Post on 24-Jun-2015
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BY:BY:BHAVIN THAKRANIBHAVIN THAKRANINITESH VARSANINITESH VARSANI
STRUCTURAL ANALYSIS-1
Columns and StrutsColumns and Struts
STRUT:STRUT:
A structural member subjected to axial compressive force is called strut.
Strut may be vertical, horizontal or inclined.
The cross – sectional dimensions of strut are small.
Normally, struts carry smaller compressive loads.
Struts are used in roof truss and bridge trusses.
COLUMN:COLUMN:When strut is vertical it is known
as column.The cross – sectional dimensions
of column are large.Normally, columns carry heavy
compressive loads.Columns are used in concrete
and steel buildings.
Q. Compare Column and StrutsA.
Radius of Gyration(k) , k = √(I/A) or I = AK²
K=radius of gyrationI = Moment of Inertia (mm4)A = Area of Section (mm2)
Slenderness ratio, Slenderness Ratio = effective length of
column/Minimum radius of gyration λ = le/kmin
LONG COLUMN :LONG COLUMN :When length of column is more as
compared to its c/s dimension, it is called long column.
Long Column Le/kmin > 50
Or,Le/d > 15 for Long
Where,Le = effective length of columnd = least lateral dimension of
column.Kmin = Minimum radius of gyration
SHORT COLUMN :SHORT COLUMN :When length of column is less as
compared to its c/s dimension, it is called Short column.
Short Column Le/kmin <50
Or, Le/d < 15
Crushing Load : The load at which, short column fails by crushing is called crushing load.
CRIPPLING LOAD OR CRIPPLING LOAD OR BUCKLINGBUCKLINGLOADLOADThe load at which, long column
starts buckling(bending) is called buckling load or crippling load.
Buckling of column depends upon the following factors.
1. Amount of load. 2. Length of column 3. End condition of column 4. C/s dimensions of column 5. Material of column.
COLUMN END CONDITION COLUMN END CONDITION AND EFFECTIVE LENTH :AND EFFECTIVE LENTH : 1.Both end hinged. 2.Both end fixed. 3.One end fixed and other
hinged. 4.One end fixed and other free.
Effective length (le)
Where l is actual length
COLUMNS HAVING VARIOUS TYPES OF SUPPORTSCOLUMNS HAVING VARIOUS TYPES OF SUPPORTS
Effective length
12
Euler’s Formula
Euler’s Crippling Load, PE = ∏²EI /le²
Where, E is Modulus of Elasticity (Mpa) I is MOI or 2nd Moment of area
(mm4) Le is Effective length (mm)
Also known as Critical Buckling Load
Eccentric LoadingEccentric Loading
Short Columnσmax = P/A + P.e/Z = P/A (1 + eyc/k2)
Z = Ak2/ yc
Long Column◦ Rankine’s Formula σc= P/A (1 + eyc/k2) (1 + αle/k)
◦ Euler’s Formula σmax = P/A + Pe v /Z
σmin = P/A – Pe v /Z
v = sec {(le/2) /√[P/(EI)]}
Euler’s Formula for both end of the columns are hingedEuler’s Formula for both end of the columns are hinged
The Bending Moment at the Section is given by
General solution is
Since x = 0 at y = 0, then C2 = 0.Since x = l at y = 0, then
02
2
yEI
P
dx
yd
x
EI
PCx
EI
PCy sincos 21
15
0sin2
L
EI
PC
IDEAL COLUMN WITH HINGED SUPPORTSIDEAL COLUMN WITH HINGED SUPPORTS
for C1 = 0, we get
Which is satisfied if
or2
2
L
EIP
LEI
P
16
0sin
LEIP
Rankine’s Formula1/P = 1/PC + 1/PE
Where, P is Rankine’s crippling Load
PC is Crushing Load
PE is Euler’s crippling Load
If A is the Cross section area of columnPC = fC . A
PE = ∏²EI /le²
I = Ak2
Where Rankine’s Constant, α = fc/(∏²E)Thus, P = PR = (fC . A) / (1 + α λ)
If a dice is thrown what is the probability of an even number or a “3”
Since three is not an even number (!) P(Even or “3”) = P(Even) + P(3)
= 3/6 + 1/6 = 4/6