Properties of Sections ERT 348 Controlled Environmental Design 1 Biosystem Engineering

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Properties of Sections  Centre of gravity or Centroid  Moment of Inertia  Section Modulus  Shear Stress  Bending Stress

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Centre of gravity  A point which the resultant attraction of the earth eg. the weight of the object  To determine the position of centre of gravity, the following method applies: 1.Divide the body to several parts 2.Determine the volume of each part 3.Assume the volume of each part at its centre of gravity 4.Take moment at convenient axis to determine centre of gravity of whole body

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Example 1:  Density=8000 kg/m 3  Thick=10 mm  Determine the position of centre of gravity x mm W 1 = 0.08m x 0.06m x 0.01m x 8000kg/m 3 x 10 N/kg = 3.84 N W 2 = 0.02m x 0.12m x 0.01m x 8000kg/m 3 x 10 N/kg = 1.92 N W 3 = 0.12m x 0.06m x 0.01m x 8000kg/m 3 x 10 N/kg = 5.76 N 80mm 60mm120mm 60mm 1 2 3

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Example 1:  Resultant, R = N = N  Rx = 3.84 (30) (60+60) ( )  x = 1555/11.52 = 135 mm

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Centroid  Centre of gravity of an area also called as centroid.

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at  x = /14400 = 135 mm 80mm Areaa (mm 2 )y (mm)∑ay 160x80= x120=2400mm x120=7200mm Total14400 mm mm 3 60mm120mm 60mm mm Example 1:

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Moment of inertia, I  Or called second moment of area, I  Measures the efficiency of that shape its resistance to bending  Moment of inertia about the x-x axis and y-y axis. xx b d y y Unit : mm 4 or cm 4

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Moment of Inertia of common shapes  Rectangle at one edge  I uu = bd 3 /3  I vv = db 3 /3  Triangle  I xx = bd 3 /36  I nn = db 3 /6 xx v v d b b d u u nn

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Moment of Inertia of common shapes  Ixx= Iyy = πd 4 /64  Ixx = (BD 3 -bd 3 )/12  Iyy = (DB 3 -db 3 )/12 xx y y B D b d xx y y

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Principle of parallel axes  Izz = Ixx + AH 2  Example: b=150mm;d=100mm; H=50mm  Ixx= (150 x )/12 = 12.5 x 10 6 mm4  Izz = Ixx + AH 2 = 12.5 x (50 2 ) = 50 x 106 mm 4 H zz xx

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Example 2:  Calculate the moment of inertia of the following structural section 400mm 200mm 24mm 12mm H= 212mm

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Solution  I xx of web = (12 x400 3 )/12= 64 x 10 6 mm 4  I xx of flange = (200x24 3 )/12= 0.23 x 10 6 mm 4  I xx from principle axes xx = 0.23 x AH 2 AH 2 = 200 x 24 x = x 10 6 mm 4 I xx from x-x axis = 216 x 10 6 mm 4  Total I xx = ( x 216) x10 6 =496 x 10 6 mm 4

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Section Modulus, Z  Second moment of area divide by distance from axis  Where c = distance from axis x-x to the top of bottom of Z.  Unit in mm 3  Example for rectangle shape:  I xx = bd 3 /12, c = d/2, Z xx = I xx /c = bd 2 /6

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Section Modulus, Z  Z = 1/y  f = M/Z = My/I  Safe allowable bending moment, M max = f.Z where f = bending stress y = distance from centroid

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Example 3  A timber beam of rectangular cross section is 150mm wide and 300mm deep. The maximum allowable bending in tension and compression must not exceed 6 N/mm 2. What maximum bending moment in N.mm?  Z= bd 2 /6 = 150 x /6 = 2.25 x 10 6 mm 3  M max = f.Z = 6 x 2.25x10 6 = 13.5x10 6 Nmm 2

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at If non-symmetrical sections M rc = 1/y 1 x compression stress M rt = 1/y 2 x tension stress Safe bending moment = f x least Z xx y2y2 y1y1

