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2007 Jawaharlal Nehru Technological University B.Tech Aeronautical APPROXIMATE METHODS IN STRUCTURAL MECHANICS Question paper

Course: B.Tech Aeronautical

University: Jawaharlal Nehru Technological University

Code No: RR412112 Set No. 1
IV B.Tech I Semester Supplementary Examinations, February 2007
APPROXIMATE METHODS IN STRUCTURAL MECHANICS
(Aeronautical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. Calculate by Rayleigh-Ritz method displacement at node 2 of a vertical bar of
length `L', supported at its upper end, due to its self weight of `w' N/m of length.
Comment on the validity of stress distribution based on this displacement ¯eld. As
shown in ¯gure 1 [16]
Figure 1:
2. (a) A solid circular shaft and a thin tube of the same material and same weight
are subjected to twist. If max. stress in both is the same, what is the ratio of
their strain energies ?
(b) Neglecting strain energy due to axial deformation, calculate strain energy and
displacements at D and J of the 2-cylinder crank shaft, with each segment of
length `L'. As shown in ¯gure 2 [6+10]
Figure 2:
3. (a) For a given column and speci¯ed axial load, which end conditions will result
in maximum Critical buckling load ?
(b) Calculate the critical load for stable equilibrium of the combination of two
rods AB and BC, connected with a spring as shown in ¯gure 3. Assume A
and C remain along a vertical line. [6+10]
1 of 3
Code No: RR412112 Set No. 1
Figure 3:
4. (a) Distinguish between collocation method and Galerkin method.
(b) Calculate the maximum de°ection in a cantilever beam, subjected to udl of
`w' N/m over its length, by 1-term collocation method and 1-term Galerkin
method. Compare the results.As shown in ¯gure 4 [6+10]
Figure 4:
5. (a) Derive ¯nite di®erence expression for d3y/dx3 using
i. forward di®erences,
ii. central di®erences and
iii. backward di®erences. [3 x 3]
(b) Calculate maximum de°ection by ¯nite di®erence method of a simply sup-
ported beam of length L, subjected to a udl of w N/m, choosing four equal
divisions along the beam. Calculate percentage error, compared to the exact
solution.As shown in ¯gure 5 [7]
Figure 5:
6. (a) Explain di®erent methods of applying boundary conditions in ¯nite element
analysis
2 of 3
Code No: RR412112 Set No. 1
Figure 6:
(b) Calculate displacements at nodes 2, 3 and 4 for the spring system shown in
¯gure6, when node 1 is ¯xed and node 5 is given a displacement of 5mm.
[6+10]
7. (a) Sti®ness matrices of four di®erent elements A, B, C and D are given by
[K]A = 104
·
2 ¡2
¡2 2
¸
; [K]B = 104
·
3 ¡3
¡3 3
¸
; [K]C = 104
·
4 ¡4
¡4 4
¸
; [K]D =
104
·
5 ¡5
¡5 5
¸
Assemble element sti®ness matrices if the truss element connectivity is
i. A (1-2) ; B (2-3) ; C (3-4) and D (4-1)
ii. A (1-2) ; B (1-3) ; C (2-3) and D (3-4)
iii. A (1-2) ; B (2-3) ; C (2-4) and D (3-4)
Draw the truss con¯gurations of the above three combinations.
(b) If a truss has 2 members A (1-2) and B (2-3) whose sti®ness matrices are given
by, [K]A = 104
·
2 ¡2
¡2 2
¸
; [K]B = 104
·
4 ¡4
¡4 4
¸
calculate nodal displacements for a load of 100 N applied at node 2 along the
direction 1-2. Assume nodes 1 and 3 are ¯xed. [8+8]
8. Calculate displacements and stress in a triangular plate, ¯xed along edge AC and
subjected to concentrated loads at its end B. Assume E = 70,000 MPa, t = 10 mm
and º = 0.3.As shown in ¯gure7 [16]
Figure 7:
? ? ? ? ?
3 of 3
Code No: RR412112 Set No. 2
IV B.Tech I Semester Supplementary Examinations, February 2007
APPROXIMATE METHODS IN STRUCTURAL MECHANICS
(Aeronautical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. Calculate by Rayleigh-Ritz method displacement at mid-point of a vertical bar,
supported at both ends, due to its self weight. Let the weight be `w' N/m of
length. Comment on the validity of stress distribution based on this displacement
¯eld.As shown in ¯gure 1 [16]
Figure 1:
2. (a) A solid circular shaft and a thin tube of the same material and same weight
are subjected to twist. If max. stress in both is the same, what is the ratio of
their strain energies ?
