mechanics of deformable bodies ii

FlexuralFlexuralMembersMembers

Department of Engineering ScienceCollege of Engineering

University of the Philippines-Diliman

Engineering Science 10Strength of Materials: Why Things Bend and Break?

AH-1Pitch Control Rod(Crash impact forces)

T-2 Wing Carry Through Beams(Inflight failure)

TH-57 Rotor Mast Note: Not a Helical (Torsional) failure, so

it is not separated due to rotational forces.

Beams

What is a beam?

Beams

A slender (usually) horizontal structural member that is subjected to a load that tends to bend it.

Examples:

Beams

Floor joists and rafters

Tree branches

Vertebral column and neck

Types:

Beams

Fixed beam (simply supported beam)

Types:

Beams

Cantilever beam

A cantilever beam can be thought of as half of a fixed beam turned upside down.

Beams

See!

Beams

Types:

Beams

Beam with overhang

Types:

Beams

Continuous beam

All materials and structures deflect, to greatly varying extents when they are loaded.

Beams

The science of elasticity is about the interactions between forces and deflections. The material of the bough is stretched near its upper surface and compressed or contractednear its lower surface by the weight of the monkey.

Bending

Forces on a cantilever beam

Bending

If the material near the neutral axis is removed, the beam collapses.

Bending

Material in the middle provides shear resistance.

Bending

Geometry of

deformation

We will consider the deformation of an ideal, isotropic prismatic beam.

Geometry of deformation

All parts of the beam that were originally aligned with the longitudinal axis bend into circular arcs.

The neutral plane/surface or axis:

Geometry of deformation

Where is the neutral plane/surface or axis?

Geometry of deformation

It is the centroid of the cross-section.

In simple circular or rectangular shapes, it is the middle.

For more complex shapes, it is the center of gravity of a cutout of uniform thickness with the object’s cross-sectional shape.

Bending stresses in a beam:

Geometry of deformation

Stress

Let’s consider a segment of a loaded beam. The segment may be deformed as shown below.

Stress

Hence we can sketch of the stress normal to the axis of the beam …

Stress

The radius of curvature:

Stress

EIM−=

ρ1

Positive ρ means the positive y-axis is on the concave side of the neutral axis.

EI is known as the flexural rigidity of the beam.

Where M is the internal bending moment acting on a section, E is the modulus of elasticity and I is the moment of inertia of the section.

The fiber stress:

Stress

Where M is the internal bending moment acting on a section, y is the distance of a point (fiber) from the neutral axis and I is the moment of inertia of the section.

IMy=σ

Cross section

Flexural rigidity:

Cross section

EI gives the resistance to bending of a structure.

Several conditions must be met in order to simply calculate it. The material must be:

Homogeneous

Isotropic

Linearly elastic

Deform equally under tension and compression

Not change shape appreciably under load

What does the I, the moment of inertia mean?

Cross section

Essentially, this says that the beam is stiffer if the material in the beam is located further away from the centroid (neutral axis).

So any small area is more effective at stiffening the beam depending on the square of the distance.

Hence, if you want to make a strong beam with little material, make sure that the material is as far as possible from the centroid.

What does the moment of inertia mean?

Cross section

Hence, we have ‘I’ beams.

I beams are very stiff in bending but not thatresistant in torsion.

Cross section

Box girders (hollow square tube)

Cross section

Very resistantto torsion

But stresses tendto concentrate at

the corners

Hollow tubes

Cross section

Best if thedirection of theload cannot be

predicted

Resistant tobending and

torsion

Typical beam cross sections and the ratio of I to the value for a solid square beam of equal cross-sectional area.

Cross section

A T-cross section can be used to decrease either the maximum tensile or the maximum compressive stress to which the beam is subjected.

Cross section

Some organisms take advantage of I to localize bending.

Cross section

Sea anemone,Metridium

Daffodil

Beam technology

The I beam:

Beam technology

Steel section terminology:

Beam technology

While we are mostly assuming beams made of steel or other metals, many are made of concrete

Beam technology

and concrete does not support a tensile stress.

For concrete beams, we assume that only the material on the compressive side of the neutral axis actually carries a load.

One solution is pre-stressed concrete.

Beam technology

Where metal bars set within the concrete are pre-stressed to provide an initial compression to the concrete beam.so it can withstand some tension, until the pre-stress is overcome

The yellow guidelines highlight the camber (upward curvature) of a pre-stressed double T. The pre-stressing strands can be seen protruding from the bottom of the beam, with the larger strands at the bottom edge. The tension is these strands produces the camber, the beam is straight when cast.`

Arches

Beam technology

How an arch works?

