Portal Method – Moments and Deflection

The portal method is an analysis method for used for finding approximate internal forces in indeterminate structures subject to a lateral load.

Portal Frame Assumptions:

• Zero moment at the center of each girder. Therefore a hinge is placed here.
• Zero moment at the center of each column, if the column has a fixed base. Therefore a hinge is placed here (or at the bottom of the frame if the frame is hinged at the base).
• The interior columns hinges have twice the shear force as the exterior columns.
  • This is due to the fact that the frame is considered to be a superposition of 2 portals.
  • I also like to envision this due to the fact that the frame acts like a short cantilevered beam where shear is the predominate deformation force. If you recall the shear stress distribution of a rectangular beam, it is greater in the middle of the beam than at the extremes. I admit this may not be techinically correct but it helps me remember the method.

Portal Method Notes:

Portal Method – Approx Deflection Using Energy Methods

I will elaborate on this method in the future. For now take a look at the notes (yes I know the handwriting is awful).

Here is a spreadsheet for single story, pin-base moment frames. Currently it only accounts for bending deflection. Not all that useful but kinda fun none the less.

Portal Method – Lateral Load – Pin Base – Single Story Deflection

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Stability – AISC’s Direct Analysis Method

Intro

In the 14th Edition of the American Steel Construction Manual the Direct Analysis Method (DAM) is moved into the main specification from the appendix. I know many are not used to this new approach and some will say that if it’s not broke don’t fix it but I like the DA Method. Like anything else you need to put some time into learning it but it’s really not so bad in the end. It is a very interesting approach in that most design codes / manuals do not get into the analysis meaning that they don’t tell you how to get the required/design forces but rather they give an allowable/capacity of the member to which you are designing. However in DAM, AISC is assisting you in your analysis to make the design simpler. Essentially we need to address two different types of deflection/deformation associated with axial load that amplify moments in the structure. The first is P-δ (I may refer to this as P-d) which are moments associated with the axial load and deflection due to column curvature (Think of these as local displacements). The second is P-Δ (I may refer to this as P-D) moments which are caused by axial load and the translation of the end of the column (i.e. interstory drift) think of these as global displacements. Neither of these moments will show up in a first order elastic analysis. Well they may…sorta. I don’t want to get off track so I will explain what I mean later (this refers to common FEM models and placing multiple nodes along a member). We will now cover the DAM for a computer based approach and a simplified hand method.

Pd and PD Sketch

Overview

The direct analysis method is basically accounting for (3) issues:

1. Effects of initial geometric imperfections
2. Second-Order effects – Axial-Displacement Moments P-D and P-d (as shown above).
3. Effects of material non-linearity – In-elasticity due to residual stresses.

AISC actually states that there are (5) requirements. Below these requirements are listed and how they are addressed (AISC C-C1.1).

Considerations:

1. Consider all deformations
   • Note that this says ‘consider’ not necessarily include, i.e. column shears deformation, in-plane ‘rigid’ diaphragm displacement.
   • The model or analysis shall ‘consider’ all deformations.
2. Consider P-d and P-D
   • Perform a rigerous second order analysis
   • Use B1, B2 Method
3. Consider geometric imperfections
   • This typically this stems from column out of plumbness
   • This may be directly modeled in the analysis
   • A notional load may be applied to the analysis
   • Use KL = L
4. Consider stiffness due to inelasticity. This is typically due to residual stresses in framing members. Therefore some elements may soften ‘inelastically’  prior to reaching their design strength.
   • Apply a stiffness reduction factor
   • Use KL = L
5. Consider uncertainty in strength and stiffness
   • Apply a stiffness reduction factor.
   • Use KL = L

 Applying the Direct Analysis Method

First we will look at applying this method in a strict sense and assuming the use of a computer model. Then we will get to a more conservative hand calc method.

