For a beam of rectangular cross-section, the resistance of top and bottom fibers with distance y = d/2 from the neutral axis is (d/2) 2. The formula for the moment of inertia I=∫y 2da reveals that the resistance of any differential area da increases with its distance y from the neutral axis squared, forming a parabolic distribution. This demonstrates the importance of correct orientation of bending members, such as beams or moment frames. The upright joist is six times stronger than the flat joist of equal cross-section. The moment of inertia I for various common shapes is given in Appendix A.Ĭomparing a joist of 2”x12” in upright and flat position as illustrated in 2 and 3 yields an interesting observation: For other shapes S can be computed as S = I /c as defined before for the flexure formula. This formula is valid for homogeneous beams of any shape but the formula S = bd 2/6 is valid for rectangular beams only. Solving M = f S for f yields the maximum bending stress as defined before: Where S = bd 2/6, defined as the section modulus for rectangular beams of homogeneous material. Substituting C = T = f bd/4 yields M = 2 (f bd/4) d/3 = f bd 2/6, or M = f S, The internal resisting moment is the sum of C and T times their respective lever arm, d/3, to the neutral axis. The magnitude of C and T is the volume of the upper and lower stress block, respectively. C and T act at the center of mass of their respective triangular stress block at d/3 from the neutral axis. The force couple C and T rotate about the neutral axis to provide the internal resisting moment. Referring to 1, the section modulus for a rectangular beam of homogeneous material may be derived as follows. The section modulus for such beams is derived here.ġ Stress block in rectangular beam under positive bending.Ģ Large stress block and lever-arm of a joist in typical upright position.ģ Small, inefficient, stress block and lever-arm of a joist laid flat. Rectangular beams of homogeneous material, such as wood, are common in practice. Assuming c as maximum fiber distance from the neutral axis yields:īoth the moment of inertia I and section modulus S define the strength of a cross-section regarding its geometric form. A simpler form is used to compute the maximum fiber stress as derived before. Which gives the bending stress f at any distance c from the neutral axis. The internal resisting moment equation M = I f/c solved for stress f yields In formal calculus the summation of area a is replaced by integration of the differential area da, an infinitely small area: Since the internal resisting moment M is the sum of all forces F times their lever arm y to the neutral axis, M = F y = (f/c) Σ y y a = (f/c) Σ y 2a, or M = I f/c, where the term Σy 2a is defined as moment of inertia (I = Σy 2a) for convenience. Substituting f y = y f / c, defined above, yields F = a y f / c. Each partial force F is the product of stress f y and the partial area a on which it acts, F = a f y. The internal resisting moment is the sum of all partial forces F rotating around the neutral axis with a lever arm of length y to balance the external moment. To satisfy equilibrium, the beam requires an internal resisting moment that is equal and opposite to the external bending moment. The bending stress f y at any distance y from the neutral axis is found, considering similar triangles, namely f y relates to y as f relates to c f is the maximum bending stress at top or bottom and c the distance from the Neutral Axis, namely f y / y = f / c. Thus stress varies linearly over the depth of the beam and is zero at the neutral axis (NA). Assuming stress varies linearly with strain, stress distribution over the beam depth is proportional to strain deformation. As illustrated by the hatched square, the top shortens and the bottom elongates, causing compressive stress on the top and tensile stress on the bottom. Referring to the diagram, a beam subject to positive bending assumes a concave curvature (circular under pure bending). It is derived here for a rectangular beam but is valid for any shape.Ģ Beam subject to bending with hatched square deformedģ Stress diagram of deformed beam subject to bending The flexure formula gives the internal bending stress caused by an external moment on a beam or other bending member of homogeneous material.
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