Beam technology

TorsionalTorsionalMembersMembers

Department of Engineering ScienceCollege of Engineering

University of the Philippines-Diliman

Engineering Science 10Strength of Materials: Why Things Bend and Break?

F-18 Engine ShaftTorsional Buckling

UH-1N Turbine: Helical shaft failure

Torsionalloads

1) Turbine exerts torque T on the shaft.

Torsional loads

2) Shaft transmits the torque to the generator.

3) Generator creates an equal and opposite torque T’.

Torque – twisting couple

Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque,

Torsional loads

Torque applied to shaft produces shearing stresses on the faces perpendicular to the axis.

Torsional loads

Conditions of equilibrium require the existence of equal stresses on the faces of the two planes containing the axis of the shaft.

Torsional loads

The slats slide with respect to each other when equal and opposite torques are applied to the ends of the shaft.

The existence of the axial shear components is demonstrated by considering a shaft made up of axial slats.

Shaft deformations

From observation, the angle of twist of the shaft is proportional to the applied torque and to the shaft length.

Shaft deformations

LT

∝∝

φφ

Shaft deformations

When subjected to torsion, every cross-section of a circular shaft remains a plane and undistorted.

Cross-sections for hollow and solid circular shafts remain plain and undistorted because a circular shaft is axisymmetric.

Shaft deformations

Cross-sections of noncircular (non-axisymmetric) shafts are distorted when subjected to torsion.

Shearing

Consider an interior section of the shaft. As a torsional load is applied, an element on the interior cylinder deforms into a rhombus.

Shearing

Since the ends of the element remain planar, the shear strain is equal to angle of twist.

It follows that

LL ρ φγρ φγ == or

Shear strain is proportional to twist and radius.

The shearing stress varies linearly with the radial position in the section.

Shearing

The elastic torsion formula,

and max JT

JTc ρττ ==

Torsional failure

Ductile materials generally fail in shear. Brittle materials are weaker in tension than shear.

Torsional failure

When subjected to torsion, a ductile specimen breaks along a plane of maximum shear, i.e., a plane perpendicular to the shaft axis.

Torsional failure

When subjected to torsion, a brittle specimen breaks along planes perpendicular to the direction in which tension is a maximum, i.e., along surfaces at 45o to the shaft axis.

Torsional failure

Angle of twist

Recall that the angle of twist and maximum shearing strain are related,

Angle of twist

In the elastic range, the shearing strain and shear are related by Hooke’s Law,

Lcφγ =max

JGTc

G== max

maxτγ

Equating the expressions for shearing strain and solving for the angle of twist,

Angle of twist

JGTL=φ

If the torsional loading or shaft cross-section changes along the length, the angle of rotation is found as the sum of segment rotations

Angle of twist

∑=i ii

iiGJLTφ

Transmission

Transmission

Design of transmission shafts:

Principal transmission shaft performance

specifications are:

power

speed

Designer must select shaft material and cross-section to meet performance specifications without exceeding allowable shearing stress.

Transmission

Determine torque applied to shaft at specified power and speed,

fPPT

fTTP

πω

πω

2

2

==

==

Where P is the power, T is the torque and ω is the angular velocity.

Transmission

Find shaft cross-section which will not exceed the maximum allowable shearing stress,

( )

( ) ( )shafts hollow2

shafts solid2

max

41

42

22

max

3

max

τπ

τπ

τ

Tcccc

J

TccJ

JTc

=−=

==

=

Transmission

The derivation of the torsion formula,

JTc=maxτ

assumed a circular shaft with uniform cross-section loaded through rigid end plates.

Transmission

The use of flange couplings, gears and pulleys attached to shafts by keys in keyways, and cross-section discontinuities can cause stress concentrations.

Experimental or numerically determined concentration factors are applied as

JTcK=maxτ

Transmission

Shaft design:

Shaft must have adequate torsional strength to transmit torque and not be overstressed.

Shafts are mounted in bearings and transmit power through devices such as gears, pulleys, cams and clutches.

Components such as gears are mounted on shafts using keys.

Shaft must sustain a combination of bending and torsional loads.

SlenderSlenderColumnsColumns

Department of Engineering ScienceCollege of Engineering

University of the Philippines-Diliman

Engineering Science 10Strength of Materials: Why Things Bend and Break?

Buckling

Columns

Columns – Straight structural members subjected to compressive axial loads.

