Shapebuilder 3 Properties Defined

ShapeBuilder 3.0 Geometric and Structural Properties

Coordinate systems: Global (X,Y), Centroidal (x,y), Principal (1,2)

Basic Symbols Units Notes: Definitions Geometric Analysis or Properties Shape Types,

Area A, L2 Non-composite Gross (or full) Cross-sectional area of the shape. Also Ag shapes. A = dA ∫ A Depth d L All Total or maximum depth in the y direction from the extreme fiber at the top to the extreme fiber at the bottom. Width b, L All Total or maximum width in the x direction from the extreme fiber at the left to the extreme fiber at Also w the right. 2 Net Area An L Non-composite Net cross-sectional area, after subtracting interior holes. shapes with A = dA − A interior holes ∫ ∑ hole A Holes Perimeter p L All The sum of the external edges of the shape. Useful for calculating the surface area of a member for painting. Center of c.g., (Point) (L,L) All, unless That point where the moment of the area is zero about any axis. This point is measured from the Gravity Also located at the global XY axes. Centroid of (Xc,Yc) global origin. xdA ydA Area ∫ ∫ x = A , y = A c A c A 4 Moments of Ix, Iy, Ixy L All Second moment of the area with respect to the subscripted axis, a measure of the stiffness of the Inertia cross section and its ability to resist bending moments. I = y 2 dA, I = x 2 dA, I = xydA x ∫ y ∫ xy ∫ A A A Principal Axes 1, 2, Θ, Coordinate Any shape The axes orientation at which the maximum and minimum moments of inertia are obtained.

Also Θp, System where Ix, Iy are Moment of inertia Ixy is zero with respect to this coordinate system. These axes are the same as Theta (Point) (L,L) NOT principal the centroidal axes (x,y) if either of the xy axes is an axis of symmetry, where Ixy = 0. Angle (A) axes 1 −  2I xy  θ = tan 1    −  2  I y I x 

4 Principal I1, I2, I12=0 L Any shape Maximum and minimum moments of inertia, defined on the principal coordinate system axes. The Moments of Also Imax, where Ix, Iy are product of inertia with respect to these axes is always zero. Inertia I NOT principal min 1  2 2  axes I = ()()I + I ± I − I + 4I 1,2 2  x y y x xy  4 Polar Moment Ip, L All, if different Moment of inertia with respect to the z axis (normal to the section plane). This is equivalent to the of Inertia Also J from the torsion constant, J, only for circular cross sections. In other cases the polar moment of inertia is Torsion larger (perhaps much larger) than the torsional constant. Constant I = r 2 dA = I + I = I + I p ∫ x y 1 2 A Torsion J, L4 All Torsional stiffness factor is a more accurate measure of the torsional rigidity than polar moment of Constant Also K, inertia. For thin-walled open cross sections it is approximated as the sum of b*t3 for each component rectangle, for thin-walled closed sections it is a function of the enclosed area. In ShapeBuilder it is calculated through a Finite Element analysis. The general definition is: T J = , Where T is the torque, G is the shear modulus, and ϕ is the angle of twist due to the Gφ torque. 3 Section Sx, Sy, L All The section modulus is useful for calculating the extreme bending stress. Defined as I/c, where I Modulus Sxt, Sxb, is the moment of inertia about the axis in question, and c is the distance from the centroid to the Syl,Syr, extreme fiber in the perpendicular direction. For unsymmetric areas, the section modulii are Sx(+y), Sx(-y), different on each side. ShapeBuilder reports up to four values for unsymmetric sections. Sy(-x), Sy(+x) Radius of rx, ry, L All Radius of gyration is the distance from a reference axis to a point at which the entire area may be Gyration r1, r2, rp concentrated and still have the same moment of inertia as the distributed area. It is used as a Also k measure of the stability of a column.

