Structural analysis 2

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Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry
STRUCTURAL ANALYSIS – 2
UNIT – 1
1. What is an arch? Explain.
An arch is defined as a curved girder, having convexity upwards and
supported at its ends. The supports must effectively arrest displacements in
the vertical and horizontal directions only then there will be arch action.
2. State the general cable theorem.
The general cable theorem helps us determine the shape of a cable
supported at two ends when it is acted upon by vertical forces. It can be
stated as: “At any point on a cable acted upon by vertical loads, the product
of the horizontal component of cable tension and the vertical distance from
that point to the cable chord equals the moment this would occur at that
section if the loads carried by the cable were acting on an simply-supported
beam of the same span as that of the cable.”
3. What are the various types of hinges in arch? (or) What are the types of
arches according to the support conditions?
 Three hinged arch
 Two hinged arch
 Single hinged arch
 Fixed or hinge less arch
4. What are the types of arches according to their shapes?
 Curved arch

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 Parabolic arch
 Elliptical arch
 Polygonal arch
5. Define horizontal thrust.
In a 3 hinged arch, the force H is calculated by equating the bending
moment at the central hinge to zero. The horizontal thrust H reduces the
beam bending moment called µx. 𝐴𝑐𝑡𝑢𝑎𝑙𝑙𝑦 𝑖𝑛 𝑎𝑛 𝑎𝑟𝑐ℎ, 𝑀 𝑥 = 𝜇 𝑥 − 𝐻 𝑦
6. What is a linear arch?
If an arch is to take loads, say W1, W2, and W3 and a vector diagram
and funicular polygon are plotted as shown in figure, the funicular polygon
is known as the linear arch or theoretical arch.
The polar distance ‘o t’ represents the horizontal thrust. The links
AC, CD, DE and EB will be under compression and there will be no bending
moment. If an arch of this shape ACDEB is provided, there will be no
bending moment.

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For a given set of vertical loads W1, W2……etc. we can have any
number of linear arches depending on where we chose ‘O’ or how much
horizontal thrust (o t) we choose to introduce.
7. Draw the influence line for horizontal reaction, H in a three hinged
stiffening girder.
8. Why stiffening girders are necessary in the suspension bridges?
 Stiffening girders enable the suspension bridge decks to remain fairly
level
 Whatever be the live load on the deck slab, the stiffening girders will
convert and transmit the load on the deck slab as a uniformly
distributed load and thereby help the cable retain the parabolic shape
during the passage of loads

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 The dead load of the girders which is a UDL is directly transmitted to
the cables and is taken up entirely by the tension in the cables
 Thus the uniformly distributed dead load will not cause any shear
force or bending moment in the stiffening girder
 The stiffening girders will have to resist the Shear force and bending
moment due to live loads
9. Write the expression for horizontal thrust in a three hinged parabolic
arch carrying UDL over entire span.
𝐻 =
𝑤 𝑙2
8𝑦𝑐
10.Write the expression for horizontal thrust of a semicircular arch.
𝐻 =
𝑊 cos2
𝛷
𝜋
If the load is applied at the centre, we get
𝐻 =
𝑊
𝜋
= 0.318 𝑊
11.A flexible cable 20m long is supported at two ends at the same level.
The supports are 16m apart. Determine the dip of the cable.
GIVEN DATA:
S = 20m
l = 16m
TO FIND:
d =?

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SOLUTION:
𝑆 = 𝑙 +
8
3
𝑑2
𝑙
𝑑 = √
( 𝑆 − 𝑙 ) × 3𝑙
8
d = 4.89m
12.State the “Eddy’s theorem” for an arch.
Eddy’s theorem states that the bending moment at any section of an
arch is equal to the vertical intercept between the linear arch and the center
line of the actual arch.
𝐵𝑀 𝑥 = 𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑂2 𝑂3 × 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟
13.What is the static indeterminacy of a three hinged parabolic arch?
For a three hinged parabolic arch, the degree of static indeterminacy is
zero. It is statically determinate.

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14.Explain with the aid of a sketch the normal thrust and radial shear in
an arch rib.
Let us take a section X of an arch. Let q be the inclination of the
tangent at X. If H is the horizontal thrust and V is the vertical shear at X,
from the free body of the RHS of the arch, it is clear that V and H will have
normal and radial components given by,
𝑅𝑎𝑑𝑖𝑎𝑙 𝑠ℎ𝑒𝑎𝑟 ( 𝑅 𝑥 ) = 𝑉𝑥 cos 𝜃 − 𝐻 sin 𝜃
𝑁𝑜𝑟𝑚𝑎𝑙 𝑡ℎ𝑟𝑢𝑠𝑡 ( 𝑁𝑥 ) = 𝑉𝑥 sin 𝜃 + 𝐻 cos 𝜃
15.Which of the two arches, viz. circular and parabolic is preferable to
carry a uniformly distributed load? Why?
Parabolic arches are preferably to carry distributed loads. Because,
both the shape of the arch and the shape of the bending moment diagram are
parabolic. Hence the vertical intercept between the theoretical arch and
actual arch is zero everywhere. Hence, the bending moment at every section
of the arch will be zero. The arch will be under pure compression which
will be economical.

