This document contains 8 questions related to the finite element method in civil engineering. The questions cover various topics including: 1) Deriving expressions for potential energy and determining displacements using Rayleigh Ritz method for a 1D rod subjected to loading. 2) Assembling stiffness and force matrices and determining displacements and stresses for a 1D rod under thermal loading using finite element discretization. 3) Evaluating shape functions and determining the Jacobian for an isoparametric triangular element. The document provides figures and equations to accompany the questions. It examines a range of finite element techniques including shape function derivation, element formulation, structural and thermal analysis, and plate bending elements.

1. JNTUW
ORLD
R07 SET-1Code No: 07A70102
B.Tech IV Year I Semester Examinations, December-2011
FINITE ELEMENT METHODS IN CIVIL ENGINEERING
(CIVIL ENGINEERING)
Time: 3 hours Max. Marks: 80
Answer any five questions
All questions carry equal marks
---
1.a) State the principle of minimum potential energy.
b) A one dimensional elastic steel rod is attached to rigid walls at each end and is
subjected to a distributed load T(x) as shown in Figure 1.
i) Write an expression for potential energy
ii) Determine u(x) using Rayleigh Ritz method. Assume a displacement field
u = a0 + a1x + a3x2
. [4+12]
Fig: 1
2.a) Write stress – strain relationship for plane stress, plane strain and axisymmetric
problems in two dimensions.
b) An axial load P = 350 ´ 103
N is applied at 250
C to the rod shown in Figure 2.
The rod is discretized using 1D FEM. The temperature is then raised to 650
C.
i) Assemble stiffness and force matrices
ii) Determine nodal displacements and element stresses. [6+10]
Fig: 2
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2. JNTUW
ORLD
3.a) State the convergence and compatibility requirements and geometric invariance
for finite element approximate functions.
b) An Isoparametric constant strain triangular element is shown in Figure 3.
i) Evaluate the shape functions N1, N2 and N3 at an intermediate point P for
the triangular element.
ii) Determine the Jacobian of transformation J for the element. [6+10]
Fig: 3
4.a) With the help of examples explain Lagrangian and Hermitian interpolation.
b) Using the shape functions of a 4 noded bilinear rectangular element derive the
shape functions for the 5 noded transition element as shown in the Figure 4.
[6+10]
Fig: 4
5.a) What are incompatible elements? State the merits and demerits of using these
elements.
b) Derive the formulation for a 4 noded improved incompatible quadrilateral element
with incompatible modes. Briefly state the requirements on quadrature rule for
such elements. [6+10]
6.a) Briefly explain what is patch test.
b) Explain with relevant expressions the static condensation procedure.
c) State the Principle of virtual work. [4+8+4]
7.a) Explain “mesh locking” and “Shear locking” in the context of plate bending finite
elements.
b) Explain in detail the Mindlins theory to include shear deformation in plates. Using
this theory derive the basic relationships of finite element formulation
(equilibrium equations) for plate bending. [6+10]
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3. JNTUW
ORLD
8.a) What are full integration, reduced integration and selective integration scheme in
evaluation of the stiffness matrices? When is each of these methods applicable?
b) Show that a tetrahedral element can be created from an eight noded element by
collapsing the sides as given in Figure 5 and the strain in the element is constant.
[6+10]
Fig: 5
******
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4. JNTUW
ORLD
SET-2R07Code No: 07A70102
B.Tech IV Year I Semester Examinations, December-2011
FINITE ELEMENT METHODS IN CIVIL ENGINEERING
(CIVIL ENGINEERING)
Time: 3 hours Max. Marks: 80
Answer any five questions
All questions carry equal marks
---
1.a) Explain steps involved in structural analysis using the Finite Element Method.
b) State the principle of virtual work.
c) The truss system shown in Figure 1 subjected to a load P =100 KN and Q = 100
KN at Joint.
Using the principle of virtual displacements derive the equilibrium conditions
for Joint 1.
Compute the deflections u and v and also the forces in members 1 and 2.