(b) Neglecting strain energy due to axial deformation, calculate strain energy and
displacement at D of the crank shaft, with AB=BC=CD=DE=EF=FG=L.As
shown in ¯gure 2 [6+10]
Figure 2:
3. Calculate de°ections at points A, B and C of the combination of beams shown below
¯gure 3, if the points B and C do not come in contact with beam DE. Assume I =
0.4m4 and E = 200 GPa. Load applied is `P' at B and C. [16]
4. (a) Discuss basic di®erences between Rayleigh-Ritz method and weighted residual
methods.
1 of 3
Code No: RR412112 Set No. 2
Figure 3:
(b) Indicate any one admissible displacement ¯eld with 1-term and 2-term options,
suitable for the following example, for use in each of the three weighted residual
methods and discuss their validity. As shown in ¯gure 4 [6+10]
Figure 4:
5. (a) Derive ¯nite di®erence expression for d3y/dx3 using
i. forward di®erences,
ii. central di®erences and
iii. backward di®erences. [3 x 3]
(b) Calculate maximum de°ection by ¯nite di®erence method of a simply sup-
ported beam of length L, subjected to a udl of w N/m, choosing four equal
divisions along the beam. Calculate percentage error, compared to the exact
solution.As shown in ¯gure 5 [7]
Figure 5:
6. (a) Explain di®erent methods of applying boundary conditions in ¯nite element
analysis
(b) Calculate displacements at nodes 2, 3 and 4 for the spring system shown in
¯gure6, when node 1 is ¯xed and node 5 is given a displacement of 5mm.
[6+10]
7. (a) What are coupled and uncoupled degrees of freedom ?
2 of 3
Code No: RR412112 Set No. 2
Figure 6:
(b) Calculate the maximum de°ection and reactions in a cantilever beam with a
concentrated load `P' at the free end of the beam, if its free end is supported
by a spring of sti®ness `k'. As shown in ¯gure 7 [4+12]
Figure 7:
8. (a) Derive elasticity matrix of a plane stress element, from basic principles.
(b) In a plane strain problem, if ¾x = 150 N/mm2, ¾y = -100 N/mm2,¾E =
200000N/mm2 and º=0.3, determine the value of stress ¾z.
(c) A long rod is subjected to loading and a temperature increase of 300C. The
total strain at a point is measured to be 1.2£10¡5. If E=200 GPa and ® =
12£10¡6/0C, determine the stress at the point. [8+4+4]
? ? ? ? ?
3 of 3
Code No: RR412112 Set No. 3
IV B.Tech I Semester Supplementary Examinations, February 2007
APPROXIMATE METHODS IN STRUCTURAL MECHANICS
(Aeronautical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. (a) Comment on the statement \All engineering solutions are approximate". [4]
(b) Determine the displacements of nodes 2 and 3 of the spring system shown
below ¯gure 1, by potential energy method. [12]
Figure 1:
2. (a) Explain the terms - resilience, proof resilience and modulus of resilience.
(b) Calculate the reactions at the three supports of a continuous beam using
Castigliano's theorem.As shown in ¯gure 2 [6+10]
Figure 2:
3. (a) What is meant by elastic instability? What factors are responsible for it?
(b) Calculate minimum length `L' for stable equilibrium of the rod shown in ¯gure
3 [6+10]
4. (a) Distinguish between collocation method and Galerkin method.
(b) Calculate the maximum de°ection in a cantilever beam, subjected to udl of
`w' N/m over its length, by 1-term collocation method and 1-term Galerkin
method. Compare the results.As shown in ¯gure 4 [6+10]
5. (a) Suggest boundary conditions in terms of derivatives of normal de°ection `w'
in a square plate with its four edges clamped and subjected to subjected to
uniform bending load over the entire plate.