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16.What is the difference between the basic action of an arch and
suspension cable?
An arch is essentially a compression member which can also take
bending moments and shears. Bending moments and shears will be absent if
the arch is parabolic and the loading uniformly distributed.
A cable can take only tension. A suspension bridge will therefore
have a cable and a stiffening girder. The girder will take the bending
moment and shears in the bridge and the cable, only tension.
Because of the thrusts in the cables and arches, the bending moments
are considerably reduced.
If the load on the girder is uniform, the bridge will have only cable
tension and no bending moment on the girder.
17.Under what conditions will the bending moment in an arch be zero
throughout?
The bending moment in an arch throughout the span will be zero, if
 The arch is parabolic
 The arch carries UDL throughout the span
18.Indicate the positions of a moving point load for maximum negative and
positive bending moments in a three hinged arch.
Considering a three hinged parabolic arch of span ‘l’ and subjected to
a moving point load W, the position of the point load for
 Maximum negative bending moment is 0.25l from end supports.
 Maximum positive bending moment is 0.211l from end supports.

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19.Draw the ILD for bending moment at a section x at a distance x from
the left end of a three hinged parabolic arch of span ‘l’ and rise ‘h’.
𝑀 𝑥 = 𝜇 𝑥 − 𝐻 𝑦
20.Distinguish between two hinged and three hinged arches.
TWO HINGED ARCHES THREE HINGED ARCHES
Statically indeterminate to first
degree
Statically determinate
Might develop temperature stresses Increase in temperature causes
increase in central rise. No stresses
Structurally more efficient Easy to analysis. But in
construction, the central hinge may
involve additional expenditure.
Will develop stresses due to sinking
of supports
Since this is determinate, no stresses
due to support sinking.
21.Explain rib shortening in the case of arches.
In a two hinged arch, the normal thrust which is a compressive force
along the axis of the arch will shorten the rib of the arch. This in turn will
release part of the horizontal thrust. Normally, this effect is not considered in
the analysis (in the case of two hinged arches).

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Depending upon the importance of the work we can either take into
account or omit the effect of rib shortening. This will be done by considering
(or omitting) strain energy due to axial compression along with the strain
energy due to bending in evaluating H.
22.What are cable structures?
Long span structures subjected to tension and uses suspension cables
for supports. Examples of cable structures are suspension bridges, cable
stayed roof.
23.Explain the yielding of support in the case of an arch.
Yielding of supports has no effect in the case of a 3 hinged arch which
is determinate. These displacements must be taken into account when we
analyze 2 hinged or fixed arches under
𝜕𝑈
𝜕𝐻
= ∆𝐻 Instead of zero
𝜕𝑈
𝜕𝑉 𝐴
= ∆𝑉𝐴 Instead of zero
Here U is the strain energy of the arch and ΔH and ΔVA are the
displacements due to yielding of supports.
24.Write the formula to calculate the change in rise in three hinged arch.
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑟𝑖𝑠𝑒 = (
𝑙2
+ 4𝑦𝑐
2
4𝑦𝑐
) × 𝛼𝑇
Where,
l = span length of the arch
yc = central rise of the arch
α = coefficient of thermal expansion

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T = change in temperature
25.In a parabolic arch with two hinges how will you calculate the slope of
the arch at any point?
𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑖𝑐 𝑎𝑟𝑐ℎ ( 𝜃 ) = tan−1
(
4𝑦𝑐
𝑙2
× ( 𝑙 − 2𝑥 ))
Where,
θ = slope at any point x (or) inclination of tangent at x
yc = central rise of the arch
26.How will you calculate the horizontal thrust in a two hinged parabolic
arch if there is a rise in temperature?
𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑡ℎ𝑟𝑢𝑠𝑡 ( 𝐻 ) =
𝛼 𝑇 𝑙 𝐸 𝐼
∫ 𝑦2 𝑑𝑥
𝑙
0
Where,
y = rise of the arch at any point x
α = coefficient of thermal expansion
T = change in temperature
E = Young’s Modulus of the material of the arch
I = Moment of Inertia
27.What is the true shape of cable structures?
Cable structures especially the cable of a suspension bridge is in the
form of a catenary. Catenary is the shape assumed by a string / cable freely
suspended between two points.

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28.What is the nature of force in the cables?
Cables of cable structures have only tension and no compression or
bending.
29.What is a catenary?
Catenary is the shape taken up by a cable or rope freely suspended
between two supports and under its own self weight.
30.Mention the different types of cable structures?
 Cable over a guide pulley
 Cable over a saddle
31.Briefly explain cable over a guide pulley.
Cable over a guide pulley has the following properties:
 Tension in the suspension cable = tension in the anchor cable
 The supporting tower will be subjected to vertical pressure and
bending due to net horizontal cable tension
32.Briefly explain cable over saddle.
Cable over saddle has the following properties:
 Horizontal component of tension in the cable = horizontal component
of tension in the anchor cable
 The supporting tower will be subjected to only vertical pressure due to
cable tension
33.What are the main functions of stiffening girder in suspension bridges?
 They help in keeping the cables in shape

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 They resist part of the Shear force and bending moment due to live
loads
34.What is the degree of indeterminacy of a suspension bridge with two
hinged stiffening girder?
The two hinged stiffening girder has one degree of indeterminacy.
35.Differentiate between plane truss and space truss.
PLANE TRUSS SPACE TRUSS
All members lie in one plane This is a three dimensional truss
All joints are assumed to be hinged All joints are assumed to be ball and
socketed
36.Give some examples of beams curved in plan.
Curved beams are found in the following structures.
 Beams in the bridge negotiating a curve
 Ring beams supporting a water tank
 Beams supporting corner lintels
 Beams in ramps
37.What are the forces developed in beams curved in plan?
Beams curved in plan will have the following forces developed in
them.
 Bending moments
 Shear forces
 Torsional moments

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38.Define tension coefficient of a truss member.
The tension coefficient for a member of a truss is defined as the pull
or tension in the member divided by its length (i.e.) the force in the member
per unit length.
39.What are the significant features of circular beams on equally spaced
supports?
 Slope on either side of any support will be zero
 Torsional moment on every support will be zero
40.Give the expression for calculating equivalent UDL on a girder.
𝑊𝑒 =
𝑡𝑜𝑡𝑎𝑙 𝑙𝑜𝑎𝑑 𝑜𝑛 𝑔𝑖𝑟𝑑𝑒𝑟
𝑠𝑝𝑎𝑛 𝑜𝑓 𝑔𝑖𝑟𝑑𝑒𝑟
41.Give the expression for determining the tension T in the cable.
The tension developed in the cable is given by,
𝑇 = √ 𝐻2 + 𝑉2
Where,
H = horizontal component
V = vertical component
42.What are cables made of?
Cables can be of mild steel, high strength steel, stainless steel, or
polyester fibres. Structural cables are made of a series of small strands
twisted or bound together to form a much larger cable. Steel cables are either
spiral strand, where circular rods are twisted together or locked coil strand,

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where individual interlocking steel strands form the cable (often with a spiral
strand core).
Spiral strand is slightly weaker than locked coil strand. Steel spiral strand
cables have a Young's modulus, E of 150 ± 10 kN/mm² and come in sizes
from 3 to 90 mm diameter. Spiral strand suffers from construction stretch,
where the strands compact when the cable is loaded.
43.Give the horizontal and vertical components of a cable structure
subjected to UDL.
The horizontal and vertical reactions are given by,
𝐻 =
𝑤 𝑆2
8𝑑
And 𝑉 =
𝑤 𝑆
2
44.What is meant by “Reaction locus” for a two hinged arch?
The Reaction locus is a line which gives the point of intersection of the two
reactions for any position of an isolated load.
45.Give the range of central dip of a cable.
The central dip of a cable ranges from 1/10 to 1/12 of the span.

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46.Give the types of significant cable structures.
 Linear structure
 Suspension bridges
 Draped cables
 Cable stayed beams or trusses
 Cable trusses
 Straight tensioned cables
 Three dimensional structure
 Bicycle wheel roof
 3D cable trusses
 Tensegrity structures
 Tensairity structures
UNIT – 2 & 3
1. Where do you get the rolling loads in practice?
Shifting of load positions is common enough in buildings. But they
are more pronounced in bridges and in gantry girders over which vehicles
keep rolling.
2. List the categories of rolling loads on beams.
 Single concentrated load
 UDL longer than the beam span
 UDL shorter than the beam span
 Two wheel axles separated by a fixed distance
 Multiple wheel axles (train of loads)

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3. What are the objectives of study on rolling loads?
 To find the load position and values of maximum shear force and
bending moment at a given section due to a given system of rolling
loads
 To find the location and values of the absolute maximum shear force
and bending moment that may occur on the span due to the given
system of rolling loads
 To find the equivalent UDL due to a given system of rolling loads to
make the designer’s work simple
4. Name the type of rolling loads for which the absolute bending moment
occurs at the mid span of a beam.
Single concentrated load, UDL longer than the span, UDL shorter
than the span Also when the resultant of several concentrated loads crossing
a span, coincides with a concentrated load then also the maximum bending
moment occurs at the centre of the span.
5. Where do you have the absolute maximum bending moment in a simply
supported beam when a series of wheel loads cross it?
When a series of wheel loads crosses a simply supported beam, the
absolute maximum bending moment will occur near mid span under the load
Wcr, nearest to mid span (or the heaviest load). If Wcr is placed to one side
of mid span C, the resultant of the load system R shall be on the other side of
C and Wcr and R shall be equidistant from C. Now the absolute maximum
bending moment will occur under Wcr. If Wcr and R coincide, the absolute
maximum bending moment will occur at mid span.