[3+3+10]
Fig: 1
2.a) What are strong and weak form of governing differential equation of a boundary
value problem?
b) Explain with the help of examples the solution of a boundary value problem by
i) Rayleigh – Ritz method
ii) Weighted residual method. [6+10]
3.a) Explain Lagrangian interpolation and hermite interpolation.
b) Using the shape functions of a 6 noded Linear strain triangular element derive the
shape functions for a 4 noded triangular element as shown in Figure 2. [6+10]
Fig: 2
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5. JNTUW
ORLD
4.a) Explain the convergence criteria for Isoparametric elements.
b) Evaluate the Jacobian matrix [J] for the elements shown in Figure 3. Discuss the
effect of distortion from the parent bisquare element. [6+10]
Fig: 3
5.a) Write stress – strain relationship for plane stress, plane strain and axisymmetric
problems in two dimensions.
b) How will you idealize and What Finite element will be used for the following
situations?(Explain with figures clearly indicating the element, number of nodes,
number of degrees of freedom per node and justify the choice of the element).
i) A rod fixed at one end and freely hanging with its own self weight.
ii) A long pipe carrying oil iii) A deep beam of a water tank
iv) A beam below a railway bridge
v) A cylindrical water tank made up of very thin steel sheet
vi) A square pile cap. vii) A square plate with a circular opening. [6+10]
6.a) With sketches explain the possible zero energy modes for a 4 noded quadrilateral
element.
b) Derive the formulation for a 4 noded improved incompatible quadrilateral element
with incompatible modes. Briefly state the requirements on quadrature rule for
such elements. [6+10]
7.a) Explain “mesh locking” and “Shear locking” in the context of plate bending finite
elements.
b) Explain in detail the Mindlins theory to include shear deformation in plates. Using
this theory derive the basic relationships of finite element formulation
(equilibrium equations) for plate bending. [6+10]
8.a) What are full integration, reduced integration and selective integration scheme in
evaluation of the stiffness matrices? When is each of these methods applicable?
b) Show that a tetrahedral element can be created from an eight noded element by
collapsing the sides as given in Figure 4 and the strain in the element is constant.
[6+10]
Fig: 4
******
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6. JNTUW
ORLD
SET-3R07Code No: 07A70102
B.Tech IV Year I Semester Examinations, December-2011
FINITE ELEMENT METHODS IN CIVIL ENGINEERING
(CIVIL ENGINEERING)
Time: 3 hours Max. Marks: 80
Answer any five questions
All questions carry equal marks
---
1.a) Briefly explain the steps in Finite Element Analysis.
b) State the principle of minimum potential energy.
c) A one dimensional elastic steel rod is attached to rigid walls at each end and is
subjected to a distributed load T(x) as shown in Figure 1.
i) Write an expression for potential energy
ii) Determine u(x) using Rayleigh Ritz method. Assume a displacement field
u = a0 + a1x + a3x2
. [4+4+8]
Fig: 1
2.a) Write stress – strain relationship for plane stress, plane strain and axisymmetric
problems in two dimensions.
b) State the principle of virtual work.
c) An axial load P = 450´ 103
N is applied at 250
C to the rod shown in Figure 2.
The rod is discretized using 1D FEM. The temperature is then raised to 650
C.
i) Assemble stiffness and force matrices
ii) Determine nodal displacements and element stresses. [4+4+8]
Fig: 2
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7. JNTUW
ORLD
3.a) State the convergence and compatibility requirements and geometric invariance
for finite element approximate functions.
b) Show that the convergence requirements for an isoparametric element can be
satisfied if the sum of shape functions is unity.
c) An Isoparametric constant strain triangular element is shown in Figure 3.
i) Evaluate the shape functions N1, N2 and N3 at an intermediate point P for
the triangular element.
ii) Determine the Jacobian of transformation J for the element. [4+4+8]
Fig: 3
4.a) With the help of examples explain Lagrangian and Hermitian interpolation.
b) Using Lagrange interpolation formula obtain the shape functions for a three noded
bar element. Also sketch the variation of the shape functions.
c) Using the shape functions of a 4 noded bilinear rectangular element derive the
shape functions for the 5 noded transition element as shown in the Figure 4.
