Thermal Stress

Temperature changes cause the body to expand or contract. The amount δT, is given by
 

$ \delta_T = \alpha L \, (T_f \, - \, T_i) = \alpha L \, \Delta T $

where α is the coefficient of thermal expansion in m/m°C, L is the length in meter, Ti and Tf are the initial and final temperatures, respectively in °C. For steel, α = 11.25 × 10-6 m/m°C.
 

If temperature deformation is permitted to occur freely, no load or stress will be induced in the structure. In some cases where temperature deformation is not permitted, an internal stress is created. The internal stress created is termed as thermal stress.
 

Solution to Problem 257 Statically Indeterminate

Problem 257
Three bars AB, AC, and AD are pinned together as shown in Fig. P-257. Initially, the assembly is stress free. Horizontal movement of the joint at A is prevented by a short horizontal strut AE. Calculate the stress in each bar and the force in the strut AE when the assembly is used to support the load W = 10 kips. For each steel bar, A = 0.3 in.2 and E = 29 × 106 psi. For the aluminum bar, A = 0.6 in.2 and E = 10 × 106 psi.
 

Figure 257

 

Derivation of Sum of Finite and Infinite Geometric Progression

Geometric Progression, GP
Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.
 

Eaxamples of GP:

  • 3, 6, 12, 24, … is a geometric progression with r = 2
  • 10, -5, 2.5, -1.25, … is a geometric progression with r = -1/2

 

Derivation of Sum of Arithmetic Progression

Arithmetic Progression, AP
Definition

Arithmetic Progression (also called arithmetic sequence), is a sequence of numbers such that the difference between any two consecutive terms is constant. Each term therefore in an arithmetic progression will increase or decrease at a constant value called the common difference, d.
 

Examples of arithmetic progression are:

  • 2, 5, 8, 11,... common difference = 3
  • 23, 19, 15, 11,... common difference = -4

 

Derivation of Formulas
Let

Sum and Product of Roots

The quadratic formula
 

$ x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} $

 

give the roots of a quadratic equation which may be real or imaginary. The ± sign in the radical indicates that
 

$ x_1 = \dfrac{-b + \sqrt{b^2-4ac}}{2a} $   and   $ x_2 = \dfrac{-b - \sqrt{b^2-4ac}}{2a} $

 

where x1 and x2 are the roots of the quadratic equation ax2 + bx + c = 0. The sum of roots x1 + x2 and the product of roots x1·x2 are common to problems involving quadratic equation.
 

Derivation of Quadratic Formula

The roots of a quadratic equation ax2 + bx + c = 0 is given by the quadratic formula
 

$ x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

 

The derivation of this formula can be outlined as follows:

  1. Divide both sides of the equation ax2 + bx + c = 0 by a.
  2. Transpose the quantity c/a to the right side of the equation.
  3. Complete the square by adding b2 / 4a2 to both sides of the equation.
  4. Factor the left side and combine the right side.
  5. Extract the square-root of both sides of the equation.
  6. Solve for x by transporting the quantity b / 2a to the right side of the equation.
  7. Combine the right side of the equation to get the quadratic formula.

See the derivation below.
 

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