CHENDU COLLEGE OF ENGINEERING AND TECHONOLOGY

DEPARTMENT OF CIVIL ENGINEERING

CE2351 – Structural Analysis – II

BY

Ms. G. BHARATHI (AP/CIVIL)

CHENDU COLLEGE OF ENGINEERING AND TECHNOLOGY

Zamin Endathur Village , Madurantakam taluk,

Kancheepuram |District – 603 311.

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UNIT I

FLEXIBILITY METHOD FOR INDETERMINATE FRAMES

1. Define Force Transformation Matrix.

The connectivity matrix which relates the internal forces Q and the external forces R is known as force transformation matrix. In matrix form,

{Q} = [b] {R} Where Q is the member force vector,

b is the force transformation matrix, and R is the external force vector.

2. What do you consider as unknowns in flexibility matrix method?

In flexibility matrix method redundant forces are considered as the unknowns.

3. What are the requirements to be satisfied in analyzing any structure?

1. Equilibrium condition 2. Compatibility condition 3. Force displacement relationship

4. Define Flexibility Coefficient.

A Flexibility coefficient fij is defined as the displacement at joint i due to a unit load at joint j while all other joints are unloaded.

5. Define kinematic indeterminacy

When a structure composed of several members is subjected to loads, the joints undergo displacements in the form of rotation and translation .The number of independent joint displacements in the structure is called the degree of kinematic indeterminacy.

6. What is the compatibility condition used in flexibility matrix method?

The deformed elements fit together at the nodal points. 7. What is the equilibrium condition used in the flexibility matrix method?

The external loads and the internal member forces must be in equilibrium at the nodal points.

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8. Define compatibility in force method.

Compatibility is a continuity condition on the displacements of the structure after the external loads are applied to the structure.

9. What is meant by static indeterminacy?

The excess number of unknown reaction components more than the maximum necessary to keep the structure in static equilibrium.

10. Can you develop flexibility matrix for an unstable structure?

No. The structure has to be stable and in equilibrium. 11. What is the relation between flexibility and stiffness matrix?

The element stiffness matrix is the inverse of element stiffness matrix and vice versa.

f = 1/k

k = 1/f

12. What is a primary structure?

The statically determinate and stable structure that remains after the removal of the extra restraints is called the primary structure.

13. What are the two general approaches to the matrix analysis of structures?

1. Force method 2. Displacement method.

14. Write the element flexibility matrix for a truss member? 15. Write the element flexibility matrix for a truss member?

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UNIT 2

MATRIX STIFFNESS METHOD

1. What do you consider as basic unknowns in stiffness method?

In stiffness method nodal displacements are considered as basic unknowns.

2. Define Stiffness Coefficient.

A Stiffness coefficient kij is defined as the force developed at joint i due to a unit displacement at joint j while all other joints are fixed.

3. What is the basic aim of stiffness method?

The aim of stiffness method is to evaluate the values of generalized coordinate’s r knowing the structure stiffness matrix k and nodal loads R through the structure –equilibrium equation.

{R}=[K] {r}

4. What do you infer if the determinant of a stiffness matrix does not exist?

The body experiences rigid body motion. It moves as a whole.

5. What is the displacement transformation matrix?

The connectivity matrix which relates the internal displacement q and the external displacement r is known as displacement transformation matrix a.

{q } = [ a] { r }

6. Under what circumstances congruent transformation method of obtaining structure stiffness matrix is advantageous over direct stiffness method?

When the number of freedom does not exceed 3, the structure stiffness matrix can be formulated using the formula

K = aT k a

7. How the basic equations of stiffness matrix are obtained?

1. Equilibrium forces 2. Compatibility of displacements 3. Force displacement relationships.

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8. What is the equilibrium condition used in the stiffness method?

The external loads and the internal member forces must be in equilibrium at the nodal points.

9. What is meant by generalized coordinates?

For specifying a configuration of a system, a certain minimum number of independent coordinates are necessary. The least number of independent coordinates that are needed to specify the configuration is known as generalized coordinates.

10. What is the compatibility condition used in flexibility matrix method?

The deformed elements fit together at the nodal points.

11. Write about the force displacement relationship

The relationship of each element must satisfy the stress strain relationship of the element material.

