• Document: Introduction to Beam. Area Moments of Inertia, Deflection, and Volumes of Beams
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Introduction to Beam Theory Area Moments of Inertia, Deflection, and Volumes of Beams What is a Beam? ƒ Horizontal structural member used to support horizontal loads such as floors, roofs, and decks. ƒ Types of beam loads ƒ Uniform ƒ Varied by length ƒ Single point ƒ Combination Common Beam Shapes p I Beam Hollow Solid Box Box H Beam T Beam Beam Terminology gy ƒ The parallel portions on an I-beam or H-beam are referred to as the flanges. The portion that connects the flanges is referred to as the web. Web Web Flanges Flanges Support pp Configurations g Source: Statics (Fifth Edition), Meriam and Kraige, Wiley Load and Force Configurations g Concentrated Load Distributed Load Source: Statics (Fifth Edition), Meriam and Kraige, Wiley Beam Geometry y ƒ Consider a simply supported beam of length, L. ƒ The cross section is rectangular rectangular, with width width, b b, and height height, h h. h L b Beam Centroid ƒ An area has a centroid, which is similar to a center of gravity of a solid body. ƒ The centroid of a symmetric cross section can be easily found by inspection. X and Y axes intersect at the centroid of a symmetric cross section, as shown on the rectangular cross section. Y - Axis Centroid h/2 X - Axis h/2 b/2 b/2 Area Moment of Inertia (I) ƒ Inertia is a measure of a body’s ability to resist movement, bending, or rotation ƒ Moment of inertia (I) is a measure of a beam’s ƒ Stiffness with respect to its cross section ƒ Abilityy to resist bending g ƒ As I increases, bending decreases ƒ As I decreases, bending increases ƒ Units of I are (length)4, e.g. e g in4, ft4, or cm4 I for Common Cross Cross-Sections Sections ƒ I can be derived for any common area using calculus. However, moment of inertia equations for common cross sections (e.g., rectangular, t l circular, i l ttriangular) i l ) are readily dil available il bl iin math th and d engineering textbooks. ƒ For a solid rectangular cross section, bh 3 h Ix = X-axis (passing 12 through centroid) b ƒ b is the dimension parallel to the bending axis ƒ h is the dimension perpendicular to the bending axis Which Beam Will Bend (or Deflect) the Most About the X-Axis? P P h = 1.00” X-Axis X-Axis h = 0.25” YA i Y-Axis Y-Axis b = 1.00” b = 0.25” Solid Rectangular g Beam #1 ƒ Calculate the moment of inertia about the X-axis b 11 h 3 2 Y-Axis I = x 2 0 . 2 5

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