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6. What is the absolute maximum bending moment due to a UDL longer
than the span of a simply supported beam?
When a simply supported beam is subjected to a moving UDL longer
than the span, the absolute maximum bending moment occurs when the
whole span is loaded.
𝑀max 𝑚𝑎𝑥 =
𝑤 𝑙2
8
7. State the location of a maximum shear force in a simple beam with any
kind of loading.
In a simple beam with any kind of load, the maximum positive Shear
force occurs at the left hand support and maximum negative Shear force
occurs at right hand support.
8. What is meant by absolute maximum bending moment in a beam?
When a given load system moves from one end to the other end of a
girder, depending upon the position of the load, there will be a maximum
bending moment for every section. The maximum of these bending
moments will usually occur near or at the mid span. The maximum of
maximum bending moments is called the absolute maximum bending
moment.
9. What is meant by influence lines?
An influence line is a graph showing, for any given frame or truss, the
variation of any force or displacement quantity (such as shear force, bending
moment, tension, deflection) for all positions of a moving unit load as it
crosses the structure from one end to the other.

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An influence line for any given point or section of structure is a curve
whose ordinates represent to scale the variation of a function, such as shear
force, bending moment, deflection etc. at the point or section, as the unit
load moves across the structure. ILD for determinate beam is linear and for
indeterminate structure is curvilinear.
10.What are the uses of influence diagrams?
 Influence lines are very useful in the quick determination of reactions,
shear force, bending moment or similar functions at a given section
under any given system of moving loads and
 Influence lines are useful in determining the load position to cause
maximum value of a given function in a structure on which load
positions can vary.
11.State Muller Breslau principle.
Muller-Breslau principle states that, if we want to sketch the influence
line for any force quantity (like thrust, shear, and reaction, support moment
or bending moment) in a structure,
 We remove from the structure the restraint to that force quantity and
 We apply on the remaining structure a unit displacement
corresponding to that force quantity.
The resulting displacements in the structure are the influence line
ordinates sought.
12.Write the uses of Muller Breslau principle.
 It is the most important tool in obtaining influence lines for statically
determinate as well as statically indeterminate structures

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 It is used as the straight application of Maxwell’s reciprocal theorem
13.Define the term “Equivalent uniformly distributed load”.
A given system of loading crossing a girder or structure can always be
replaced by a uniformly distributed load longer than the span. Such that the
B.M or S.F due to the static load everywhere is at least equal to the caused
by the actual system of moving loads. Such a static load is known as
Equivalent Uniformly Distributed Load.
14.A single load of W rolls along a girder of span ‘l’. Draw the diagrams of
maximum bending moment and shear force.
15.What is meant by maximum shear force diagram in influence line?
Due to a given system of rolling loads the maximum shear force for
every section of the girder can be worked out by placing the loads in
appropriate positions. When these are plotted for all the sections of the

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girder, the diagram that we obtain is the maximum shear force diagram. This
diagram yields the ‘design shear’ for each cross section.
16.What do you understand by the term reversal of stresses?
In certain long trusses the web members can develop either tension or
compression depending upon the position of live loads. This tendency to
change the nature of stresses is called reversal of stresses.
17.Draw the ILD for shear force shear force at a point x in a simply
supported beam AB of span l.
18.State Maxwell – Betti’s theorem.

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In a linearly elastic structure in static equilibrium acted upon by either
of two systems of external forces, the virtual work done by the first system
of forces in undergoing the displacements caused by the second system of
forces is equal to the virtual work done by the second system of forces in
undergoing the displacements caused by the first system of forces.
Maxwell Betti’s theorem helps us to draw influence lines for structures.
19.Draw the influence line for radial shear at a section of a three hinged
arch.
𝑅𝑎𝑑𝑖𝑎𝑙 𝑠ℎ𝑒𝑎𝑟 ( 𝑅 𝑥 ) = 𝑉𝑥 cos 𝜃 − 𝐻 sin 𝜃
Where, θ is the inclination of tangent at x
20.What is the necessity of model analysis?
 When the mathematical analysis of problem is virtually impossible
 Mathematical analysis though possible is so complicated and time
consuming that the model analysis offers a short cut
The importance of the problem is such that verification of mathematical
analysis by an actual test is essential

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21.Draw the ILD for bending moment at any section x of a simply
supported beam and mark the ordinates.
22.Sketch the ILD for the normal thrust at a section X of a symmetric
three hinged parabolic arch.
𝑁𝑜𝑟𝑚𝑎𝑙 𝑡ℎ𝑟𝑢𝑠𝑡 ( 𝑁𝑥 ) = 𝑉𝑥 sin 𝜃 + 𝐻 cos 𝜃
Where, θ is the inclination of tangent at x
23.Define Maxwell’s reciprocal theorem or Bette’s theorem.
The work done by the first system of loads due to displacements
caused by a second system of loads equals to the work done by the second
system of loads due to displacements caused by the first system of loads.

