx=0
at x=L=> Mx=-WL
15. SF and BM formulas
Cantilever with uniform distributed load
w Per unit
length x Fx= Shear force at X
Mx= Bending Moment at X
A B
L
Fx=+wx
BM at x=0 Fx=0
wL at x=L Fx=wL
Mx=-(total load on right portion)*
Distance of C.G of right portion
wL2/2 Mx=-(wx).x/2=-wx2/2
at x=0=> Mx=0
at x=L=> Mx=-wl2/2
16. SF and BM formulas
Cantilever with gradually varying load
Fx= Shear force at X
w wx/L
Mx= Bending Moment at X
A B
x
wx 2
L Fx =
2L
at x=0 Fx=0
at x=L Fx=wL/2
wL/2 €
Parabola
C
Mx=-(total load for length x)*
Distance of load from X
wx 3
Cubic Mx =
6L
at x=0=> Mx=0
at x=L=> Mx=-wl2/6
€
17. SF and BM formulas
Simply supported with point load
W
x Fx= Shear force at X
C
Mx= Bending Moment at X
A B
W W
RA =
2 L RB =
2
Fx=+W/2 (SF between A & C)
€ € Resultant force on the left portion
W/2 SF
Baseline B ⎛ W ⎞ W Constant force
A C ⎜ − W ⎟ = −
⎝ 2 ⎠ 2 between B to C
SF W/2
€
BM
WL/4
B C B
18. SF and BM formulas
Simply supported with point load
W
x Fx= Shear force at X
C
Mx= Bending Moment at X
A B
W W
RA = L RB = for section
2 2
between A & C
W
€ € M x = RA x = x
W/2 SF 2
Baseline B at A x=0=> MA=0 W L
A C at C x=L/2=> MC = ×
SF W/2 2 2
€ for section
between C & B
BM €W × ⎛ x − L ⎞ = W x − Wx + W L
M x = RA x − ⎜ ⎟
WL/4 ⎝ 2 ⎠ 2 2
W L
B C B = − x +W
2 2
WL W
MB = − L =0
2 2
€
19. SF and BM formulas
Simply supported with uniform distributed load
w Per unit length
x Fx= Shear force at X
Mx= Bending Moment at X
A B
RA C RB
L
wL
RA = RB =
BM 2
wL/2 wL
Fx = RA − w.x = − w.x
A C B 2
wL w.0 wL
x = 0 ⇒ FA = − =
wL/2 2 2 2
L wL wL
x = ⇒ FC = − =0
wL2 2 2 2
wL2/2
8 wL wL
x = L ⇒ FB = − wL =
2 2
€
€
20. SF and BM formulas
Simply supported with uniform distributed load
w Per unit length
x Fx= Shear force at X
Mx= Bending Moment at X
A B
C x
RA RB M x = RA x − w.x
L 2
wL w.x 2
BM = x−
wL/2 2 2
A C B wL w.0
x = 0 ⇒ MA = .0 − =0
2 2
wL/2 L
2
wL L w ⎛ L ⎞ wL2 wL2 wL2
x = ⇒ Mc = . − ⎜ ⎟ = − =
2 2 2 2 ⎝ 2 ⎠ 4 8 8
wL2/2 wL2
wL w
8 x = L ⇒ MB = L − L2 = 0
2 2
€
€
21. SF and BM diagram
P Constant Linear
Load
Constant Linear Parabolic
Shear
Linear Parabolic Cubic
Moment
22. SF and BM diagram
0 0 Constant
Load M
Constant Constant Linear
Shear
Linear Linear Parabolic
Moment
23. Relation between load, shear
force and bending moment
1 2
x w/m run
dF
A B = −w
dx
1 C 2
L The rate of change of shear force is equal
to the rate of loading
M M+dM €
F F+dF dM
dx =F
dx
The rate of change of bending moment
is equal to the shear force at the section
€
