N-W.F.P University of Engineering & Technology Peshawar: Advanced Structural Analysis-1
N-W.F.P University of Engineering & Technology Peshawar: Advanced Structural Analysis-1
N-W.F.P University of Engineering & Technology Peshawar: Advanced Structural Analysis-1
Subject CE-51111
Topics to be Covered
Overview of Bernoulli-Euler Beam Theory
Overview of Theory of Torsion
Static Indeterminancy
Kinematic Indeterminancy
b
d
P
V + dv
M + dM
dx
V w dx ( V + dV) = 0
dx
M V . dx w dx
( M dM ) 0
2
Neglect
dV
w
dx
dM
V
dx
w = P/dx
V1
M + dM
dx
V1
dx
V1 V P
V P V1 = 0
dx
M V . dx w dx
( M dM ) 0
2
Neglect
dM
V
dx
M + dM
Theory of Torsion
Derivation
Theory of Torsion
Derivation
Theory of Torsion
Derivation
Theory of Torsion
Derivation
Theory of Torsion
Derivation
Torsion Formula
We want to find the maximum shear stress max which
occurs in a circular shaft of radius c due to the
application of a torque T. Using the assumptions above,
we have, at any point r inside the shaft, the shear stress
is r = r/c max.
rdA r = T
r2/c max dA = T
max/cr2 dA = T
Now, we know,
J = r2 dA
is the polar moment of intertia of the cross sectional
area J = c4/2 for Solid Circular Shafts
Theory of Torsion
Derivation
= /G
For a shaft of radius c, we have
c=L
where L is the length of the shaft. Now,
is given by
= Tc/J
so that
= TL/GJ
Theory of Torsion
Theory of Torsion
Torsional Constant for an I Beam
For an open section, the torsion constant is as
follows:
J = (bt3 / 3)
So for an I-beam
J = (2btf3 + (d - 2tf)tw3) / 3
where
b = flange width
tf = flange thickness
d = beam depth
tw = web thickness
Static Determinancy
Equilibrium of a Body
y
x
z
Px 0
Py 0
Mz 0
Static Determinancy
ra 3
Static Determinancy
ra = 3, Determinate, Stable
ra > 3, Determinate, Stable
ra =3, Unstable
KI = 5