Masonry Columns, Piers and Pilasters

How to analyze and design reinforced masonry columns, piers and pilasters and clarifying effective spacing.

For analysis of masonry members utilizing flanges and compression reinforcement see Masonry – Compression Reinforcement and Effective Flanges

References

Ref 1: MSJC – 2005, TMS 402 (AKA ACI 530) – 05 (2008 version is very similar). Found here

Ref 2: Masonry Structures Behavior and Design 2nd Edition by Robert Drysdale, Hamid, and Baker.  (the Third edition found here)

Ref 3: Reinforced Masonry Engineering Handbook Clay and Concrete Masonry 6th Edition, Max L Porter (James E. Amrhein original author). Found here

Definitions

Based on MSJC 05 (all should be similar for 08)

Column

(MSJC-05 Section 1.6) – An isolated vertical member whose horizontal dimension measured at right angles to its thickness does not exceed 3 times its thickness and whose height is greater than 4 times its thickness.

Pier

(MSJC-05 Section 1.6) - An isolated vertical member whose horizontal dimension measured at right angles to its thickness is not less than 3 times its thickness nor greater than 6 times its thickness and whose height is less than 5 times its length. In reference to MSJC these are typically part of wall frames (see Ref 3 page 250) and is why there dimensions are defined as such. This however may be confusing as many times a pier is referred to a column which is built integrally with a wall (see pg 403 of Ref 2). Having said that I believe if the column is built integrally with the wall it would be considered a “flush pilaster” (see Ref 3).

Pilaster

-  While Ref 1 does not give an explicit definition of a pilaster, it does infer that a pilaster is built integrally with the wall. Ref 2 states that a pilaster is a column that is built integrally with a wall and interacts with the wall to resist an out-of-plane lateral load, it is called a pilaster. Pilasters maybe ‘flush’ (in the plane of the wall) or project out from the wall in one or both directions.

Pedestal

– Upright compression member with a ratio of unsupported height to average least lateral dimension not exceeding 3. This term is typically found in concrete design and

refers to short foundation elements. Many may refer to pedestals as piers or foundation piers.

Foundation Piers

- (Ref 1) An isolated vertical foundation member whose horizontal dimension measured at right angles to its thickness does not exceed 3 times its thickness and whose height is equal to or less than 4 times its thickness.

Wall

- (Ref 1) A vertical element with a horizontal length to thickness ratio greater than 3, used to enclose space.

Loadbearing Wall

- (Ref 1) a Wall carrying vertical loads greater than 200 plf in addition to its own weight.

Discussion

As you can see these elements may be referred to one another as they all are used to resist axial and flexure. I would need to check the 2011 version of the MSJC but it should also be noted that Piers are found in the Strength Design of Masonry only and Pilaster are only found in the ASD method. However lets look at how the analysis and design differs for each. I think some main points to consider are – Is the element isolated or part of wall construction? If the element is isolated it would generally be considered a column or pier. If it is part of a wall system then it would generally be considered a pilaster or flush pilaster. The main difference then between these is that for column/pier elements there is – added vulnerability due to isolation, typically compression reinforcement and potential for bi-axial bending. Another key to remember is that if the reinforcement is going to be considered to be effective in compression, the enforcement must be tied (confined) for any of the above mentioned elements.

Now lets look at how each classification effects the design.

Design

Dimensional Constraints / Effective Widths

This can be hard to keep track of as well. How much of the wall should be taken as effective to resist axial and flexural loads? What are the limits on effective flange widths? It is interesting to note that Ref 3 pg 177 limits the effective flange for pilasters to 3x the wall thickness each side of the pilaster however allows for 6x wall thickness for effective flange width on flanged masonry shear walls.

