mechanics of deformable bodies ii
TRANSCRIPT
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