chapter 5: column

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Chapter 5: Column

5.1 Introduction

Columns are usually considered as vertical structural elements, but they can be positioned in any

orientation (e.g. diagonal and horizontal compression elements in a truss).

Columns are used as major elements in trusses, building frames, and sub-structure supports for

bridges (e.g. piers).

• Columns support compressive loads from roofs, floors, or bridge decks.

• Columns transmit the vertical forces to the foundations and into the subsoil.

The work of a column is simpler than the work of a beam.

• The loads applied to a column are only axial loads.

• Loads on columns are typically applied at the ends of the member, producing axial

compressive stresses.

• However, on occasion the loads acting on a column can include axial forces, transverse

forces, and bending moments (e.g. beam-columns).

Columns are defined by the length between support ends.

• Short columns (e.g. footing piers).

• Long columns (e.g. bridge and freeway piers).

Virtually every common construction material is used for column construction.

• Steel, timber, concrete (reinforced and pre-stressed), and masonry (brick, block, and stone).

The selection of a particular material may be made based on the following.

• Strength (material) properties (e.g. steel vs. wood).

• Appearance (circular, square, or I-beam).

• Accommodate the connection of other members.

• Local production capabilities (i.e. the shape of the cross section).

Columns are major structural components that significantly affect the building’s overall

performance and stability.

• Columns are designed with larger safety factors than other structural components.

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• Failure of a joist or beam may be localized and may not severely affect the building’s

integrity (e.g. there is redundancy with girders and beams, but not with columns).

• Failure of a strategic column may be catastrophic for a large area of the structure.

• Failure may be due to overstressed, loss of section (deterioration),

accident/sabotage (terrorism).

Safety factors for columns are used to account for the following.

• Material irregularities (e.g. out of straightness).

• Support fixity at the column ends.

• Construction inaccuracies (e.g. out of plumbness).

• Workmanship.

• Unavoidable eccentric (off-axis) loading.

5.2 Short and Long Columns – Modes of Failure

Column slenderness and length greatly influence a column’s ability to carry load.

• Very short, stout columns fail by crushing due to material failure.

- Failure occurs once the stress exceeds the elastic (yield point) limit of the material.

• Long, slender columns fail by buckling – a function of the column’s dimensions and its

modulus of elasticity.

- Buckling is the sudden uncontrolled lateral displacement of a column at which point no additional load can be supported.

- Failure occurs at a lower stress level than the column’s material strength due to buckling

(i.e. lateral instability).

Short columns

Short columns fail by crushing at very high stress levels that are above the elastic limit of the

column material.

Compressive stress for short columns is based on the basic stress equation developed at the

beginning of Chapter 5.

• If the load and column size (i.e. cross-sectional area) are known, the compressive stress may

be computed as

fa = Pactual/A ≤ Fa

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where

fa = actual compressive stress (psi or ksi) A =

cross-sectional area of the column (in2)

Pactual = actual load on the column (pounds or kips)

Fa = allowable compressive stress per code (psi or ksi)

• This stress equation can be rewritten into a design form to determine the required short

column size when the load and allowable material strength are known.

Arequired = Pactual/Fa

where

Arequired = minimum cross-sectional area of the column

5. 3 Long Columns – Euler Buckling

Long columns fail by buckling at stress levels that are below the elastic limit of the column

material.

• Very short column lengths require extremely large loads to cause the member to buckle.

• Large loads result in high stresses that cause crushing rather than buckling.

Buckling in long, slender columns is due to the following.

• Eccentricities in loading.

• Irregularities in the column material.

Buckling can be avoided (theoretically) if the loads were applied absolutely axially, the column

material was totally homogeneous with no imperfections, and construction was true and plumb.

A Swiss mathematician named Leonhard Euler (1707 – 1783) was the first to investigate the

buckling behavior of slender columns within the elastic limit of the column’s material.

• Euler’s equation shows the relationship between the load that causes buckling of a (pinned

end) column and the material and stiffness properties of the column.

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The critical buckling load can be determined by the following equation.

Pcritical = π2EImin/L2

where

Pcritical = critical axial load that causes buckling in the column (pounds or kips) E =

modulus of elasticity of the column material (psi or ksi)

Imin = smallest moment of inertia of the column cross-section (in2)

(Most sections have Ix and Iy; angles have Ix, Iy and Iz.) L =

column length between pinned ends (inches)

• As the column length increases, the critical load rapidly decreases (since it is proportional to

L2), approaching zero as a limit.

• The critical load at buckling is referred to as Euler’s critical buckling load.

Euler’s equation is valid only for long, slender columns that fail due to buckling.