1 of 3
Code No: RR412112 Set No. 3
Figure 3:
Figure 4:
(b) Calculate de°ections in a propped cantilever of length `L' by ¯nite di®erence
method, subjected to udl of `w' N/m, choosing four equal divisions along the
beam.As shown in ¯gure 5 [4+12]
Figure 5:
6. Determine the nodal displacements, element stresses and reactions of the truss,
shown below ¯gure 6, by FEM. Assume A= 100 mm2, L = 2 m and E = 200GPa
for all the three members. [16]
7. (a) Sti®ness matrices of four di®erent elements A, B, C and D are given by
[K]A = 104
·
2 ¡2
¡2 2
¸
; [K]B = 104
·
3 ¡3
¡3 3
¸
; [K]C = 104
·
4 ¡4
¡4 4
¸
; [K]D =
104
·
5 ¡5
¡5 5
¸
Assemble element sti®ness matrices if the truss element connectivity is
i. A (1-2) ; B (2-3) ; C (3-4) and D (4-1)
ii. A (1-2) ; B (1-3) ; C (2-3) and D (3-4)
iii. A (1-2) ; B (2-3) ; C (2-4) and D (3-4)
Draw the truss con¯gurations of the above three combinations.
(b) If a truss has 2 members A (1-2) and B (2-3) whose sti®ness matrices are given
by, [K]A = 104
·
2 ¡2
¡2 2
¸
; [K]B = 104
·
4 ¡4
¡4 4
¸
2 of 3
Code No: RR412112 Set No. 3
Figure 6:
calculate nodal displacements for a load of 100 N applied at node 2 along the
direction 1-2. Assume nodes 1 and 3 are ¯xed. [8+8]
8. (a) If a displacement ¯eld is described by u = 3x2-2xy+6y2 ; v = 4x2+6xy-8y2
,determine normal strains and shear strain at point x = -1 and y = 1.
(b) What are the di®erent types of elasticity matrices used for 2-D elements ?
Give one example for each type. Derive any one of them from Hooke's law.
[4+12]
? ? ? ? ?
3 of 3
Code No: RR412112 Set No. 4
IV B.Tech I Semester Supplementary Examinations, February 2007
APPROXIMATE METHODS IN STRUCTURAL MECHANICS
(Aeronautical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. (a) Comment on the statement “All engineering solutions are approximate”. [4]
(b) Determine the displacements of nodes 2 and 3 of the spring system shown
below figure 1, by potential energy method. [12]
Figure 1:
2. (a) A solid circular shaft and a thin tube of the same material and same weight
are subjected to twist. If max. stress in both is the same, what is the ratio of
their strain energies ?
(b) Calculate load shared by the three bars AD, BD and CD, using Castiglianos
theorem. Take BD = L. Assume same moment of inertia for all the beams.As
shown in figure 2 [6+10]
Figure 2:
3. (a) Which factors influence Critical buckling load in a column ?
(b) What is the effective length w.r.t. maximum buckling load in a column. Show
the critical length, by simple sketches of deformed shape, for the column end
conditions - fixed-free, fixed-fixed, hinged-hinged and fixed-hinged. [6+10]
1 of 3
Code No: RR412112 Set No. 4
4. (a) Distinguish between collocation method and Galerkin method.
(b) Calculate the maximum deflection in a cantilever beam, subjected to con-
centrated load ‘P’ at its free end, by 1-term collocation method and 1-term
Galerkin method. Compare the results.As shown in figure 3 [6+10]
Figure 3:
5. (a) Derive finite difference expression for d3y/dx3 using
i. forward differences,
ii. central differences and
iii. backward differences. [3 x 3]
(b) Calculate maximum deflection by finite difference method of a simply sup-
ported beam of length L, subjected to a udl of w N/m, choosing four equal
divisions along the beam. Calculate percentage error, compared to the exact
solution.As shown in figure 4 [7]
Figure 4:
6. For the eight bar truss shown in figure 5, determine the displacements of all the
nodes and stresses in all the elements, assuming that nodes 1 and 2 are fixed and
load P is applied at node ‘A’. (Hint : Use symmetry conditions wherever applicable)
[16]
7. Two beams AB of length ‘2L’ and BC of length ‘L’ are connected as shown in
figure 6. Calculate the displacements and reactions, when a concentrated load ‘P’
is applied at B at an angle of 450 to BC. [16]
8. (a) If a rectangular plate is modeled by
i. a rectangular element,
ii. an assembly of two triangular elements and
iii. an assembly of four triangular elements, what are the differences in the
assembled stiffness matrix ?
2 of 3
Code No: RR412112 Set No. 4
Figure 5:
Figure 6:
(b) Which polynomials do you suggest for representing displacement field, for the
following cases
i. Constant strain 2-D axisymmetric triangular element.
ii. Linear strain 2-D plane stress quadrilateral element .
iii. Triangular plate bending element .
iv. Triangular thin shell element. [6+10]
? ? ? ? ?
3 of 3

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