1. Model your structure and apply all loads. Set up your load combinations according to LRFD or ASD (Most likely see IBC load combo’s).
2. Run a first-order analysis and determine deflections.
   • Amplify the ASD loads x 1.6
   • Modify the stiffness of all members. For a first trial run use a 0.8 factor. This would be applied to axial (0.8*EA) and flexural (0.8EI) stiffness.
     • AISC states that the stiffness reduction need only be applied to members that contribute to the stability of the structure however they can be applied to all members to prevent artificial distortion.
3. Run a second-order analysis.
   • Amplify the ASD loads x 1.6
   • Modify the stiffness of all members. For a first trial run use a 0.8 factor. This would be applied to axial (0.8*EA) and flexural (0.8EI) stiffness.
     • AISC states that the stiffness reduction need only be applied to members that contribute to the stability of the structure however they can be applied to all members to prevent artificial distortion.
   • So this is just a mouse click away right? Well not quite. You should really know what your analysis software is doing. It is difficult if not impossible in some situations for software programs to perform a rigorous second-order analysis. For the program to perform this analysis it usually needs to run an iterative process on many nodes which may not be realistic. Therefore the program may use a geometric stiffness method which only accounts for P-D moments. Therefore P-d moments are still unaccounted for. However these moments may be “semi” captured if the column element is broken into several nodes. This way the deflection between nodes is captured in the analysis. AISC recognizes this practical problem and states that the P-d effects on the structure may be neglected if the second order drift to first order drift ratio (also known as B2) is equal or less than 1.7, also no more than 1/3 of the total gravity load on the structure is supported by columns that are part of the moment-resisting frames in the direction of translation being considered.
   •  In the commentary they equate the 1.7 to a 1.5 limit with no stiffness reduction.
   • Furthermore P-d effects must be considered to individual members subject to compression and flexure. In this case B-1 could be used.
4. Find the drift ratio (B2) of second order to first order drift. This will be used to determine what sort of notional loads will need to be applied.
5. Notional loads – Initial Imperfections
   • Initial imperfections may be directly applied in the model. Typically an out of plumbness of 1/500 is used the maximum specified in the Code of Standard Practice.
   • If not modeled directly notional loads may be applied. These are lateral loads Ni =0.002*α*Yi. These loads are distributed over the level in the same maner as the gravity load.
     • Ni = notional load at level i
     • α = 1.0 (LRFD); α = 1.6 (ASD)
     • Yi = gravity load applied at level i under each respective load combination
   • If B2 (drift ratio) is <= 1.7 then the notional loads may be applied as a minimum. Meaning that they are applied to gravity only load combinations but are not applied if the ‘actual’ lateral loads i.e. wind/EQ forces are greater than the notional load.
6. Adjust stiffness
   • For all stiffness that contribute to the stability of  the structure a 0.8 factor shall be applied i.e. EI, AE, etc.
   • Additionally for flexure the stiffness should be multiplied by 0.8:
     • For
     • For
       • α = 1.0 (LRFD); α = 1.6 (ASD)
       • Pr = required axial compressive strength of the member
       • Py = axial yield strength = Fy*Ag (yield stress x gross area)
   • In lieu of using taub a notional load of 0.001*α*Yi may be applied to the structure in similar fashion as the notional loads for initial imperfections. However these notional loads are additive for all load combinations.
7. Rerun second order analysis and check drift ratio, B2. Update any parameters based on the new drift ratio.
8. Design members using K=1. No Alignment Chart Required, Yeah!!!

 Applying the Direct Analysis Method

Now for a simplified hand calc to demonstrate the use of B1 and B2.
This post got to be a tad long so I’m going to break this into a separate post here.

References:

AISC 14th Edition CSC’s “Simple Guide to Direct Analysis” and webinar. Note that CSC’s Fastrak software does perform a rigorous second order analysis. RISA’s Practical Analysis with the AISC 13th Edition by Josh Plummer AISC Engineering Journal 3th Q 2008 “A comparison of Frame Stability Analysis Methods”

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UPDATED 4/26/12

Maybe I missed that day of class when diaphragm design was covered but for me I do not recall having done any diaphragm design in school. So here is a some basics just to get started and I will cover more details for different materials later.