Effective Width

1. Walls – (ref 1 Sect. 2.3.3.3)
   1. Out-of-plane Forces – Running Bond Effective width shall be the lesser of:
      1. Center – Center spacing of bars
      2. 6 x wall thickness
      3. 72 inches
      4. H/6 in the case of a flanged shear wall. Where H is the total wall height (between lateral supports transferring load to the load to the shear wall).
   2. Axial Concentrated Load – Effective width
      1. 4 x wall thickness + bearing width or length of wall
   3. Design for Axial and Flexure
      1. The reinforcement is rarely tied and typically only resists tension forces.
      2. For ASD if the load is concentrated then the 4x the wall thickness limit would apply otherwise the limits presented in point 1 would apply and the reinforcement would need to be tied if it going to be relied upon to carry compression forces.
   4. In-Plane Forces – Flanged Shear Walls
      1. ASD – The flange is limited to 6x wall thickness each side of the web (Ref 3 pg 199)
      2. Wall intersections shall meet Ref 1 – Section 1.9.4.1
2. Flush Pilasters  
   1. Effective Wdith
      1. Tpcially taken as 4 x wall thickness (Ref 3 page 178)
3. Pilasters
   1. Effective width
      1. Ref 1 and 2 – 6 x Wall thickness each side of web
      2. Ref 3 – 3 x Wall thickness each side of web
   2. Consideration for flanges to be effective (see Ref 1 – 1.9.4.2).
      1. Flanges  (wall intersections) must be capable to transfer the required shear stress
      2. Atleast 50% of the masonry units at the interface shall interlock; or Steel connectors grouted into the wall (1/4″x1.5″ x28″ inculding 2″ long 90 deg bend at each end to form a U or Z shape; or Intersecting bond beams shall be provided in intersecting walls at a maximum spacing of 48″ oc (As = 0.1 in^2 and shall be developed on each side of intersection)
4. Piers
   1. Limits on Width
      1. The width should be greater than 3 x thickness but less than 6 times its thickness
         1. I believe if it were less than 3 x thickness it would then be a column and if it were greater than 6 it would then be a wall.
   2. Design limitations – Strength Design
      1. Max factored axial load (per Ref 1 3.3.4.3.1)
      2. Max effective height should not exceed 25 x nominal thickness unless the pier is designed according to provisions of Ref 1 Section 3.3.5 (walls, must consider P-Delta effects)
      3. Nominal thickness should not exceed 16″
      4. Provide (1) bar in the end cells and minimum area of longitudinal reinforcement should be 0.0007bd
      5. Uniformly distribute reinforcement
5. Column
   1. Limit on Width (Ref 1 – 2.1.6)
      1. Horizontal dimension measured at right angles to its thickness does not exceed 3 times its thickness. Minimum side dimension should be 8″
   2. Limits on Height
      1. Height is greater than 4 times its thickness
      2. Ratio of Effective Height / least nominal dimension < 25 (Ref 1 – 2.1.6.2)
   3. ASD Requirements (Ref 1 Section 2.1.6
      1. Min Side Dimension – 8″
      2. Height / least nominal dimension <= 25
      3. Min Moment should be 0.1 x each side dimension
      4. Min longitudinal reinforcement and shall at least 4 bars
      5. Max long reinf
      6. Compression reinforcement shall met lateral tie requirements (as always)
   4. SD (strength design) Requirements (Ref 1 Section 3.3.4.4)
      1. Same as ASD except that the provisions limit the distance between lateral support should be limited to 30 x the nominal width where as ASD uses ‘effective height’ (meaning considering end restraints) should be < 24 x thickness. Also max reinforcement shall be per Ref 1 Section 3.3.5 but not to exceed  
6. Lateral Ties
   1. Requirements (Ref 1 Section 2.1.6.5) If reinforcement is to be considered effective in resisting compressive forces lateral ties must be provided.
      1. 1/4″ diameter (min)
      2. Vertical spacing < smaller of -
         1. 16″
         2. 48 lateral tie bar diameters
         3. least cross-sectional dimension
      3. Every corner and alternate longitudinal bar shall have lateral support provided by the corner of a lateral tie (angle should be less than 135 deg). Except for a circle arrangement which is permitted. No bar bar shall be farther than 6″ (each side) from a laterally supported bar. (i.e. the middle bar should be less than 6″ from 2 bars that are laterally supported by a lateral tie ‘corner’)
      4. Place lateral ties in mortar joint or grout
      5. Lap length shall be 48 tie diameters (min)
      6. Locate the first and last lateral tie at 1/2 spacing.  Also above and below single beams framing into the vertical member.
      7. Where beams or brackets frame into a column from four directions ties may be terminated 3″ (max) from the lowest reinforcement in the shallowest of the beam members framing into the column.

