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.