12. Write the element stiffness matrix for a truss member.

13. Write the element stiffness matrix for a beam member

14. Compare flexibility method and stiffness method. In flexibility method, the redundant forces are treated as basic unknowns. Thus the number of equations involved is equal to the degree of static indeterminacy of the structure. This method is the generalization of consistent deformation method. In stiffness method, the joint displacements are treated as basic unknowns. The number of displacements involved equals the number of degrees of freedom of the structure .This method is the generalization of the slope deflection method. In flexibility method, different procedures are used for determinate and indeterminate structures.

UNIT-3

FINITE ELEMENT METHOD

1. What is FEM?

FEM is a numerical technique for solving boundary value problems in which one divides the domain of the problem in to little pieces over which the solution is approximated using polynomials. The little pieces are finite elements and the polynomials are called shape functions.

2. List out the advantages in FEM? Since the properties of each element are evaluated separately we can

incorporate different material properties for each element. There is no restriction to the shape of the medium.

Any type of boundary condition can be accommodated.

3. List out the disadvantages in FEM?

The computational cost is more.

The solution is approximate.

4. Mention the various coordinates in FEM?

Local (or) element coordinates

Natural coordinates Simple natural coordinates

Area coordinates (or) triangular coordinates

Generalized coordinates

5. What are the basic steps in FEM?

Discretization of the structure

Selection of displacement function Finding element properties

Assembling the element properties Applying the boundary conditions

Solving the system of equations

Computing additional results

6. What is meant by discretization?

Discretization is the process of subdividing the given body in to a number of elements which results in an equivalent body of finite elements.

7. What are the factors that govern the selection of type of finite element?

The geometry of the body The number of independent space coordinates

The nature of stress variations expected

8. Define displacement function?

Simple functions which are assumed to approximate the displacements for each element are called displacement functions. It may be a function of polynomials, trigonometry, or others.

9. Define nodes?

Nodes are points at which different elements are connected to generate the structure (or) system.

10. What are the types of finite elements used for plane stress/strain problems?

Quadrilateral elements

Rectangular elements

Triangular elements

11. What are the most obvious locations for nodes?

Points of application of concentrated loads

Locations where there are changes in intensity of loads Locations where there are discontinuities in the geometry of the structure

Interfaces between materials of different properties

12. What is meant by element aspect ratio?

The element aspect ratio is defined as the ratio of the largest dimension of the element to the smallest dimension.

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13. What is meant by displacement models?

The simple functions that are assumed to approximate the displacements of each element are called displacement models.

14. What are the properties of shape function?

There will be as many shape functions as there are nodes. It will have unit value at the node in question and zero values at other nodes.

The sum of all the shape functions is unity.

15. What are the primary unknowns in displacement based finite element analysis?

The primary unknowns in displacement based finite element analysis are the nodal displacements.

16. What are the secondary unknowns in displacement based finite element analysis?

The secondary unknowns in displacement based finite element analysis are the stresses.

17. What are the characteristics of displacement functions?

The displacement field must be continuous. The displacement must be compatible between adjacent elements.

The displacement field must represent constant strain states of the elements.

The displacement function must represent the rigid body displacement of the element.

18. Define shape function?

Shape function is defined as the one whose value at a particular node is unity and other nodes zero.

19. What are the types of finite elements?

One dimensional element

Two dimensional element

Three dimensional element

20. What is meant by plane strain?

It is the state of strain in which normal strain and shear strain directed perpendicular to the plane of the body are assumed to be zero.

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UNIT IV

PLASTIC ANALYSIS OF STRUCTURES 1. What is a plastic hinge?

When a section attains fully plastic moment Mp, then it acts like a hinge called a plastic hinge. It is defined as a yielded zone due to bending at which large rotations can take place at a constant plastic moment, Mp.

2. Define shape factor?

It is defined as the ratio of the plastic moment of the section to the yield

moment of the section.

3. Define collapse load.

The load that causes the (n +1)th hinge to form a mechanism is known as a collapse load where n is the degree of statical indeterminacy. Because once a structure becomes a mechanism, it will collapse.

4. What is a mechanism?

When an n-degree indeterminate structure develops n plastic hinges, it becomes determinate; formation of an additional hinge will reduce the structure to a mechanism. Once a structure becomes a mechanism, it will collapse.

5. What is the difference between plastic hinge and mechanical hinge?

Plastic hinges modify the behavior of structures much in the same way as mechanical hinges. The only difference is that the plastic hinges permit rotation offering a constant resisting moment, Mp to the rotation. At mechanical hinges, the resisting moment is zero.

6. List out the assumptions made for the plastic section.

Plane transverse sections remain plane and normal to the longitudinal axis, bending and the effect of shear being neglected.

Modulus of elasticity has the same value in tension and compression.

The material is homogeneous and isotropic in both the elastic and plastic state.

There is no resultant axial force on the beam. 9

The cross-section of the beam is symmetrical about an axis through its

centroid parallel to the plane of bending. 7. List out the shape factors for various sections.

Rectangular section, S =1.5

Triangular section, S =2.346

Circular section, S =1.697

Diamond section, S =2

8. Which section is having the maximum shape factor?

The section with the maximum shape factor is the triangular section, S =2.345.

9. Define load factor

Load factor is defined as the ratio of the collapse load and the working load. It is denoted as

10. Enumerate different types of mechanism.

Beam mechanism

Column mechanism Panel or sway mechanism

Cable mechanism

Combined or composite mechanism

11. State upper bound theory

Upper bound theory states that a load computed on the basis of an assumed mechanism is always greater than or at best equal to the true ultimate load.

12. State lower bound theory

Lower bound theory states that a load computed on the basis of an assumed

equilibrium bending moment diagram in which the moments are not greater than Mp is less than or at the worst equal to the true ultimate load.

13. Define static indeterminacy.

When the reactions and internal forces of a structure can not be determined

from the equilibrium equations alone, this condition is known as static indeterminacy.

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14. When a structure is called as statically externally indeterminate.

A structure becomes statically externally indeterminate when the number of reaction components exceeds the number of equilibrium equations.

UNIT-5

CABLES AND SPACE STRUCTURES

1. What is the true shape of suspension cable which is supported between two pegs?

Catenary 2. What is a catenary?

The shape that is taken up by a cable (or) rope freely suspended between two supports and under its own weight is called a catenary.

3. Differentiate between cable passed over a guide pulley and cable passed over a saddle?

Cable passed over a guide pulley

Tension in the suspension cable= Tension in the anchor cable

The supporting tower will be subjected to vertical pressure and bending due to the net horizontal cable tension.

Cable passed over a saddle

Horizontal component of tension in the suspension cable= Horizontal component of tension in the anchor cable.

The supporting tower will be subjected to only vertical pressure due to the

cable tension.

4. What is the degree of static indeterminacy of a suspension bridge with two hinged stiffening girder?

The two hinged stiffening girder is one degree indeterminate.

5. What are the main functions of stiffening girders in suspension bridges?

They help the cables keep their shape.

They resist a part of shear force and bending moment due to live loads.

6. Differentiate plane truss and space truss?

Plane truss

All the members lie in one plane.

All joints are assumed to be hinged. Space truss

This is a three dimensional truss.

All joints are assumed to be ball and socketed. 7. Define tension coefficient of a truss member?

The tension coefficient for a member of a truss is defined as the pull or tension

in that member divided by its length, ie the force in the member per unit length.

8. Give some examples for beams curved in plan?

Beam in a bridge negotiating a curve. Ring beams supporting water tanks.

Beam supporting corner lintels.

Beam in ramp in traffic interchanges. 9. What are the forces developed in a beam curved in plan?

Bending moments

Shears and Torsional moments

10. In a circular beam on several equally spaced supports what are the quantities zero?

Slope on either side of any support will be zero. Torsional moment on every support will be zero.

11. Give the expression for calculating the equivalent udl on a girder?

We = (Total load on the girder) / (Span of the girder)

12. What are cables?

Cables are usually flexible which carry their loads in tension only.

13. What is the range of the central dip of a cable?

The central dip of a cable ranges from 1/10 to 1/12 of the span.

14. Where is the suspension cable bridges used?

Suspension cable bridges are used only for pedestrian traffic