Related links (design per specific material)
https://howtoengineer.com/wood-diaphragm-design/
https://howtoengineer.com/steel-deck-diaphragm-design/
https://howtoengineer.com/concrete-diaphragm-design/

The diaphragm can be thought of as a horizontal beam or as a plate element. It is usually constructed of wood sheathing, steel deck or concrete. Just as the floor (or roof) is checked for vertical load capacity, it is considered a diaphragm in the plane of the floor and check for shear when designing the Lateral Force Resisting System. Generally there are two different types of diaphragms. Rigid and Flexible. There is also semi-rigid which acounts for the stiffness of the diaphragm and the shearwalls and is similiar to a continuous beam supported on springs. We will cover this later. Rigid assumes that the diaphragm is infinitely rigid. Rigid diaphragms are generally concrete diaphragms which are very stiff compared to the LFRS supporting elements (supporting elements might be – Moment Frame, Braced Frame, Shear Walls, etc). Flexible diaphragms are more flexible when compared to the LFRS supporting members. (Side note – technically speaking the diaphragm is part of the LFRS.) So, how to know which to use? Well generally if the diaphragm deflects twice as much or more than the supporting vertical elements (shearwalls, moment frame, braced frame, etc.) of the LFRS than a flexible diaphragm may be assumed.

Summary :

Flexible – A horizontal simple span or continuous beam analogy is typically used. The shear walls act as supports and simple span or continuous beam and shear moment diagrams are used. The sides of the diaphragm transmit shear to the shearwalls and the top and bottom of the diaphragms are commonly supported by chord members. These members are subject to tension and compression forces and are usually designed by taking the moment of the diagram and dividing by the depth of the diaphragm/beam.

Rigid – Assumes the diaphragm is rigid and distributes in-plane forces to supporting members based on stiffness of the supporting members. When analyzing the diaphragm it is assumed to be perfectly rigid. When drawing the shear and moment diagrams the applied lateral load can be uniform or triangularly varying to represent accidental torsion. The vertical supporting elements (shearwalls, etc) can be thought of as applying opposing point loads. Therefore for the case of stiff end walls relative to the interior shear walls, the moment diaphragm is comparable to a simply supported beam spanning between the end walls.

To analyze the deflection of diaphragms bending, shear and slip must be accounted for. The deck will have deflection similar to typical beam deflection. Also because the diaphragm/beam is deep relative to its span it will also have shear deflection. There will also be slip. There will also be slip. The slip may occur in the diaphragm panel connection to the substrate (i.e. nail slip in wood panel to joist connection) or in chords (i.e. nail splice of 2x top plate for wood framed construction).

See the attached examples and video to really help better understand general diaphragm design better. It should be noted that the flexible diaphragm presented in the example is a conservative approach as it assumes two simple spans. The actual behavior may more closely resemble a continuous beam with intermediate supports. However finding the tension based on simple span will result in a conservative yet practical design.
General Diaphragm Design Video Tutorial
General Flexible Diaphragm Tutorial PDF

Here is some more food for thought- Several good references:
http://www.nehrp-consultants.org/publications/download/nistgcr10-917-4.pdf
http://www.hnd.usace.army.mil/techinfo/ti/809-04/ch7d.pdf

Below is an interesting thread where a semi-rigid diaphragm might want to be used. Basically it is a situation were there are different diaphragm spans and diaphragm/shearwall stiffness on the same floor:

http://www.eng-tips.com/viewthread.cfm?qid=318931

I will try to work up and example for this situation in the future in the mean time here is a decent referance:

http://communities.bentley.com/products/structural/structural_analysis___design/w/structural_analysis_and_design__wiki/ram-ss-semirigid-diaphragms-tn.aspx

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I was following a discusion found here and thought that the pdf that was posted would be helpful to some.

It lists equation to deflection or sidesway of braced frame configurations. It can be verified with the method of virtual work.

Braced Frame Story Shear Deflection

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