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Stable Feature Behind MSE Wall / Narrow Wall

This will be one of several  posts regarding MSE walls with complex geometry. The main references for this post and the posts to follow will be the research conducted by The Federal Highway Administration (FHWA) and mainly the publications listed below. The methods presented are simplified / preliminary methods. Ultimately a limit equilibrium analysis should be run. However this is beyond the scope… for now.

References

FHWA-NHI-10-024 Design and Construction of MSE walls and Reinforced Soil Slopes

FHWA-NHI-00-043 MSE Walls and Reinforced Soils Slopes Design and Construction Guidelines (Now superseded by publication above)

FHWA-CFL/TD-06-001 Shored MSE Wall Systems (SMSE) Design Guidelines

All of which can be found here on FHWA website.

Overview

I am posting my notes that correlate to the spreadsheet for now and a brief overview. We will elaborate more on the procedure in the near future.

The stable feature prevents external lateral earth pressures from exerting force on the reinforced soil mass. Therefore traditional sliding and overturning are not generally a design concern. For the trial wedge we define the geomtry and solve for the forces in the horizontal and vertical directions. We have two unknowns and two equations of equilibrium which we solve using matrix algebra (in excel) to determine the active force on the wall.

Required checks are as follows:

1. Pullout of geogrid
2. Reinforced soil and stable feature interface failure plane stability
3. Bearing
4. Global / Slope stability

Calculation

Force Diagram – Stable Feature Behind MSE Wall

Calculation assumes that the stable features extends up or very near the top grade surface. The wall face, top grade and stable feature slopes may all be adjusted. Currently the failure angle must be ‘manually’ adjusted to find the maximum active pressure acting on the back of the wall facing units.

Notes

Trial Wedge with Stable Feature Behind Wall Notes HTE

Spreadsheet

Trial Wedge for Stable Feature

As always please use with caution and check for errors!

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Torsion in Rectangular Prism

I hope to expand on this post in the future and relate the equations that are derived for find stresses and strains of a rectangular prism (bar) to that of an I shaped section. Then relating that to AISC’s design guide 9 (see here for AISC design guides).

Explanation of Stress – Strain Behavior

We recall that for circular shafts that section planes remain plane (and circular) and that all radii lines remain straight; no bulging or cupping. Therefore shear stress occurs along a set of concentric circles. This however is not possible with a rectangular shaft as if the shear stress was tangent to a set of concentric circles the shear stress would not be perpendicular to the boundary. The shear stress must be perpendicular to the boundary. Why? Because if it is not than components perpendicular to the boundary will exist which creates a ‘complementary’ shear stress on the outside free edge of the prism (see Figure below). From this we conclude that the shear stress must be perpendicular to the periphery (boundary) of the section.

Torsion – Shear Stress in Non-Circular Prism

We can also conclude that the shear stress must also be zero at the corners as neither one of the two perpendicular components can exist (see Figure below).

Torsion – Shear Stress in Non-Circular Prism Corner Stress

For a circular section it was also shown that the plane cross sections remained undistorted in their own plane. Meaning there was no distortion of the concentric circles and radii, the radii remained straight.  We cannot directly prove this to be true for a non-circular prism but we can look at what it means if we assume there is some distortion.

For a non-circular section, we imagine a perpendicular grid pattern across a section. These grids then become distortion in the plane of the section. These shear stress then cause shear stresses to appear on sections parallel to the longitudinal axis. These stresses do not add up to a twisting moment and are ‘useless’. This does not constitue a proof but it should be noted that nature opposes an applied action (Torque) through the minimum elastic energy i.e. the simplest way possible or via the simplest stresses possible. It is shown via energy methods that these stress do NOT occur and or reasoning is correct.

Torsion – In Plane Distortion

 Calculating Stress

Too be continued….. Refer to Roark for now

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Mechanics of Materials – Basics

I would like to describe the process or logic or algorithm if you will in which must be followed in order to develop theories for mechanics of materials.  The process / logic / algorithm can go in either direction but you CANNOT skip steps. This idea is fact but was nicely presented in Dr. Madhukar Vable in his textbook Intermediate Mechanics of Materials.

