Beam Stresses Bending and Shear

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Beam Stresses Bending and Shear Notation: A = name or area A web = area o the web o a wide lange setion b = width o a retangle = total width o material at a horizontal setion = largest distane rom

1 Beam Stresses Bending and Shear Notation: A = name or area A web = area o the web o a wide lange setion b = width o a retangle = total width o material at a horizontal setion = largest distane rom the neutral axis to the to or bottom edge o a beam d = alulus symbol or dierentiation = deth o a wide lange setion d y = dierene in the y diretion between an area entroid ( y ) and the entroid o the omosite shae ( ŷ ) E = modulus o elastiity or Young s modulus b = bending stress = omressive stress max = maximum stress t = tensile stress v = shear stress F b = allowable bending stress F onnetor = shear ore aaity er onnetor h = height o a retangle = moment o inertia with reset to neutral axis bending x = moment o inertia with reset to an x-axis L = name or length M = internal bending moment = name or a moment vetor n = number o onnetors aross a joint n.a. = shorthand or neutral axis (N.A.) O = name or reerene origin = ith o onnetor saing P = name or a ore vetor q = shear er length (shear low) Q = irst moment area about a neutral axis Q onneted = irst moment area about a neutral axis or the onneted art R = radius o urvature o a deormed beam S = setion modulus S req d = setion modulus required at allowable stress t w = thikness o web o wide lange = internal shear ore longitudinal = longitudinal shear ore T = transverse shear ore w = name or distributed load x = horizontal distane y = vertial distane y = the distane in the y diretion rom a reerene axis (n.a) to the entroid o a shae ŷ = the distane in the y diretion rom a reerene axis to the entroid o a omosite shae = alulus symbol or small quantity = elongation or length hange = strain = ar angle = summation symbol Pure Bending in Beams With bending moments along the axis o the member only, a beam is said to be in ure bending. 1

2 Normal stresses due to bending an be ound or homogeneous materials having a lane o symmetry in the y axis that ollow Hooke s law. y x Maximum Moment and Stress Distribution n a member o onstant ross setion, the maximum bending moment will govern the design o the setion size when we know what kind o normal stress is aused by it. For internal equilibrium to be maintained, the bending moment will be equal to the M rom the normal stresses the areas the moment arms. Geometri it hels solve this statially indeterminate roblem: 1. The normal lanes remain normal or ure bending.. There is no net internal axial ore. 3. Stress varies linearly over ross setion. 4. Zero stress exists at the entroid and the line o entroids is the neutral axis (n. a)

Relations or Beam Geometry and Stress Pure bending results in a irular ar deletion. R is the distane to the enter o the ar; is the angle o the ar (radians); is the distane rom the n.a. to the extreme iber; max is the maximum normal stress at the extreme iber; y is a distane in y rom the n.

3 Relations or Beam Geometry and Stress Pure bending results in a irular ar deletion. R is the distane to the enter o the ar; is the angle o the ar (radians); is the distane rom the n.a. to the extreme iber; max is the maximum normal stress at the extreme iber; y is a distane in y rom the n.a.; M is the bending moment; is the moment o in zertia; S is the setion modulus. L R M i A i My b Now: or a retangle o height h and width b: RELATONS: L M M R E R max y i A i E y 1 * My b A y max S ½ S R L y ½ M max 3 bh bh S 1 h 6 M S b max M M S S required M F b *Note: y ositive goes DOWN. With a ositive M and y to the bottom iber as ositive, it results in a TENSON stress (we ve alled ositive). Transverse Loading in Beams We are aware that transverse beam loadings result in internal shear and bending moments. We designed setions based on bending stresses, sine this stress dominates beam behavior. There an be shear stresses horizontally within a beam member. t an be shown that horizontal vertial 3

Equilibrium and Derivation n order or equilibrium or any element CDD C, there

D da C da Q is a moment area with reset to the neutral axis o the area above or

Q is a maximum when y = 0 (at the neutral axis).

4 Equilibrium and Derivation n order or equilibrium or any element CDD C, there needs to be a horizontal ore H. D da C da Q is a moment area with reset to the neutral axis o the area above or below the horizontal where the H ours. Q is a maximum when y = 0 (at the neutral axis). q is a horizontal shear er unit length shear low Shearing Stresses v = 0 on the beam s surae. Even i Q is a maximum at y = 0, we ave don t know that the thikness is a minimum there. v A b x vave Q b longitudinal q longitudinal x T Q x T Q Retangular Setions ours at the neutral axis: vmax then: 3 bh 1 Q Ay Q 1 8 bh 3 v 3 b 1 bh b bh 1 b h 1 h bh 3 v A 8 4

5 Webs o Beams n steel W or S setions the thikness varies rom the lange to the web. We neglet the shear stress in the langes and onsider the shear stress in the web to be onstant: vmax 3 A A web vmax t d web Webs o beams an ail in tension shear aross a anel with stieners or the web an bukle. Shear Flow Even i the ut we make to ind Q is not horizontal, but arbitrary, we an still ind the shear low, q, as long as the loads on thin-walled setions are alied in a lane o symmetry, and the ut is made erendiular to the surae o the member. Q q The shear low magnitudes an be skethed by knowing Q. 5

6 Connetors to Resist Horizontal Shear in Comosite Members Tyial onnetions needing to resist shear are lates with nails or rivets or bolts in omosite setions or slies. The ith (saing) an be determined by the aaity in shear o the onnetor(s) to the shear low over the saing interval,. where = ith length nf onnetor Q x onneted area y y a n = number o onnetors onneting the onneted area to the rest o the ross setion F = ore aaity in one onnetor Q onneted area = A onneted area y onneted area longitudinal Q y onneted area = distane rom the entroid o the onneted area to the neutral axis longitudinal Q 1 8 Connetors to Resist Horizontal Shear in Comosite Members Even vertial onnetors have shear low aross them. The saing an be determined by the aaity in shear o the onnetor(s) to the shear low over the saing interval,. nf Q onnetor onneted area Unsymmetrial Setions or Shear the setion is not symmetri, or has a shear not in that lane, the member an bend and twist. the load is alied at the shear enter there will not be twisting. This is the loation where the moment aused by shear low = the moment o the shear ore about the shear enter. 6

7 Examle 1 ALSO: Determine the minimum nail saing required (ith) i the shear aaity o a nail (F) is 50 lb

8 Examle Q = Ay = (9")(½")(4.5")+(9")(½")(4.5")+(1.5")(3.5")(8.5") = 83.8 in 3 3 (, 600# )( 83. 3in. v max. 4 ( 1, 0. 6in. )( 1 1 " ) 180 si ") (n) (n)f (n)f 8

Beam Bending Stresses and Shear Stress

Beam Bending Stresses and Shear Stress Notation: A = name or area Aweb = area o the web o a wide lange section b = width o a rectangle = total width o material at a horizontal section c = largest distance