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E LEMENTS OF A RCHITECTURAL S TRUCTURES : F ORM, B EHAVIOR, AND D ESIGN ARCH 614 D R. A NNE N ICHOLS S PRING 2019 lecture eight beam sections - geometric properties Sections 1 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

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 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

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 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

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 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Centroid • for a line, sum up length     x L  x L     y L  y L  L Sections 5 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

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 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Symmetric Areas • symmetric about an axis • symmetric about a center point • mirrored symmetry Sections 7 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Composite Areas • made up of basic shapes • areas can be negative • (centroids can be negative for any area)  + (-) - Sections 8 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

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 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Area Centroids • Figure A.1 – pg 598 b h 3 right triangle only b Sections 10 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Moments of Inertia • 2 nd moment area – math concept – area x (distance) 2 • need for behavior of – beams – columns Sections 11 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Moment of Inertia • about any reference axis • can be negative y      2 2 I x A x dA y i dA = y  dx dx      2 2 I y A y dA x x i   x 2 a dx ( or I z ) el  x x • resistance to bending and buckling Sections 12 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Moment of Inertia • same area moved away a distance – larger I x x x x Sections 13 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

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 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Radius of Gyration • measure of inertia with respect to area I x  x r A Sections 15 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Parallel Axis Theorem • can find composite I once composite centroid is known (basic shapes)  o  2 I I Az 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 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

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 y 5. Fill in table and get x and x for composite 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 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614

Area Moments of Inertia • Figure A.11 – pg. 611: ( bars refer to centroid) – x, y – x’, y’ – C Sections 18 Elements of Architectural Structures S2019abn Lecture 8 ARCH 614