Structural Analysis Algorithm (Logic)

1. Displacements
   • Kinematics
2. Strains
   • Material Models
3. Stresses
   • Static Equivalency
4. Internal Forces and Moments
   • Equilibrium / Energy Methods
5. External Forces and Moments

So, you may start with external forces then using equilibrium or energy methods you may determine the internal forces and moments. From here you may determine stresses by static equivalency. Then strains may be determined by the material model and finally displacements by use of kinematics. You may start at any step in the above algorithm and can proceed in any direction but you cannot skip a step. Meaning, if you know the internal forces and moments you can not determine the displacements with out considering the stresses and strains.

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AISC’s Approximate Second-Order Analysis B1 B2 Method

Lets look at a very simple building with a simple moment frame to resist lateral loads. We will complete the analysis using AISC’s approximate second-order analysis more commonly known as the B1 – B2 method. We will demonstrate this method with the use of an example on a very simple building. Before we begin lets discus the B1, B2 method. The B1-B2 Method is an approximate second-order analysis using the multipliers B1 and B2 (no way!). The procedure can be found in Appendix 8 of the 2010 AISC 350 Steel Construction Manual.

B1 Discussion:

     See AISC Eqn A-8-3

Where:

• coefficient assuming no lateral translation of the frame. See Eqn A-8-4 . However I would conservatively use 1.0 and be done with it!
• 1 (LRFD), 1.6 (ASD)
• = may be determined from a first-order estimate is permitted (for use in eqn A-8-3 only).
  • = axial load (using ASD/LRFD Load Combo’s) assuming the structure is “restrained against lateral translation” (first order).
  • = axial load due to lateral translation of the structure only (using ASD/LRFD Load Combo’s) (first order).
• = coefficient assuming no lateral translation of the frame.
  • = Flexural rigidity required by the analysis i.e because we are using DAM =
  • where is defined in AISC Section C2.3 (adjustments to stiffness). I prefer to avoid adjusting as it seems to become an iterative process. Therefore I add an additional notional load =
    • is defined in Section C2.2b (essentially the gravity load at level ‘i’) and is as defined above.
• is based on the assumption of no lateral translation so we conservatively use 1.0.

B1 accounts for for effects in compression non-sway compression members. These are the moments caused by local displacements due to axial load. The AISC commentary suggests that if B1 is > than 1.2 than a rigorous second order analysis should be undertaken. This is due to the fact that B1 captures the local effects of second order forces/deformation but does not capture what effect these ‘local’ deformations may have on the overall structure. This is semi obvious in the fact that the we are using results from a first order analysis and also none of the variables relate to the rest of the structure.

B2 Discussion:

      See AISC Eqn A-8-6

Where:

• is as defined above.
• total vertical load supported by the story (using ASD/LRFD load Combos) including loads in columns not part of the lateral force resisting system. This is essential the total gravity load on the story under evaluation.
• the “elastic critical buckling strength for the story in the direction of translation being considered, determined by buckling analysis” or
  • 
    • 
      • = total vertical load in columns (in the story under evaluation) that are part of the moment frames (=0 for braced frame systems).
    • L = height of story
    •  Inter-story drift. Use first order analysis and stiffness as required by analysis i.e For DAM use reduced stiffness as discussed above (see the B1 discussion). Where drift varies across the story, the maximum drift may be used conservatively or a weighted average based on vertical load. It is important to realize here that this drift or deflection should include the deflection of the columns in the frame and ALSO the diaphragm deflection. The commentary words this as any “horizontal framing system that increases over-turning effect”. This makes sense as columns that are not part of the frame (usually called leaning columns) will displace greater than the columns that are part of the moment frame. This displacement coupled with gravity load will increase the demand on the frame.
    • H = Story shear (lateral force), produced by the lateral force used to compute the inter-story drift. Once again, coordinate the use of total story shear or individual force on the frame. As mentioned for .
    • AISC provides a user note that says H and “may be based on any lateral loading that provides a representative value of the story lateral stiffness.” As you can see the equation is really using the lateral stiffness of the structure (kip/in).

B2 accounts for effects on forces and moments in all members. These effects are due to lateral displacement of the structure. We also notice that B2 uses several variables which related back to the overall structure, mainly the the story shear, gravity load and deflection. Furthermore we see that the deflection is based on not only on the deflection of the frame but also of the diaphragm which indirectly accounts for “leaning columns”.

