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Steel Design Plastic Analysis Designing means to fabricate structures which will be able to support the given load conditions.
These may be either steel structures or RCC structures.
There are three methods of steel design:
a) Working Stress method: The worst combination of working (service) loads is ascertained and
the members are proportioned on the basis of working stress. The worst combination should
not exceed the permissible stresses. The permissible stresses are generally some fraction
(factor of safety) of the yield stress.
b) Plastic Method: We find that steel can withstand higher loads than the elastic limit. In this
method the reserve strength of the structure is used. This method is vased on failure
conditions and not working stresses. This method extends the structural usefulness of the
material upto the ultimate strength. <explain that steel is ductile and therefore can
withstand a bit more of stress than elastic. Bends a bit until it collapses>. In this method the
loads are primarily supported by resistance to bending as deformation is small in case of
bending. So continuous beams and rigid frames are analysed using this method. Gen not
used for statically determinate frames or pin connected beams.
c) Limit State method: this is based on limit strength and serviceability. That is the acceptable
limit for the safety and serviceability before failure. The objective is to achieve a structure
which will not become unfit for the period design like say 50 years.
Stress Strain curve of steel
Fig 2.1 shows the stress strain curve of annealed(i.e it has been given metallurgical treatment) mild
steel specimen in tension.
A is the limit of proportionality.
B is the elastic limit.
C represents the upper yield point at which there is a definite increase in strain without any further
increase in stress.
Cβ is the lower yield point as the stress drops abruptly, then strain increases on constant stress up to
D. Beyond D strain increases further and the material is said to be strain hardening at E. F is failure
point.
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The strain scale is enlarged considerably in Fig 2.1 (b), in order to study the yield range. fu and fL are
the upper and lower yield point in fig 2.1(b). For the purpose of plastic design the elastic limit and
the lower yield point may be assumed to be numerically identical. The yield strength is fixed at fy
and the strain hardening region is ignored. Ignoring the strain hardening adds to the safety.
In fig 2.1 (c) the stress strain behaviour is idealized and it is assumed that extension is unlimited at
constant stress fy
So during Bending of beams the following picture illustrates what happens assuming the idealised
stress-strain curve:
Collapse takes place when plastic moment is reached at the highly stressed points.
Shape factor:
Shape factor is the ratio of plastic moment to yield moment of the section. It depends on the cross
section of the shape. It is a measure of the reserve strength available in the section after bending.
S=Mp/My =ππ¦ππ
ππ¦ππ=
ππ
ππ
Shape Moment of inertia about Cg Moment of inertia about base
Rectangle ππ3
12
ππ3
3
Triangle πβ3
36
πβ3
12
Circle ππ4
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NA
Shape factor of rectangle
Shape factor π =ππ
ππ¦=
ππ¦ππ
ππ¦ππΈ=
ππ
ππΈ=
ππ2
4ππ2
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= 1.5
Zp =π΄
2(π¦1Μ Μ Μ + π¦2Μ Μ Μ ); A is the area of the whole cross-section and π¦1Μ Μ Μ and π¦2Μ Μ Μ are the distances from the
centroidal axis to the centroids of the area in compression and tension respectively. In fig 2.5 C is
compression and T is tension.
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Some shape factors are as givrn below
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Load factor:
Ratio of collapse load to the working load.
F=Pu/Pw=Mp/Mw=fyZp/(f x Ze)=fy/f x S= FOS x S(Shape factor)
What is the load factor that is generally assumed in case of steel design?
1.7 to 2
What is the reduction in load factor when the structures are subject to winds?
25 %
<Explain that the maximum stressed points attain plastic state>
<Pn 70: Fig:2.12>
For any structure to fail a number of hinges need to formed:
There is no relation between the number of hinges and no. of redundancies. You have to find by
experience.
Redundancy is the static indeterminacy of the structure.
What is partial collapse of structure?
No of plastic hinges in the collapse mechanism are less than r+1
What is complete collapse of structure?
No of plastic hinges in the collapse mechanism are equal to r+1
What is over complete collapse of structure?
No of plastic hinges in the collapse mechanism are more than r+1
For calculating the failure load there are three theorems:
Static or lower bound theorem; Kinematic or upper bound theorem; Uniquness theorem.
What is the Static or lower bound theorem?
For a given frame and loading if there exists a set of loads which is both safe and statically admissible
with a set of load P, then the value of the load P must be less than or equal to the collapse load Pu.
Lower limit to the true load
What is Kinematic or upper bound theorem?
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For a given frame subjected to a set of loads P, the value of P which is found to correspond to any
assumed mechanism must be either greater than or equal to the collapse load Pu. Upper limit to the
true load
What is Uniqueness theorem?
