nine beam sections - geometric properties Sections 1 - PowerPoint PPT Presentation

A RCHITECTURAL S TRUCTURES : F ORM, B EHAVIOR, AND D ESIGN A RCH 331 D R. A NNE N ICHOLS S PRING 2018 lecture nine beam sections - geometric properties Sections 1 Architectural Structures S2018abn Lecture 9 ARCH 331

Center of Gravity • location of equivalent weight • determined with calculus y  W z  W 4  W 1  W 3  W 2 x   • sum element weights W dW Sections 2 Architectural Structures S2018abn Lecture 9 ARCH 331

Center of Gravity • “average” x & y from moment y  W z  W 4  W 1  W 3  W 2 x     n x W        M x W x W x y i i W  1 i   “bar” means average   n y W        M y W y W y x i i W  1 i Sections 3 Architectural Structures S2018abn Lecture 9 ARCH 331

Centroid • “average” x & y of an area • for a volume of constant thickness      – where is weight/volume W t A – center of gravity = centroid of area     x A  x A     y A  y A Sections 4 Architectural Structures S2018abn Lecture 9 ARCH 331

Centroid • for a line, sum up length     x L  x L     y L  y L  L Sections 5 Architectural Structures S2018abn Lecture 9 ARCH 331

1 st Moment Area • math concept • the moment of an area about an axis Q x  y y A A (area)   y   y A Q y   x A y x x A Sections 6 Architectural Structures S2018abn Lecture 9 ARCH 331

Symmetric Areas • symmetric about an axis • symmetric about a center point • mirrored symmetry Sections 7 Architectural Structures S2018abn Lecture 9 ARCH 331

Composite Areas • made up of basic shapes • areas can be negative • (centroids can be negative for any area)  + (-) - Sections 8 Architectural Structures S2018abn Lecture 9 ARCH 331

Basic Procedure 1. Draw reference origin (if not given) 2. Divide into basic shapes (+/-) 3. Label shapes Component Area x y x A y A 4. Draw table 5. Fill in table  6. Sum necessary columns ˆ ˆ 7. Calculate x and y x y Sections 9 Architectural Structures S2018abn Lecture 9 ARCH 331

Area Centroids • Table 6.1 – pg. 304 b h 3 right triangle only b Sections 10 Architectural Structures S2018abn Lecture 9 ARCH 331

Moments of Inertia • 2 nd moment area – math concept – area x (distance) 2 • need for behavior of – beams – columns Sections 11 Architectural Structures S2018abn Lecture 9 ARCH 331

Moment of Inertia • about any reference axis • can be negative y   2 I y x dA dA = y  dx dx   2 x I x y dA x dx el • resistance to bending and buckling Sections 12 Architectural Structures S2018abn Lecture 9 ARCH 331

Moment of Inertia • same area moved away a distance – larger I x x x x Sections 13 Architectural Structures S2018abn Lecture 9 ARCH 331

Polar Moment of Inertia • for roundish shapes • uses polar coordinates (r and  ) • resistance to twisting r    o 2 J o r dA pole Sections 14 Architectural Structures S2018abn Lecture 9 ARCH 331

Radius of Gyration • measure of inertia with respect to area I x  x r A Sections 15 Architectural Structures S2018abn Lecture 9 ARCH 331

Parallel Axis Theorem • can find composite I once composite centroid is known (basic shapes)   2 I I Ad y  x cx y dA B  axis through centroid  I  B 2 Ad at a distance d away y from the other axis d x y A  axis to find moment of A inertia about     2 I I Ad   2 I I Ad Sections 16 Architectural Structures S2018abn Lecture 9 ARCH 331

Basic Procedure 1. Draw reference origin (if not given) 2. Divide into basic shapes (+/-) 3. Label shapes 4. Draw table with A, x, xA, y, yA, I ’s, d’s, y y A I x x A and Ad 2 ’s ˆ ˆ 5. Fill in table and get x and x for composite y x 6. Sum necessary columns   ˆ ( d x x ) 7. Sum I ’s and Ad 2 ’s I x   ˆ ( d y y ) y Sections 17 Architectural Structures S2018abn Lecture 9 ARCH 331

Area Moments of Inertia • Table 6.2 – pg. 315: ( bars refer to centroid) – x, y – x’, y’ – C Sections 18 Architectural Structures S2018abn Lecture 9 ARCH 331