A RCHITECTURAL S TRUCTURES : Equilibrium F ORM, B EHAVIOR, AND D ESIGN ARCH 331 • rigid body D R. A NNE N ICHOLS – doesn’t deform F ALL 2018 C B – coplanar force systems A lecture five • static: 0 R F x x R F 0 y y rigid body M 0 M equilibrium Rigid Body Equilibrium 1 Architectural Structures S2010abn Rigid Body Equilibrium 2 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Free Body Diagram Free Body Diagram • determine body • FBD (sketch) 100 lb • FREE it from: • tool to see all forces on a body or a + weight – ground point including – supports & connections – external forces • draw all external forces 100 lb – weights acting ON the body – force reactions m g – reactions – external moments – applied forces – moment reactions – gravity – internal forces Rigid Body Equilibrium 3 Foundations Structures S2010abn Rigid Body Equilibrium 4 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 1
Free Body Diagram Free Body Diagram • sketch FBD with relevant geometry • solve equations • resolve each force into components – most times 1 unknown easily solved – known & unknown angles – name them – plug into other equation(s) – known & unknown forces – name them – known & unknown moments – name them • are any forces related to other forces? • common to have unknowns of • for the unknowns – force magnitudes • write only as many equilibrium equations as – force angles needed – moment magnitudes • solve up to 3 equations Rigid Body Equilibrium 5 Foundations Structures S2010abn Rigid Body Equilibrium 6 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Reactions on Rigid Bodies Supports and Connections • result of applying force • unknown size • connection or support type – known direction – related to motion prevented no translation no vertical motion no rotation no translation Rigid Body Equilibrium 7 Foundations Structures S2010abn Rigid Body Equilibrium 8 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 2
Supports and Connections FBD Example • 500 lb known • pin – A x , A y • smooth surface – B at 4:3 • 3 equations • sum moments at – A? – B? (B x ) Rigid Body Equilibrium 10 Foundations Structures S2010abn Rigid Body Equilibrium 9 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Moment Equations Recognizing Reactions • sum moments at intersection where the F F most forces intersect • multiple moment equations may not be 3 unknowns useful • combos: F x F M 1 0 0 0 3 F y M 1 M 2 unknowns 0 0 0 m g weight 1 2 3 M 0 M 0 M 0 Rigid Body Equilibrium 11 Foundations Structures S2010abn Rigid Body Equilibrium 12 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 3
Recognizing Reactions Constraints • completely constrained 3 – doesn’t move unknowns m g – may not be statically determinate weight • improperly or partially constrained F 1 F 2 F 1 F 2 6 – has unknowns unknowns for not independent 2 bodies – can’t solve 2 m g weight unknowns Rigid Body Equilibrium 13 Foundations Structures S2010abn Rigid Body Equilibrium 14 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Constraints Partial Constraints • overconstrained – won’t move B B 0.75 m 0.75 m – can’t be solved with statics 30 30 A A – statically indeterminate to n th degree 1 m 1 m 100 N 100 N 60 lb 60 lb 500 mm 200 lb-ft 200 lb-ft 200 mm 55 55 B A x A B A B A B C C 5’ 9’ M RA W W A C y y Rigid Body Equilibrium 15 Foundations Structures S2010abn Rigid Body Equilibrium 16 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 4
Cable Reactions Two Force Rigid Bodies • equilibrium: • equilibrium: – more reactions (4) than equations – forces in line, equal and opposite – but, we have slope relationships – x component the same everywhere F 1 F 1 F 1 B B B d d F 2 A A A B a 4 m 6 m A 2 m F 2 F 2 C 45 kN ( no) ( no) A B C 45 kN Rigid Body Equilibrium 17 Foundations Structures S2010abn Rigid Body Equilibrium 18 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Three Force Rigid Bodies Concentrated Loads • equilibrium: – concurrent or parallel forces beams! F 2 F 2 F 2 F 3 F 3 F 3 B d 1 B B C C C a d 2 A A A F 1 F 1 F 1 ( no) A B C Rigid Body Equilibrium 19 Foundations Structures S2010abn Rigid Body Equilibrium 20 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 5
Distributed Loads Beam Supports • statically determinate L L L simply supported overhang cantilever (most common) • statically indeterminate L L L L Propped Restrained continuous (most common case when L 1 =L 2 ) Rigid Body Equilibrium 21 Foundations Structures S2010abn Rigid Body Equilibrium 22 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 Equivalent Force Systems Load Areas • replace forces by resultant • area is width x “height” of load • place resultant where M = 0 • w is load per unit length • using calculus and area centroids • W is total load L W wdx dA A loading 0 loading 2w w x W w x W w 2 2 w w y 0 w(x) x x x W W W/2 W/2 dx x x x/2 x/2 2x/3 x/3 x/2 x/6 x/3 L x el dx Rigid Body Equilibrium 23 Foundations Structures S2010abn Rigid Body Equilibrium 24 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 6
Method of Sections Method of Sections • relies on internal forces being in • joints on or off the section are good to equilibrium on a section sum moments • cut to expose 3 or less members • quick for few members • coplanar forces M = 0 too • not always obvious where to cut or sum P P P P . C C AC AC D D F F A . A . E B E B B A AB AB A B y B y Rigid Body Equilibrium 25 Foundations Structures S2010abn Rigid Body Equilibrium 26 Foundations Structures S2010abn Lecture 5 ARCH 331 Lecture 5 ARCH 331 7
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