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24.Define similitude.
Similitude means similarity between two objects namely the model
and the prototype with regard to their physical characteristics.
 Geometric similitude is similarity of form
 Kinematic similitude is similarity of motion
 Dynamic and / or mechanical similitude is similarity of masses and /
or forces
25.State the principle on which indirect model analysis is based.
The indirect model analysis is based on the Muller Breslau principle.
Muller Breslau principle has led to a simple method of using models
of structures to get the influence lines for force quantities like bending
moments, support moments, reactions, internal shears, thrusts, etc.
To get the influence line for any force quantity,
 Remove the restraint due to the force
 Apply a unit displacement in the direction of the force
 Plot the resulting displacement diagram
This diagram is the influence line for the force.
26.What is the principle of dimensional similarity?
Dimensional similarity means geometric similarity of form. This
means that all homologous dimensions of prototype and model must be in
some constant ratio.
27.What is Begg’s deformeter?
Begg’s deformeter is a device to carry out indirect model analysis on
structures. It has the facility to apply displacement corresponding to

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moment, shear or thrust at any desired points in the model. In addition, it
provides facility to measure accurately the consequent displacements all
over the model.
28.Name any four model making materials.
Perspex, Plexiglas, acrylic, plywood, sheet araldite and Bakelite are
some of the model making materials. Micro – concrete, mortar and plaster
of Paris can also be used for models.
29.What is dummy length in models tested with Begg’s deformeter?
Dummy length is the additional length (of about 10 to 12mm) left at
the extremities of the model to enable any desired connection to be made
with the gauges.
30.What are the three types of connections possible with the model used
with Begg’s deformeter?
 Hinged connection
 Fixed connection
 Floating connection
31.What are the uses of a micrometer microscope in model analysis with
Begg’s deformeter?
Micrometer microscope is an instrument used to measure the
displacement of any point in the x and y directions of a model during tests
with Begg’s deformeter.

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UNIT – 4
1. What is meant by yield stress?
Most structural materials have under gradually increasing strain an
elastic and plastic stage.
Plastic stage mark the stage at which increased strain does not
produce in stress.
The stress consequent to stretching stabilize at a value is known as
yield stress.
2. What are the basic conditions to be satisfied for plastic analysis?
 Mechanism condition
The ultimate load or collapse load is reached when a mechanism is
formed. There must, however, be just enough plastic hinges that a
mechanism is formed.
 Equilibrium condition
The summation of the forces and moments acting on a structure must
be equal to zero.
 Plastic moment condition
The bending moment anywhere must not exceed the fully plastic
moment.
3. What are the basic conditions to be satisfied for elastic analysis?
 Continuity equation
 Equilibrium condition

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 Limiting stress condition
4. List out the shape factors for the rectangular, triangular, circular and
diamond section.
 Rectangular section, S = 1.5
 Triangular section, S = 2.346
 Circular section, S = 1.697
 Diamond section, S = 2
5. Mention the types of frames.
 Symmetric frames
 Un-symmetric frames
6. What are symmetric frames and how they analyzed?
Symmetric frames are frames having the same support conditions,
lengths and loading conditions on the columns and beams of the frame.
Symmetric frames can be analyzed by,
 Beam mechanism
 Column mechanism
7. What are unsymmetrical frames and how they analyzed?
Un–symmetric frames have different support conditions, lengths and
loading conditions on its columns and beams. These frames can be analyzed
by,
 Beam mechanism
 Panel or sway mechanism

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 Combined mechanism
8. What is the effect of axial force on plastic moment when a section is
subjected to axial force?
Thus far the cross sections considered are only carrying moment. In
the presence of axial force, clearly some material must be given over to
carry the axial force and so is not available to carry moment, reducing the
capacity of the section. Further, it should be apparent that the moment
capacity of the section therefore depends on the amount of axial load being
carried. Considering a compression load as positive, more of the section
will be in compression and so the neutral axis will drop.
9. Draw a stress strain curve for a perfectly plastic material.
10.What is a mechanism?
When an n-degree indeterminate structure develops n plastic hinges, it
becomes determinate and the formation of an additional hinge will reduce
the structure to a mechanism. Once a structure becomes a mechanism, it will
collapse.

