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Fredrik Lagerström 5.1 1 = {{id name="Area properties of concrete cross-section"/}}Area properties of concrete cross-section =
Fredrik Lagerström 3.1 2
3 A general integral over an area of arbitrary shape is given as
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5 H,,mn,, = ∫ y^^m^^ z^^n^^ dA
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7 with **m** and **n** as positive integers. Setting **m** and **n** equal to **0**, **1** or **2**, the integral will give values of the area integrals **A**, **S,,y,,**, **S,,z,,**, **I,,y,,**, **I,,z,,** and **I,,yz,,** which are defined as area properties.
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Fredrik Lagerström 4.1 9 During calculation all section types (open as well as closed) are converted to polygon shape and after that the section properties are easily calculated using numeric integration (see [[article>>https://www.pci.org/PCI_Docs/Publications/PCI%20Journal/1996/May-June/Time-Dependent%20Stresses%20in%20Prestressed%20Concrete%20Sections%20of%20General%20Shape.pdf]]).
Fredrik Lagerström 3.1 10
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12 [[image:1589356542701-107.png]]
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Fredrik Lagerström 5.1 14 = {{id name="Area properties of reinforcement in cross-section"/}}Area properties of reinforcement in cross-section =
Fredrik Lagerström 3.1 15
16 Section properties for reinforcement composed of **N** bars with cross-section area **A,,sj,,** and positions (**y,,j,,**, **z,,j,,**) are calculated as a summation:
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18 N,,mn,,
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20 H,,mn,, = Σ y,,j,, z,,j,, A,,sj,,
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22 j = 1
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24 With **m** and **n** equal to **0**, **1** or **2**, the summation will give values of the area integrals **A**, **S,,y,,**, **S,,z,,**, **I,,y,,**, **I,,z,,** and **I,,yz,,** which are defined as area properties.
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27 [[image:1589356752521-520.png]]
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Fredrik Lagerström 5.1 29 = {{id name="Area properties of composite cross-section"/}}Area properties of composite cross-section =
Fredrik Lagerström 3.1 30
31 Section properties for the composite section are calculated using the section properties of net concrete section (i.e. with area corresponding to reinforcement subtracted) and section properties of reinforcement considering the difference in modulus of elasticity. This section is also referred to as the transformed section.
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33 The difference in modulus of elasticity is expressed by the factor:
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35 α = E,,s,, / E,,c,,
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37 Summation of areas and static moments around coordinate axes gives property values of the transformed section:
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39 A = A,,cn,, + α A,,s,,
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41 S,,y,, = S,,cny,, + α S,,sy,,
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43 S,,z,, = S,,cnz,, + α S,,sz,,
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45 Coordinates for the centroid of the transformed area are then easily found by
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47 y,,cg,, = S,,y,, /A z,,cg,, = S,,z,, /A
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49 Moments of inertia are then calculated using Steiner’s theorems:
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51 I,,ycg,, = I,,cny,, + α I,,sy,, – z,,cg2,, A
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53 I,,zcg,, = I,,cnz,, + α I,,sz,, – y,,cg2,, A
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55 I,,yzcg,, = I,,cnyz,, + α I,,syz,, – y,,cg,, z,,cg,, A
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