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2E4: SOLIDS & STRUCTURES Lecture 2

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2E4: SOLIDS & STRUCTURESLecture 2

Dr. BidishaGhosh

Notes: http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/Solids & Structures

Statics or the bahaviour of the rigid bodies under external load is studied using the three Newton’s Laws (friction is considered as well)

Mechanics of solids deals with the internal changes/effects of a body(deformable) to the application of external load.

Mechanics of Solids/Strength of materials/ Mechanics of Materials/ Mechanics of deformable bodies

There are 6 DOF for a rigid body;

When multiple rigid bodies are connected to develop a system, then the number of DOF increases according to the number of bodies in the system.

At the point of connection of two rigid bodies some DOF are suppressed : ‘The concept of CONSTRAINTS’

http://www.beam-wiki.org/wiki/Degree_of_freedom

http://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)

If a support prevents translation of a body in a given direction, a reaction force is developed on the body in that direction.

If a support prevents rotation of a body in a given direction, a reaction moment is developed on the body in that direction.

3 common types: roller, pinned/hinged, fixed

When a body is rigid, it does not deform.

When the system of forces acting on a body has a zero resultant, then the body is in equilibrium.

Equilibrium of forces and moments:

6 DOF: There can be 3 unknown displacements along axes and 3 unknown rotations about axes. (Consider Cartesian Axis, 3D)

Number of unknown displacements and rotations = 6

Sums of forces along each axis = 0 give rise to 3 equations.

Sums of moments about each axis = 0 give rise to 3 equations.

Number of available equations = no of unknowns: Unique solution.

For (2D) planar problems,

Force: Sum of forces in each direction is 0

Moment: Sum of moments about a point is 0

3 equations and 3 unknowns

If a structure/machine can be solved using the equations of equilibrium, then the structures are statically determinate.

http://wwwtw.vub.ac.be/werk/Mechanicasite/Statica/equilib/iv-vi.htm

http://www.iowadot.gov/subcommittee/bridgeterms.aspx

What external loads are given?

What are the support reactions?

Typically, the reactions are the unknowns of the problem.

Draw Free Body Diagram

The rigid bodies should be in force balance. Check if there is any unbalanced force and sort it!

Write down the equations of force equilibrium

Define positive direction

Sum up all forces in x-direction,

Sum up all forces in y-direction,

Take a point and sum up the moments about it,

Solve the equations to find the unknowns.

What happens when bodies are not rigid but deformable??

Three main concepts:

Stress

Strain

Hooke’s Law

What happens to deformable bodies when load is applied?

Next two slides on what are the different types of loadings possible!

There are four basic types of loading (in order of complexity).

Tension

Compression

Torsion

Bending

Sometimes, two or more basic types of loading can act simultaneously on a member of a structure or machine.

This is a compression testing machine.

The different members are under different types of loading.

The specimen tested is under compression.

The two side bars (N) are under tension.

The screw is subjected to twist or torsion.

The crosshead is under bending.

25th May, 2010

A body responds to the application of external forces by deforming and by developing an internal force system, the body keeps changing shape or deforming until equilibrium is reached between the external and internal forces.

The intensity of deformation is called STRAIN.

Deformation per unit length is called strain.

25th May, 2010

Strain is the most significant factor in deformation analysis.

Strain is involved in the experimental technique of measuring stress. Stress is not measurable, but strain is.

Strain is a dimensionless and often expressed in 10-6 (microstrain)

Deformation should be compatible. Each deformed portion of the member must fit together with adjacent portion.

A body responds to the application of external forces by deforming and by developing an internal force system to hold together the particles which forms the body.

The intensity of internal force is called STRESS.

The bar subjected to force P

From equilibrium, along the section aa, the particles/fibres are subjected to force P in aggregate.

Assuming uniform cross-section, force per unit area

(total area A) ,

Diagram : Mechanics of materials (T.A. Philpot)

25th May, 2010

A material which regains its shape when the external load is removed is considered as ‘perfectly elastic’.

From tensile tests, it can be seen within the range of elastic behaviour of a material the elongation is proportional to both the external load and the length of the bar.

For linearly elastic materials, this Stress is proportional to strain.

The factor of proportionality between stress and strain is called, ‘Modulus of Elasticity’ or Young’s modulus.

E has the dimension of stress

How do we analyse the behaviour of deformable bodies when external loads are applied on them??

THEORY OF ELASTICITY

This course is an introduction to the ‘Theory of Elasticity’

The principals of analysis of deformable bodies depend on:

Equilibrium Conditions: Things should be in equilibrium

Material Behaviour: Things should follow a force-deformation relation

Compatibility: Deformations from all sides must match