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11.What are the different types of mechanism?
 Beam mechanism
 Panel or sway mechanism
 Gable mechanism
 Combined or composite mechanism
12.What are the methods of plastic analysis?
 Static method (Lower Bound Theorem)
 Kinematic method (Upper Bound Theorem)
13.State the lower bound theorem or static theorem of plastic collapse.
Lower bound theory states that the collapse load is determined by
assuming suitable moment distribution diagram.
The moment distribution diagram is drawn in such a way that the
conditions of equilibrium are satisfied.
14.State upper bound theorem of plasticity.
Upper bound theory states that of all the assumed mechanisms the
exact collapse mechanism is that which requires a minimum load.
15.Define shape factor.
The shape factor (S) is defined as the ration of the plastic moment of a
section to the yielded moment of the section. The shape factor is also the
ratio of plastic modulus of the section to the elastic modulus of the section.
𝑆 =
𝑀 𝑃
𝑀 𝑦
=
𝑍 𝑃
𝑍

P a g e | 226
16.Define the term load factor.
Plastic analysis can tell us at what load (or load combination) a
structure will collapse. This will help us design structures for a desired
safety factor on limiting loads. This safety factor is usually termed as load
factor. It is also defined as the ratio of collapse load to the working load.
𝜆 =
𝑊𝑐
𝑊
17.Define collapse load.
The load that causes the (n + 1) the hinge to form a mechanism is
called collapse load where n is the degree of statically indeterminacy. Once
the structure becomes a mechanism it will collapse.
18.What are the limitations of load factor?
 The analysis procedure does not give us any clue if at a load𝑊𝑢 ÷
𝑙𝑜𝑎𝑑 𝑓𝑎𝑐𝑡𝑜𝑟, the structure behaves well, meaning, whether the
stresses are within limit. So we have to check the stresses at crucial
points by conventional elastic method
 The assumption of monotonic increase in loading is a simplistic,
native assumption. But it is convenient and so we stick to it.
19.Explain the term plastic hinge.
When a section attains full plastic moment Mp, it acts as hinge which
is called a plastic hinge.
It is defined as the yielded zone due to bending at which large
rotations can occur with a constant value of plastic moment Mp.

P a g e | 227
20.What is plastic moment?
When the moment is further increased, there will be a stage at which
all fibres from top to bottom of the section will completely yield and the
section would not be able to take any further moment. The resisting moment
corresponding to this fully plastic stage is called the plastic moment Mp.
21.Define plastic modulus of a section.
The plastic modulus of a section is the first moment of the area above
and below the equal area axis. It is the resisting modulus of a fully
plasticized section.
𝑍 𝑝 =
𝐴
2
( 𝑦1 + 𝑦2 )
22.List the possible locations of plastic hinges in a structure.
 Plastic hinges occurs under the loads
 Plastic hinges occurs at joints
23.Define moment redistribution.
Moment redistribution refers to the behavior of statically
indeterminate structures that are not completely elastic, but have some
reserve plastic capacity.
24.How is the shape factor for a hollow circular section related to the shape
factor of an ordinary circular section?
The shape factor of the hollow circular section = a factor K * shape
factor of ordinary circular section.

P a g e | 228
𝑆ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 ℎ𝑜𝑙𝑙𝑜𝑤 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑠𝑒𝑐𝑡𝑖𝑜𝑛
= 𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 ×
( 1 − 𝑐3
)
( 1 − 𝑐4 )
25.Give the governing equation for bending.
The governing equation for bending is given by,
𝑀
𝐼
=
𝜎
𝑦
Where,
M = bending moment
I = moment of inertia
σ = stress
y = CG distance
26.What is meant by plastic analysis of structure?
The analysis of beams and structures made of such flexural members
are called plastic analysis of structure.
27.What is the difference between plastic hinge and mechanical hinge?
Plastic hinges modify the behavior of structures in the same way as
mechanical hinges. The only difference is that plastic hinges permit rotation
with a constant resisting moment equal to the plastic moment Mp. At
mechanical hinges, the resisting moment is equal to zero.
28.List out the assumptions made for plastic analysis.
 Plane transverse sections remain plane and normal to the longitudinal
axis before and after bending

P a g e | 229
 Effect of shear is neglected
 The material is homogeneous and isotropic both in the elastic and
plastic state
 Modulus of elasticity has the same value both in tension and
compression
 There is no resultant axial force in the beam
 The cross section of the beam is symmetrical about an axis through its
centroid and parallel to the plane of bending
UNIT – 5
1. State the principle of super position of forces?
When a body is subjected to a number of external forces, the forces
are split up, and their effects are considered on individual sections. The
resulting deformation, of the body is equal to the algebraic sum of the
deformations of the individual sections. Such a principle of finding the
resultant deformation is called the principle of superposition.
2. Define statically determinate structure.
If the conditions of equilibrium (i.e.) ΣH=0, ΣV=0 and ΣM=0 alone
are sufficient to find either external reactions or internal forces in a structure,
the structure is called a statically determinate structure.
3. Define statically indeterminate structure.
If the conditions of equilibrium (i.e.) ΣH=0, ΣV=0 and ΣM=0 alone
are not sufficient to find either external reactions or internal forces in a
structure, the structure is called a statically indeterminate structure.

