Basics of 3 Dimensional Geometry and the SDS2

Basics of 3 Dimensional Geometry and the SDS/2 Model Presented by Bruce Vaughan 2: 00 -3: 20 Thursday, September 17 Track 3

Basic Geometry Elements • Point – Exact location and no size • Line – Extends infinitely in either direction, no width or height, one dimensional • Plane – Extends infinitely in two dimensions, no thickness

SDS/2 Coordinates • 3 Dimensional Cartesian Coordinate System • Points are defined by X, Y, and Z coordinates with respect to an origin • Lines are defined with two points. A line segment is the portion of a line between two points. • Planes are defined with three non-collinear points.

Vectors • A vector is an object that represents a direction and magnitude. • Commonly represented by the difference between 2 point locations. • A load on a steel structure is an example of a vector.

Unit Vectors • Contains direction information only • A unit vector multiplied by a scalar value produces a vector with magnitude equal to that scalar in the direction of the unit vector.

Unit Vectors >>> from macrolib import Geom 3 D >>> from Point 3 D import Point 3 D >>> from point import * >>> from member import * >>> mem = Member. Locate() >>> mem. left. location -195, -85. 678511, 16 >>> mem. right. location -195, -11. 5, 219. 875 >>> v = mem. right. location - mem. left. location >>> v 0, 74. 178511, 203. 875 >>> uv = Geom 3 D. unit. V(v) >>> uv Point 3 D(0, 0. 34191463864218158, 0. 93973101464311926) >>> Geom 3 D. mag(v) 216. 95037922891734 >>> uv*Geom 3 D. mag(v) Point 3 D(0, 74. 178510517339532, 203. 875)

Spherical Coordinates • Defined by radial coordinate (r), zenith angle (theta), azimuth angle (phi) • Zenith angle, also known as the inclination angle, is with respect to the positive "Z" axis. • Azimuth angle is the angle in plan • Points can be converted from Cartesian coordinates to spherical coordinates and back to Cartesian coordinates.

Spherical Coordinates >>> pt = Point 3 D(120, 41, 212) >>> spt = Geom 3 D. point. To. SPt 3 D(pt) >>> spt SPt 3 D(247. 032387, 0. 539069, 0. 329232) >>> spt. to. Point 3 D() Point 3 D(119. 9999999, 41, 212) >>> pt = Point 3 D(-120, -41, 212) >>> spt = Geom 3 D. point. To. SPt 3 D(pt) >>> spt SPt 3 D(247. 032387, 0. 539069, -2. 812361) >>> spt. to. Point 3 D() Point 3 D(-119. 9999999, -40. 999999979, 212)

Orthonormal Basis • A set of vectors whose lengths are all 1 and are orthogonal to each other is called an orthonormal basis. • The standard basis for 3 D space and the SDS/2 model is: (1, 0, 0), (0, 1, 0), (0, 0, 1)

Vector Operations • Addition >>> p 1 = Point 3 D(1, 2, 3) >>> p 2 = Point 3 D(4, 5, 6) >>> p 1+p 2 Point 3 D(5, 7, 9) • Scalar Multiplication >>> p 1*10 Point 3 D(10, 20, 30)

Vector Operations • Dot Product – Result is a scalar value. – The dot product operation is commutative. – The dot product of 2 unit vectors is the cosine of the angle between the vectors.

Vector Operations – Dot Product >>> p 1 = Point 3 D(1, 2, 3) >>> p 2 = Point 3 D(4, 5, 6) >>> dot_product(p 1, p 2) 32. 0 >>> dot_product(unit. V(p 1), unit. V(p 2)) 0. 9746318461970762 >>> degrees(acos(Geom 3 D. dot_product(unit. V(p 1), unit. V(p 2)))) 12. 933154491899135

Vector Operations • Cross Product – The result is a vector that is perpendicular to both. – The calculation is non-communitive, that is cross_product(p 1, p 2) is not equal to cross_product(p 2, p 1).

Vector Operations – Cross Product >>> v 1 = Point 3 D(1, 0, 0) >>> v 2 = Point 3 D(0, 1, 0) >>> cross_product(v 1, v 2) Point 3 D(0, 0, 1) >>> v 3 = Point 3 D(100, 0, 0) >>> cross_product(v 1, v 3) Point 3 D(0, 0, 0)

Member and Material Coordinates Every SDS/2 model object has its own defined orthonormal basis. Example member “xform” of a beam framing left to right: ref_xform: ((1. 0, 0. 0), (0. 0, -1. 0, 0. 0), (-6. 0, 637. 0, 794. 75, 1. 0))

Define a Plane From 3 Points >>> import Point 3 D >>> from macrolib. Geom 3 D import * >>> from point import * >>> p 1 = Point. Locate("") >>> p 2 = Point. Locate("") >>> p 3 = Point. Locate("") >>> plane 1 = Plane(p 1, p 2, p 3) >>> plane 1. trans. XYZTo. Global(10, 0, 0) Point 3 D(9. 8480775301220795, 1. 5133884835558331, 0. 85148356328538122) >>> Point 3 D(p 1)+plane 1. trans. XYZTo. Global(10, 0, 0) Point 3 D(739. 84807753012205, 172. 31121931185208, 709. 43676950380836)

