Column Buckling Analysis of Wood Stud Members due to Reduced Stiffness

over Partial Member Lengths

Thesis

Presented in Partial Fulfillment of the Requirements for the Degree Master of

Science in the Graduate School of The Ohio State University

By

Joseph Edward Scott

Graduate Program in Civil Engineering

The Ohio State University

2018

Thesis Committee

Natassia Brenkus, Advisor

Anthony Massari, Committee Member

Nan Hu, Committee Member

Copyrighted by

Joseph Edward Scott

2018

Abstract

The design of residential structures has changed and evolved throughout years of research based on numerical modelling of real-world conditions. Wood design is controlled heavily by member use, baseline material load capacities, and wood species, all of which determine material properties. Extensive work has been done to determine the effects of altering material properties due to environmental stimuli, however, certain types of decay and wood rot have yet to be fully tested and understood.

Certain microbial organisms, given the right conditions, can cause irreversible damage to wood structures. A particularly critical mode of failure is premature column collapse which is driven in part by degraded cross sectional and material properties. Using stiffness relationships smartly programmed into matrix form, it is possible to calculate reduced buckling capacities for these degraded members to draw some important conclusions. Two important items are to determine what degrees of wood decay are critical and what column heights are particularly susceptible to this type of premature failure.

Finally, specifications and code should be developed further to assist engineers make choices about the criticality of wood decay without the use of personalized software.

This can be done using pre-generated aids to help designers make smarter and more cost effective choices. ii

Dedication

This thesis is dedicated to my lovely wife, Nicole, and our daughter Eloise. Without your continual support and love, this achievement would not be possible. Also to my family and friends who have provided endless emotional support.

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Acknowledgments

I would like to thank my advisor, Dr. Natassia Brenkus for support and direction on this project. I would also like to thank Dr. Anthony Massari and Dr. Nan Hu who both provided intellectual support and expert guidance. Without their leadership this would not have been possible.

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Vita

May 2013 B.S., Civil Engineering,

Washington State University, Pullman, WA

May 2014 to January 2016 Project Engineer,

49th Civil Engineer Squadron, Holloman AFB, NM

January 2016 to July 2016 Design Engineer,

577th Expeditionary Prime BEEF Sq, Al Udeid AB, Qatar

Fields of Study

Major Field: Civil Engineering

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Table of Contents

Abstract ...... ii Dedication ...... iii Acknowledgments...... iv Vita ...... v List of Tables ...... viii List of Figures ...... ix Chapter 1 Introduction ...... 1 Wood Degradation Sources ...... 3 Geographical Locations of Concern ...... 5 Problem Statement ...... 7 Organization ...... 8 Chapter 2 Literature Review ...... 10 NDS Overview ...... 11 NDS Adjustment Factors ...... 12 Wet Service Factor ...... 13 Temperature Factor ...... 14 Size Factor ...... 14 Incising Factor ...... 15 Column Stability Factor ...... 15 LRFD Specific Factors ...... 19 Compression Stress Parallel to the Grain ...... 20 Decay in ASTM D245 ...... 21 Importance of Wood Decay ...... 22 Chapter 3 Mathematical Methodology ...... 23 vi

Typical Construction Member Modelled ...... 23 Computer Programs Used ...... 24 Modelling using Matrix Structural Analysis Techniques ...... 25 Derivation of the Material Stiffness Matrix ...... 26 Geometric Matrix Derivation ...... 30 Global Stiffness Matrix Compilation ...... 33 Member Modelling ...... 34 Model Discretization ...... 35 Artificial Stiffness and Instability ...... 37 Column Stability Modification ...... 39 Chapter 4 Results ...... 42 Original Column Stability Factor ...... 43 Overall Effect of Stiffness Reduction ...... 48 Modified Interaction Diagrams ...... 50 Capacity Reduction ...... 54 Chapter 5 Discussion ...... 57 Wood Rot Capacity ...... 57 Critical Member Lengths ...... 59 Additions to NDS ...... 62 Chapter 6 Conclusion ...... 64 Limitations ...... 64 Future Work ...... 65 Bibliography ...... 66 Appendix A : Beam Stiffness Derivation ...... 68 Appendix B : Column Stability Factor Derivation ...... 73 Appendix C : Geometric Stiffness Matrix Derivation ...... 75

