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, 𝑀𝑥 = 𝜇𝑥 − 𝐻𝑦

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.

𝐵𝑀𝑥 = 𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑂2𝑂3 × 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 ( 𝑅𝑥 ) = 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’.

𝑀𝑥 = 𝜇𝑥 − 𝐻𝑦

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

𝜕𝑉𝐴= ∆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 𝑑𝑥𝑙

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 ( 𝑅𝑥 ) = 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

Column 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

Column 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 = 𝑀𝑃

𝑀𝑦=

𝑍𝑃

𝑍

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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.

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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.

𝑍𝑝 = 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.

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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

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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.

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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.

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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.

https://en.wikipedia.org/wiki/Linear_algebra

https://en.wikipedia.org/wiki/Matrix_%28mathematics%29

https://en.wikipedia.org/wiki/Rotation_%28mathematics%29

https://en.wikipedia.org/wiki/Euclidean_space

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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.

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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.

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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

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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

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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.

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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.

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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

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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.

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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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