me 4175 – chap-4-part-3
DESCRIPTION
Slides for Chap 4 of ME 4175 - Columns Beams in Axial CompressionTRANSCRIPT
ME 4175 – Machine Design
Chapter 4 Stress, Strain, & Deflection Part 3
columns
Structural elements that are subjected to axial compressive forces only are called columns.
centroid
Columns/Beams in Axial Compression
Columns in Axial Compression• Columns or beams in axial
compression have a buckling failure mechanism in addition to a max compressive stress failure• For long columns or beams,
buckling will be the primary failure mechanism
Comments on the Pinned-Pinned Buckling Solution• Assumptions:
• The column geometry is perfectly straight and without defects• The load passes through the centroid of the beams cross section
exactly
• For the area moment of inertia in the radius of gyration equation, always use the smallest value for a non-circular cross section• In general, buckling must be considered for columns with a
slenderness ration of greater than 10• Buckling failure depends on the elastic modulus of the
material and not the compressive failure stress
Failure modes• Short columns –crushing
(materials failure)
P
P
Long columns –buckling (loss of stability)
Axial loads cause lateral deformations (bending-like deformations)
Buckling
Buckling Load for a Pinned-Pinned Column
Beam equation:
M=P(-y)
Differential equation:
Solution:
Bucking Load for a Pinned-Pinned Column
Solve for C1 and C2 using the following boundary conditions:
y(0)=0 and y(l)=0
y(0)=0 C2=0
y(l)=0
Taking C1 as non-zero:
for n= 0, 1,2…
Bucking Load for a Pinned-Pinned ColumnThe critical load required to buckle the beam is
2
22
lEInPcr
where n defines the buckling mode shapesFirst mode of buckling
Second mode of buckling
Third mode of buckling
2
2
1 LEIP
2
2
24LEIP
2
2
39LEIP
P1 P1
P2P2
P3 P3
First mode of buckling
Second mode of buckling
Third mode of buckling
2
2
1 LEIP
2
2
24LEIP
2
2
39LEIP
P1 P1
P2P2
P3 P3
First mode of buckling
Second mode of buckling
Third mode of buckling
2
2
1 LEI
P
2
2
24LEIP
2
2
39LEIP
P1 P1
P2P2
P3 P3
• First mode of buckling
Bucking Load for a Pinned-Pinned Column
Define:
Radium of Gyration
Slenderness Ratio
The critical load can now be expressed as:
Buckling Solution for Other End Conditions
Buckling Solution for Other End Conditions
where
Column failure criteria
Solution Strategy for Beams Loaded Exactly at the Centroid
• Calculate the slenderness ratio and compare to the transition between Johnson and Euler failure regions
If then:
Else:
Radium of Gyration
Slenderness Ratio
Eccentrically Loaded Columns • The offset load causes a net moment
before the beam is deflected
• The differential equation defining the deflection of the beams becomes
• From this differential equation, the deflection and the maximum moment of the beam can be calculated
Peak Compressive Stress for an Eccentrically Loaded Beam
Eccentricity ratio
Optimal Geometry for Columns under Compression
Buckling Load factor
From statics and mechanics of Materials by P Beer et al.