• Euler’s equation contains no safety factors.

• Euler’s equation results in compressive stresses developed in columns that are well below the

elastic limit of the material.

Slenderness Ratios

The radius of gyration is a geometric property of a cross section that was first introduced in Chapter

6.

I = Ar2

and r = (I/A) 1/2

where

r = radius of gyration of the column cross section (in) I = least

(minimum) moment of inertia (in4)

A = cross-sectional area of the column (in2)

The radius of gyration is geometric property that is used in the analysis and design of columns.

Using the radius of gyration, the critical stress developed in a long column at buckling can be

expressed by the following equation.

fcritical = Pcritical/A = π2EImin/AL

2 = π

2E(Ar

2)/AL

2 = π

2E/(L/r)

2

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The term “L/r” is known as the slenderness ratio.

• A higher slenderness ratio means a lower critical stress that will cause buckling.

• Conversely, a lower slenderness ratio results in a higher critical stress (but still within the

elastic range of the material).

Column sections with large r-values are more resistant to buckling.

• Compare the difference in rmin values and slenderness ratios for the three column cross

sections shown below.

- All three cross sections have relatively equal cross-sectional areas but very different radii

of gyration about the critical buckling axis.

- All three columns are assumed to be 15 feet in length and pin-connected at both ends.

Comparison of steel cross sections with equivalent areas

The most efficient column sections for axial loads are those with almost equal rx and ry values.

• Circular pipe sections and square tubes are the most effective shapes since the radii of

gyration about both axes are the same (rx = ry).

• Circular pipe sections and square tubes are often used as columns for light to moderate loads.

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Wide-flange shapes may be preferred despite the structural advantages of closed cross-sectional

shapes (like tubes and pipes).

• The practical considerations of wide-flange shapes include the following.

- Wide-flange sections support heavy loads.

- Wide-flange sections accommodate beam connections.

Special wide-flange sections are specifically manufactured to provide relatively symmetrical

columns (rx/ry ratios approaching 1.0) with large load-carrying capability.

• Most of these column sections (generally W8, W10, W12, and W14) have depth and flange

widths approximately equal (i.e. a “boxy” configuration).

5.4 End Support Conditions and Lateral Bracing Previously, each column was assumed to have pinned ends in which the member ends were free to

rotate (but not translate) in any direction at their ends.

• When the column buckles, it will do so in one smooth curve.

• The length of this curve is referred to as the effective length.

In practice, a column may not be pinned at the ends.

• The column length free to buckle is greatly influenced by its end support conditions.

• The load-carrying capacity of a column is affected by the end support conditions.

- Restraining the ends of a column with a fixed support increases the load- carrying

capacity of a column.

- Allowing translation as well as rotation (i.e. free end) at one end of a column generally

reduces its load-carrying capacity.

Column design formulas generally assume a condition in which both ends are pinned.

• When other conditions exist, the load-carrying capacity is increased or decreased and the

allowable compressive stress is increased or decreased.

• A factor K is used as a multiplier for converting the actual column length to an effective

buckling length based on end conditions.

The American Institute of Steel Construction (AISC) provides recommended effective length

factors when ideal conditions are approximated.

• The six cases are presented as follows.

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Case A: Both ends are pinned.

The structure is adequately braced against lateral forces (e.g. wind and earthquake forces).

Theoretical K-value: K = 1.0

Effective length: Le = L


Examples: Timber column nailed top and bottom; steel

column with simple clip angle connection top and

bottom.

Case B: Both ends are fixed.


Theoretical K-value: K= 0.5 Effective

length: Le = 0.5 L Pcritical

= π2

EImin/(0.5L)2

= 4π2EImin/L

2

Examples: Concrete column rigidly (monolithically

cast) connected to large beams top and bottom; steel

column rigidly connected (welded) to large steel beams

top and bottom.

Case C: One end is pinned and one end is fixed.


Theoretical K-value: K = 0.7 Effective


= π2

EImin/(0.707L)2

= 2π2EImin/L

2

Examples: Concrete column rigidly connected to

concrete slab at the base and attached to light- gauge

roofing at the top.

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Case D: One end is free and one end is fixed.

Lateral translation is possible.

• An eccentric column load is developed.

Theoretical K-value: K = 2.0 Effective


= π2

EImin/(2L)2

= π2

EImin/4L2

Examples: Water tank mounted on a simple pipe

column; flagpole.

Case E: Both ends are fixed with some lateral translation.


Effective length: Le = 1.0 L


Examples: Flexible column attached to a rigid beam

and supported by a fixed base.

Case F: The base is pinned and the top is fixed with some lateral translation.


