# 7.63 and 7.64 Use Castigliano’s second theorem to determine the slope and deflection at point D of the beam shown in Figs. P7.34 and P7.35. FIG. P7.35, P7.64

### Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931

Chapter
Section

### Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 7, Problem 64P
Textbook Problem
120 views

## 7.63 and 7.64 Use Castigliano’s second theorem to determine the slope and deflection at point D of the beam shown in Figs. P7.34 and P7.35.FIG. P7.35, P7.64

To determine

Find the slope and deflection at point D of the beam using Castigliano’s second theorem.

### Explanation of Solution

Given information:

The beam is given in the Figure.

Value of E is 30,000 ksi, I is 4,000in.4 for span AB, and 3,000in.4 for span BD.

Apply the sign conventions for calculating reactions, forces and moments using the three equations of equilibrium as shown below.

• For summation of forces along x-direction is equal to zero (Fx=0), consider the forces acting towards right side as positive (+) and the forces acting towards left side as negative ().
• For summation of forces along y-direction is equal to zero (Fy=0), consider the upward force as positive (+) and the downward force as negative ().
• For summation of moment about a point is equal to zero (Matapoint=0), consider the clockwise moment as negative and the counter clockwise moment as positive.

Calculation:

Let apply a load P and couple M¯ at point D in the desired direction to find the deflection and slope.

The value of load P is 35 k and couple M¯ is zero.

Sketch the beam with load P and couple M¯ as shown in Figure 1.

Let the equation for bending moment at distance x in terms of load P be M, the derivative of M with respect to P is MP.

The derivative of M with respect to M¯ is MM¯.

Find the reactions and moment at the supports:

Consider portion BCD, Summation of moments about B is equal to 0.

MB=08CyP(16)M¯=08Cy=16P+M¯Cy=2P+M¯8

Summation of forces along y-direction is equal to 0.

+Fy=0Ay+Cy2.5(16)P=0Ay+2P+M¯82.5(16)P=0Ay=40PM¯8

Summation of moments about A is equal to 0.

MA=0MAP(32)M¯+Cy(24)2.5(16)(162)=0MA32PM¯+48P+3M¯320=0MA=32016P2M¯

Find the equations for M, MM¯, and MP for the 3 segments of the beam as shown in Table 1.

 Segment x-coordinate M ∂M∂M¯ ∂M∂P Origin Limits (ft) DC D 0−8 −Px−M¯ −1 −x CB D 8−16 −Px−M¯+(2P+M¯8)(x−8) −1+18(x−8) x−16 AB A 0−16 −(320−16P−2M¯)+(40−P−M¯8)x−1.25x2 2−x8 16−x

The expression for slope at D using Castigliano’s second theorem (θD) is shown as follows:

θD=0L(MM¯)MEIdx (1)

Here, L is the length of the beam.

Rearrange Equation (1) for the limits 08, 816, and 016 as follows.

θD=1E[1I08(MM¯)Mdx+1I816(MM¯)Mdx+1I016(MM¯)Mdx]

Substitute the value of load P as 35 k and couple M¯ as 0 in the column 4 of Table 1.

Substitute 1 for MM¯, 35x for M for the limit 08, 1+18(x8) for MM¯, 35x+70(x8) for M for the limit 816, 2x8 for MM¯, 4I3 for I, and 240+5x1.25x2 for M for the limit 016.

θD=1E[1I08(1)(35x)dx+1I816(1+18(x8))(35x+70(x8))dx+143I016(2x8)(240+5x1.25x2)dx]=1E[1I08(35x)dx+1I816(70x140x+1,12035x28+70x28560x8)dx+34I016(480+10x2

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts
Give examples of properties that engineers consider when selecting a material for a product or an application.

Engineering Fundamentals: An Introduction to Engineering (MindTap Course List)

operating system

Enhanced Discovering Computers 2017 (Shelly Cashman Series) (MindTap Course List)

What steps are required to create an ODBC data source name?

Database Systems: Design, Implementation, & Management

Define security and privacy. How are the two concepts related?

Database Systems: Design, Implementation, & Management

List three methods of applying cutting fluids.

Precision Machining Technology (MindTap Course List)

What range of leakage current must trip an IDCI?

EBK ELECTRICAL WIRING RESIDENTIAL

If your motherboard supports ECC DDR3 memory, can you substitute non-ECC DDR3 memory?

A+ Guide to Hardware (Standalone Book) (MindTap Course List)