P a g e | 230
4. Define compatibility in force method of analysis.
Compatibility is defined as the continuity condition on the
displacements of the structure after external loads are applied to the
structure.
5. Differentiate the statically determinate structures and statically
indeterminate structures.
S.
NO
STATICALLY
DETERMINATE
STRUCTURES
STATICALLY
INDETERMINATE
STRUCTURES
1. Conditions of equilibrium are
sufficient to analyze the structure
Conditions of equilibrium are
insufficient to analyze the
structure
2. Bending moment and shear
force is independent of
material and cross sectional area
Bending moment and shear
force is dependent of material
and independent of cross
sectional area
3. No stresses are caused due to
temperature change and lack of
fit
Stresses are caused due to
temperature change and lack of
fit
4. Extra conditions like
compatibility of displacements
are not required to analyze the
structure.
Extra conditions like
compatibility of displacements
are required to analyze the
structure along with the
equilibrium equations.

P a g e | 231
6. Write down the rotation matrix for 2D truss element.
In linear algebra, a rotation matrix is a matrix that is used to perform a
rotation in Euclidean space. For example the matrix
7. Write down the compatibility equation used in flexibility matrix
method.
𝑅 × ∆ = {𝐹}
8. Define force transformation matrix.
The connectivity matrix which relates the internal forces Q and the
external forces R is known as the force transformation matrix. Writing it in a
matrix form,
{Q} = [b] {R}
Where,
Q = member force matrix / vector
b = force transformation matrix
R = external force / load matrix / vector
9. What is transformation matrix?
If, A and B are the matrices of two linear transformations, then the
effect of applying first A and then B to a vector x is given by: (This is called
the associative property.) In other words, the matrix of the combined
transformation A followed by B is simply the product of the individual
matrices.

P a g e | 232
10.Write down the stiffness matrix for 2D beam element.
The stiffness matrix for a 2 D beam element is given by,
11.Describe the uses of force method. What are the basic steps in the force
method to find internal forces in statically indeterminate structure?
With the advent of computers, matrix methods of solving structures
have become very popular. The behavior of a structure can largely be
defined by defining the force – displacement relationship in the form of a
matrix.
Steps:
 Applying a force on the structure
 Working out the internal forces and moments
 Computing displacement (and rotations) at specific locations making
use of the values in the above step.
12.What are the basic unknowns in stiffness matrix method?
In the stiffness matrix method nodal displacements are treated as the
basic unknowns for the solution of indeterminate structures.
13.Define stiffness coefficient kij.
Stiffness coefficient ‘kij’ is defined as the force developed at joint ‘i’
due to unit displacement at joint ‘j’ while all other joints are fixed.

P a g e | 233
14.What is the basic aim of the stiffness method?
The aim of the stiffness method is to evaluate the values of
generalized coordinates ‘r’ knowing the structure stiffness matrix ‘k’ and
nodal loads ‘R’ through the structure equilibrium equation.
{R} = [K] {r}
15.What is the displacement transformation matrix?
The connectivity matrix which relates the internal displacement ‘q’
and the external displacement ‘r’ is known as the displacement
transformation matrix ‘a’.
{q} = [a] {r}
16.How are the basic equations of stiffness matrix obtained?
The basic equations of stiffness matrix are obtained as:
 Equilibrium forces
 Compatibility of displacements
 Force displacement relationships
17.What is the equilibrium conditions used in the stiffness method?
The external loads and the internal member forces must be in
equilibrium at the nodal points.
18.What is meant by generalized coordinates?
For specifying a configuration of a system, a certain minimum no of
independent coordinates are necessary. The least no of independent
coordinates that are needed to specify the configuration is known as
generalized coordinates.

P a g e | 234
19.Write the element stiffness for a truss element.
The element stiffness matrix for a truss element is given by,
20.Write about the force displacement relationship.
The relationship of each element must satisfy the stress-strain
relationship of the element material.
21.What is the compatibility condition used in the flexibility method?
The deformed elements fit together at nodal points.
22.Write the element stiffness matrix for a beam element.
The element stiffness matrix for a beam element is given by,
23.Compare flexibility method and stiffness method.
FLEXIBILITY MATRIX
METHOD
STIFFNESS MATRIX METHOD
The redundant forces are treated as
basic unknowns.
The joint displacements are treated
as basic unknowns
The number of equations involved is
equal to the degree of static
indeterminacy of the structure.
The number of displacements
involved is equal to the no of
degrees of freedom of
the structure