Effective length: Le = 2.0 L

Pcritical = π2EImin/(2L)

2 = π

2EImin/4L

2

Examples: Steel column with a rigid connection to a

beam above and a simple pin connection at the base.

M

E

y I R

Chapter – 6 Bending Stress in Beam

6.1 Euler Bernoulli’s Equation or (Bending stress formula) or Bending

Equation

Where = Bending Stress

M = Bending Moment

I = Moment of Inertia

E = Modulus of elasticity

R = Radius of curvature

y = Distance of the fibre from NA (Neutral axis)

6.2 Assumptions in Simple Bending Theory

All of the foregoing theory has been developed for the case of pure bending i.e constant B.M

along the length of the beam. In such case

The shear force at each c/s is zero.

Normal stress due to bending is only produced.

Beams are initially straight

The material is homogenous and isotropic i.e. it has a uniform composition and its

mechanical properties are the same in all directions

The stress-strain relationship is linear and elastic

Young’s Modulus is the same in tension as in compression

Z = I

y

Sections are symmetrical about the plane of bending

Sections which are plane before bending remain plane after bending

6.3

max t Mc1

I

min c Mc2

I

(Minimum in sense of sign)

6.4 Section Modulus (Z)

Z is a function of beam c/s only

Z is other name of the strength of the beam

The strength of the beam sections depends mainly on the section modulus

The flexural formula may be written as,

Rectangular c/s of width is "b" & depth "h" with sides horizontal, Z = bh

2

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M

Z

Square beam with sides horizontal, Z = a

6

Square c/s with diagonal horizontal, Z = 6

d 3

Circular c/s of diameter "d", Z =

32

A log diameter "d" is available. It is proposed to cut out a strongest beam

from it. Then

b(d 2 b

2 )

Z = 6

bd 3

d Therefore, Zmax = for b =

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6.5 Flexural Rigidity (EI)

Reflects both

Stiffness of the material (measured by E)

Proportions of the c/s area (measured by I )

6.6 Axial Rigidity = EA

6.7 Beam of uniform strength

It is one is which the maximum bending stress is same in every section along the longitudinal

axis.

For it M bh2

Where b = Width of beam

h = Height of beam

To make Beam of uniform strength the section of the beam may be varied by

Keeping the width constant throughout the length and varying the depth, (Most widely

used)

Keeping the depth constant throughout the length and varying the width

By varying both width and depth suitably.

a3 2

3

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6.8 Bending stress due to additional Axial thrust (P).

A shaft may be subjected to a combined bending and axial thrust. This type of situation arises in various machine elements.

If P = Axial thrust

Then direct stress ( d ) = P / A (stress due to axial thrust)

This direct stress ( d ) may be tensile or compressive depending upon the load P is tensile or

compressive.

And the bending stress ( ) = My

is varying linearly from zero at centre and extremum b

I

(minimum or maximum) at top and bottom fibres.

If P is compressive then

At top fibre

At mid fibre

P

My

A I

P

A

(compressive)

(compressive)

At bottom fibre

P

A -

My

I

(compressive)

6.9 Load acting eccentrically to one axis

max

min

P P e y

Where ‘e’ is the eccentricity at which ‘P’ is act.

A I

P P e y

A I

Chapter 7

Torsional Loads

7.1 Torsion of circular shafts

Definition of Torsion: Consider a shaft rigidly clamped at one end and twisted at the other end by a torque T

= F.d applied in a plane perpendicular to the axis of the bar such a shaft is said to be in torsion.

Effects of Torsion: The effects of a torsional load applied to a bar are

(i) To impart an angular displacement of one end cross – section with respect to the other end.

(ii) To setup shear stresses on any cross section of the bar

perpendicular to its axis.

(iii) GENERATION OF SHEAR STRESSES

The physical understanding of the phenomena of setting up of shear stresses in a shaft subjected to a torsion

may be understood from the figure 1-3.

Fig 1: Here the cylindrical member or a shaft is in static equilibrium where T is the resultant external torque

acting on the member. Let the member be imagined to be cut by some imaginary plane ‘mn'.

Fig 2: When the plane ‘mn' cuts remove the portion on R.H.S. and we get a fig 2. Now since the entire

member is in equilibrium, therefore, each portion must be in equilibrium. Thus, the member is in

equilibrium under the action of resultant external torque T and developed resisting Torque Tr .

Fig 3: The Figure shows that how the resisting torque Tr is developed. The resisting torque Tr is produced

by virtue of an infinites mal shear forces acting on the plane perpendicular to the axis of the shaft. Obviously such shear forces would be developed by virtue of sheer stresses.

