Problem $12\mathrm{.}1$

Construct the envelope of maximum positive and negative moments and determine the

absolute maximum positive and negative moments for a moving uniformly distributed

live load of $8$kN$//$m acting on the continuous beam in the figure.

Draw the influence line diagram for the vertical reaction at support B$.$

Draw the influence line diagram for the bending moment at point B in the first

span. Apply the method of sections by cutting the beam at B$.$ Sketch the FBD for the

left portion of the beam and write an equilibrium equation for M$\_($B$)$ for one position of

the unit load. Compute the positive and negative areas under the IL diagram.

Tip: The direction of the M$\_($B$)$ curved arrow in the FBD should correspond to the

positive sign convention for bending moment.

Draw the influence line diagram for the bending moment at point C in the second

span. Apply the method of sections by cutting the beam at C$.$ Sketch the FBD for the

left portion of the beam and write an equilibrium equation for M$\_($C$)$ for one position of

the unit load. Compute the positive and negative areas under the IL diagram.

Tip: The direction of the Mc curved arrow in the FBD should correspond to the

positive sign convention for bending moment. In the equilibrium equation use the

value of By from the influence line of the vertical reaction at B$.$

Determine the values of each influence line for positions of the unit load every eight

meters. Report the influence line values for steps $1),2)$ and $3)$ in the table.

$5)$ Draw the bending moment diagram for load case $1($uniformly distributed live load acting on the first span$).$

Tip: The bending moment at B can be computed by multiplying the magnitude of the uniformly distributed live load by the negative area under the IL diagram of $\backslash (\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$B$\}\}\backslash ).$

$6)$ Draw the bending moment diagram for load case $2($uniformly distributed live load acting on the second span$).$

Tip: The bending moment at C can be computed by multiplying the magnitude of the uniformly distributed live load by the positive area under the IL diagram of $\backslash (\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$C$\}\}\backslash ).$