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

where α is the coefficient of thermal expansion in m/m°C, L is the length in meter, T_i and T_f 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.
 

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.
 

Problem 256
Three rods, each of area 250 mm², jointly support a 7.5 kN load, as shown in Fig. P-256. Assuming that there was no slack or stress in the rods before the load was applied, find the stress in each rod. Use E_st = 200 GPa and E_br = 83 GPa.
 

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

Arithmetic Progression, AP
Definition

Pythagorean Theorem
In any right triangle, the sum of the square of the two perpendicular sides is equal to the square of the longest side. For a right triangle with legs measures $a$ and $b$ and length of hypotenuse $c$, the theorem can be expressed in the form
 

The quadratic formula
 

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

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.
 

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

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.
 

Problem 255
Shown in Fig. P-255 is a section through a balcony. The total uniform load of 600 kN is supported by three rods of the same area and material. Compute the load in each rod. Assume the floor to be rigid, but note that it does not necessarily remain horizontal.
 

Problem 254
As shown in Fig. P-254, a rigid bar with negligible mass is pinned at O and attached to two vertical rods. Assuming that the rods were initially stress-free, what maximum load P can be applied without exceeding stresses of 150 MPa in the steel rod and 70 MPa in the bronze rod.