Columns

Types of columns:

Columns

Short

Long

Failure of Modes of Columns:

Columns

Crushing

Failure of Modes of Columns:

Columns

Buckling

Crushing failure governs in short columns while buckling failure governs in long columns.

Columns

Buckling

When a structure (subjected usually to compression) undergoes visibly large displacements transverse to the load then it is said to buckle.

Buckling

Buckling may be demonstrated by pressing the opposite edges of a flat sheet of cardboard towards one another.

For small loads the process is elastic since buckling displacements disappear when the load is removed.

Buckling

Types:

Buckling

Euler buckling

Column is smoothly bent from end to end.

Compressive forces on the concave side and tension forces on the convex

Types:

Buckling

Euler buckling

The critical Euler load:

2

2

(KL)EIπP =

Where E is the Young’s modulus, I is the moment of inertia, K is the effective length factor and L is the length of the column.

Types of column and the Euler buckling:

Buckling

Pinned ends:

Types of column and the Euler buckling:

Buckling

Fixed and free ends:

Types of column and the Euler buckling:

Buckling

Fixed ends:

Types of column and the Euler buckling:

Buckling

Pinned and Fixed ends:

Types of column and the Euler buckling:

Buckling

2

2

(KL)EIπP =

Types:

Buckling

Local buckling

The way empty cans fail.

Indicated by growth of bulges, waves or ripples.

Buckling proceeds in manner which may be either :

Buckling

Stable - in which displacements increase in a controlled fashion as loads are increased, ie. the structure's ability to sustain loads is maintained, or

Unstable - in which deformations increase instantaneously, the load carrying capacity nose-dives and the structure collapses catastrophically.

Neutral equilibrium is also a theoretical possibility during buckling – this is characterized by deformation increase without change in load.

Buckling

Compression structures

Advantage: so long as they are put together correctly, gravity will do the work of keeping everything in its proper place (no specific fasteners needed). Traditionally, large compression structures have been built of brick/stone.

Compression structures

Disadvantage: susceptible to failure by bending; stone has little resistance to tension.

Theoretical limit: the point at which the vertical load becomes so great that the material/s are crushed beneath the weight of the material/s above them.

Compression structures

The crushing strength of stone and brick is

> 40 MN/m2

We should be able to build a column/wall several miles high.

How high can we build a column/wall?

The wall will fail long before the theoretical limit is reached. Why?

Compression structures

Important point: mortar between bricks does not glue the bricks together (it carries no tensile load). Mortar fills in the surface irregularities in the bricks’ surfaces, so that the compressive load is not carried by a few high spots (which would increase local stress).

How high can we build a column/wall?

Assumptions:

Compression structures

Compressive stresses are so low that the material will not be broken by crushing.

Analysis of column/wall stability?

Mortar is used, so the fit between the stones is so good that the compressive force is distributed over the whole area of the joint.

Friction in the joints is so high that failure will not occur because of the stones sliding over one another.

The joints (mortar) have no tensile strength.

Compressive stresses are spread evenly over the entire width of the joint

Compression structures

Symmetrical vertical force,P

Compressive stresses are no longer uniform.

Compression structures

What if P is eccentric?

Stress will vary linearly across the width of the column/wall if the material is Hookean.

Whole joint is still in compression.

Stress at the outside edge of the wall is zero.

Compression structures

P is now at the edge of the middle third of the column/wall.

We still have compressive forces across the entire width of the column/wall.

The outside edge of the wall is now in tension.

Compression structures

P is outside of the middle third of the column/wall.

But mortar cannot carry tension stresses!

What will happen?

We are still okay, because the remaining width of the wall can still carry the load.

Compression structures

The column/wall cracks.

If P’s line of action moves outside of the wall, the whole joint is put into tension.

Compression structures

Disaster!

The wall tips up and falls over.

We need to know the point at each successive joint at which we can consider the weight to be acting.

Compression structures

We can connect the points and plot a thrust line.

What if the vertical force is applied to the wall obliquely?

Compression structures

Thrust line

Compression structures

Thrust line and stability

If the thrust line comes in contact with the edge of the wall at any point, the wall is liable to fall over.

How can one insure that the thrust line stays within the middle third of the wall?

Compression structures

One of the most effective strategies is to simply add more weight to the top of the wall

Compression structures

That is the practical reason behind the decorative statues adorning so many large stone buildings

Compression structures

cornice or gargoyle

Next Meeting:

Project

Bridge building competition, 9/14.

Bring toothpicks