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AC/AA 1.2: Advance Strength of Materials

Multiple Choice Type Questions

(1) Castigliano’s theorem is applicable to determine the deformation for

[a] both determinate and indeterminate structures [b] only determinate structures

[c] unstable structures [d] all of these

(2) The Degree of Static Indeterminacy for the given fixed beam in figure.1 with two

intermediate hinges is

Figure-1

[a] 0 [b] 1

[c] 2 [d] None of these

(3) What will be the deflection of a propped cantilever beam at its propped end if any of the

supports is not yielding or undergoes rigid body translation?

[a] 0 [b] infinity

[c] can’t be determined due

to data inadequacy

[d] same as deflection at Mid length of the propped cantilever

beam

(4) Volumetric strain is

[a] sum of three elastic strains [b] cubical dilatation

[c] Change in volume/unit Volume [d] all of these

(5) Among the following who represents a scalar or zero order tensor

[a] Mass [b] Body forces

[c] Surface Traction [d] all of these

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(6) In The Plane stress state the stress(es) is/are zero

[a] σz [b] σz , τxz, τyz

[c] τxz, τyz [d] None of these

(7) A sphere under diametral compression or a cone under a load at the apex is the case of

[a] Plane stress [b] Un-symmetrical bending

[c] Axisymmetric case [d] Plane strain

(8) For the pure shear case where σx = σy = σz =0 in a particular coordinate system the value

of first invariant, I1 is

[a] 0 [b] 1

[c] 1/2 [d] None of the above

(9) In order to apply St. Venant’s principle following points/point are/is important

[a] The actual loading & the loading

used to compute stresses must be

statically equivalent

[b] Stresses can not be computed in the immediate

vicinity of the point of application of loads

[c] All of the above [d] None of the above

(10) Shown in Fig.2 given below is an element of an elastic body, which is subjected to pure

shearing stresses xyτ. The absolute value of the magnitude of the principle stresses is

Figure-2

(a) zero (b) 2

xyτ

(c) xyτ (d) xyτ

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(11) The strain energy stored in a curved member subjected to load w as shown in Fig. 5

Figure-5

(a) EI

Rw

24

32π

(b) EI

Rw

16

32π

(c) EI

Rw

8

32π

(d) EI

Rw

4

32π

(12) Lame’s ellipsoid is also known as

(a) Deformation ellipsoid (b) Strain ellipsoid

(c) Stress ellipsoid (d) volume ellipsoid

(13) A continuous beam 12 m long, supported over two spans 6m each, carries a concentrated load

of 40 kN each at the centre of each span. The bending moment at the centre of two supports is

[a] 30 kNm [b] 45 kNm

[c] 90 kNm [d] 150 kNm

(14) If in a pin-jointed plane frame (m + r) > 2j then the frame is (a) stable and statically determinate (b) stable and statically indeterminate

(c) unstable (d) none of the above

where m is number of members, r is the reaction components and j is number of joints

(15) A pin-jointed plane frame is unstable if (a) (m + r) < 2j (b) m + r = 2j

(c) (m + r) > 2j (d) none of the above

(16) A rigid jointed plane frame is stable and statically determinate if (a) (m + r) = 2j (b) (m + r) = 3j

(c) (3m + r) = 3j (d) (m + 3r) = 3j

where m is number of member, r is reaction components and j is number of joints

(17) The carryover factor in a prismatic member whose far end is hinged is

(a) 0 (b) 2

1

(c) 3/4 (d) 1

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(18) The moment required to rotate the near end of a prismatic beam through a unit angle

without translation, the far end being simply supported, is given by

(a) L

EI3 (b)

L

EI4 (c)

L

EI2 (d)

L

EI

where EI is flexural rigidity and L is span of beam.

(19) Which of the following is matched correctly

(Real beam) (Conjugate beam)

(a) Free end (a) Free end of cantilever

(b) Fixed end

(b) Continuous intermediate

(c) Simple support at end

(c) Simple support at end

(d) Internal hinge (d) Fixed support

(20) The M/EI diagram of a real beam becomes the (a) Deflection diagram of the conjugate beam (b) Load diagram of the conjugate beam

(c) Shear force diagram of the conjugate beam (d) none of the above

(21) Slope deflection method is

(a) equilibrium method (b) deflection method

(c) stiffness coefficient method (d) all of the above

(22) Which diagram is/are true to the loading diagram as shown in the figure 7 .

Figure-7

(i) Free – body diagram :

(ii) Shear – Force diagram

(iii ) Bending Moment Diagram :

(a) only (i) is true

(b) only (ii)&(iii) are true

(c) (i)&(iii) are true

(d) All are true.