Line and Plane Operations • After defining lines and planes, several calculations can be performed that relate to the SDS/2 model. • Plane methods: – – – – – on. Plane(point, tolerance=EPS) -> boolean dist. Plane(point) -> signed float (distance from point to plane) angle. Between. Planes(other) -> float in radians is. Parallel(other, tol=EPS) -> boolean inters. Plane(self, other, u 1=1, u 2=100) inters. Line. Plane(line, tol=EPS) -> Point or None is. Line. Parallel(line, tol=EPS) -> boolean inters. Line. Seg. Plane(line, tol=EPS) -> Point or None inside(point, tolerance=EPS) -> boolean

Line and Plane Operations • Plane methods continued: – trans. XYZTo. Global(self, X, Y, Z, base. Pt=Point 3 D()) – trans. To. Global(self, local. Pt)

Line and Plane Operations • Line methods: – closest. Pt. Line(self, pt): – dist. Pt. Line(self, pt): – pt. Near. Segment(self, pt, tol=EPS): – pt. On. Segment(self, pt, tol=EPS): – inters(self, other) – is. Parallel(self, other) – angle(self, other)

Line and Plane Operations • Other operations: – Rotate a point about an arbitrary axis – Three point construction circle (intersection of 3 planes) – Determine if a point is inside a Box. The box is defined with 6 planes.

Solving Hip And Valley Roof Geometry • A roof composed of a series of intersecting planes that form a ridge and slope down to the eave on all sides is a hip or hipped roof. • A valley is formed where two rectangular hipped roof areas intersect. • The detailer needs the following information to prepare shop drawings for a hipped roof: 1. Ridge work point elevation at bottom of deck or elevation at top of roof and roof thickness dimension 2. Roof slope 3. Locating dimensions of ridge lines 4. Eave work point elevation at bottom of deck or elevation at top of roof and roof thickness dimension 5. Locating dimensions of eave lines • The plan bevel of the hip beam is a function of the adjacent roof slopes and angle between the directions of the roof slopes.

Solving Hip And Valley Roof Geometry Preparation of geometry plans can sometimes be helpful on complicated hipped roofs.

Solving Hip And Valley Roof Geometry Five Forks Library Partial Geometry Plan

Solving Hip And Valley Roof Geometry Five Forks Library Sample Hip and Valley Details The angle between the roof plane and rafter flange plane is L 10.

Solving Hip And Valley Roof Geometry • The roof beam that frames along a hip line is known as a hip rafter or hip beam. • Roof beams framing perpendicular to the roof slope are also known as purlins. • The geometry of the roof beams is straightforward for a framing member whose web is vertical, which is the case when sloping parallel to the roof. This member supports the purlins and will be referred to as the “truss”. • The geometry presented on the following slides is based on the book “Hip and Valley Design” by H. L. Mc. Kibben and L. E. Gray. Mc. Kibben and Gray were engineers with American Bridge Company. The book was originally published in 1912.

Solving Hip And Valley Roof Geometry A = Roof slope B = Angle between hip line and truss R = Slope of rafter = atan(A) * cos(B))

Solving Hip And Valley Roof Geometry L 1 = Angle on PURLIN WEB PLANE made by intersection of RAFTER WEB PLANE L 1 = atan(sin(A) * tan(B))

Solving Hip And Valley Roof Geometry L 2 = Angle on ROOF PLANE made by intersection of RAFTER WEB PLANE L 2 = atan(cos(A) * tan(B))

Solving Hip And Valley Roof Geometry L 3 = Angle on RAFTER WEB PLANE made by intersection of PURLIN WEB PLANE L 3 = atan(sin(A) * cos(A) * sin(B) * tan(B))

Solving Hip And Valley Roof Geometry L 4 = Angle on RAFTER FLANGE PLANE made by intersection of PURLIN WEB PLANE L 4 = atan(cos(A)² * tan(B) / cos(R))

Solving Hip And Valley Roof Geometry L 5 = Complement of angle between PURLIN WEB PLANE and RAFTER WEB PLANE L 5 = atan(cos(L 3) * tan(L 4))

Solving Hip And Valley Roof Geometry L 6 = Complement of angle between PURLIN WEB PLANE and RAFTER FLANGE PLANE L 6 = atan(L 3) * cos(L 4))

Solving Hip And Valley Roof Geometry L 7 = Angle on PURLIN WEB PLANE made by RAFTER FLANGE PLANE L 7 = atan(B) * sin(R) * cos(L 2))

Solving Hip And Valley Roof Geometry L 8 = Angle between PURLIN WEB PLANE and a plane perpendicular to both RAFTER FLANGE PLANE and RAFTER WEB PLANE L 8 = atan(B) * cos(A))

Solving Hip And Valley Roof Geometry L 9 = Bevel on a plane perpendicular to both RAFTER WEB PLANE and RAFTER FLANGE PLANE made by intersection of PURLIN WEB PLANE L 9 = atan(B) * sin(R))

Solving Hip And Valley Roof Geometry L 10 = Angle between ROOF PLANE and RAFTER FLANGE PLANE L 10 = atan(B) * sin(R)) Vert Rafter Drop Dist = (bf/2 * tan(L 10) + thk / cos(L 10)) / cos(R) where “thk” is the hip plate thickness

Solving Hip And Valley Roof Geometry Vertical Rafter Drop Distance = (bf/2 * tan(L 10) + brg / cos(L 10)) / cos(R) where “brg” is the joist bearing depth