# Designing the Optimum Cast Iron T-Beam

• hatchelhoff
In summary, the homework statement is saying that cast iron has a strength in compression of about three to four times the strength in tension, depending upon the grade. The Attempt at a Solution section provides a diagram of a T section, which is designed to optimize the compressive stress by relating it to the tensile stress.
hatchelhoff

## Homework Statement

Cast iron has a strength in compression of about three to four times the strength in tension, depending upon the grade.
using the stress as a prime consideration, design the optimum T section for a cast iron beam using a uniform section thickness such that the compressive stress will be related to the tensile stress by a factor of 4

2. Homework Equations
TStress = (M*c1)/I
CStress = -(M*c2)/I

## The Attempt at a Solution

TSectionStress
Please see attached Diagram of T Beam

Let 1 = Top Rectangle
Let 2 = Bottom Rectangle
Let T = Thickness = 10
Let H1 = T = Height of top
Let H2 = height of bottom
Let HT = Total height of T section = T + H2
Let B1 = Length of Top
Let B2 = length of bottom = T
Let A1 = Top Area = B1 * T
Let A2 = Bottom area = H2 * T
Let AT = Total Area = A1 + A2
Let C1 = centroid from the top of 1 = ((T * B1 * (T / 2)) + ((T * H2) * (T + (H2 / 2)))) / AT
Let C2 = Centroid from the bottom of 2 = HT - C1
Let I1 = Second moment of area of top rectangle = (((B1 / 10) * ((H1 ^ 3) / 10)) / 12) / 100
Let I2 = Second moment of area of bottom rectangle = (((B2 / 10) * ((H2 ^ 3) / 10)) / 12) / 100
Let D1 = Distance from C1 to 1 axis = C1 - (T / 2)
Let D2 = Distance from C2 to 2 axis = C2 - (H2 / 2)
Let IX1 = Second moment of area about any parallel axis to the c1 axis a distance d1 removed = (I1 + ((A1 / 100) * ((D1 / 10) ^ 2)))
Let IX2 = Second moment of area about any parallel axis to the c2 axis a distance d2 removed = (I2 + ((A2 / 100) * ((D2 / 10) ^ 2)))
let I = IX1 + IX2
let StressT = Tensile Stress = ((1600 * (C1 / 10)) / I)
Let StressC = Compressive Stress = ((1600 * (C2 / 10)) / I)

Attempt at a solution

StressC/StressT = 4

((1600 * (C2 / 10)) / I)/((1600 * (C1 / 10)) / I) = 4
Therefore
C1 = C2/4 and C2 =4*C1

But from above C2 = HT-C1
So
((T * B1 * (T / 2)) + ((T * H2) * (T + (H2 / 2)))) / AT= HT-C1/4

The above equation should give me H2
but I can only solve it to get 0=0
What am I doing wrong.

#### Attachments

• T Diagram.png
3.9 KB · Views: 483
Solution
((T * B1 * (T / 2)) + ((T * H2) * (T + (H2 / 2)))) / (B1 * T) + (H2 * T)
Put the above equation for C1 into the form of a quadratic equation in terms of H2
Then put in values for T and B1.
Use the Quadratic formula the find H2.

## What is a cast iron T-beam?

A cast iron T-beam is a structural member commonly used in construction to support heavy loads. It consists of a horizontal top flange, a vertical web, and a horizontal bottom flange, giving it a T-shaped cross section.

## What are the advantages of using cast iron for T-beams?

Cast iron has high compressive strength and is able to withstand heavy loads, making it a suitable material for T-beams. It is also relatively inexpensive and has good corrosion resistance.

## What factors should be considered when designing a cast iron T-beam?

The design of a cast iron T-beam should take into account the loads it will be supporting, the span and length of the beam, and the type of connections used. Other factors include the type and grade of cast iron used, the shape and size of the cross section, and the presence of any pre-stressing.

## How can the optimum design for a cast iron T-beam be determined?

The optimum design for a cast iron T-beam can be determined by considering the load and span requirements, as well as the desired strength and stiffness of the beam. This can be achieved through structural analysis and calculations, taking into account the various design factors mentioned above.

## What are some common design guidelines for cast iron T-beams?

Some common design guidelines for cast iron T-beams include maintaining a minimum thickness for the top and bottom flanges, providing adequate reinforcement in the web, and ensuring a proper connection between the beam and its supports. It is also important to consider the potential for cracking and deflection in the design process.

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