Short Summary

We can see that B2 applies to all members part of the Lateral Force Resisting System (LFRS), meaning any member with (members not part of the LFRS will not have these forces) and B1 applies only to compression members of the LFRS. We see that by using the DA method we eliminate having to use sidesway alignment chart (fig C-A-7-2) to try and determine K (effective length). However the B1-B2 method can be trick when when B1 gets large and multiple members frame into a column. Why? Well because the moments should be balanced and thus the column (compression member) will be multiplied by B1 and the beams would not be. So this moment then needs to be distributed to these connecting elements. I would suggest reading the Summary at the end of the commentary to the B1-B2 method  (Comm 8 pg 16.1-526) they discuss the how to apply the method in more ‘global’ terms and will give you a better feel for applying the method. It is too long to repeat here.

Example

Alright so lets define some parameters and loads.

Lets use a 1-story, 3-bay x 4-bay rectangular building. Bay size is 25′x25′. The columns are pinned at the base and have a moment connection from beam to column. For this analysis we will use a “Wind Only” moment frame or flexible moment connection. There is definitely some debate on using this type of system. Essentially beams are designed as simply supported for gravity loads and fixed for lateral loads. For a more complete discussion on “wind only” or flexible moment connection- moment frames see - Wind Only Moment Frames Discussion.

Size: 1-Story- 3×4 – 25′x25′ Bays (75′ x 125′ Building). Height is 15′ columns with 5′ parapet.

Gravity Loads:

• Dead load: Say 30 psf just to give it some weight.
• Live load: Say 100 psf again, too add some weight.

Lateral load:

• Check seismic, but lets use wind for now.
• Wind: say 20 psf. Most people will forget that there is a 1.5 multiplier on the parapet when designing for the LFRS so lets use 30 psf on  the parapet.

Analysis

We have 2 separate analysis to perform.

• Gravity load on analysis
• Lateral load only analysis

Gravity Load Analysis

Lets place the moment frame on grid lines 2 and 4 in the north-south direction.
For the gravity load analysis we would use all load combinations and assume the frame is restrained against lateral movement. Therefore we would not have moments due to lateral forces at the moment connections. For simplicity we will use our dead and live load. This would typically be snow load as this is a one story roof and remember to account for drift load as well.

Typical Beam;

Dead load;
Live load;
Total load;
Shear;
Moment;
Unbraced length say 5ft. (practically fully braced for positive moment)

Required moment of inertia – dead load;

Required moment of inertia – live load; (Controls)

Typical Column – Exterior;

Dead load =
Live load =
Required axial load =

Typical Column – Interior;

Dead load =
Live load =
Required axial load =

To Be Cont….

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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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Strip Load Surcharge Analysis using Elastic Methods

Here we will dig deeper into analyzing strip loads with elastic methods. First to avoid confusion between seeing similar equations which use different reference angles, we will set our own nomenclature.

Strip Load – Elastic Methods RSF Nomenclature

Nomenclature

K = Constant typically between 1 and 2, depending on the stiffness of the wall. It may be appropriate to use K=1 for flexible walls such as cantilevered sheet pile, segmental walls. It may be appropriate to use K=2 for rigid walls. Refer to CivilTech Software manual (http://www.civiltechsoftware.com/downloads/sh_manu.pdf).

Below are (2) common equations which result in the same solution but are shown differently. I will refer to these equations as the integrated method or equations. Most codes and design guides refer to these equations but it seems too differ on who is credited. Some refer to them as Boussinesq as modified by Tang or Spangler, others Mindlin or Terzaghi (although Terzaghi is mostly responsible for modifying point and line load elastic equations).

Note that β is located at half of θ and not half the width of the strip load.

The other

In Joseph Bowles’ “Foundation Analysis and Design” he discuses different methods of analyzing offset surcharges using elastic methods. He discuss the work of Spangler and the factor of two suggested by Mindlin. In summary he suggests that Spangler’s experiments were not accurate and possibly flawed due to the setup geometry. Also the equations derived by Spangler use a Poisson’s ratio of 0.5 which may not be correct for all soils. However Spangler’s equation for the strip load results in the same equation as shown above. Bowles also states that Mindlins reasoning for the factor 2 (Mindlin says that a rigid wall produced a mirror effect) is wrong.
Bowles suggests to only use the original Boussinesq equation.

