superposition method

Problem 715 | Distributed loads placed symmetrically over fully restrained beam

Problem 12
Determine the moment and maximum EIδ for the restrained beam shown in Fig. RB-012. (Hint: Let the redundants be the shear and moment at the midspan. Also note that the midspan shear is zero.)
 

715-restrained-beam-symmetrical-uniform-loads.gif

 

Problem 714 | Triangular load over the entire span of fully restrained beam

Problem 714
Determine the end moments of the restrained beam shown in Fig. P-714.
 

714-restrained-beam-triangular-load.gif

 

Solution
$\delta_A = 0$

$\delta_{triangular\,\,load} - \delta_{fixed\,\,end\,\,moment} - \delta_{reaction\,\,at\,\,A} = 0$
 

Problem 713 | Fully restrained beam with symmetrically placed concentrated loads

Problem 713
Determine the end moment and midspan value of EIδ for the restrained beam shown in Fig. PB-010. (Hint: Because of symmetry, the end shears are equal and the slope is zero at midspan. Let the redundant be the moment at midspan.)
 

713-fully-restrained-beams-symmetrical-point-loads.gif

 

Problem 712 | Propped beam with initial clearance at the roller support

Problem 712
There is a small initial clearance D between the left end of the beam shown in Fig. P-712 and the roller support. Determine the reaction at the roller support after the uniformly distributed load is applied.
 

712-propped-beam-with-clearance.gif

 

Problem 711 | Cantilever beam with free end on top of a simple beam

Problem 711
A cantilever beam BD rests on a simple beam AC as shown in Fig. P-711. Both beams are of the same material and are 3 in wide by 8 in deep. If they jointly carry a load P = 1400 lb, compute the maximum flexural stress developed in the beams.
 

The ends of cantilever beam rests on top of simple beam at the third point.

 

Problem 708 | Two Indentical Cantilever Beams

Problem 708
Two identical cantilever beams in contact at their ends support a distributed load over one of them as shown in Fig. P-708. Determine the restraining moment at each wall.
 

Two cantilever beams.

 

Problem 706 | Solution of Propped Beam with Decreasing Load

Example 03
The propped beam shown in Fig. P -706 is loaded by decreasing triangular load varying from wo from the simple end to zero at the fixed end. Find the support reactions and sketch the shear and moment diagrams
 

Propped with decreasing load from w at simple support to zero at the fixed end.

 

Problem 705 | Solution of Propped Beam with Increasing Load

Problem 705
Find the reaction at the simple support of the propped beam shown in Fig. P-705 and sketch the shear and moment diagrams.
 

Propped beam loaded with triangular or uniformly varying load

 

Problem 704 | Solution of Propped Beam

Problem 704
Find the reactions at the supports and draw the shear and moment diagrams of the propped beam shown in Fig. P-704.
 

704-propped-beam-uniform-load.gif

 

Application of Double Integration and Superposition Methods to Restrained Beams

Superposition Method

There are 12 cases listed in the method of superposition for beam deflection.

  • Cantilever beam with...
    1. concentrated load at the free end.
    2. concentrated load anywhere on the beam.
    3. uniform load over the entire span.
    4. triangular load with zero at the free end
    5. moment load at the free end.
  • Simply supported beam with...
    1. concentrated load at the midspan.
    2. concentrated load anywhere on the beam span.
    3. uniform load over the entire span.
    4. triangular load which is zero at one end and full at the other end.
    5. triangular load with zero at both ends and full at the midspan.
    6. moment load at the right support.
    7. moment load at the left support.

See beam deflection by superposition method for details.
 

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