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List of Tables

Table 2.1: Sawn Lumber Adjustment Factors (American Wood Council 2015) ...... 12 Table 2.2: Wet Service Factor for SYP (American Wood Council 2015) ...... 13 Table 2.3: Time Duration Factor, λ (American Wood Council 2015) ...... 19

Table 4.1: CP Factor for Different Values of Fcr, 36-in. Tall 2x4 SYP ...... 46

Table 4.2: CP Factor for Different Values of Fcr, 20-in. Tall 2x4 SYP ...... 46

Table 4.3: Capacity Ratios for a 36-in. Tall Column, EIp=20% ...... 55 Table 4.4: Capacity Ratios for a 36-in. Tall Column, EIp=80% ...... 56 Table 5.1: Critical Member Height Capacity ...... 61

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List of Figures

Figure 1.1: Permanent Wood Foundation (American Wood Council 2001) ...... 2 Figure 1.2: New Orleans, LA Flooding post-Hurricane Katrina (Zimmermann 2015) ...... 3 Figure 1.3: Cripple Wall Dry Rot (McCormick 2017) ...... 5 Figure 1.4: Best track positions for Hurricane Katrina, 23-30 August 2005 (Knabb, Rhome, and Brown 2005)...... 7 Figure 2.1: Recent Lumber Column Data from Various Sources (Zahn 1993) ...... 16 Figure 2.2: General Interaction Diagrams for Different Values of "c" ...... 17 Figure 3.1: Typical Cripple Wall Application (WRCM 2014) ...... 23 Figure 3.2: Wood stud in a typical cripple wall application ...... 25 Figure 3.3: Simple Model Under Unit Displacement (Kassimali 2012) ...... 27 Figure 3.4: Beam Member Bending Model (Cook, Malkus, and Plesha 1989)...... 31 Figure 3.5: Beam Member Differential Element (Cook, Malkus, and Plesha 1989) ...... 31 Figure 3.6: Simply Supported Beam including DOFs ...... 35 Figure 3.7: Analysis of Model Sub-Elements Quantity ...... 36 Figure 3.8: Degraded Column Model ...... 40 Figure 4.1: Column Stability Factor for a 2x4 SYP as Prescribed by the NDS ...... 44 Figure 4.2: Column Stability Factor Varied with Member Length and Critical Stress .... 45 Figure 4.3: Critical Buckling Load Ratio for Variable 퐻 ∗ and 퐸퐼푝, 36-in. Member ...... 49 Figure 4.4: 2x4 Column Interaction Diagram, 20% Affected Member Length ...... 51 Figure 4.5: 2x4 Column Interaction Diagram, 60% Affected Member Length ...... 52 Figure 4.6: Wood Decay Design Aids ...... 53 Figure 5.1: 2x4 Column Interaction Diagram, 60% Affected Member Length ...... 58 Figure 5.2: Critical Member Height for Various Levels of Wood Decay ...... 60 Figure 5.3: Wood Decay Design Aids ...... 63 Figure A.1: Beam Member with Internal Forces ...... 68 ix

Figure A.2: Degree of Freedom 2 Unit Rotation ...... 69 Figure A.3: Degree of Freedom 3 Unit Displacement ...... 71 Figure A.4: Degree of Freedom 4 Unit Rotation ...... 72

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Chapter 1 Introduction

Wood construction and timber utilization has been widely used for most of known human history, but the practice of engineering and design have been largely developed during the last and current centuries. Wood construction continues to get cheaper and more efficient with new construction materials, design requirement, building strategies, and a better overall understanding of the material’s many strengths and weaknesses. Timber members are used abundantly in residential construction utilizing light-frame stud walls reinforced with thin yet strong shear envelopes. For these reasons wood construction continues to dominate small, low-rise construction of all kinds.

Wood material is extremely susceptible to environmental stimuli, however, and this is one if it’s biggest weaknesses. Composed of organic material wood can be a food source for insects and rodents, a place for shelter for some larger avian animals, and can decay due to fungus and microbial organisms if not prevented properly. Wood decay, or wood rot, is a serious problem leaving countless buildings susceptible to structural degradation and possible catastrophic collapse.

Observed all over the United States, wood construction is well equipped to resist the various loading scenarios that it would experience. It is strong and flexible for resisting gravity loads such as equipment and material weight, snow, and rainwater. It can also be 1

fitted with plywood or other shell members to resist lateral loads such as wind and earthquake. It can be made rigid enough to stand-up through hurricane winds in the southeast, flexible enough to withstand earthquake forces along the Pacific shoreline, as well as, resist a vast array of environmental and gravity load combinations it would see in the Midwest. There are not many locations where wood construction would not be suitable and it is even used in some foundation systems or near the interface between construction materials and the earth.

Wood members are often protected with membranes and waterproof flashings to resist moisture seepage and subsequent material degradation and rot. Even given these extensive protective procedures this phenomenon is still apparent in the flood-prone areas of the southeastern United States and can degrade structures to a catastrophic level.

A typical permanent wood foundation detail is shown in Figure

1.1. Gravel is installed at the base of the footing for drainage, and the finished grade is sloped away from the plywood sheathed wall.

Polyethylene film is usually installed inside the crawlspace for Figure 1.1: Permanent Wood Foundation (American Wood Council 2001) 2

moisture protection, but there are still substantial conditions for moisture to infiltrate the space and deteriorate both the sheathing and the stud. More moisture protections and membranes are prescribed for full basements, but moisture seepage and potential fungal growth is also existent in these scenarios, especially coupled with intense rain or storm surge events.

Wood Degradation Sources

There have been many areas of investigation with respect to timber design and construction, however, compression member buckling action is of particular interest.

Figure 1.2: New Orleans, LA Flooding post-Hurricane Katrina (Zimmermann 2015) Furthermore, susceptibility to buckling can be worsened by weathering effects of wood

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members including rot caused by fungi carried by floodwater. The southeastern

Continental United States are affected annually by hurricanes, high winds, and floodwaters that bring with them an array of unknown microbial organisms. During such events, buildings or parts of buildings can be submerged in floodwater potentially laced with deteriorating microbial life. The CDC estimates that after Hurricane Katrina the presence of microbial contaminants were elevated and consistent with levels detected historically in typical storm-water discharges in the area (CDC 2011).

There are many different types of wood rot, all caused by different or differing types of microbial organisms. Brown rot is one such type and is generally considered the most aggressive in terms of overall breakdown of the cell structure within a wood member

(Valá, Correspondence, and Baldrian 2006). Characterized by the destruction and almost complete removal of cellulose and hemicellulose in wood organisms that cause brown rot leave behind almost exclusively the lignin structure within the cell and an otherwise porous material. This degradation severely reduces the ability for the structure to resist any sort of loading leaving an affected structure susceptible to failure and, in some instances, complete collapse.

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Figure 1.3: Cripple Wall Dry Rot (McCormick 2017) Geographical Locations of Concern

The southeastern portion of the United States and the islands in the Caribbean Sea experience numerous tropical storms, depressions, and hurricanes each year. Each one brings with it wind-driven rain, storm surges and excess precipitation that can cause flooding to a catastrophic level.

One such storm system is the well-known Hurricane Katrina, which struck the southern tip of Florida and later made landfall near the Louisiana/Mississippi border.

Katrina is considered by many professionals as being the costliest and one of the five deadliest hurricanes to hit the United States (“Hurricanes in History” 2018). She produced category I wind speeds in Florida, strengthening to category III wind speeds over the Gulf

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of Mexico before its eventual landfall in Louisiana and Mississippi. Along with the high wind speeds, severe amounts of rain fell on the Florida Keys and storm surges affected costal Louisiana, Mississippi, Georgia, and the panhandle of Florida (Knabb, Rhome, and

Brown 2005). These storm surges had the potential to carry the microbes that are needed for the rotting effect of wood to take place and the potential future failure of affected buildings and homes.

This region utilizes a large number of light-framed residential construction atop cast-in-place or masonry stem walls and footings foundation systems. The interface between these two material types creates a zone of importance where short wood compression members are subjected to floodwaters for short periods of time potentially degrading their ability to act as columns as prescribed.

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Figure 1.4: Best track positions for Hurricane Katrina, 23-30 August 2005 (Knabb, Rhome, and Brown 2005).

Problem Statement

There are many conclusions that can be made about wood rot and structural degradation, most not fully understood. The apparent reduction in structural capacity can be investigated through experimentation and numerical modelling. This thesis focuses on determining the mathematical boundaries of 2x4 wood studs given a variable amount of stiffness degradation over a variable member length. These boundaries can be checked

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against experimental data to determine if wood rot through fungal decay is a structural problem or a minor aesthetic nuisance.

Using the reduced critical buckling stress on these members, new design charts were developed to supplement the design criteria currently found in the National Design

Specifications for Wood Construction (NDS). This document provides well defined interaction between a short member that fails in pure buckling and a tall slender member that fails in pure buckling, but does little to reduce the buckling capacity of a member with degraded values of stiffness due to rot.

Organization

The following chapters explore the interaction between crushing and buckling failure in a simply supported timber column and how its capacity can be affected by changes in its perceived stiffness.

Chapter 2 explores the National Design Specifications for Wood Construction

(NDS) and how the committee that compiled that specification handled buckling and crushing interaction in wood members. It also overviews multiple studies on buckling phenomenon focusing on timber and some other well defined mathematical modelling.

Chapter 3 builds the foundation needed for the derivation of the model and some background knowledge needed to program and run MATLAB scripts. It describes the fundamental idea behind local and global stiffness matrices and how to compile them for the overall structure. Finally, it touches on geometric stiffness and how the combination

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of geometric stiffness and member stiffness of a beam member can be used to determine the buckling loads on the modelled member using eigenvectors and eigenvalues.

Chapter 4 describes the results achieved from running the model and how we can use these results to evaluate existing wood framed structures given an unknown yet real degree of stiffness loss. Furthermore, this chapter provides some visual design guides that can be used to analyze the sensitivity of certain choices engineers use in the field.

Chapter 5 concludes the study and presents the research alongside current NDS design procedures. The results are presented in relation to historical designs and modifications to the procedures are provided.

Chapter 6 provides ideas for future work that would be interesting and necessary for a complete investigation of buckling behavior.

The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the

U.S. Government.

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Chapter 2 Literature Review

Light frame wood construction tends to be used extensively in residential construction. Wood construction materials have a very efficient strength to weight ratio, comparable to that of steel, making them very efficient in construction where the load is dominated by the structures self-weight. Wood members are susceptible to dry shrinkage which can be minimized by using small cross-sections and using simple connection methods that can accommodate the resulting shrinkage. Furthermore, the low mass of wood construction can help to reduce the seismic weight effectively lessening seismic forces (Ramage et al. 2017). All of these aspects contribute to wood construction being ideal for low-rise residential construction in many areas throughout the United States and abroad.

Wood design is largely governed by the American Forest & Paper Association

(AF&PA) and the American Wood Council (AWC). These two organizations work with industry specialists to publish public codes with the intent to provide safe structural systems and sustainable building practices through the publication of the National Design

Specification for Wood Construction (NDS).

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NDS Overview

The NDS was first issued in 1944 and focused mainly on stress-graded lumber products and the fasteners utilized as connections. The scope of the publication changed in 1971 when the committee added additional structural systems, including shear members and some other engineered members. The AF&PA first gained accreditation by the

American National Standards Institute (ANSI) in 1991 and published the ANSI/NFoPA

NDS-1991 in October of 1992. This addition was reorganized and many of the sections re-written for clarity and ease of use. In 2010 AWC was separately incorporated and the current edition is the ANSI/AWC-2015. (American Wood Council 2015)

The developers of the code relied heavily on the Forest Products Laboratory (FPL) and the U.S. Department of Agriculture for data and publications. Many articles written in collaboration with scientists and engineers from the FPL were key to the background and direction of this paper.

The current NDS is organized into chapters by type of wood product. Within those chapters, certain modification factors affect different stresses that a wood member could experience. Sawn lumber represents its own chapter within the code with applicable adjustment factors shown in Table 2.1.

The strategy laid out in the NDS to find member capacities given a loading scenario is based on statistical data of lab tested capacities for different wood species and grades called “Reference Design Values”. The procedures to test for these values are outlined in

ASTM D245 and are based on the lower 5% exclusion limit. This limit directs engineers 11

to the near lowest probable value for strength parameters leaving a very low probability

that any piece of wood has strength properties below the published values (Anthony and

Nehil 2018). The reference design values found in the NDS supplement have a 95%

confidence that any random piece of lumber has strength properties larger than those found

in the tables.

Table 2.1: Sawn Lumber Adjustment Factors (American Wood Council 2015)

NDS Adjustment Factors

The reference design values are based on a standard wood size with general

conditions as specified in ASTM D245. To deviate from these member conditions, 12

correction factors are provided in the NDS and must be applied to reduce or increase the member’s capacity. The factors that are applicable to the compression force parallel to the grain are wet service (CM), temperature (Ct), size (CF), incising (Ci), and column stability

(CP), along with format conversion factors (KF), LRFD resistance factors (), and time dependency factors (λ).

Wet Service Factor

Allowable design limits for bending strength, tensile and compressive strength parallel to the grain, and shear strength for wood specimens are determined from test methods prescribed in ASTM D2555 using clear, straight-grained wood in a green condition. It has been determined that the strength and stiffness of wood members increase as moisture content decreases below the fiber saturation point. For larger members thicker than 4-in. nominal, this effect is offset by shrinkage and seasoning defects, resulting in negated correction values (“Standard Practice for Establishing Structural Grades and

Related Allowable Properties for Visually Graded Lumber 1” 2018).

The NDS captures this modification by assuming wood with a moisture content below 19% in service and applying a reduction if the MC is higher for an extended period of time. Furthermore, the wet service factor is applied to all sizes of dimension lumber and effects of members thicker than 2- Table 2.2: Wet Service Factor for SYP (American Wood Council 2015) in. are captured by the size factor described below. Typical wet

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service factor values for Southern Yellow Pine from the NDS are shown in Table 2.2.

Temperature Factor

Lumber strength is prescribed when operating under ordinary ranges of temperatures. This range of temperatures is defined in the NDS as anything below a temperature of 150℉. Generally, as wood is heated the strength decreases and as wood is cooled the strength increases and the change in strength is reversible up to the maximum operating temperature. Beyond 150℉ prolonged heating will cause permanent loss of strength.

Due to the correlation that an increase in temperature (decreasing strength) causes a decrease in moisture content (increased strength), temperature effects are less pronounced and ordinary reference design values are used for temperature fluctuations up to 150℉. At the engineer’s discretion, strength reduction factors will affect highly heated wood structures for extended time-periods. (American Wood Council 2015)

Size Factor

The size adjustment factor is built into the reference design values for Southern

Yellow Pine. For differing types of wood, however, strength needs to be increased or decreased to coincide with the thickness and width of a member’s cross-section. Members are tested using a standard 2x4 and the NDS converts these values of strength to equivalent values of strength for a 2x12 by multiplying it by the factor

1⁄9 퐶퐹 = (2⁄푑) 2.1

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where CF is the size factor and d is the depth of the member. This formula takes a similar form to what’s presented in the NDS as

1⁄9 퐶퐹 = (12⁄푑) 2.2

which is based on a 12-in. deep reference wood member. The NDS supplement clearly specifies which factor can be used for each type of lumber used. Tables 4A-4F tabulate the reference design values for sawn lumber and describe the circumstances in which the factor may be utilized.

Incising Factor

Many different strategies may be utilized to treat wood, depending on the desired effect. ASTM D245 suggests that when high temperatures and high pressures are used to treat and preserve wood, these temperatures and pressures should be kept to a minimum to not adversely affect the design strength. Incising may have an adverse effect on strength by effectively decreasing the surface area at the outermost fibers.

Column Stability Factor

Unique to compression parallel to the grain is the column stability factor (CP).

Columns of all materials undergo either buckling, crushing, or a combination of both at failure. Members that have sufficiently short lengths will experience yielding across their cross-section before out-of-plane instability or buckling will occur. Alternatively, sufficiently tall members will undergo buckling as described by Euler buckling theory

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before cross-sectional yielding occurs. Each material has their own methods for determining this buckling stress limitations and wood is no different.

Column stability for wood structures is based on the idea of interaction between pure buckling and pure crushing. Interaction is never pure crushing or pure buckling, there is always some portion of the critical load contributing to member crushing and the rest contributes to buckling. The plot in Figure 2.1 show the straight line interaction between the crushing interaction ratio and the buckling interaction ratio as the dotted line labelled

Figure 2.1: Recent Lumber Column Data from Various Sources (Zahn 1993) 16

Eq 1. along with raw column failure data drawn from various sources (Zahn 1993). Using a straight line interaction as a baseline and fitting the data using a cross product multiplier engineers can model the presumed actual behavior of a wood column loaded in compression.

푃 푃 + = 1 2.3 푃푐 푃푒

푃 푃 푃 푃 + = 1 + × × 푐 ; where 0 ≤ 푐 ≤ 1 2.4 푃푐 푃푒 푃푐 푃푒

Figure 2.2: General Interaction Diagrams for Different Values of "c" 17

휋2퐸 2.5 푃푒 = × 퐴 푘퐿 2 ( ⁄푟)

In these equations Pc is the crushing strength of the wood column, Pe is the Euler buckling strength defined by equation 2.5, P is the actual load that the column experiences at failure, and c is a modifier to fit the curve to the data.

An easy conversion from force (P) to stress (F) can be accomplished by dividing the force by cross-sectional area. Through some elementary algebra a formula relating F to Fc can be derived (see Appendix B). This ratio, when multiplied by the unmodified compression force found using the NDS factors (Fc) results in the expected column capacity of the member before buckling occurs

퐹푒 퐹푒 퐹푒 퐹 1 + ⁄ 1 + ⁄ ⁄ 퐹푐 퐹푐 퐹푐 2.6 퐶푃 = = − √( ) − 퐹푐 2푐 2푐 푐

The column stability factor is the key, in wood design and construction, to determining the load carrying capacity of a compression member. It includes slenderness, buckling strength, and the unmodified crushing strength of a column and their overall interaction. Fitted to data taken from other studies, this formula allows for the variation of material type using a variable “c” term which can be modified to fit data of different types of wood members such as sawn lumber (c=0.8), round timbers and poles (c=0.85), and glued laminated timbers (c = 0.9). This increases the versatility of the factor to be applied to different wood varieties in a variety of building scenarios.

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LRFD Specific Factors

There are three factors found in Table 2.1 that are specific to the LRFD design methodology. The three factors are the Format Conversion Factor, the Resistance Factor, and the Time Dependency Factor. Wood design has been historically dominated by

Allowable Stress Design (ASD) methods and thus all of the reference design values found in the NDS supplement include the appropriate factor of safety built-in to them. Converting these known ASD stresses to LRFD stresses is done by multiplying the tabulated reference design values by the pre-applied factor of safety to get the raw strength value. This is done using the term that the NDS denotes as the Format Conversion Factor (KF).

Unique to the LRFD method, calculated member strengths must be multiplied by an adjustment factor less than one to determine the engineered capacity of that member or structure. The NDS tabulates these resistance factors for each stress type. Using KF and  the engineer can take an allowable stress design value to the equivalent strength design value for use with LRFD ultimate loads. Finally, the strength of wood is directly related

Table 2.3: Time Duration Factor, λ (American Wood Council 2015)

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to the duration of the loading acting on the member. That is, a member that experiences a short load such as wind or earthquake resists that load with a lower capacity than a load of longer duration such as dead load. Member strengths are determined from testing using a load duration of 10 minutes and other load combinations are given a factor based on which loads they include and how much those load types contribute. Appropriate load duration factors (λ) for each LRFD load combination are shown in Table 2.3. Load combinations that include large values of dead load get a larger reduction than combinations that include wind and earthquake loadings.