I I y I I I p r = x , r = ,r = 1 ,r = 2 ,r = x A y A 1 A 2 A p A

Polar Radius of ro L Advanced The polar radius of gyration about the shear center is defined in AISC LRFD Appendix E as: Gyration About Stress Analysis + + 2 + 2 I x I y A(xs ys ) Shear Center r = o A

Where xs, ys represent the distance from the centroid to the shear center. This property is only shown if the shear center does not coincide with the ceintroid. AISC Flexural H unitless Advanced Derived from the polar radius of gyration about the shear center is the AISC flexural constant H = 2 2 2 Constant Stress Analysis 1 - (xs + ys )/ro . Refer to AISC LRFD Appendix E. H=1 if the shear center and centroid coincide. Plastic Neutral PNA Point (L,L) All, if different Location of the plastic neutral axes divides the area of the section into equal halves and is not Axes from centroid. always located at the centroid. © Copyright 2002, IES, Inc. All rights reserved. 2 Basic Symbols Units Notes: Definitions Geometric Analysis or Properties Shape Types,

3 Plastic Modulus Zx, Zy L All The plastic section modulus is the arithmetical sum of the statical moments about the plastic neutral axis of the parts of the section above and below that axis. This term is used to calculate

the plastic moment capacity of a section Mpx = σyZx, where σy is the yield stress of the material. 2 Shear Areas SAx, SAy L Advanced Shear area, represents the area of the cross section that is effective in resisting shear Stress Analysis deformation. These areas can be defined in terms of the shear correction factors: (FEM) αA A SA = , SA = x y α x y

Material Related Symbols Units Analysis Types or Definitions Properties Shape Types

Density δ F/L3 All The material density is the weight per unit volume. Weight W F/L All The weight per unit length of the cross sectional area. This is a function of the material density (or densities in a composite section). Transformed Area A* L2 Any shape with For composite area, A*, (with different modulus values, the areas are ìweightedî by (And other transformed components the modular ratio of Ei/Ebase. properties) including multiple n E * = i materials A ∑ Ai i=1 Ebase This property is shown as an example of transformed properties. The following properties are calculated by ShapeBuilder as transformed and will be denoted with an asterisk (*) to indicate the transformation to Ebase. Area, Centroid, Moment of Inertia, Section Modulus, Radius of Gyration, Torsion Constant, Warping Constant, First Sectorial Moment, Shear Center, Plastic

Neutral Axis (Reinforced Concrete: Force balance point: bβ10.85fíc and Asfy) For equations, please refer to Pilkey, Formulas for Stress, Strain and Structural Matrices, pp 38-41. For concrete shapes these properties are calculated on the uncracked section. 2 Modulus of Elasticity E, Ei, Ebase F/L All (May be Modulus of elasticity or Youngís Modulus of the material is a measure of the rate of different for parts of change of normal stress to normal strain. For some materials this may be different in the section) different directions and may be different for tension and compression. Often, the value is approximated as a constant, although it is not constant. For composite sections, composed of multiple materials, ShapeBuilder will calculate many properties as modulus-weighted properties. These properties are called ìTransformedî properties and are calculated upon a specified base modulus, Ebase. Transformed properties are denoted with an asterisk (*) in ShapeBuilder. Shear Modulus, G F/L2 All Shear modulus is the rate of change of shear stress with respect to shear strain within Also Modulus of the elastic range. For some materials, like wood, it is different in different directions. Rigidity E G = , Where ν is Poissonís ratio. 2(1+ν ) Poissonís Ratio ν, unitless All An experimentally defined material coefficient that is the ratio of lateral to longitudinal Also v strain due to uniaxial longitudinal stress in the elastic range. For metals Poissonís ratio is usually in the range of 0.25 to 0.35. It ranges from 0.1 (for some concrete) to 0.5 (for rubber).

Torsion Properties Symbols Units Analysis Types Definitions or Shape Types

Shear Center SC, (Point) (L,L) Advanced Stress Shear center is the point on the cross section where an applied shear force will Also, Flexural Center Also S, Analysis cause no twisting of the cross section as it bends. In general this is not the Also C, Shown only if centroid. If the section is symmetric, the shear center will lie on the axis of (xs,ys) different from the symmetry, for doubly symmetric sections, the shear center will coincide with the centroid. centroid. This point is located with respect to the global origin. The notation (xs, ys) locates the point with respect to the centroid (If different). 6 Warping Constant, Cw, L Advanced Stress Warping constant is calculated as: Γ Also: Warping Stiffness Also , Analysis Γ = ω 2 dA Factor Also Iw ∫ A Also: Sectorial Moment Reference Pilkey, Ch. 2, Ch. 14, Ch. 15. of Inertia Angle of Twist Phi, A Advanced Stress T φ Analysis φ = , Where T is the Torque, G is the shear modulus for the material. GJ Warping Function Omega L2 Advanced Stress Varies over the cross section. Plotted graphically in ShapeBuilder. ω Analysis

Reinforced Concrete Symbols Units Analysis Types Definitions Properties or Shape Types 4 Cracked Moment of Icrx,Icry L Concrete Represents the cracked stiffness of the section, where concrete in Inertias Analysis tension is neglected. Interaction Diagrams -NA- -NA- Concrete A diagram showing the relationship between axial load and bending in a Analysis column. The surface represents an upper bound before failure. Points lying inside the curve represent acceptable load combinations.

Capacity Forces Pn, Mnx, Mny F, FL, FL Concrete A set of loads (P, Mx, My), that produce concrete strain levels that would Analysis cause failure (0.003 usually means failure). These are the extreme points on the concrete interaction diagram.

Applied Loads and Symbols Units Analysis Types Definitions Stresses or Shape Types

Load Application (xL, yL), γ (L,L), A Advanced Stress The location and orientation of the applied loads are specified at an Analysis arbitrary coordinate system (a,b), γ where a and b are coordinates in the global system and angle γ is the rotation of that system with respect to the global X axis. Counter-clockwise angles are positive. Load application may be with respect to the following coordinate systems: Global Origin & Orientation, Centroidal Origin & Orientation Principal Origin & Orientation, Shear Center (Global Orientation), or Any Specified Location & Orientation Axial Force P F Advanced Stress Applied axial force, where compression is negative, tension is positive. If Analysis the load is applied at a point other than the centroid, it will also induce moments on the cross section.

Bending Moments Ma, Mb FL Advanced Stress Applied bending moments in the ab axes, as defined above. Moment Analysis sign conventions follow the right-hand-rule. For example, a positive moment Mx, would cause tension on the top of the section. Shear Force Va, Vb F Advanced Stress Applied shear forces in the a and b directions, respectively. Positive Analysis shear is in the direction of the positive axis. Shear forces applied at a point other than the shear center will induce torsional loads as well. Torque T, FL Advanced Stress Applied torsional moment about the shear center. Counterclockwise is Also Mz Analysis positive, following the right-hand-rule. 2 Axial Stress fa F/L Advanced Stress Normal axial stress due to axial load only, P/A. Normal stress is positive Analysis for tension, negative for compression. 2 Flexural Stress fbx, fby, fb (total) F/L Advanced Stress Normal stress due to bending moments only, Mc/I. For biaxial bending Also: Bending Stress Analysis on unsymmetric shapes, the total bending stress is calculated according to: + +  M x I y M y I xy   M y I x M x I xy  f =  y −  x b  − 2   − 2   I x I y I xy   I x I y I xy  2 Flexural Shear Stress fvx, fvy, fv (total) F/L Advanced Stress Shear stress due to shear force only. This equation is given as follows, Also: Bending Shear Analysis where b is the width of the cross section at the point where shear is Stress calculated:  + +  1  Qx I y Qy I xy   Qy I x Qx I xy  τ =  V −  V  b  I I − I 2  y  I I − I 2  x  x y xy   x y xy   2 St. Venant Shear ft F/L Advanced Stress Shear stress due torque only, Tr/J. This is also called the pure torsional Stress Analysis shear stress. 2 Warping Normal Stress fw F/L Advanced Stress Normal stress due to warping, Bω/Cw. This component of normal stress is Analysis included in the total normal stress if there is warping of the cross section under the given loads.