P a g e | 235
The method is the generalization of
consistent deformation method.
The method is the generalization of
the slope deflection method.
Different procedures are used for
determinate and indeterminate
structures
The same procedure is used for both
determinate and indeterminate
structures.
24.Is it possible to develop the flexibility matrix for an unstable structure?
In order to develop the flexibility matrix for a structure, it has to be
stable and determinate.
25.What is the relationship between flexibility and stiffness matrix?
The element stiffness matrix ‘k’ is the inverse of the element
flexibility matrix ‘f’ and is given by f = 1/k or k = 1/f.
26.What are the types of structures that can be solved using stiffness
matrix method?
Structures such as simply supported, fixed beams and portal frames
can be solved using stiffness matrix method.
27.Give the formula for the size of the global stiffness matrix.
The size of the Global Stiffness Matrix (GSM) = number of nodes *
degrees of freedom per node.
28.List the properties of the stiffness matrix.
 It is a square matrix and always it should be a square matrix.
 It is a symmetric matrix
[𝑘] = [𝑘] 𝑇

P a g e | 236
 The sum of elements in any column must be equal to zero.
 It is an unstable element therefore the determinant is equal to zero.
 The order of stiffness is equal to the number of co – ordinates.
29.List the properties of flexibility matrix.
 Flexibility matrix is a square matrix of order nd nd
 Flexibility matrix is a symmetrical matrix
 Elements of flexibility matrix may be positive or negative except
leading diagonal element which is always positive
 Elements of flexibility matrix are displacements and they can be
computed only if the structure is stable. If structure is unstable
internally or externally, then displacements are indefinitely large and
flexibility matrix does not exist.
30.Why the stiffness matrix method is also called equilibrium method or
displacement method?
Stiffness method is based on the superposition of displacements and
hence is also known as the displacement method. And since it leads to the
equilibrium equations the method is also known as equilibrium method.
31.Define a primary structure.
A structure formed by the removing the excess or redundant restraints from
an indeterminate structure making it statically determinate is called primary
structure. This is required for solving indeterminate structures by flexibility
matrix method.

P a g e | 237
32.If the flexibility matrix is given as[𝑭] = [
𝟐 −𝟏
−𝟏 𝟒
]. Write the
corresponding stiffness matrix.
𝑆𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠 𝑚𝑎𝑡𝑟𝑖𝑥 =
1
𝐹𝑙𝑒𝑥𝑖𝑏𝑖𝑙𝑖𝑡𝑦 𝑚𝑎𝑡𝑟𝑖𝑥
(i.e.) [𝐾] = [𝐹]−1
33.Define degree of kinematic indeterminacy (or) Degree Of Freedom.
It is defined as the least no of independent displacements required to
define the deformed shape of a structure. There are two types of DOF
 Joint type DOF
 Nodal type DOF
34.Briefly explain the two types of DOF.
 Joint type DOF
This includes the DOF at the point where moment of inertia changes,
hinge and roller support, and junction of two or more members.
 Nodal type DOF
This includes the DOF at the point of application of concentrated load
or moment, at a section where moment of inertia changes, hinge
support, roller support and junction of two or more members.

P a g e | 238
35.Name any two force methods to analyze the statically indeterminate
structures.
 Column analogy method
 Flexibility matrix method
 Method of consistent deformation
 Theorem of least work
36.What are the different methods used to analyze indeterminate
structures?
 Finite element method
 Stiffness matrix method
37.Write the formulae for degree of indeterminancy.
 Two dimensional in jointed truss (2D truss)
𝑖 = (𝑚 + 𝑟) − 2𝑗
 Two dimensional rigid frames/plane rigid frames (2D frame)
𝑖 = (3𝑚 + 𝑟) − 3𝑗
 Three dimensional space truss (3D truss)
𝑖 = (𝑚 + 𝑟) − 3𝑗
 Three dimensional space frame (3D frame)
𝑖 = (6𝑚 + 𝑟) − 6𝑗
Where,
m = number of members
r = number of reactions

P a g e | 239
j = number of joints
38.Write the element flexibility matrix for a truss member.
The element flexibility matrix (f) for a truss member is given by,
39.Briefly mention the two types of matrix methods of analysis of
indeterminate structures.
It is defined as the deformation produced for unit load. It is denoted
by the symbols[𝑎] 𝑜𝑟 [𝑓] 𝑜𝑟 [𝛼]. This method is also called the force
method in which the forces in the structure are treated as unknowns.
The no of equations involved is equal to the degree of static
indeterminacy of the structure.
 Stiffness matrix method
It is defined as the force required for unit deformations. It is denoted
by the symbol[𝑘]. This is also called the displacement method in
which the displacements that occur in the structure are treated as
unknowns. The no of displacements involved is equal to the no of
degrees of freedom of the structure.

P a g e | 240
40.Define flexibility influence coefficient.
Flexibility influence coefficient (fij) is defined as the displacement at
joint ‘i’ due to a unit load at joint ‘j’, while all other joints are not load.
41.Define element co – ordinates.
Each element having a displacement along two directions (x and y) is
said to be a element coordinates.
42.Define global co – ordinates.
For the whole structure having a displacement along the two
directions (x and y) is said to be a global coordinates.
43.Write the element flexibility matrix for a beam element.
The element flexibility matrix (f) for a beam element is given by,

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