For a given frame if one load is found which is both statically admissible and safe and if in this dist
the bending moment is such as to cause fully plastic moment at sufficient cross section to cause
failure as a mechanism due to rotation of plastic hinges then the load will be equal to the collapse
load.
Conditions to be satisfied in the plastic moment of analysis:
Equilibrium condition: Summation of all moments and forces are zero
Mechanism condition or Continuity condition: Ultimate load is reached when a mechanism forms
Yield Condition: Bending moment at any section must not exceed full plastic moment at section.
Note: How the area is calc to find
ext. work done.
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For static method you have to know the BMD well. And for kinematic method the principle of virtual
work is used. The main aim is to find the maximum Mp and minimum Ultimate load Wu which
causes this maximum Mp in the beam. Thus we donβt equate ππ =ππΏ
12 in the above Fig 2.16 as this
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is not the maximum Mp but we equate 2ππ =ππΏ
8. This is because the middle hinge has to form to
complete the mechanism and it requires 2ππ.
What is the principle of virtual work?
Work done by the external forces equal the work done by the internal forces.
We=Wi
Find the reaction at A in the following figure. Ans =5wL/12. Hint β ππ΅ = 0
Hinge length for a point load on a simple beam with rectangular cross-section.
Hinge length for uniformly distributed load with simple beam is L/β3;
/m
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The independent mechanisms are
first formed, using only single
loads and then these mechanisms
are combined. They are combined
such that the geometry remains
the same, if you canβt keep the
geometry same you do not need
to do those mechanisms. The
geometry remains the same
meaning that the mechanisms
should be combined such that the
geometry of the independent
mechs remain (i.e the angles
should be same)
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What are the number of independent mechanisms?
n=N-r; n = possible no. of mechanisms, N= number of plastic hinges; r= number of redundancies
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A gable is the generally triangular portion of a wall between the edges of a sloping roof.
What is the max yield stress of grade of steel for plastic design?
450 Mpa
What are the classification of cross Sections and their char?
Plastic(class 1): Cross Section which can develop plastic hinges. Only these section are used for
plastic analysys
Compact(class 2): Plastic moment developed but does not form plastic hinges
Semi β Compact(class 3): Extreme fibre in compression can reach yield stress, but due to local
buckling doesnβt develop plastic hinges
Slender(class 4): Buckle locally even before attainment of yield stress.
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ππ =5π€πΏ
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Connections The three different types of connections are: Riveted, bolted and pinned connections: These
connections are alike while the welded connection is different.
There are certain terms associated with riveted connections: These are as explained below:
What is the grip of the rivet?
The distance between the undersides of the two heads of rivet.
What is the nominal diameter?
Diameter of shank.
What are hot driven field rivets?
When the rivets are heated before driving
What is the gross diameter of rivets?
The diameter of the rivets when it is hot is equal to the diameter of hole is called gross diameter.
Where are cold driven rivets used?
These are inserted into the rivet hole at room temperature, and very high pressures are required to
form the heads. Only limited to applications where there is high pressure
Is riveted connection practiced nowadays?
No
The main advantages of bolted connection over riveted connections are:
a) Erection of structure can be speeded up.
b) Less skilled persons are required.
c) More economical due to lesser equipment and labour costs.
Holes can be either punched or drilled. Punched holes are used in materials with yield stress less
than 360 MPa and thickness should not exceed 5660/fy mm.
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Disadvantages:
1) Cost of material is high
2) Tensile strength of bolts is reduced because of area reduction at root of threads
3) Loose fitting so less strength
4) May get loose due to vibrations
Joints may be of different kinds:
Lap joint; Butt joint Explain:
Load transfer is either by bearing and shearing or by friction.
Bearing: Bearing is the stress on the plate due to the bolt being forced onto the plate
Shearing: This is the stress on the bolt which tries to shear away or cut away the bolt at the point of
contact. This is shown in Fig.1
Fig. 1: Shearing
Explain that for bolting certain terms like diameter of bolts, spacing of bolts are reqd to be known.
Few large diameter bolts are more economical than a more number of small dia bolts.
Pitch:
It is the distance between two consecutive bolts measured parallel to the direction of stress.
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Why is a minimum amt of pitch required?
1) Prevent bearing failure of members
2) Permit efficient installation of bolts
Why is the maximum pitch required?
1) To reduce length of connection and gusset plate
2) To have uniform stress in bolts.
What is the minimum ptch?
2.5 times the nominal diameter.
What is the maximum pitch?
Tension Members: 16t or 20mm, whichever is less
Compression members: 12t or 200mm
Pitch of tacking bolts should not exceed 32t or 300mm, but when exposed to weather 16t or
200mm.
Tacking bolts are used to make members act in unison also known as stitch bolts. These bolts are
not for being subjected to stresses like tension or compression.
