CE 6501 STRUCTURAL ANALYSIS I UNIT I

1. Calculate degree of indeterminacy of propped cantilever beam. [M/J-15]

For beams degree of indeterminacy is given by

i = r – e (a) i = r – e

where, r = no of reactions, e = no of equilibrium conditions

r = 4 and e = 3

i = 4 – 3 = 1

2. Write the difference between static and kinematic indeterminacies.[M/J-15]

Static Indeterminacy:-

When the number of unknown forces is more than the number of equilibrium equations required to find the forces, then the system is said to be statically indeterminate;

Let,

N1 = Number of unknown forces,

N2 = Number of equilibrium equations to be solved;

Then, “Q” is called the Degree of (Static) Indeterminacy,

where, “Q” is defined as follows; Q = N1-N2

Kinematic indeterminacies

Degree of freedom is defined as the least no of independent displacements required to define the deformed shape of a structure.

There are two types of DOF: (a) Nodal type DOF and (b) Joint type DOF.

3. Brief method of consistent deformation for the analysis of a propped cantilever.[M/J-16]

The force method is used to calculate the response of statically indeterminate structures to loads and/or

imposed deformations. The method is based on transforming a given structure into a statically determinate

primary system and calculating the magnitude of statically redundant forces required to restore the

geometric boundary conditions of the original structure. The force method (also called the flexibility method

or method of consistent deformation) is used to calculate reactions and internal forces in statically

indeterminate structures due to loads and imposed deformations.

The basic steps in the force method are as follows:

(a) Determine the degree of static indeterminacy, n of the structure.

(b) Transform the structure into a statically determinate system by releasing a number of static constraints

equal to the degree of static indeterminacy, n. This is accomplished by releasing external support

conditions or by creating internal hinges. The system thus formed is called the basic determinate structure.

(c) For a given released constraint j, introduce an unknown redundant force corresponding to the type and

direction of the released constraint.

(d) Apply the given loading or imposed deformation to the basic determinate structure. Use suitable method

to calculate displacements at each of the released constraints in the basic determinate structure.

(e) Solve for redundant forces (j =1 to n) by imposing the compatibility conditions of the original structure.

These conditions transform the basic determinate structure back to the original structure by finding the

combination of redundant forces that make displacement at each of the released constraints equal to zero.

It can thus be seen that the name force method was given to this method because its primary

computational task is to calculate unknown forces, i.e. the redundant forces through.

4. Find the degree of static indeterminacy for the following structures and specify whether the structure is stable or not. [N/D-16]

(a) S.I = I.I + E.I (b) S.I = I.I + E.I = 3 = 0

5. Determine the prop reaction of a propped cantilever using energy method when

it is subjected to a uniformly distributed load over the entire span.[N/D-16]

Let a statically indeterminate structure has degree of indeterminacy as n . On the selected basic

determinate structure apply the unknown forces , ..... and . Using the Eq. (4.16) the displacement in the

direction of is expressed by ( j = 1, 2, .. ……n) (5.1) The equations (5.1) will provide the n linear

simultaneous equations with n unknowns , ..... and . Since the is known, therefore, the solution of

simultaneous equations will provide the desired ( j =1, 2,…., n ). For structures with members subjected

to the axial forces only (i.e. pin-jointed structures), the equation (5.1) is re-written as (5.2) where P is

the force in the member due to applied loading and unknown ( j =1, 2,…., n ); and L and AE are length

and axial rigidity of the member, respectively. For structures with members subjected to the bending

moments (i.e. beams and rigid-jointed frames), the equation (5.1) is re-written as (5.3) where M is the

bending moment due to applied loading and unknown ( j =1, 2,…., n ) at a small element of length dx ;

and EI is the flexural rigidity.

1. (a) Find the forces in the members of the truss shown in figure 11(a). The cross sectional area and young’s modulus of all the members are the same.

[M/J15]

SOLUTION:

The forces in the members are found by using the following methods given below

Step 1 - virtual force

Step 2 – real forces

Step 3 - virtual work equation

2. (b) Analyse the truss shown in figure 11(b) by consistent deformation method. Assume that the

cross sectional area of all the members a same.

(16) [M/J-15]

SOLUTION:

The forces in the members are found by using the following methods given below

Step 1 - virtual force

Step 2 – real forces

Step 3 - virtual work equation

3. (a) A fixed beam of span 6 m carries a uniformly distributed load of 4 kN/m over the left half span. Analyze the beam using energy method and draw the bending moment diagram. [M/J-16]

4. A Continuous beam ABC of uniform section is fixed at A and simply supported at B and C. The

spans AB and BC are 6 m and 4 m respectively. The span BC carries a uniformly distributed load of 6

kN/m and the span AB carries a central concentrated load of 12 kN. Analyze the beam by consistent

deformation method and draw the shearing force and bending moment diagrams.

[M/J-16]

5. (a) The frame shown in figure Q.11 (a) is pin jointed to rigid supports at A and B and the joints C and D are also pinned. The diagonals AD and BC act independently and the members are all of the same cross sectional area and material. ABC and BCD are equilateral triangles. Using energy method, find the forces in all the members if a load of 5 kN is hung at D.

[N/D-16]

Solution:

6. (b) Using Consistent deformation method, determine the horizontal reaction at the support C for the

frame shown in figure Q.11 (b). Flexural rigidity E1 is constant for both the members.

[N/D-16]

SOLUTION: