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HW$-12$: Ch$.12-$ Deflection

Slope and Displacement by Integration

$2$ of $11$

Review

Learning Goal:

To determine the general equation for the elastic curve for a cantilevered beam that is subjected to a distributed load w

and a concentrated load P

$.$

A cantilevered beam is loaded as shown. Because of the discontinuities from the different types of loading on the beam, three coordinate systems, x$1$

$,$ x$2$

$,$ and x$3$

$,$ are used to define the locations of the internal moments. Assume EI

is constant. Note:that the coordinate systems are at arbitrary distances and not necessarily at the locations shown in the figure.$($Figure $1)$

Figure$1$ of $1$

The figure shows horizontal beam A B fixed to a vertical wall at A$.$ Length of the beam is $2$ times a$.$ Point C is marked on the beam at distance a halved from A$.$ Point D is marked on the beam at distance a from A$.$ Vertical downward force P acts on the beam at C$.$ Vertically downward distributed load w acts on the beam in segment D B$.$ Coordinate x subscript $1$ is measured along the beam from point A in segment A C$.$ Coordinate x subscript $2$ is measured along the beam from point A in segment C D$.$ Coordinate x subscript $3$ is measured along the beam from point A in segment D B$.$

Part A $-$ Moment reaction at the wall

Before finding the equations of the moment, the moment reaction at A must be determined.

Express the moment reaction in terms of a

$,$ w

$,$ and P

in simplified form.

View Available Hint$($s$)$for Part A

Activate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value type

MA

$=$

nothing

Part B $-$ Equation of the elastic curve using x$1$ over section AC

To find the elastic curve for the entire beam, the equation that uses the coordinate system x$1$

must be determined for section AC of the beam. Write this equation in terms of EI

$,$ w

$,$ a

$,$ and P

$.$ Let C$1$

and C$2$

denote the constants of integration.

Express your answer in terms of EI

$,$ w

$,$ a

$,$ P

$,$ C$1$

$,$ and C$2$

$.$

View Available Hint$($s$)$for Part B

Activate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value type

v$($x$1)$

$=$

nothing

Part C $-$ Equation of the elastic curve using x$2$ over section CD

To find the elastic curve for the entire beam, the equation that uses the coordinate system x$2$

must be determined for section CD$.$ Write this equation in terms of EI

$,$ w

$,$ and a

$.$ Let C$3$

and C$4$

denote the constants of integration.

Express your answer in terms of EI

$,$ w

$,$ a

$,$ C$3$

$,$ and C$4$

$.$

View Available Hint$($s$)$for Part C

Activate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value type

v$($x$2)$

$=$

nothing

Part D $-$ Equation of the elastic curve using x$3$ over section DB

To find the elastic curve for the entire beam, the equation that uses the coordinate system x$3$

must be determined for section DB$.$ Write this equation in terms of EI

$,$ w

$,$ and a

$.$ Let C$5$

and C$6$

denote the constants of integration.

Express your answer in terms of EI

$,$ w

$,$ a

$,$ C$5$

$,$ and C$6$

$.$

View Available Hint$($s$)$for Part D

Activate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value type

v$($x$3)$

$=$

nothing

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