Gauge:
Distance between adjacent bolt lines; The gauge is such as to accommodate the bolts properly as
shown in fig 4.8.
Edge distance explain
Distance at right angle to the direction of stress from the centre of the bolt hole to the adjacent
edge of the member.
End Distance: From the centre of hole to the end of element along the direction of stress.
What is the maximum edge distance?
12tπ πππ β= β250
ππ¦ and t is the thickness of thinner outer plate
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N.B.: While designing generally two or three bolts are provided , even if one bolt hole is required,
this is a safety measure,as one bolt may fail to live upto its strength
The minimum of bearing, shearing, tensile strength of bolt and tensile strength of plate is taken as
strength of bolted joint, now we calculate them:
Tell we use a factor of safety of πΎππ =1.25 for ultimate strength and for yield strength πΎπ0 = 1.1.
The best bolting pattern is βdiamondβ.
How is class designated?
4.6 Ultimate stress 4 x 100 =400N/mm2; Yiels dtrength =0.6 x 400 =240N/mm2
How is the cover plate thickness selected in case of butt joint?
Plate thickness not less than 5/8t
Cover plates are used in butt joints to connect the two members.
Formulae for bolt connection design Shearing strength of bolts:
πππ π =ππ’π
β3(πππ΄ππ + ππ π΄π π)
πππ π Nominal shear strength of bolt
ππ’π ultimatre tensile stress of bolt
ππ number of shear planes with threads intercepting the shear plane
ππ number of shear planes without threads intercepting the shear plane
π΄π π nominal shank Area of bolt=ππ2
4; d is dia of bolt.
π΄ππ net tensile stress area. =0.78π΄π π
Now for long joints and other factors we use some reduction factors:
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πππ π =ππ’π
β3(πππ΄ππ + ππ π΄π π)π½πππ½πππ½πππ
π½ππ = reduction factor used in long bolts = 1.075βππ
200π; 0.75 β€ π½ππ β€ 1; ππ is the length of joint
between the first and last row of bolts. Used when ππ > πππ
π½ππ =reduction factor for large grip length = 8π
3π+ππ; Used when total thickness of connected plates
exceed five times the nominal dia of boltπ½ππ β― π½ππ πππ π½ππ β― 8π
π½πππ= reduction factor when packing plate thickness exceeds 6mm. =(1 β 0.0125π‘πππ). π‘πππ is the
thickness of thicker packing
ππ π β€πππ π
πΎππ
πΎππ = 1.25 (ππππ‘πππ π ππππ‘π¦ ππππ‘ππ)
ππ π is the shear strength of bolt used for designing. When finding bolt value use:
ππ π =πππ π
πΎππ
Remember whenever there is ultimate strength involved use partial safety factor of 1.25, when yield
strength is involved use 1.1
Bearing strength of bolt:
ππππ = 2.5ππππ‘ππ’;
ππ =smaller of π
3π0;
π
3π0β 0.25;
ππ’π
ππ’; πππ 1.0;
π = nominal dia of bolt
π0 = hole dia
π‘ = aggregate thickness for connected plates (will be explained later) experiencing stress in the
same direction. If bolts are countersunk then thickness of plate minus half of the depth of
countersunk.
ππππ =nominal bearing strength of bolt.
πππ β€ππππ
πΎππ
πΎππ = 1.25 (ππππ‘πππ π ππππ‘π¦ ππππ‘ππ)
When finding bolt value use: πππ =ππππ
πΎππ
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Tensile strength of bolt
πππ = 0.9ππ’ππ΄ππ < ππ¦ππ΄π π
πΎππ
πΎπ0
πππ = nominal tensile strength of bolt.
πππβ€
πππ
πΎππ
Tensile strength of plate: (not in syllabus) π΄π = (π΅ β ππβ)π‘; for chain bolting
π΄π = [π΅ β ππβ + βπ2
4π] π‘; for staggered bolting
πππ = 0.9π΄π
ππ’
πΎπ1
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Efficiency of joint (not in syllabus)
π =ππ‘πππππ‘β ππ ππππ‘ππ πππππ‘ πππ πππ‘πβ πππππ‘β
ππ‘πππππ‘β ππ π ππππ ππππ‘π πππ πππ‘πβ πππππ‘β Γ 100
If pitch is staggered then take two pitch lengths and take them from the middle, such that two rivets
are included. Thus two rivets have to fail, but only one hole in a pitch (because it is (p-d) x t and no
breadth term is included).
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Which plate thickness to be used? 1) Double cover butt joint. If thickness of cover plates = tp and thickness of thinner member tm
then if
2tp>tm use tm
Else 2tp
2) If 2 angle sections or I-sections connected back to back . If thickness of gusset plates = tp and
thickness of thinner member tm then if
2tm>tp use tp else 2tm
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