*Answer the following (10 x 2)*

**What for Castigliano’s first theorem is used?**

- Castigliano’s first theorem: The first partial derivative of the total internal energy (strain energy) in a structure with respect to any particular deflection component at a point is equal to the force applied at that point and in the direction corresponding to that deflection component.
- This first theorem is applicable to linearly or non-linearly elastic structures in which the temperature is constant and the supports are unyielding.
- It is given as ∂U/∂Δ
_{i}= P_{i}

**Discuss the concept of indeterminacy for steel structures.**

- A structure that cannot be analysed by using equations of equilibrium only is called a
**statically indeterminate structure,**e.g., fixed beams, continuous beams, propped cantilevers. - To analyse indeterminate structures, apart from using equations of equilibrium, one has to determine the various deformations and make use of compatibility conditions.
- Indeterminate structures are also called redundant structures.

**What assumptions are made in the solution of plane frames/trusses?**

The following assumptions are made in the analysis of truss

- Truss members are connected only at their ends
- Truss members are connected by frictionless pins
- Trusses are loaded only at the joints
- The weight of the members may be neglected

**What are influence line diagrams? Discuss their use.**

The steps involved in determining S.F. and B.M. at different sections of a beam as the rolling loads move from one end to the other are rather cumbersome.

**Influence lines**are interesting and are a very useful tool in dealing with rolling loads.*Definition*

An influence line is defined as a function whose value at a point represents some structural quantity as a unit load is placed at that point.

The following figure shows an example of an influence line diagram.

**Discuss the functions of cables in designing bridges.**

The main function of towers of cable-stayed bridges or suspension bridges is to safely support bridge loads and traffic loads.

**Discuss the method of consistent deformation.**

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:

- Determine the
*degree of static indeterminacy*, n of the structure. - 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
- For a given released constraint j, introduce an unknown redundant force R
_{f}corresponding to the type and direction of the released constraint. - Apply the given loading or imposed deformation to the basic determinate structure. Use a suitable method to calculate displacements at each of the released constraints in the basic determinate structure.
- Solve for redundant forces R
_{f}( 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.

**Application of Moment Distribution method for analysing plane frames.**

The moment distribution method, also known as the Hardy Cross method.

This is basically an iterative process.

*Steps:*

The procedure, in general, involves artificially restraining temporarily all the joints against rotation and writing down the fixed end moments for all the members. The joints are then released one by one in succession. At each released joint the unbalanced moments are distributed to all the ends of the members meeting at that joint. A certain fraction of these distributed moments is carried over to the far end of members.

A released joint is again restrained temporarily before proceeding to the next joint.

The same set of operations are carried out at each joint till all the joints are completed. This completes one cycle of operations.

The process is iterated for a number of cycles till the values obtained are within the desired accuracy].

**Use of three-hinged arches as compared to fixed arches.**

- In the case of
**beams**supporting uniformly distributed load, the maximum bending moment increases with the square of the span, and hence they become uneconomical for long-span structures. In such situations,**arches**could be advantageously employed, as they would develop horizontal reactions, which in turn reduce the design bending moment. - A three-hinged arch is a geometrically stable and statically determinate structure. It consists of two curved members connected by an internal hinge at the crown and is supported by two hinges at its base. Sometimes, a tie is provided at the support level or at an elevated position in the arch to increase the stability of the structure.
- The hinges on the two fixed supports absorb vertical and horizontal forces and are known as abutment hinges.
- The three-hinged arch is a statically determinate structure, and its reactions/internal forces are evaluated by static equations of equilibrium.
- Two-hinged arch and fixed-fixed arch are statically indeterminate structures. The indeterminate reactions are determined by the method of least work or by the flexibility matrix method.

**What is the load factor? How does it compare with a factor of safety?**

*Load Factor*

The ratio of the load causing collapse to the working load is called the load factor. The load factor is dependent upon the shape of the section as the working load is dependent upon the I and Z values and the collapse load is dependent upon the shape of the section. The load factor for a rectangular beam is 2.25.

*Factor of safety*

A Factor of Safety (FS) ensures that a reasonable margin exists between the component’s maximum operating load and its load capacity. It accounts for the possibility of unanticipated loads that could occur during operation. The factor of safety is defined as the factor by which the yield stress of the

material is divided to give the working stress (permissible stress) in the material

**Use of laces and battens in steel towers.**

*Lacing*

The lacing is also termed as ‘latticing’ and it is most commonly used. The rolled steel flats, angles and channels are used for lacing. The rolled steel sections or tubes of equivalent length may also be used instead of flats. The lacing is of two types: (i) Single lacing (ii) Double lacing.

*Battens*

The batten plates are also called the plates, and these are also used in the lateral systems. The angle sections, channels and I-sections are also used as battens. The components of the built-up column sharing the load are connected together by batten plates.

Battened plates

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