Engineering

Civil EngineeringSteel Design (Activate Learning with these NEW titles from Engineering!)A W 18 × 40 floor beam supports a 4-inch-thick reinforced concrete slab with an effective width b of 81 inches. Sufficient anchors are provided to make the beam fully composite. The 28-day compressive strength of the concrete is f c ' = 4 ksi. a . Compute the moment of inertia of the transformed section. b . For a positive service load moment of 290 ft-kips, compute the stress at the top of the steel (indicate whether tension or compression), the stress at the bottom of the steel, and the stress at the top of the concrete.BuyFind*launch*

6th Edition

Segui + 1 other

Publisher: Cengage Learning

ISBN: 9781337094740

Chapter 9, Problem 9.1.1P

Textbook Problem

A
*b* of 81 inches. Sufficient anchors are provided to make the beam fully composite. The 28-day compressive strength of the concrete is

**a**. Compute the moment of inertia of the transformed section.

**b**. For a positive service load moment of 290 ft-kips, compute the stress at the top of the steel (indicate whether tension or compression), the stress at the bottom of the steel, and the stress at the top of the concrete.

Expert Solution

To determine

**(a)**

The moment of inertia of the transformed section.

**Given:**

A W 18 X 40 floor beam, the 28-day compressive strength of the concrete is

**Calculation:**

We have the modulus of elasticity of concrete as follows:

Where, the modulus of elasticity of concrete is

unit weight of concrete is

the 28-day compressive strength of concrete is

Substitute

Modular ratio by using the following formula:

Where, the modulus of elasticity of concrete is

the modulus of elasticity of steel is

and n is the modular ratio.

Substitute

Now the section will be transformed.

Since the modulus of elasticity of concrete can only be approximated, the usual practice of rounding n to the nearest whole number is sufficiently accurate. Thus,

The transformed width of the section is as following:

Where,

Substitute

Following is the section for the given set of conditions:

Data from the steel code:

DesignationImperial (in x lb/ft) |
Depthh (in) |
Widthw (in) |
Web Thicknesst_{w} (in) |
Flange Thicknesst_{f} (in) |
Sectional Area (in^{2}) |
Weight (lb_{f}/ft) |
Static Parameters |
|||

Moment of Inertia |
Elastic Section Modulus |
|||||||||

I_{x} (in^{4}) |
I_{y} (in^{4}) |
S_{x} (in^{3}) |
S_{y} (in^{3}) |
|||||||

W 18 x 40 | 17.9 | 6.02 | 0.315 | 0.525 | 11.8 | 40 | 612 | 19.1 | 68.4 | 6.4 |

The transformed section is shown in the above figure. The neutral axis is not known yet whether it lies in the steel or the concrete.

The location of the neutral axis can be found by applying the principle of moments with the axis of moments at the top of the slab. The computations are summarized in Table, and the distance from the top of the slab to the centroid is:

Component | |||

Concrete | |||

W 18 X 40 | |||

Sum |

The distance of centroid of the section is as follows:

Substitute

Since this is greater than 4 inches (the thickness of the slab) the neutral axis lies below the slab in the web. Applying the parallel axis theorem and tabulating the computations in table, we obtain the moment of inertia of the transformed section as:

Component | |||||

Concrete | |||||

W 18 X 40 | |||||

Sum |

**Conclusion:**

Therefore, total moment of inertia of the transformed section is

Expert Solution

To determine

**(b)**

The stress at the top of the steel (indicate whether tension or compression), the stress at the bottom of the steel, and the stress at the top of the concrete.

**Given:**

Positive service load moment of 290 ft-kips.

**Calculation:**

For the computation of the stress at top of the steel is as following:

Where, the stress at the top of the steel is

The distance from neutral axis to top of steel section is as follows:

Here the thickness of the concrete slab is t.

Compute the stress at top of the steel as:

As, the centroid lies below the top of the steel, the stress is compressive.

Now, the stress at the bottom of the steel is as follows:

The distance from the neutral axis to bottom of steel section.

Where,

As, the centroid lies above the top of steel, the stress is tensile.

Stress at the top of the slab is as follows:

Where, n is the modular ratio is n.

As the concrete slab is above the neutral axis, hence the stress is compressive.

**Conclusion:**

Therefore, the stress at top of steel section is

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