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Example 4  Figure shows an old type cast iron joist (in mm) with a tension flange of 9600 mm 2, a compression flange 2880 mm 2 and a web of 7200 mm 2.  The safe stress in compression is 5 N/mm 2 and in tension 2.5 N/mm 2. What is the safe bending moment for the section? What safe uniform load will the beam carry on a 4.8m span. xx

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Solution  x =∑ay/∑a = (2880x x x24) ( ) = 137 mm  Total I xx = (120x24 3 / x223 2 )+ (24x300 3 / x61 2 ) + (200x48 3 / x113 2 ) = x 10 6 mm 4  Mrt = 2.5 x x 10 6 = 6.36x10 6 Nmm  Mcr= 5.0 x x 10 6 = 7.42x10 6 Nmm  WL/8 = 6.36x10 6 Nmm  W= 6.36 x 8/4.8 = 10.6 kN

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Elastic Shear Stress Distribution  The shear forces induced in a beam by an applied load system generate shear stresses in both the horizontal and vertical directions.  At any point in an elastic body, the shear stresses in two mutually perpendicular directions are equal to each other in magnitude.

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Shear Stress

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Example 5: Elastic Shear Stress  The rectangular beam shown in Figure is subject to a vertical shear force of 3.0 kN.  Determine the shear stress distribution throughout the depth of the section. A 200mm 50mm xx y y

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Solution: y=50mm y=25mm y=0mm

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Answer:

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Answer:  Average τ = V/A = 3 x 10 3 /(50x200) = 0.3 N/mm 2  Maximum = 1.5 V/A = 1.5x0.3 = 0.45 N/mm 2

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Bending Stress Distribution  f=σ= bending stress = My/I

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at 200mm 50mm xx M= 2.0 kNm

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Material Properties Concrete

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Concrete  Concrete compressive strength: f cu  C30,C35,C40,C45 and C50  Where the number represent compressive strength in N/mm 2

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Modulus Elasticity, E E s (Modulus Elasticity of steel reinforcement) = 200 kN/mm 2

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Poisson ratio υ c  Refer to Clause BS 8110  The value = 0.2

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Steel Reinforcement Strength: f y  Refer to Clause BS 8110 (Table 3.1)  f y = 250 N/mm 2 for hot rolled mild steel (MS)  f y = 460 N/mm 2 for hot rolled or cold worked high yield steel (HYS)

Material Properties Steel

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Design strength, p y

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Modulus Elasticity, E  Modulus elasticity, E = N/mm2 Poisson ratio υ c  The value = 0.3 Shear Modulus,G  G=E/[2(1+υ)] = x10 3 N/mm 2

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Section classification  Refer to Table 11 & Table 12 BS5950-1:2000 Clause 3.5  Class 1 - Plastic Sections  Class 2 - Compact Sections  Class 3 - Semi-compact Sections  Class 4 - Slender Sections

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Aspect ratio  ε=(275/p y ) 0.5

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Types of sections  I sections  H sections  Rectangular Hollow Sections (RHS)  Circular Hollow Sections (CHS)  Angles (L shape or C shapes)

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Types of sections

Material Properties Timber

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Moisture content m 1 =mass before drying m 2 =mass after drying Unit in %  The strength of timber is based on its moisture content.  In MS 544, the moisture content – 19% >19% - moisture <19% - dry

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Material Properties  Elastic Modulus E = 4600 – N/mm 2  Poisson’s Ratio υ = 0.3

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Grade of timber  Timber can be graded by Visual Inspection Machine strength grading  3 grade only Select Standard Common Less defect

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Group of timber  We have Group A B C D Lower strength

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Defects by nature

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Defects in timber  In addition to the defects indicated in Figure 7.1 there are a number of naturally occurring  defects in timber. The most common and familiar of such defects is a knot

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Typical sawing pattern

ERT 348 Controlled Environment Design 1 Siti Kamariah Binti Md Sa’at Reference to design  MS 544 : Pt.1-Pt.11 : Code of Practice for Structural Use of Timber

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