[EQN - 1]

With these equations one can “discretize” the strip load. Meaning you divide the strip load into a series of concentrated loads. Then apply the Boussinesq equation to each point load. Then the sum the results to find the resulting pressure at a certain elevation on the wall. You then find these resulting pressures at a certain number of elevations on the wall and sum these to find the resulting total force. It should be noted that Poisson’s ratio should be modified so that it represents a plan strain condition.

where represents a triaxial condition
Refer to Excavating Systems, Planning, Design and Safety, 2009 for the following suggested values of
Moist clay soils: 0.4-0.5
Saturated clay soils: 0.45-0.5
Cohesionless, medium dense: 0.3-0.4
Cohesionless, loose to medium: 0.2-0.35

Here are some notes on the discretized method (semi-large file – about 10MB, I will try to shrink later):
Boussinesq Strip Load Discritization Method 2

Now my problem is this – when I compare the “discretized” approach to the integrated I do not get similar values, not even close. I have to think that I am in error somewhere, but I cannot find where. I have compared spreadsheets to other programs to hand calcs. I will continue to search, but for now, I have attached a couple hand calcs showing the discrepancy. I will also attach a spreadsheet comparing techniques.
Attached is a hand calc comparison between the “integrated” and discretized approach.
Boussinesq Integrated vs Discretized
Elastic Method – Compare Integrated to Discretized

Elastic Methods Nomenclature Notes

Elastic Methods Spangler Derives Integrated Method
Offset Surcharge Strip Load Comparison Spreadsheet

Another problem when using elastic methods is that all equations assume that the load is applied at the same elevation as the top of the wall. How can we handle a slope atop the wall with a load applied at the top of the slope? Well the railroad design manuals (See UPRR Shoring Manual) distribute the load down through the soil at a certain angle (they use a 2V:1H) and find a new uniform load applied at this elevation.

Elastic Methods – Slope Diagram

The post Surcharge Analysis – Elastic Methods – Strip Load appeared first on How To Engineer.

Helpful Links for Determining Minimum Design Loads

Hopefully these links can save you some time and help get you more accurate design loads. A quick heads-up – you will usually need to search the town/county/state to see if the Authority Having Jurisdiction (AHJ) has a specific requirement.

Wind Load

A favorite for Wind Loads in accordance w/ ASCE 7

http://www.atcouncil.org/windspeed/index.php

Seismic

A favorite for determining your ‘base acceleration’ coefficients:

http://earthquake.usgs.gov/hazards/designmaps/

Snow

This site is no longer free but when I used it, it was useful:

http://www.groundsnowbyzip.com/

This is a little dated and really not that useful but I’ll mention it anyway:

http://www.fs.fed.us/t-d/snow_load/states.htm

Others

This is a ‘pay-for’ site but some may use it:

http://www.groundsnowbyzip.com/

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Analysis of Offset Surcharges on Retaining Walls Using Elastic Methods

Offset surcharges are always up for some debate. Which method to use? What is the line of influence? Is the wall flexible or rigid? Well in this segment we will cover the analysis of surcharge loads on retaining walls using elastic methods. You may commonly here this as a Boussinesq analysis or a Boussinesq as modified by Terzaghi or Tang. Also other equations have been produced by Spangler and Mindlin. Trying to wrap you head around all the different equations and who did what can be confusing. We will try to straighten some of this out and present some of the different approaches and how most of the equations are similar but the equations are presented slightly differently.

Constraints and Assumptions

There are generally 4 types of surcharges considered – Point load, Line load, Strip load and Area load. Point loads and area loads have a finite length. Line and strip loads are assumed to be of infinite length parallel to the wall. The back slope is generally considered to be flat. Elastic methods do not consider any soil parameters such as effective shear strength or wall friction. The only way to try and account for different soil types is by adjusting Poisson’s ratio or adjusting the factor of safety. However most of the theories assume a Poisson’s ratio (u) of 0.5 which simplifies the analysis, but more on this latter.

The basis of the elastic methods discussed below is Boussinesq’s equation for a surcharge on a semi infinite mass

Equation 1:

Boussinesq Equation

Boussinesq Equation Diagram

Where:

P = point load

v = Poisson’s ratio

History

Most of this is from Bowles 4th Edition Foundation Analysis and Design.

1936 Spangler – Performed experiments to measure the lateral pressure on a wall due to point loads form a truck behind a rigid retaining wall. Spangler used Boussinesq’s equation with u=0.5 and found that the actual lateral pressure was approx 2x the pressure found by equation 1 with u=0.5. Bowles points out some flaws of this experiment – loose soils with a wall of ‘finite’ length and ‘old’ technology used in the load cells.

1936 Mindlin – Discusses the results of Spangler. He explains that the 2x factor could be due to the rigid wall producing a mirror load effect. This reasoning is disputed by Bowles.

1954 Terzaghi – suggests 0.4*H (wall height) is a critical distance behind the wall where two different equations should be used for point loads and line loads. Terzaghi proposed (2) different modified boussinesq equations for point and line loads both using v=0.5.

1962 Teng – Similar modification claiming that the pressure on rigid walls for strip loads should be 2x the results obtained by the ‘integrated’ Boussinesq equation. Some also credit Teng the strip load equation shown below as a plastic’solution as opposed to the Boussinesq solution which is an elastic solution (See Civiltech Software Design Manual and Foundation Design by Wayne C. Teng 1962).

1972 – Rehnman and Broms showed that when the soil behind the wall was dense the lateral pressure from point loads was much less than with loose soils. Also gravelly backfill’s produced larger lateral pressures than finer-grained soils. This observation would mean soil state and Poisson’s ratio are significant.

 When and How to Apply Elastic Methods

First the how – The equations presented below give pressures at a certain height on the wall. Therefore it is suggested that a program or spreadsheet is made that uses these equations to find pressures at set intervals or segments of the wall (sigmah). Then the force is found by multiplying the height of this segment by the pressure (Fi=force at elevation ‘i’ = sigmah * hseg). Then you solve for the height above the bottom of the wall to this segment (yi = height to force, Fi). Then you solve for the resultant force and it elevation. Fh=sum(Fi’s) Then find the elevation Ybar = Sum(yi’s * Fi’s)/Sum(Fi’s). This is similar to finding the centroid of an area by parts. This pressure/force is then superimposed on the soil pressure.

Zone of Influence

Second the when – This is a much more difficult question to answer. There are many different ways that an offset surcharge maybe handled see here for a more indepth look. So if we are specifically looking at when to apply elastic methods the answer would be – anytime. We can superimpose the results of the elastic method results on to the soil pressure as would be found using Rankine or Coulomb equations (see here for refresher). However elastic methods will yield a horizontal force for any surcharge applied at any distance behind the retaining wall. This may be overly conservative however it is up to the engineer to determine what distance beyond the wall that the surcharge would have no affect on the retaining wall. Usually this information is given in the governing code. Some examples – many use the Rankine or Coulomb failure plane as the determining distance however a trial wedge may give different results when finding the failure plane angle. Edward White has suggested that pressures may distribute down at 1V:1H (See Foundation Engineering Handbook by Hans F. Winterkorn and Fang). Sometimes a 2V: 1H is used as in a Meyerhoff or Boussinesq bearing pressure distribution. While a 2V:1H has been found accurate at short depths (See Bowles 3rd Edition p172) and is usually conservative when finding vertical stress at a certain depth but can be unconservation when finding horizontal stresses. Most rail road design manuals suggest at 1.5H:1V influence line. The NCMA recognizes a 2H:1V as a conservative estimate to find the horizontal distance to which you may disregard surcharge loads.

Elastic Method Equations

Below are the ‘typical’ equations used in most design manuals for using elastic methods. There are links to other posts which will give further discussion, examples and spreadsheet calculations. The equations assume rigid walls (pressures maybe less for flexible walls, Civiltech Software recommends 0.5 for flexible, 0.75 for semi-flexible and 1.0 for rigid – see further discussion), a Poisson’s ratio of 0.5, and the pressure maybe combined by method of superposition.
Nomenclature:
σh= horizontal pressure
H = Height of wall (excavation)
x1 = m*H = distance to surcharge (point load, line load, strip load)
zi = n*H = distance from top of wall to elevation under consideration
Q = point load or line load surcharge
q = uniform surcharge

Point Loads

Point Load Section

Point Load Plan View

Boussinesq equation as modified by Terzaghi.

To evaluate a point at an angle to the point load along the wall:

See here for a thorough discussion

Line Loads

Line Load Section

See here for a thorough discussion

Strip Loads

Strip Load Section

See here for further discussion.

Area Loads

Will update shortly

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