Compression Stress Parallel to the Grain

The factors that are prevalent to the differing loading scenarios are outlined in Table

2.1. Since the mathematical model that is being developed is concerned with column stability as it relates to buckling-crushing interaction, it is necessary to separate pure crushing stress (퐹′′푐) with actual column capacity (퐹′푐) Specific to this study, vertical compression members loaded in pure compression experience the stresses denoted as 퐹′′푐 and 퐹′푐 which are influenced by the factors per the equations below.

퐹′′푐 = 푓푐 × 퐶푀 × 퐶푡 × 퐶퐹 × 퐶푖 × 퐾퐹 × 휙 × 휆 2.7

퐹′푐 = 퐹′′푐 × 퐶푃 2.8

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The factors that only affect material strength not including column stability shall be taken as 1.0. These factors include CM, Ct, CF, and Ci. This will be used to determine our baseline engineered member capacity.

The assumed load duration factor for this project is 0.8 which roughly correlates to a live occupancy loading scenario. The experimental data that we are attempting to correlate with tests the capacity of the specimens over a time period of approximately ten minutes. The two remaining LRFD specific factors should be taken as their value from

Table 2.1.

Decay in ASTM D245

Decay in the field is difficult to detect and even more difficult to quantify. Age, weathering, rot, and many other factors play role in the type of decay, rate, and overall structural degradation. All of these uncertainties can lead to designs being over- conservative which may not be necessary by using a small amount of math and some less conservative assumptions.

The materials testing standard for wood strength acknowledges the difficulty in determining the numerical effect on the strength of wood. For this reason, strength reduction due to decay is omitted from most grades of wood (ASTM D245). Wood design omits effects due to wood decay which can lead to older buildings that are susceptible to collapse, endangering owners and occupants.

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Importance of Wood Decay

Overall, there is a fundamental lack in the inclusion of decay in wood members and structures. Decay can occur very easily in structures that are not properly protected from moisture infiltration or structures where moisture protection has broken down. Residential construction is dominated by timber framing and, paired with flood prone areas, these structures are extremely susceptible to structural degradation from decay. Over- conservative engineering is a technique to overcome uncertainties in decay, but in an effort to fully understand all aspects of wood design, it is not the best course of action. Through experimentation and mathematical modelling, a new technique to analyze these structures is possible.

(“Standard Practice for Establishing Structural Grades and Related Allowable

Properties for Visually Graded Lumber 1” 2018)

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Chapter 3 Mathematical Methodology

Building a mathematical model of the column buckling behavior was crucial to understanding visually how external forces are applied and transferred through a compression member. To do this, Matrix Structural Analysis (MSA) methods were utilized and the model was built inside the Matrix Laboratory (MATLAB) interface. Certain assumptions were also used to allow the model to be more straightforward and easier to understand; furthermore, it afforded the program inputs were generalized, allowing greater flexibility to changes to loading scenarios, boundary conditions, and section and material properties allowing for a more complex future analysis. (WRCM 2014)

Typical Construction Member Modelled

The type of member that was of particular interest was a short 2x4 used primarily in the foundation wall in typical residential construction.

Nicknamed a cripple wall, this structure acts near the interface between the concrete or concrete masonry unit (CMU) strip footing and the floor Figure 3.1: Typical Cripple Wall Application (WRCM 2014) joists above. The structure is represented by the

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member labelled “Permanent Wood Foundation” shown in Figure 3.1. Essentially, this structural element elevates the first floor off the foundation wall a short distance separating it from the ground below and utilizing cheap, abundant, and lightweight timber framing.

This type of construction creates a small crawl space under the first floor and are frequent in areas where wood is easily attainable.

These small crawl-spaces accumulate moisture from the ground surface, but can be protected from typical elemental exposure using moisture barriers. During flood events, extensive moisture and subsequent microbial organisms may be introduced to the system, which in turn can cause wood rot potentially decreasing the member’s stiffness and stability in the long-term. Usually sheathed with plywood and protective membranes, these members are extremely susceptible to floodwater damage and perpetual wood rot strictly based on where they are in relation to the ground surface.

