five moments ELEMENTS OF ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SPRING 2014 lecture ARCH 614

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1 ELEMENTS OF ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SPRING 2014 lecture five moments Moments 1

2 Moments forces have the tendency to make a body rotate about an axis same translation but different rotation Moments 2

3 Moments Moments 3

4 Moments a force acting at a different point causes a different moment: Moments 4

5 Moments defined by magnitude and direction units: N m, k ft direction: + cw (!) F - ccw C A value found from F B d and distance M F d d also called lever or moment arm Moments 5

6 Moments with same F: M A F d M F d 1 A 2 (bigger) Moments 6

7 Moments additive with sign convention can still move the force along the line of action B A d d F = B A d d M A = F d M B = F d - F M A = F d M B = F d Moments 7

8 Moments Varignon s Theorem resolve a force into components at a point and finding perpendicular distances calculate sum of moments equivalent to original moment makes life easier! geometry when component runs through point, d=0 Moments 8

9 Moments of a Force moments of a force introduced in Physics as Torque Acting on a Particle and used to satisfy rotational equilibrium Moments 9

10 Physics and Moments of a Force my Physics book (right hand rule): Moments 10

11 Moment Couples 2 forces same size opposite direction distance d apart cw or ccw d d 2 d 1 F A M F d F not dependant on point of application M F d F d 1 2 Moments 11

12 Moment Couples equivalent couples same magnitude and direction F & d may be different 300 N 100 mm 300 N 200 N 120 N 120 N 200 N 150 mm 250 mm Moments 12

13 Moment Couples added just like moments caused by one force can replace two couples with a single couple 300 N 100 mm 300 N N 200 N 240 N 240 N = 150 mm 250 mm Moments 13

14 Moment Couples moment couples in structures Moments 14

15 Equivalent Force Systems two forces at a point is equivalent to the resultant at a point resultant is equivalent to two components at a point resultant of equal & opposite forces at a point is zero put equal & opposite forces at a point (sum to 0) transmission of a force along action line Moments 15

16 Force-Moment Systems single force causing a moment can be replaced by the same force at a different point by providing the moment that force caused A F F A d F A -F A moments are shown as arched arrows Moments 16

17 Force-Moment Systems a force-moment pair can be replaced by a force at another point causing the original moment A A F F A -F d A F F A M=F d A Moments 17

18 Parallel Force Systems forces are in the same direction can find resultant force need to find location for equivalent moments A a A B ( A B) x R=A+B C B b a b D C x D Moments 18

19 Equilibrium rigid body doesn t deform coplanar force systems A C B static: R F x x 0 ( H) R F y y 0 ( V) M M 0 Moments 19

20 Free Body Diagram FBD (sketch) tool to see all forces on a body or a point including external forces weights force reactions external moments moment reactions internal forces Moments 20

21 Free Body Diagram determine body FREE it from: ground supports & connections draw all external forces acting ON the body reactions applied forces gravity 100 lb + weight 100 lb m g Moments 21

22 Free Body Diagram sketch FBD with relevant geometry resolve each force into components known & unknown angles name them known & unknown forces name them known & unknown moments name them are any forces related to other forces? for the unknowns write only as many equilibrium equations as needed solve up to 3 equations Moments 22

23 Free Body Diagram solve equations most times 1 unknown easily solved plug into other equation(s) common to have unknowns of force magnitudes force angles moment magnitudes Moments 23

24 Reactions on Rigid Bodies result of applying force unknown size connection or support type known direction related to motion prevented no vertical motion no translation no translation no rotation Moments 24

25 Supports and Connections Moments 25

26 Supports and Connections Moments 26

27 Moment Equations sum moments at intersection where the most forces intersect multiple moment equations may not be useful combos: F x 0 F y 0 M 1 0 F 0 M 1 0 M 0 2 M 1 0 M 2 0 M 3 0 Moments 27

28 Concentrated Loads Moments 28

29 Distributed Loads Moments 29

30 Beam Supports statically determinate L L L simply supported (most common) overhang cantilever statically indeterminate L continuous (most common case when L 1 =L 2 ) L L L Propped Restrained Moments 30

31 Equivalent Force Systems replace forces by resultant place resultant where M = 0 using calculus and area centroids W 0 L wdx da loading A loading y w(x) x el dx x x dx L Moments 31

32 Load Areas area is width x height of load w is load per unit length W is total load w x W x W w 0 w x 2 x W 2 W/2 w w x W W/2 2w x/2 x/2 2x/3 x/3 x/2 x/6 x/3 Moments 32

five moments Moments Moments Moments 8 a force acting at a different point causes a different moment:

five moments Moments Moments Moments 8 a force acting at a different point causes a different moment: ELEENTS O RCHITECTURL STRUCTURES: OR, EHVIOR, ND DESIGN DR. NNE NICHOLS SPRING 2014 forces have the tenency to make a boy rotate about an ais lecture five moments http://www.physics.um.eu same translation

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