Monday, April 2, 2012

Structural Theory: Introduction

Overall, engineers design bridges, buildings, ships, machine parts, equipment and other structural installations. To design such it is a must to determine first the layout of the structural, its dictated future shape and constituent members. Estimation or determination of loads that the structure will carry will then proceed.

  • The theory of structure concerns about direct stress, shear area and bending moment and deflection at any section of the structure's constituents. After acquiring such, it is vital to design each members proportion to the allowable working stresses of the materials while complying to other requirements and limits for the proper function of the engineered structure.
Four Stages of Design:
  1. layout of the structure
  2. loadings that consists of dead load (own weight) and live load (which may include snow load, wind load, etc.)
  3. direct stresses that the members will carry
  4. sizes of the members that will bear the loads and stresses most costly efficiently within the bound limit
  • Layout and Classification. Structures layout largely depends on its function, loading conditions and the properties of the materials that will be used. The determination of the layout requires field experience and expertise, judgment and vast knowledge. After considering all the necessary elements in the preliminary design it will be test to determine the preferred design that meet even the unanticipated conditions that the structure might encounter during the stages of creation. Basic structure classification are inclusive but not limited to beams, rigid frames, trusses or combination of two or more of these elements. Generally, a beam is a structural member subjected to transverse loads only that bears shear and bending moment, hence, horizontally oriented. Rigid frames on the other hand is a structure composed of several members connected by rigid joints whether welded or bolted that is completely analyzed when all variations in direct stress, shear and bending moment along  the entire lengths of all members is acquired. Finally, a truss is a structure consists of several members connected by frictionless hinges that is completely analyzed with determined direct stresses. Other structures with members or machine parts are also subjected to react on direct stresses, shear and bending and twisting moments.
  • Loads. Structures are subjected to dead loads, live loads and dynamic outside force like nature and impact of live loads. Dead Load is the weight of the structure itself, it has to be first assumed since most of its part cannot be determined until the members have actually been design, then checked after the sizes of the members are finalized although changes are at minimum that in routine design it is seldom modified. Live Load also known as the moving loads are maximum loads to be carried by the structure. Since it is moving, the Impact or its Dynamic Effect is also considered because it is usually more critical than stagnant live loads only since there are times when the live load comes on rather suddenly as a moving, passing or rolling load.
  • Methods. Theory of Structures deals with the principles and methods wherein direct stress, shear and bending moment at any section of the member may be found under the conditions of loading. It is then assumed that the forces acting on each structural elements are on the same plane and in equilibrium.
    • General coplanar-force system:
      • Summation of Forces on X-axis = 0
      • Summation of Forces on Y-axis = 0
      • Summation of Moment at a certain joint or point = 0

Saturday, March 17, 2012

Economics: Interest

Interest: amount of money earned by given principal money.
Borrower's Viewpoint - interest is the amount of money to be paid for the use of a borrowed capital.
Lender's Viewpoint - interest is the income generated by the capital that was lent.

  • Simple Interest - varies directly with time since it is computed after or at the end of the invested period
    • Ordinary Simple Interest - base on one banker's year
    • Exact Simple Interest - based on exact number of days of a year, which considers ordinary year of 365 days and leap years that have 366 days
Formula: Future Worth = Present Worth + Interest Earned 
(F = P + i)

  • Compound Interest - computed every end of each interest period which is called compounding interest and the interest is added to the principal (which is now interest plus principal) that will produce a bigger interest 

Formula: Future Worth = Present Worth (1 + Interest Earned) ^total number of compounding period 
(F = P [1 + i]^n) 

Friday, March 9, 2012

Engineering Mechanics: Introduction

Engineering Mechanics - is a field of science that deals with forces and its effect on rigid bodies. Engineering Mechanics are subdivided into two parts:

  •  Statics - rigid bodies remains at rest while absorbing the effects and distributions of forces
    • Force System - any arrangement where in two or more forces act on a body or on group of related bodies. 
      • Three Major Divisions of Force System
        • Concurrent - all forces pass through a common point
        • Non-Concurrent - all of the lines of action of the forces in this system do not meet at one point
        • Parallel - forces whose line of action are parallel, in same or opposite direction
    • Applications - trusses, centroids, friction
  • Dynamics - rigid bodies motion caused by the force applied to it, it deals with bodies in motion
    • Kinematics - is the geometry of motion, the motion of a particle without considering the forces causing the motion
      • Motion of Particles
        • Translation - motion of rigid bodies where in a straight line pass through any two of its particles always remain to be parallel on its original position
        • Rotation - motion of rigid bodies where in the particles move in circular paths with centers or axis of rotation on a fixed straight line
        • Plane Motion - motion of rigid bodies where in all particles in the body remain at a constant distance form a fixed reference plane
    • Kinetics - relates the force action on the body to its mass and acceleration
      • Newton's Law of Motion
        • a body at rest will remain to be at rest or in motion will remain in motion along a straight path unless acted upon by unbalanced force
        • a particle acted upon by an unbalanced force system has an acceleration in line with and directly proportional to the resultant of the force system and inversely proportional to its mass
        • in every action there is always an equal and opposite reaction
      • D'Alembert's Principle
        • the resultant of the external forces applied to a body rigid or non-rigid composed of a system of particles is equivalent to the vector summation of the effective forces acting on all particles


Wednesday, March 7, 2012

Solid Geometry: Polyhedrons

Polyhedrons - solids whose faces are plane polygons.

