ME 201 Engineering Mechanics: Statics

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1 ME 01 Engineering Mechanics: Statics Unit 5.3 Reduction of a Simple Distributed Loading

2 Distributed Loads Thus far we ve been working with loads that are concentrated at a point: Many times in engineering we need to be concerned with another type of loading referred to as distributed loading: 10 N/m 6 kn/m

3 Distributed Loads Instead of being concentrated at a point, a distributed load is spread out over a distance It can be thought of as a collection of smaller loads To find F R, we need to sum an infinite number of small forces

4 Distributed Loads Consider a small differential element, df with a width of d w df and a height of w() w() The area of the element is df w( ) d d Since infinite number of forces, need to integrate to find F R F R w( ) d l d

5 Centroids for Simple Shapes Where is the centroid for these common shapes? h h c y c b h b b b c 3 h y c 3

6 Given: trapezoid Find: F R, Solution: FD F R Eample Problem Solution 100 lb/ft 9 ft F tri F rect 50 lb/ft 1 F tri (100 50) lb / ft9 5lb F rect 50lb / ft9 ft 450lb ft 9 ft F R 9 ft FR Ftri Frect 675lb

7 Given: trapezoid Find: F R, Solution: FD F R 1 tri ft 1 rect ft ft ft Eample Problem Solution F tri 5lb F rect 450lb F R 675lb ft 100 lb/ft 9 ft F tri F rect 9 ft F R 9 ft 50 lb/ft

8 Given: trapezoid Find: F R, Solution: F R integral Eample Problem Solution R F w( ) d L 60 d N w=60 N/m m

9 Eample Problem Solution Given: Find: trapezoid F R, Solution: F R integral integral L L 0 0 w( ) d w( ) d d d w=60 N/m m 1.5m

13 Distributed Loads Thus far we ve been working with loads that are concentrated at a point: Many times in engineering we need to be concerned with another type of loading referred to as distributed loading: 10 N/m 6 kn/m

14 Distributed Loads Instead of being concentrated at a point, a distributed load is spread out over a distance It can be thought of as a collection of smaller loads

15 Distributed Loads To compute the resultant, F R, of a distributed load, consider the following: To find F R, we need to sum an infinite number of small forces

16 Distributed Loads Consider a small differential element, df with a width of d and a height of w() w df w() d

17 Distributed Loads w df w() The area of the element is df w( ) d Since infinite number of forces, need to integrate to find F R F R w( ) d l d d

18 Distributed Loads Where does F R act? Can be determined by equating the moments of the force resultant and the force distribution F w( ) d R l l w ( ) F R d This is also the centroid of the area l w( ) d l w( ) d F w( ) d R l d d

19 Centroids For simple shapes, centroid can be found in a table (see back cover of tetbook) Where is the centroid for these common shapes? h h c y c b h b b b c 3 h y c 3

20 Centroids of Simple Shapes Where does F R act? h. c = b/ 1 h. c > b/ b C. c < b/ D. Cannot tell from information given

21 Eample Problem Given: w=400 lb/ft w lb/ft Find: F R, 10 ft

22 Given: w=400 lb/ft Find: F R, Eample Problem Solution w lb/ft 10 ft

23 Eample Problem Solution Given: w=400 lb/ft Find: F R, Solution: FD F R F R 400lb / ft10 ft 4000lb or 4k 10 ft 5 ft w lb/ft 10 ft F R 10 ft

24 Eample Problem Given: 600 N/m w=100 N/m Find: F R, 6 m

25 Given: w=100 N/m Find: F R, Eample Problem Solution 6 m 600 N/m

26 Eample Problem Solution 600 N/m Given: w=100 N/m Find: F R, Solution: FD F R 1 F R 600 N / m6m 1800 Nm or 1. 8kNm 6 m 6 m F R 6 m 3 4 m

27 Eample Problem Given: 100 lb/ft trapezoid Find: 50 lb/ft F R, 9 ft

28 Given: trapezoid Find: F R, Eample Problem Solution 100 lb/ft 50 lb/ft 9 ft

29 Given: trapezoid Find: F R, Solution: FD F R Eample Problem Solution 100 lb/ft 9 ft F tri F rect 50 lb/ft 1 F tri (100 50) lb / ft9 5lb F rect 50lb / ft9 ft 450lb ft 9 ft F R 9 ft FR Ftri Frect 675lb

30 Given: trapezoid Find: F R, Solution: FD F R 1 tri ft 1 rect ft ft ft Eample Problem Solution F tri 5lb F rect 450lb F R 675lb ft 100 lb/ft 9 ft F tri F rect 9 ft F R 9 ft 50 lb/ft

31 Eample Problem Given: function Find: F R, w=60 N/m m

32 Given: function Find: F R, Eample Problem Solution w=60 N/m m

33 Given: trapezoid Find: F R, Solution: F R integral Eample Problem Solution R F w( ) d L 60 d N w=60 N/m m

34 Eample Problem Solution Given: Find: trapezoid F R, Solution: F R integral integral L L 0 0 w( ) d w( ) d d d w=60 N/m m 1.5m

35 In Class Eercise

36 Solution

37 In Class Eercise

38 In Class Eercise

Quizzam Module 1 : Statics

Quizzam Module 1 : Statics Structural Steel Design Quizzam odule : Statics NAE Draw shear and moment diagrams for the following loading conditions. Note the reactions. Calculate the maximum amount of internal bending moment. 0 500