MOVING LOADS

MECHANICS - CASE STUDY
		Introduction
A truck moving across a simply-supported bridge		After watching a truck driving across a bridge, Daniel, a new engineering student, wonders if it is possible to determine the worst bending moment. He knows how to find the bending moment when the load location is given, but not when it can be at any location.
Load Distribution on Wheels		What is known: Bridge is assumed to b simply-supported Bridge length is 50 m Wheel locations are 5 m, 10 m, and 12 m from the front wheel as shown in the diagram Total truck weight is 140 kN Truck weight is unevenly distributed to the wheels; front wheels, F1, carries 15%, second set of wheels, F2, carries 25%, and remaining weight is equally carried on the back two wheel sets, F3 and F4.
		Question:
		What is the maximum bending moment that occurs when the truck moves across the bridge? Note, the location of the truck that causes the maximum moment is not given and needs to be determined.
		Approach
		Determine the weight on each wheel. Find the position of total resultant force, 140 kN, with respect to one of the wheels. Find the position of the truck where maximum bending moment will occur. Using static equilibrium equations      Σ F = 0 and Σ M = 0 calculate reaction forces occurred at the support for a particular position of the truck. Again use the static equilibrium equations to calculate the bending moment at various sections in the beam. Then calculate the bending moments by taking forces at various positions

MECHANICS - CASE STUDY SOLUTION
		The fixed distances between the loads are determined by the wheel base of the vehicle but the location of the resultant force needs to be determined. Then the position of the truck where maximum bending moment will occur can be calculated.
Actual Wheel Force		The total truck weight of 140 kN is distributed 15% on front wheels, 25 % on second wheels and remaining equally on third and fourth wheel. The actual load on each wheel is    F1 = 15% of 140 kN = (140) (15) / 100 = 21 kN    F2 = 25% of 140 kN = (140) (25) / 100 = 35 kN    F3 = 30% of 140 kN = (140) (30) / 100 = 42 kN    F4 = 30% of 140 kN = (140) (30) / 100 = 42 kN
Resultant Force Location		Since the load on each wheel is known, the resultant force position can be found by summing moments about the first wheel, giving      ΣM1 = 0      35 (5) + 42 (15) + 42 (17) - 140 xR = 0      140 xR = 1519      xR = 10.85 m The resultant force is located at 10.85 meters away from first wheel.
Beam Support Reaction		The end reactions on the beam cannot be determined since the location of the truck is still unknown. However, the reactions can be modeled as a function of the truck location. Letting x represent the truck location, and summing the moments gives      ΣMB= 0      140 (50 - x - 10.85) - Ay (50) = 0      Ay = 109.62 - 2.8 x kN One of the major difficulties in solving this type of problem is the maximum moment location is not known. However, since the bridge is simply supported, and the loads are all point force loads, the maximum moment will occur under one of the point loads.
		To help minimize the number of calculations, it can be further assumed that the maximum moment will not occur under the outside force loads since they are equal to or less than the inside two force loads. This means, only the second wheel and third wheel will be examined for the maximum bending moment. This assumption will be verified at the end of the problem.
Location of Maximum Moment under Wheel 2		First, it is assumed that the maximum will occur at wheel 2 where the load is 35 kN. Cutting the beam at wheel two and summing moment at the cut, gives      M2 = Ay (x + 5) - 21 (5)      M2 = 95.62 x - 2.80 x2 + 443.1 Remember, the truck can be at any location along the beam, and thus this equation needs to be maximized as a function of x. This gives,      dM2/dx = 0 = 95.62 - 5.60 x      x = 17.075 m Using this value of x, M2 becomes,      M2 = 17.075 (95.62) - 2.80 (17.075)2 + 443.1      M2 = 1,259.5 kN-m
Location of Maximum Moment under Wheel 3		Second, assume the maximum moment occurs at wheel 3. The moment is      M3 = Ay (x + 15) - 21 (10 + 5) - 35 (10)      M3 = (109.62 - 2.8 x) (x + 15) - 665      M3 = -2.80 x2 + 67.62 x + 1,644.3 - 665 Again, the truck can be at any location. The maximum will occur when,      dM3/dx = 0 = 67.62 - 5.60 x      x = 12.075 m Using this value of x, M3 becomes,      M3 = -2.8 (12.075)2 + 67.62 (12.075)                           + 1,644.3 - 665      M3 = 1,387.5 kN-m Comparing M2 and M3, bending moment is maximum under wheel 3 when wheel 1 is at 12.075 meters away from support A.
Free-body Diagram of the System		For a complete understanding of the bending moment across the beam, a moment diagram can be constructed. Taking moments about support A and assuming counter clockwise (CCW) moment as positive gives,      ΣMA= 0      By (50) - (21) (12.075) - (35) (17.075)       - (42) (27.075) - (42) (29.075) = 0      By = 64.19 kN Equating forces gives,      ΣFy = 0      Ay + 64.19 - 21 - 35 - 42 - 42 = 0      Ay = 75.81 kN
Final Moment Diagram		Next, summing moment below each point load gives,      M1 = 75.81 (12.075)           = 915.4 kN-m      M2 = 75.81 (12.075 + 5) - 21 (5)           = 75.81 (17.075) - 21 (5)           = 1,189.45 kN-m      M3 = 75.81 (12.075 + 5 + 10) - 21 (5 + 10)                - 35 (10)           = 75.81 (27.075) - 21 (15) - 35 (10)           = 1,387.5 kN-m      M4 = 75.81 (12.075 + 5 + 10 + 2)                - 21 (5 + 10 + 2) - 35 (10 + 2) - 42 (2)           = 75.81 (29.075) - 21 (17) - 35 (12) - 84           = 1,343.2 kN-m Points A and B are supports so the bending moment at A and B is zero.     As determined previous, the maximum bending moment occur under wheel 3 and will have a magnitude of 1,387.5 (kN-m) and it occurs when truck (first wheel) is at 12.075 meters away from support A. The moments under each wheel can be plotted to give the moment diagram, as shown on the left. Notice the bending moment under the first and fourth wheel is less than the maximum moment under the third wheel, as expected. Rarely is the maximum bending moment largest under one of the outside point loads

The effect of moving loads is better demonstrated by a simulation. The bending moment diagram for a particular positions of forces can be observed by dragging the pink rectangle across the beam. This also updates maximum bending moment and position where the maximum bending moment occurs. The position is marked by red cross.  The effect of changes in the magnitude of forces and distances between the forces on bending moment diagram can also be seen. Notice, the maximum bending moment always occurs under one of the point loads. It also generally occurs under one of the inside point loads.