Therefore we can say that when a particular member (say shaft in this case) is subjected to a torque, the

result would be that on any element there will be shear stresses acting. While on other faces the

complementary sheer forces come into picture. Thus, we can say that when a member is subjected to torque,

an element of this member will be subjected to a state of pure shear.

Shaft: The shafts are the machine elements which are used to transmit power in machines.

Twisting Moment: The twisting moment for any section along the bar / shaft is defined to be the algebraic

sum of the moments of the applied couples that lie to one side of the section under consideration. The

choice of the side in any case is of course arbitrary.

Shearing Strain: If a generator a – b is marked on the surface of the unloaded bar, then after the twisting

moment 'T' has been applied this line moves to ab'. The angle ‘ ' measured in radians, between the final and

original positions of the generators is defined as the shearing strain at the surface of the bar or shaft. The

same definition will hold at any interior point of the bar.

7.2 Modulus of Elasticity in shear: The ratio of the shear stress to the shear strain is called the modulus of

elasticity in shear OR Modulus of Rigidity and in represented by the symbol

Angle of Twist: If a shaft of length L is subjected to a constant twisting moment T along its length, than the

angle through which one end of the bar will twist relative to the other is known is the angle of twist.

Despite the differences in the forms of loading, we see that there are number of similarities between bending and torsion, including for example, a linear variation of stresses and strain with position. In torsion the members are subjected to moments (couples) in planes normal to their axes. For the purpose of desiging a circular shaft to withstand a given torque, we must develop an equation giving the relation between twisting moment, maximum shear stress produced, and a quantity representing the size and shape of the cross-sectional area of the shaft.

Not all torsion problems, involve rotating machinery, however, for example some types of vehicle

suspension system employ torsional springs. Indeed, even coil springs are really curved members in torsion

as shown in figure.

Many torque carrying engineering members are cylindrical in shape. Examples are drive shafts, bolts and screw drivers

7.3 Simple Torsion Theory or Development of Torsion Formula : Here we are basically interested to derive an equation between the relevant parameters

7.4 Relationship in Torsion:

1 st Term: It refers to applied loading ad a property of section, which in the instance is the polar

second moment of area. 2 nd Term: This refers to stress, and the stress increases as the distance

from the axis increases.

3 rd Term: it refers to the deformation and contains the terms modulus of rigidity & combined term (

l) which is equivalent to strain for the purpose of designing a circular shaft to with stand a given torque we

must develop an equation giving the relation between Twisting moments max m shear stain produced and a

quantity representing the size and shape of the cross – sectional area of the shaft.

Refer to the figure shown above where a uniform circular shaft is subjected to a torque it can be shown that

every section of the shaft is subjected to a state of pure shear, the moment of resistance developed by the

shear stresses being every where equal to the magnitude, and opposite in sense, to the applied torque. For

the purpose of deriving a simple theory to describe the behavior of shafts subjected to torque it is necessary

make the following base assumptions.

Assumption:

(i) The materiel is homogenous i.e of uniform elastic properties exists throughout the material.

(ii) The material is elastic, follows Hook's law, with shear stress proportional to shear strain.

(iii) The stress does not exceed the elastic limit.

(iv) The circular section remains circular

(v) Cross section remain plane.

(vi) Cross section rotate as if rigid i.e. every diameter rotates through the same angle.

Consider now the solid circular shaft of radius R subjected to a torque T at one end, the other end being

fixed Under the action of this torque a radial line at the free end of the shaft twists through an angle ,

point A moves to B, and AB subtends an angle ‘ ' at the fixed end. This is then the angle of distortion of

the shaft i.e the shear strain.

Since angle in radius = arc / Radius

arc AB = R

= L [since L and also constitute

the arc AB] Thus, = R / L(1)

From the definition of Modulus of rigidity or Modulus of elasticity in shear

Stresses: Let us consider a small strip of radius r and thickness dr which is subjected to shear stress '.

The force set up on each element

= stress x area

The total torque T on the section, will be the sum of all the contributions.

Since ' is a function of r, because it varies with radius so writing down ' in terms of r from the equation

(1).

ere

T = applied external Torque, which is constant over Length L;

J = Polar moment of Inertia

[ D = Outside diameter ; d = inside diameter ] G = Modules of rigidity

(or Modulus of elasticity shear)

= It is the angle of twist in radians on a length L.

Tensional Stiffness: The tensional stiffness k is defined as the torque

per radius twist i.e, k = T / = GJ / L

Power Transmitted by a shaft : If T is the applied Torque and is the angular velocity of the shaft, then the power transmitted by the shaft is

chapter 5: column

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