Regular Polyhedrons - are polyhedrons that have identical faces. There are only five known kinds of polyhedrons.

Let:
a = length of the edge
n = number of vertices
f = number of faces
m = number of polygons meeting at a vertex

  • Tetrahedron - four faces - a pyramid
    • faces: 4
    • edges: 6
    • vertices: 4
    • number of polygons meeting at a vertex: 3
    • surface area: a square * square root of 3
    • volume: a cube / 6 * square root of 2 
  • Hexahedron - six faces - a cube
    • faces: 6
    • edges: 12
    • vertices: 8
    • number of polygons meeting at a vertex: 3
    • surface area: 6 * a square
    • volume: a cube
  • Octahedron - eight faces
    • faces: 6
    • edges: 12
    • vertices: 8
    • number of polygons meeting at a vertex: 4
    • surface area: 6 * a square
    • volume: a cube
  • Dodecahedron - twelve faces
    • faces: 12
    • edges: 30
    • vertices: 20
    • number of polygons meeting at a vertex: 3
  • Icosahedron - twenty faces
    • faces: 6
    • edges: 12
    • vertices: 8
    • number of polygons meeting at a vertex: 5
For any Polyhedron:
Number of Edges: nf/2
Number of Vertices: nf/m

Tuesday, March 6, 2012

Tips and Tricks: Multiplication


The multiplication table of 1 is a no-brainer, BUT, let's face it! even adults sometimes are having a hard time doing numbers in their heads!

I can mentally calculate tables 2 to 5, and table of 10 is as no-braner as the multiplication of 1. :D
Then comes, 6, 7, 8 and 9.

You might already knew this, and forgotten about is, so this will help you refresh a Math skill!
  • The Multiplication by Hands - applicable for 6, 7, 8, 9 and 10 as multiplier and multiplicand
    • Step 1 - look at your hands, your thumbs represents number 10, the index finger as 9 and so on, hence, your pinky is number 6.
    • Step 2 - let's set an example. So lets say its 7 x 8. (No calculators pls! that's cheating!) What you do is connect the fingers representing 7 and 8.
    • From the connecting fingers, count each fingers downwards by ten. So thats 10, 20, 30, 40, 50.
    • Step 3 - Count the remaining left fingers and add it to the sum of the remaining right fingers. so that's 3 x 2 = 6
      • Step 4 - Add the sum of step 5 and 6, so that will give you 50 + 6 = 56. Awesom I    know! :D
Take a look:
Photo Courtesy of helpingwithmath.com
I first observed that the digits of the answers if added will always be 9.
And aside from tables 1 and 10, 9 is a no-brainer too! you just have to know the trick!

Example: 9x7. Instead of counting 9 + 9 + 9 + 9 + 9 + 9 + 9... All you need to do is add zero to the digit you are multiplying to 9, in this case 7 will become 70. Then we have to minus that digit to the new set of number, 70 - 7 = 63.

In short: 8 x 9 = 80 - 8 = 72.

Get it, got it!

Do you have more math tricks to share? Comment down below, its time to share yours! Click like for more math tricks! :P

Monday, March 5, 2012

Algebra: Laws and Theorems of Equality and Inequality

Equality - two equations or quantities are of equal and the same value.
Inequality - two equations or quantities are lesser or greater than each other.


Basic Laws of Equality
  • Reflexive Property: x = x
  • Symmetric Property: if x = y, then y = x
  • Transitive Property: 
    • if x = y, and y = z, then, x = z. (x = y = z)
    • if w = x, and y = z, then, w + y = x + z.
    • if w = x, and y = z, then, wy = xz.
  • First Conclusion: things equal to the same thing are equal to each other.
  • Second Conclusion: equals added to equals are equals
  • Third Conclusion: equals multiplied to equals are equals

Basic Laws of Inequality
  • x > y;        x is greater than y
  • x < y;        x is less than y
  • x </= y;     x is less than or equal to y
  • x>/=y;       x is greater than or equal to y

Basic Theorems on Inequalities
  • x > y if and only if -x < -y
  • if x > 0, then -x < 0 
  • if -x > 0, then x < 0
  • if x > y, z < 0, then ac < bc
  • if x > y, w > z, then (x + w) > (y + z)
  • if x > y, w> z, and w, x, y, z > 0, then xw > yz
  • if x > 0, y > 0, x > y, then 1/x < 1/y

Saturday, March 3, 2012

Algebra: Law of Natural Numbers


Natural numbers refer to members of the set of positive integers or set of non-negative integers.

In Law of Natural Numbers, let x, y and z be any number, hence,

Law 1 - closure for addition: 
x + y

Law 2 - commutative law for addition: 
x + y = y + x

Law 3 - associative law for addition:
(x + y) + z = x + (y + z) = x + y + z

Law 4 - closure for multiplication:
x * y


Law 5 -  commutative law for multiplication:
x * y = y * x


Law 6 - associative law for multiplication:
x(y*z) = (x*y)z

Law 7 - distributive property of addition and multiplication:
x(y +z) = x*y + x*z