**Structure**

A combination of units constructed and so interconnected in an organized way, so as to provide rigidity between its elements.

**Theory of Structures **

Deals with the principles and methods by which direct stresses, shear and bending moment at any section may be found under given conditions of loading.

**Importance of Theory of Structures**

it is the basis of the design or determination of sizes of the structural members.

Table of Contents

## Structural Elements

**1)- Beam **

A structural member whose primary function is to carry transverse loads such as joist, girder, rafter or purlins.

**A) Statically determinate beams**

those beams that can be analyzed by simply using the three equations of static equilibrium or those beams that have no external redundant.

**Cantilever beam –**supported at one end (fixed support).**Simple beam –**a one span beam supported on a pin or hinged on one end and a roller at the other end**Simple beam with overhang**– simply supported beams with overhang either on one end or both ends.

**B) Statically indeterminate beams**

Those beams whose external reactions exceeds the number of equations of equilibrium.

**Restrained beam –**those with fixed ends.**Propped beam –**fixed at one end and supported by a hinged or roller at the other end.**Continuous beams –**those that have more than two supports.

**c) Compound beam**

The beam stated above may be connected by internal hinges or rollers to form a compound beam.

**2) Tie rod**

It is a structural member subjected to tensile force. These members are slender, and are often chosen from rods, bars, angles or channels.

**3) Columns**

Members that are generally vertical and resist axial compressive loads.

## Types of Structures

## 1) Arches and Cables

**A) Arches**

It is characterized by relatively low bending moment and large compressive forces with supports capable of resisting horizontal forces due to the tendency of the arch to flatten out under load.

**B) Cables **

a very efficient method for supporting loads over long spans. It acts in tension and tends to lengthen under load.

**2) Rigid frameworks**

A structural framework in which all columns and beams are rigidly connected.

**A) Trusses**:

They are composed of slender rods usually arrenged trintriangular fashion. Trusses are suitible for constructions with large span when the depth is not an important criterion for desing. Plane trusses are composed of members that lie in the same plane and are frequantly used for bridge and roof support.

**B) Three-hinged frame**:

This structure is simple determinate frame used generally for base element for complicated frame structures.

**C) Frames**:

They are often used in buildings and are composed of beams and columns which are with hinge or rigid connections. These structures are usually indeterminate and the load causes generally bending of its members.

**3) Plane structures**:

Plates, walls and etc. These structures have two significant dimensions and one small called thickness. The theory of elasticity is capable to analyze such structures.

**4) Surface structures**:

Shells and etc. These structures can be made from flexible or rigid material and has a three-dimensional shape like a cylinder hyperbolic paraboloid etc. The analysis of these structures is also aim of theory of elasticity.

**5) Composite structures**

structures in which some of the members are subjected to axial forces, while some of the members are subjected to bending.

# Building Loads

**1) Dead loads**

These are loads that are constant in magnitude and fixed in location throughout the lifespan of the structure. These include the weight of the structure itself including the weight of the fixtures and equipment permanently attached to it.

- floor, floor finishes, piping, and roof materials
- beams, girders and columns, including the footings
- permanently placed interior walls
- ceiling materials, including heating and air-conditioning ducts and electrical power installations
- mechanical equipment, including elevators and escalators

## 2) Live loads

These consist mainly of transient occupancy loads in buildings and traffic loads in bridges. The magnitude and distribution at any given instant are variable. Building Code usually assign minimum values, depending upon the type of occupancy.

## 3) Environmental loads

These consist of snow loads, wind pressure or suction, soil pressures, fluid pressures, earthquake or seismic loads, ponding and thermal stress. Their magnitude, distribution as well as their time of occurrence is highly variable.

**4) Displacement load**:

load displacement is caused from displacement of some point or points of the structure.

# Types of Connections and Supports

Idealized structure is needed to the engineer to perform a practical force analysis of the whole frame and its member. This is the reason in this section to show different member connections and supports and there idealizations. If one know these models may compose idealized model of each real structure after all perform the analysis and design.

**Rigid (fixed) connections:**

- This connection carry moment, shear and axial forces between different members.
- In addition, in this case all members including in such a connection have one and the same rotation and displacements – the nodal rotation and displacements.
- Typical rigid connections between members in metal and in reinforced concrete constructions and there idealized models are shown in the following figure:

**Hinged (pin) connections:**

- This connection carry shear and axial forces but not moment between different members. Hinged connection allow to the jointed members to have different rotations but the same displacements.
- Typical hinged connections between members in metal and in reinforced concrete constructions and there idealized models are shown in the next figure:

**Fixed support:**

This support carry moment, shear and axial forces between different members. This kind of support doesn’t allow any displacements of the support point. So if the displacement along the x axis is u, the displacement along y axis is y and the rotation is called (phi)then we can say that: uA = 0; vA = 0 and (Phi)A = 0.

**Hinged (pin) support:**

This support carry shear and axial forces but not moment between different members. The hinged support allows rotation of the support point but the two displacement are equal zero or: uA = 0; vA = 0 and (phi)A is not equal to 0.

**Roller support:**

This support carry only shear forces between jointed members. The roller support allows rotation and one displacement of the support point: uA is not equal to 0; vA = 0 and (phi)A is not equal to 0.

**Spring supports:**

These supports are like the previous but with the difference that they are not ideally rigid but with some real stiffness. The spring has a stiffness constant c equals to the force caused by displacement d = 1.

**Structure idealization:**

The main idea of this idealization is to made a mathematical model of the real construction to be convenient for analysis and calculation. After we know the idealization of different joints and supports, we will take care about whole structure idealization. To make this we follow the middle axis of the elements of the structure. In the following figure are shown some real and idealized structures: