Architectural Structures

10-7 DESIGN METHODS Conceptual Design

Portal method

The Portal Method for rough moment frame design is based on these assumptions:

•Lateral forces resisted by frame action

•Inflection points at mid-height of columns

•Inflection points at mid-span of beams

•Column shear is based on tributary area

•Overturn is resisted by exterior columns only

1Single moment frame (portal)

2Multistory moment frame

3Column shear is total shear V distributed proportional to tributary area: Va = (V/B) L1 / 2

Vb = (V/B) (L1+L2) / 2 Vc = (V/B) (L2+L3) / 2 Vd = (V/B) L3 / 2

4Column moment = column shear x height to inflection point Ma = Va h / 2

Mb = Vb h / 2 Mc = Vc h / 2 Md = Vd h / 2

5Exterior columns resist most overturn, the portal method assumes they resist all

6Overturn moments per level are the sum of forces above the level times lever arm

of each force to the column inflection point at the respective level:

Column axial force = overturn moment divided by width B

N = M / B

7Beam shear = column axial force below beam minus column axial force above beam Level 1 beam shear:

V = N1 - N2 Roof beam:

V = N2 - 0 = N2

Beam bending = beam shear times distance to inflection point at beam center M = V L / 2

Beam axial force is negligible and assumed 0

10-15 DESIGN METHODS Conceptual Design

10-16 DESIGN METHODS Conceptual Design

Moment frame

Eight story steel moment frame, high strength steel, Fy = 50 ksi Fa = Fb = 30 ksi Live load 50 psf, load reduction R in percent per IBC

R = 0.08 (A-150) A = tributary area

Max. reduction: 40% for members supporting a single level, 60% for other members

10-17 DESIGN METHODS Conceptual Design

Ground floor

Column shear

4th floor

Column shear

10-18 DESIGN METHODS Conceptual Design

Test models

Static models are useful to test structures for strength, stiffness, and stability. They may have axial resistance (truss), bending resistance (beam), or both axial and bending resistance (moment frame). Static models have three scales: geometric scale, force scale, and strain scale. The geometric scale relates model dimensions to original dimensions, such as 1:100. The force scale relates model forces to the original structure. For a force scale of 1:100 one pound in the model implies 100 pounds in the original structure. The force scale should be chosen to keep model forces reasonable (usually under 50 pounds). The strain scale relates model strain (deformation) to strain in the original structure. A strain scale of 1:1 implies model strain relates to original strain in the geometric scale; given a geometric scale of 1:10 a model strain of 1 inch implies 10 inch original strain. For structures with small deformations may a strain scale of 5:1, for example, helps to visualize strain. However, structures with large strain like membranes, require a strain scale of 1:1 to avoid errors (see 2). Scales are defined as:

Geometric Scale: SG = Lm/Lo = model dimension / original dimension Force Scale: SF = Pm/Po = model force / original force

Strain Scale: SS = εm/εm = model strain / original strain The derivation for axial and bending resistance models assumes:

A = Cross-section area E = Modulus of elasticity I = Moment of inertia

k = Constant of integration for deflection; for cantilever beams with point load ∆ = kPL3 / (EI) where k = 1/3

m = Subscript for model

o = Subscript for original structure

Axial resistance

1Axial strain ∆L = P L/(AE); unit strain ε = ∆L/L

2Structures with large deformations, such as membranes, yield errors if the strain scale SS is not 1:1; as demonstrated in the force polygon

3Structures like trusses, with small deformations, may require a strain scale SS >1 to better visualize deformations.

10-21 DESIGN METHODS Conceptual Design

Bending resistance

Since the model and original have the same load and support conditions the constants of integration km = ko, hence the term ko/km may be eliminated. The term Lo2/Lm2 = 1/SG2 = 1/geometric scale squared, and εm/εo = SS = strain scale. Therefore the force scale is:

SF = EmIm/(EoIo) 1/SG2 SS

In the simplest form the force scale is equal to the geometric scale squared for both axial and bending resistant models. Thus a model with a geometric scale of 1:100 has a force scale of 1:10,000 if it is made of the same material or modulus of elasticity as the original structure.

Combined axial and bending resistance

Models with both axial and bending resistance, such as moment frames, should be of the same material or elastic modulus as the original in order to avoid errors. Referring to diagrams 3, if, for example, the elastic modulus of a model is half as much as in the original structure and the cross-section area is doubled to compensate for it, then the moment of inertia is four times greater, assuming area increase is perpendicular to the bending axis. For small adjustments this can be avoided by increasing the area parallel to the bending axis. Large differences in stiffness, such as wood simulating steel, with an elastic modulus about 20 times greater, are not possible. In such a case the strain scale could be 20:1 to amplify deflection rather than adjusting the cross-section area.

Since k is the same in the model as in the original, for equal load and support conditions, it may be eliminated from the force scale equation

3Correlation between cross-section area A and moment of inertia I

demonstrates incompatibility between A and I since they increase at different rates, unless the increase is only in width direction

10-22 DESIGN METHODS Conceptual Design

Test stand

•Light gauge steel frame 3’x5’

•Frame to support test models

•Adjustable platform for loads below the test model (crank mechanism lowers platform to apply load)

•Blocking deice holds load platform at any position

•Support frame for measure gauges above test model

Test procedure

•Position model with open base to allow loads below

•Suspend load cups filled with lead or sand from model (support loads on load platform before loading)

•Attach measure gauges above model

•Lower load platform with crank to apply load

•Measure deformations and stress

•Apply alternate loads (half load, etc.)

•Record deformations and stresses for all load conditions

Note

•Apply loads briefly to avoid creep deformation

•Apply loads gradually to avoid rupture

•Test all load conditions that may cause critical deformation or stress

•Adjust design if deformation or stress exceeds acceptable limits

10-23 DESIGN METHODS Conceptual Design