Mechanics of solids/Strength of materials


by: ranjan317
Solid mechanics is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and other external or internal agents.



IBeams
 

Beam

statically determinate beam, bending (sagging) under an evenly distributed load.

beam is a structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.

Beams are traditionally descriptions of building or civil engineering structural elements, but smaller structures such as truck or automobile frames, machine frames, and other mechanical or structural systems contain beam structures that are designed and analyzed in a similar fashion.

Overview

Historically beams were squared timbers but are also metal, stone, or combinations of wood and metal such as a flitch beam. Beams generally carry vertical gravitational forces but can also be used to carry horizontal loads (e.g., loads due to an earthquake or wind or in tension to resist rafter thrust as a tie beam or (usually) compression as a collar beam). The loads carried by a beam are transferred to columnswalls, or girders, which then transfer the force to adjacent structural compression members. In light frame construction joists may rest on beams.

In carpentry a beam is called a plate as in a sill plate or wall plate, beam as in a summer beam or dragon beam.

Types of beams

In engineering, beams are of several types:

  1. Simply supported - a beam supported on the ends which are free to rotate and have no moment resistance.
  2. Fixed - a beam supported on both ends and restrained from rotation.
  3. Over hanging - a simple beam extending beyond its support on one end.
  4. Double overhanging - a simple beam with both ends extending beyond its supports on both ends.
  5. Continuous - a beam extending over more than two supports.
  6. Cantilever - a projecting beam fixed only at one end.
  7. Trussed - a beam strengthened by adding a cable or rod to form a truss.

Structural characteristics

Moment of inertia

The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis. Therefore, it encompasses not just how much mass the object has overall, but how far each bit of mass is from the axis. The farther out the object's mass is, the more rotational inertia the object has, and the more force is required to change its rotation rate.

Diagram of stiffness of a simple square beam (A) and universal beam (B). The universal beam flange sections are three times further apart than the solid beam's upper and lower halves. The second moment of inertia of the universal beam is nine times that of the square beam of equal cross section (universal beam web ignored for simplification)

Stress in beams

Internally, beams experience compressivetensile and shear stresses as a result of the loads applied to them. Typically, under gravity loads, the original length of the beam is slightly reduced to enclose a smaller radius arc at the top of the beam, resulting in compression, while the same original beam length at the bottom of the beam is slightly stretched to enclose a larger radius arc, and so is under tension. The same original length of the middle of the beam, generally halfway between the top and bottom, is the same as the radial arc of bending, and so it is under neither compression nor tension, and defines the neutral axis (dotted line in the beam figure). Above the supports, the beam is exposed to shear stress. There are some reinforced concrete beams in which the concrete is entirely in compression with tensile forces taken by steel tendons. These beams are known as prestressed concrete beams, and are fabricated to produce a compression more than the expected tension under loading conditions. High strength steel tendons are stretched while the beam is cast over them. Then, when the concrete has cured, the tendons are slowly released and the beam is immediately under eccentric axial loads. This eccentric loading creates an internal moment, and, in turn, increases the moment carrying capacity of the beam. They are commonly used on highway bridges.

The primary tool for structural analysis of beams is the Euler–Bernoulli beam equation. Europe has superseded Euler-Bernoulli equations with the Perry Robertson formula. Other mathematical methods for determining the deflection of beams include "method of virtual work" and the "slope deflection method". Engineers are interested in determining deflections because the beam may be in direct contact with a brittle material such as glass. Beam deflections are also minimized for aesthetic reasons. A visibly sagging beam, even if structurally safe, is unsightly and to be avoided. A stiffer beam (high modulus of elasticity and high second moment of area) produces less deflection.

Mathematical methods for determining the beam forces (internal forces of the beam and the forces that are imposed on the beam support) include the "moment distribution method", the force or flexibility method and the direct stiffness method.

General shapes

Most beams in reinforced concrete buildings have rectangular cross sections, but a more efficient cross section for a beam is an I or H section which is typically seen in steel construction. Because of the parallel axis theorem and the fact that most of the material is away from the neutral axis, the second moment of area of the beam increases, which in turn increases the stiffness.

An I-beam is only the most efficient shape in one direction of bending: up and down looking at the profile as an I. If the beam is bent side to side, it functions as an H where it is less efficient. The most efficient shape for both directions in 2D is a box (a square shell) however the most efficient shape for bending in any direction is a cylindrical shell or tube. But, for unidirectional bending, the I or wide flange beam is superior.

Efficiency means that for the same cross sectional area (volume of beam per length) subjected to the same loading conditions, the beam deflects less.

Other shapes, like L (angles), C (channels) or tubes, are also used in construction when there are special requirements.

IIFormulae for deflection of different types of beams
BeamFormulas.pdf
IIIDeflection of cantilever Beam with a couple moment at the free end

Cantilever Beam – Couple moment M at the free end
IVDeflection of beam simply supported at ends with a couple moment at the right end

Beam Simply Supported at Ends – Couple moment M at the right end
VDeflection of Beam Simply Supported at Ends – Concentrated load at the center

Deflection of Beam Simply Supported at Ends – Concentrated load at the center
VIDetermination of slopes and maximum deflection of a simply supported beam with a uniformly varying load (UVL)

Determination of slopes and maximum deflection of a simply supported beam with a uniformly varying load (UVL)
VIIDetermination of slope and deflection at free end of a cantilever beam with uniformly distributed load from free end

Determination of slope and deflection at free end of a cantilever beam with uniformly distributed load from free end
VIIIDetermination of maximum slope and maximum deflection of a simply supported beam with a uniformly distributed load (UDL)

Determination of maximum slope and maximum deflection of a simply supported beam with a uniformly distributed load (UDL)
IXMaximum slope and maximum deflection of a simply supported beam with load at centre (midpoint)

Maximum slope and maximum deflection of a simply supported beam with load at centre (midpoint)
XDetermination of deflection and slope at the end of a cantilever beam carrying a uniformly varying load ( U.V.L)

Determination of deflection and slope at the end of a cantilever beam carrying a uniformly varying load ( U.V.L)
XIDeflection of Beam Simply Supported at Ends-Triangular load

Deflection of Beam Simply Supported at Ends-Triangular load
XIIStrength of Materials: Beam Deflection
http://www.youtube.com/watch?v=8rzdO8i9u2Y