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(23) Shear flow is

[a] shear stress per unit cross sectional area [b] shear force per unit length

[c] shear force per unit volume of the shaft [d] none of these

(24) Given following statements:

(i) If originally plane sections remained plane after twist , the torsional rigidity can be

calculated simply as the product of the polar moment of inertia multiplied by the shear

modulus.

(ii) When the resultant of forces act away from the shear centre axis , then the beam will

not only bend but also twist.

[a] Statement (i) is correct only [b] Statement (ii) is correct only

[c] both the statements are correct [d] none of these are correct

(25) Given following statements:

(i) Non-Uniform Torsional resistance at any non-circular section is the sum of St.

Venant’s torsion and warping torsion.

(ii) In case of a Non-circular section , EГ is termed as the warping rigidity of the

section, analogous to GJ, the St. Venant’s torsional stiffness..

[a] Statement (i) is correct only [b] Statement (ii) is correct only

[c] both the statements are correct [d] none of these are correct

(26) When the torsional rigidity(GJ) is very large compared to the warping rigidity,EГ, then the

section will effectively be in

[a] Uniform torsion [b] Non-uniform section

[c] Warping torsion [d] Total torsion

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(27) Given following statements:

(i) Uniform Torsion ( St. Venant’s torsion ) applied to a beam would cause a twist.

(ii) Non-uniform torsion will cause both twisting and warping of the cross section.

[a] Statement (i) is false [b] Statement (ii) is false

[c] both the statements are false [d] both (i) & (ii) are true.

(28) A rectangular tube has outside dimensions 62mm x 42mm and has a wall 2 mm

thick.What will be the maximum shear stress when a torque of 2400 N-m is applied.

[a] 200 MPa [b] 250 MPa

[c] Data inadequate [d] 500 MPa

(29) What will be the maximum shear stress for the solid rectangular section having

dimensions 100mm x 10 mm when a torque of 5 Nm is applied ?

[a] 1.5 MPa [b] 1.0MPa

[c] 0.75MPa [d] None of these

(30) Given following statements in relation to torsion neglecting warping

(i) The Polar Moment of inertia for solid circular section is πr4/2 , where r is the radius

(ii) The Polar Moment of inertia for solid rectangular section is ht3/3 for h/t greater than

equal to 10.

[a] Statement (i) is false [b] Statement (ii) is false

[c] both the statements are false [d] both (i) & (ii) are true.

(31) The theory of curved beam was postulated by

[a] Rankline [b] Mohr

[c] Castigliano [d] Winkler- Bach

(32) In curved beams the distribution of bending stresses is

[a]linear [b] parabolic

[c] uniform [d] hyperbolic

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(33) The neutral axis in curved beams

[a]lies at the top of beam [b] lies at the bottom of beam

[c] does not coincides with geometric axis of the section [d] coincides with the geometric axis

(34) Which of the following assumption is made in the analysis of curved beam theory

[a] limit of proportionality is not exceeded [b] radial strain is negligible

[c] the material is considered isotropic and obeys Hooke’s law [d] all of these

(35) For a crane hook the most suitable section is

[a] triangular [b] trapezoidal

[c] circular [d] rectangular

(36) Analysis of pure bending can be done when

[a] members subjected to bending couples acting

in a plane of symmetry

[b]members remain symmetric and bend in a

plane of symmetry

[c] the neutral axis of the cross section coincides

with the axis of couple

[d] All of these

(37) Analysis of Unsymmetrical bending can be done when

[a] members subjected to bending couples do not act in a

plane of symmetry

[b]members will not bend in a plane of

the couples

[c] the neutral axis of the cross section will not coincides

with the axis of couple

[d] All of these

(38) A cast iron machine part is subjected to a couple of 3 KN-m. Knowing the EI=300 KN-m2 and

neglecting the effect of fillet what will be the radius of curvature ?

[a] 100m [b] 0.01m

[c] 50m [d] 10 m

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(39) Which is true among the following statements made in favour of “ shear centre” ?

[a] It is the point in the cross-section through which the

lateral (or transverse) loads must pass to produce bending

without twisting.

[b] It is also the centre of rotation,

when only pure torque is applied.

[c] The shear centre and the centroid of the cross section

will coincide, when section has two axes of symmetry and

The shear centre will be on the axis of symmetry, when the

cross section has one axis of symmetry.

[d] All are true.

(40) The shear centre for the section shown in figure 8 is at point

Figure-8

[a] A [b] C

[c] D [d] B

(41) Given following statements in relation shear centre

(i) Depending on the beam's cross-sectional shape along its length, the

location of shear center may vary from section to section.

(ii) A line connecting all the shear centers is called the elastic axis of the beam.

[a] Statement (i) is false (ii) is true [b] Statement (ii) is false (i)is true

[c] both the statements are false [d] both (i) & (ii) are true.

(42) Energy stored in a closed coil helical spring when subjected to an axial twist is given by

[a] (σb2/6E) x volume of spring [b] (σb

2/8E) x volume of spring

[c] (σb2/4E) x volume of spring [d] (σb

2/2E) x volume of spring

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(43) Two springs of stiffness k1 and k2 respectively are connected in series, the stiffness of

composite spring will be given by

[a] k=k1+k2 [b] k=k1k2

[c] k=k1k2/ k1+k2 [d] k=k1+k2/ k1k2

(44) The resilience of flat spring is given by

[a] σmax/24E [b] σ2

max/24E

[c] σ2

max/12E [d] σ2

max/8E

(45) A closed coil helical spring absorbs 80 N of energy while extending by 4 mm. The

stiffness of the spring is

[a] 5 N/mm [b] 10 N/mm

[c] 16 N/mm [d] 20 N/mm

(46) The deformation of a spring produced by a unit load is called

(a) Stiffness (b) flexibility

(c) influence coefficient (d) none

(47) Shear stress in a closed coiled spring under axial load is given by

(a) 3

8

d

WD

π (b)

3

8

d

WR

π

(c) 3

8

R

WD

π (d)

3

8

d

WR

π

(48) In the case of rotating disc of uniform strength which of the following statements is

correct

[a] circumferential stress is constant [b] radial stress is constant

[c] circumferential and radial stress are equal to each other

and are constant

[d] none of these

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(49) In the case of rotating long cylinder

[a] shear stress is zero at central cross

sectional plane

[b] shear stress is maximum at central cross

sectional plane

[c] shear stress is zero at end cross sectional

planes

[d] none of these

(50) A disc of uniform strength must have

[a] constant thickness [b] varying thickness

[c] varying cross sectional area [d] none of these

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True/False Type Questions

1. As per Castigliano’s First Theorem ,”for a linearly elastic structure, the derivative of the strain energy of a structure with respect to a load gives the

displacement of the load in all directions.”

2. The bulk modulus (K) describes volumetric elasticity, or the tendency of an

object to deform in all directions when uniformly loaded in all directions.

3. Dilation of material is change in area per unit area.

4. Shear stresses tend to deform the material without changing its volume, and are

resisted by the body's shear modulus.

5. A rigid jointed plane frame is stable and statically determinate of 3m+r=3j.

The shear force on a deflected beam is given by 3

3

1dx

ydEV =

6. The Bending Moment on a deflected beam is given by M = -EI (d2y/dx2 ).

7. The slope of the normal stress-strain graph is equal to the bulk modulus of

elasticity.

8. The following beam(as shown in figure-3) is statically indeterminate to the second degree.

Figure-3

9. The slope of a cantilever beam at the free end will be zero if the beam is loaded

with uniformly distributed load throughout.

10.The point of transition on the elastic curve into reverse curvature is called point

of contraflexure.

11. If the member is not allowed to warp freely, the applied torque is resisted by

St. Venant's torsional shear stress and warping torsion and this behaviour is

called non-uniform torsion.

12. If the section is doubly symmetric , the centre of flexure coincides with the centroid of the section.

13. In uniaxial or symmetrical bending the Neutral axis is not normal to the

plane of loading.

14. Symmetric bending: Where the member possesses at least one plane of

symmetry and is subjected to couples acting in that plane.

15. A neutral surface must exist in the member in the analysis of curved beam.

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16.In case of curved beam analysis σ = My/I no longer applies.

17.If there is also an axial force present, Winkler’s formula can be written as

follows.

Where, Z=Curved beam factor

A = Cross-sectional area

R = Radius of curvature to the centroidal axis

y = Distance from centroidal axis to point of interest

18.The normal stresses in a curved beam due to bending is same as normal

stress due to bending of a straight beam.

19.Whal’s correction factor is applied to accommodate the effect of

curvature of spring and direct shear stresses.

20.Leaf springs are also called carriage spring.

21.The wire of open coiled helical spring subjected to axial force P, is

subjected to direct shear only.

22.Shocker used in car is laminated spring of high stiffness.

23.Circumferential stresses in rotating disc or cylinders are compressive in

nature.

24.Circumferential stress on the outer edge of rotating disc is zero.

25.The direction of axis in case of rotating disc is the direction of zero

principal stress

+++=

)(1

yRZ

y

AR

M

A

Pθ

σ

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Short Answer Type Question

1. What are the conditions required for applicability of principle of

superposition?

2. In 3-D stress the three principal stresses are obtained as the three real

roots of the equation:

Where I1,I2 and I3 are called stress invariants. Why the name called

stress invariant?

3. State “ St. Venant’s Principle”.

4. State Castigliano’s first Theorem .

5. Define Plane stress. Give an example

6. What is the difference between Direct stress and shear stress with

respect to volume of a body?

What is the significance of Airy’s Stress function ?

7. What is Clapeyron’s Theorem of three moments for a continuous beam ?

8. State the assumptions of Euler-Bernoulli’s beam bending theory .

9. Using strain energy theorms / Castigliano’s theorem calculate the

deformation at end A of the cantilever beam.

10.State Mohr’s Moment-area theorem .

11.Find out the degree of Static indeterminacy of the 2-D rigid jointed

frame as shown below.

12.What is Prandtl’s Stress function ?What is wraping ?When it is considered

along with twisting ?

13.Using Membrane Analogy find out the maximum shear stress in thin-

walled cross section.

14.The annualar ring having inner diameter of 2.3 m and outer diameter of

2.5m subjected to Torque of 400 Kg.cm. Find out the average shear

stress in thin-walled analysis.

15.What do you mean by “Shear Centre”?

16.What do you mean by unsymmetrical bending?

17.Where the shear center lies for the beam having two axes of symmetry?

18.What will be the strain energy stored in a spring material due to bending.

Also write the stiffness for a spring .

3 2

1 2 3 0I I Iσ σ σ− + − =

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19.Write down the expressions for bending stress and energy stored in a

Helical springs subjected to 'axial twist'.

20.Write down the expressions for shear stress and angle of twist for a

closed coil helical springs subjected to 'axial load'.

21.Write the expression for bending of curved beams as per Winkler-Bach formula. State the nature of normal stress variation across the depth.

22.What are the assumptions for flexure in curved beams to determine

distribution of stress?

23.Write the expression for the distance of the centroidal axis from neutral

axis in case of a curved beam with triangular cross section.

24.State the expressions for the circumferential and radial stresses in a solid cylinder.

25.State the expressions for the radial stresses in a hollow cylinder rotating

with uniform speed .

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Long Answer Type Questions

1. A structure is shown in the figure (No.1)below. Assuming the member to be of

uniform cross section through out find the strain energy stored by the structure and

determine the vertical deflection of end A.

figure (No.1)

2. The framework shown in figure No.2 below is pin-jointed to the ground at A and D and is

loaded along AB with a distributed load w. If the flexural rigidity EI is constant throughout,

obtain expressions for the reaction at A and D.

Figure (No.2)

3. The following state of stress exists at a point P

In the direction PQ having direction cosines nx= 0.6 , ny = 0 and nz =0.8.

(a) determine εPQ

(b) cubical dilatation at point P

4. A two span continuous beam ABC fixed at the ends is loaded as shown in figure

No.3. Find :

i. Moments at the supports

ii. Reactions at the supports

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Draw BM and SF diagrams also.

Figure (No.3)

5. A propped cantilever beam AB supports a concentrated load P acting at the midpoint

C (see figure No.4).Beginning with the second-order differential equation of the

deflection curve (the bending-moment equation), determine all reactions of the beam

and draw the shear-force and bending-moment diagrams for the entire beam. Also,

obtain the equations of the deflection curves for both halves of the beam, and draw

the deflection curve for the entire beam.

figure (No.4)

6. A continuous beam ABCDE carrying a udl of w/unit length rests on three supports

B,C and D, all at the same level. It has two equal overhangs of length L0 on either

sides. Assuming EI constant, find the ratio of L0 /L for the three support reactions to

be equal.

7. A rectangular steel shaft 50mmX 25mm is subjected to a torque of 2KNm. Find:

i. Maximum shear stress developed in the shaft.

ii. Angular twist per Meter length. Assume modulus of rigidity = 80GN/m2.

8. A shaft of elliptical section is subjected to torque of 2.5KNm. If the maximum shear

stress in the shaft is not to exceed 80MN/m2 determine:

i. The major and minor axis , if major axis = 1.5 minor axis

ii. The angular twist per metre length. Assume modulus of rigidity = 80GN/m2.

9. An I section with flanges 10 cm X 2 cm and web 28cm X 1 cm is subjected to torque

6 KNm. Find:

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i. Maximum shear stress

ii. Angle of twist per unit length. Assume modulus of rigidity = 80 GN/m2

10. A double celled cross section is shown in figure No.5. When a torque of 5 KNm is

applied find:

i. Shear stress in each part

ii. Angular twist per metre length

Take Shear modulus = 80 GN/m2.

Figure (No.5)

11. Define the term “ Centre of twist “ ?

Determine the position of the shear centre of each of the three thin walled sections shown

below in figure No.6.

Figure (No.6)

12. A channel section has flanges 12 cm x 2 cm and web 16 cm x 1 cm. Determine the shear

centre of the channel which is shown below in Figure No. 7.

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Figure (No.7)

13. Determine the position of the shear centre of the section of a beam shown in Figure No.8.

Figure (No.8)

14. A curved bar of square section, 3 cm sides and mean radius of curvature 4.5 cm is initially

unstressed. If a bending moment of 300 Nm is applied to the bar tending to straighten it,

find the stresses at the inner and outer faces.

15. A crane hook whose horizontal cross-section is trapezoidal, 50 mm wide at the inside and

25 mm wide at the outside, thickness 50 mm, carries a vertical load of 1000 kg whose line

of action is 38 mm from the inside edge of this section. The centre of curvature is 50 mm

from the inside edge. Calculate the maximum tensile and compressive stresses set up.

16. A steel tube having outside diameter 5 cm, bore 3 cm, is bent into a quadrant of 2 m radius.

One end is rigidly attached to a horizontal base plate to which a tangent to that end is

perpendicular, and the free end supports a load of 100 kg . Determine the vertical and

horizontal deflections of the free end under this load. E =208,000 N/mm2.(Refer Figure

No.9)

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Figure No.9

17. A central horizontal section of hook is a symmetrical trapezium 60 mm deep the inner

width being60 mm and the outer being 30 mm . Estimate the extreme intensities of stress

when the hook carries a load of 30 kN. The load line passing 40 mm from the inside edge of

the section and centre of curvature being in the load line. Also plot the stress distribution

across the section. (Refer Figure No. 10 )

Figure (No.10)

18. A curved beam has a T-section as shown in figure No.11 below. The inner radius is 300 mm.

What is the eccentricity of the section?

Figure (No.11)

19. A close-coiled helical spring is to have a stiffness of 900 N/m in compression, with a

maximum load of 45 N and a maximum shearing stress of 120 N/mm2. The "solid" length of

the spring (i.e. coils touching) is 35 mm. Find the wire diameter, mean coil radius, and

number of coils. G =40,000N/mm2.

20. A laminated steel spring, simply supported at the ends and centrally loaded, with a span of

0.75 m, is required to carry a proof load of 750 kg, and the central deflection is not to

exceed 50 mm; the bending stress must not exceed 380 Nlmm2. Plates are available in

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multiples of1 mm for thickness and 4 mm for width. Determine suitable values for width,

thickness and number of plates, and calculate the radius to which the plates should be

formed. Assume width = 12 x thickness. E = 208,000 N/mm2.

21. A laminated spring of the quarter-elliptic type, 0.6 m long, is to provide a static deflection

of 75 mm under an end load of 200 kg. If the leaf material is 60 mm wide and 6 mm thick,

find the number of leaves required and the maximum stress. From what height can the

load be dropped on to the un-deflected spring to cause a maximum stress of 750 N/mm2?

E =208,000 N/mm2.

22. A close-coiled helical spring is made of a round wire having “n” turns and the mean coil

radius R is 5 times the wire diameter. Show that the stuffiness of such a spring is R/n x

constant. Determine the constant when the modulus of rigidity C of the spring wire is

82000 N/ mm2

.

If the above spring is to support a load of 1-2 kN with 120 mm compression and the

maximum shear stress 250N/mm2

.Calculate:

i) Mean radius of the coil

ii) Number of turns.

iii) Weight of the spring.

Assume density of the material to be: 76.5 KN/m3.

23. Determine the intensities of principal stresses in flat steel disc of uniform thickness having

a diameter of 1 m and rotating at 2400 r.p.m. What will be stresses if the disc has a central

hole of 0-2m diameter?

Take Poisson’s ratio = 1/3 and ρ = 7850Kg/m3.

24. A steel disc of uniform thickness and of diameter 400 mm is rotating about its axis at 2000

r.p.m. The density of the material is 7700 kg/m3 and Possion’s ration is 0.3 . Determine the

variations of circumferential and radial stresses.

25. Calculate the principal moment of inertia and direction of principal axes for the cross-

section of a beam show in fig.(No.12) (b) Calculate the polar moment of inertia and radius

of gyration for the cross-section of beam shown in fig.

Figure (No.12)

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