Moving Loads

Moving Loads From the previous section, section, we see that the maximum moment occurs at a point of zero shears. For beams loaded with concentrated loads, the point of zero shears usually occurs under a concentrated load and so the maximum moment. Beams and girders such as in a bridge or an overhead crane are subject to moving concentrated loads, which are at xed distance with each other. !he problem here is to determine the moment under each load when each load is in a position to cause a maximum moment. !he largest value of these moments governs the design of the beam. "ingle Moving Load For a single moving load, the maximum moment occurs when the load is at the midspan and the maximum shear occurs when the load is very near the support #usually assumed to lie over the support$.

!wo Moving Movin g Loads For two moving loads, the maximum shear occurs at the reaction when the larger load is over that support. !he maximum moment is given by

where %s is the smaller load, % b is the bigger load, and % is the total load #% & % s ' %b$. !hree or more moving loads (n general, the bending moment under a particular load is a maximum when the

center of the beam is midway between that load and the resultant of all the loads then on the span. )ith this rule, we compute the maximum moment under each load, and use the biggest of the moments for the design. *sually, the biggest of these moments occurs under the biggest load. !he maximum shear occurs at the reaction where the resultant load is nearest. *sually, it happens if the biggest load is over that support and as many a possible of the remaining loads are still on the span. (n determining the largest moment and shear, it is sometimes necessary to chec+ the condition when the bigger loads are on the span and the rest of the smaller loads are outside.

%roblem - A truck with axle loads of 40 kN and 60 kN on a wheel base of 5 m rolls across a 10-m span. Compute the maximum bendin moment and the maximum shearin force. "olution - !ideClick here to show or hide the solution

For maximum moment under / +0 wheel1

For maximum moment under 2/ +0 wheel1

 "hus#

answer 

!he maximum shear will occur when the 2/ +0 is over a support.

 "hus#

answer 

"olution -

Maximum moment under / +0 wheel1

Maximum moment under -/ +0 wheel1

 "hus#

answer 

!he maximum shear will occur when the -/ +0 is over a support.

 "hus#

answer 

Solution 455 HideClick here to show or hide the solution

When the midspan is midway between W r  and R, the front wheel W f   will be outside the span (see gure). n this case, only the rear wheel W r  ! "#$$ lb is the load. %he ma&imum moment for this condition is when the load is at the midspan.

Maximum moment under W r 

Maximum moment under W f 

%hus, answer 

The maximum shear will occur when the rear wheel (wheel of greater load) is directly over the support.

%hus, answer 

"olution -2

the resultant $ is 1% ft from & 1

Maximum moment under %3

Maximum moment under %4

Maximum moment under %

 "hus# answer 

!he maximum shear will occur when % 3 is over the support.

 "hus#

answer 

"olution -5

Maximum moment under 3/ +0 wheel load

Maximum moment under 4/ +0 wheel load

Maximum moment under / +0 wheel load 'hen the centerline of the beam is midwa( between reaction $ ) 60 kN and %0 kN# the 10 kN comes o* the span.

from +0 kN wheel load

 "hus# the maximum moment will occur when onl( the +0 and %0 kN loads are on the span. answer 

Maximum "hear  "he maximum shear will occur when the three loads are on the span and the %0 kN load is

directl( o,er the support.

 "hus# answer