[4+4+8]
Fig: 4
5.a) What are incompatible elements? State the merits and demerits of using these
elements.
b) Derive the formulation for a 4 noded improved incompatible quadrilateral element
with incompatible modes. Briefly state the requirements on quadrature rule for
such elements. [6+10]
6.a) Briefly explain what is patch test.
b) Explain with relevant expressions the static condensation procedure.
c) State the Principle of virtual work. [4+8+4]
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8. JNTUW
ORLD
7.a) Derive the general expression of stiffness matrix for a four noded quadrilateral
element.
b) Explain in detail the Mindlins theory to include shear deformation in plates. Using
this theory derive the basic relationships of finite element formulation
(equilibrium equations) for plate bending. [6+10]
8.a) What are full integration, reduced integration and selective integration scheme in
evaluation of the stiffness matrices? When is each of these methods applicable?
b) Show that a tetrahedral element can be created from an eight noded element by
collapsing the sides as given in Figure 5 and the strain in the element is constant.
[6+10]
Fig: 5
******
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9. JNTUW
ORLD
SET-4R07Code No: 07A70102
B.Tech IV Year I Semester Examinations, December-2011
FINITE ELEMENT METHODS IN CIVIL ENGINEERING
(CIVIL ENGINEERING)
Time: 3 hours Max. Marks: 80
Answer any five questions
All questions carry equal marks
---
1.a) Explain the principle of virtual displacments and principle of virtual forces.
b) What are strong and weak form of governing differential equation of a boundary
value problem?
c) Explain with the help of examples the solution of a boundary value problem by
i) Collocation method
ii) Galerkin method. [4+4+8]
2.a) Explain how finite element method is different from finite difference method.
b) State the principle of virtual work.
c) The truss system shown in Figure 1 subjected to a load P =200 KN and Q = 200
KN at Joint.
Using the principle of virtual displacements derive the equilibrium conditions for
Joint 1.
Compute the deflections u and v and also the forces in members 1 and 2.
[3+3+10]
Fig: 1
3.a) Explain Lagrangian interpolation and hermite interpolation.
b) Obtain the shape functions for any one corner nodes and midside node of a eight
nodded quadrilateral element.
c) Using the shape functions of a 6 noded linear strain triangular element derive the
shape functions for a 4 noded triangular element as shown in Figure 2. [4+4+8]
Fig: 2
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10. JNTUW
ORLD
4.a) With sketches explain the possible zero energy modes for a 4 noded quadrilateral
element.
b) Derive the formulation for a 4 noded improved incompatible quadrilateral element
with incompatible modes. Briefly state the requirements on quadrature rule for
such elements. [6+10]
5.a) Explain the convergence criteria for Isoparametric elements.
b) Evaluate the Jacobian matrix [J] for the elements shown in Figure 3. Discuss the
effect of distortion from the parent bisquare element. [6+10]
Fig: 3
6.a) Write stress – strain relationship for plane stress, plane strain and axisymmetric
problems in two dimensions.
b) How will you idealize and What Finite element will be used for the following
situations?(Explain with figures clearly indicating the element, number of nodes,
number of degrees of freedom per node and justify the choice of the element).
i) A rod fixed at one end and freely hanging with its own self weight.
ii) A long pipe carrying oil
iii) A deep beam of a water tank
iv) A beam below a railway bridge
v) A cylindrical water tank made up of very thin steel sheet
vi) A square pile cap.
vii) A square plate with a circular opening. [6+10]
7.a) Explain “mesh locking” and “Shear locking” in the context of plate bending finite
elements.
b) Explain in detail the Mindlins theory to include shear deformation in plates. Using
this theory derive the basic relationships of finite element formulation
(equilibrium equations) for plate bending. [6+10]
8.a) What are full integration, reduced integration and selective integration scheme in
evaluation of the stiffness matrices? When is each of these methods applicable?
b) Show that a tetrahedral element can be created from an eight noded element by
collapsing the sides as given in Figure 4 and the strain in the element is constant.
[6+10]
Fig: 4
******
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