Computer Programs Used

A simple finite element model was used to create the virtual member used for performing capacity analysis. The idea behind Finite Element Modelling is that a member is broken up into small subsections with homogeneous material properties and dimensions.

Each subsection’s stiffness properties and loading conditions are analyzed simultaneously with the adjacent subsection allowing a complete picture of the member’s physical deformations along the length, as well as the internal stresses. MATLAB was used to solve these sets of linear equations. This program is an excellent tool to perform this type of analysis because of its ability to perform complex calculations repeatedly and in a 24

systematic manner. It is possible to program generic design equations into MATLAB while allowing users to easily manipulate material and section property variables, and end conditions, allowing the model to be expanded for future use.

MATLAB also has a powerful set of mathematical functions which is useful when performing matrix algebra, compiling stiffness matrices, and determining eigenvalues.

Taking advantage of these functions greatly reduces the need to write model specific programs, allowing more focus on theory and increasing the flexibility for future work.

Another benefit is the program’s ability to easily produce visual comparisons and design aids to analyze the actual stiffness degradation and P determine if the effect is of concern to designers. The following figures were all developed using MATLAB. P 2x4 SYP Modelling using Matrix Structural Analysis Techniques Pinned End The idea behind MSA is to organize stiffness values Conditions corresponding to different member degrees of freedom 36” (dof) in matrix form. These values of stiffness, when y multiplied by their corresponding value of deflection, x 36” produce a force vector with a value of force for each degree of freedom in the system, explained later.

Certain degrees of freedom are mathematically constrained by forcing their corresponding value in the Figure 3.2: Wood stud in a typical displacement vector equal to zero. This technique allows cripple wall application 25

the programmer to easily decide how end conditions are treated. The stiffness values for the constrained degrees of freedom are not compiled in the global stiffness matrix eliminating a set of variables that are needed to be solved for initially. This process is described in more detail further on.

A 2x4 stud resists both internal bending moment and the internal shear forces acting on it. This wall member carries mainly axial force parallel to the grain, however, the determining factor for buckling failure is unbounded nodal displacement in the local y- direction only; this occurs mathematically when the stiffness matrix is not invertible. For this reason, axial stiffness ignored because it was not a large concern. Therefore, the compression member was modelled as a beam rotated 90 degrees, as in Figure 3.2, with resistive degrees of freedom as moment and shear.

Derivation of the Material Stiffness Matrix

The principle of virtual work is used to determine member stiffness properties. For each of the four degrees of freedom per member, small unit displacements are applied and the resulting forces are calculated using the relationship.

푀 푑2푢 3.1 = 2 퐸 ∗ 퐼 푑푥

Differentiating displacement (u) with respect to x and applying boundary conditions, the stiffness values can be calculated. Standard beam orientation and nodal stiffnesses are shown in Figure 3.3.

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Figure 3.3: Simple Model Under Unit Displacement (Kassimali 2012)

Rearranging equation 3.1 to isolate the curvature value and differentiate to get

1 푢′′ = 푀 ∗ 3.2 퐸 ∗ 퐼

This is a fundamental equation used to determine the values of stiffness in a beam member. Using this equation, summing forces a distance “x” from the beginning node, and integrating curvature with respect to the x, this relationship can be rewritten in the sequence as follows

푀 = −푘21 + 푘11 ∗ 푥 3.3

(−푘 + 푘 푥) 푢′′ = 21 11 3.4 퐸 ∗ 퐼

푘11 2 (−푘21푥 + 푥 ) ∫ 푢′′푑푥 = 푢′ = 2 + 퐶 3.5 퐸 ∗ 퐼 1

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−푘 푘 ( 21 푥2 + 11 푥3) ∫ 푢′푑푥 = 푢 = 2 6 + 퐶 푥 + 퐶 3.6 퐸 ∗ 퐼 1 2

Our known boundary conditions for our beam are below. The beam is assumed to be fixed-fixed with a released first degree of freedom (dof) allowing a unit displacement only in the local y-direction.

푢′(0) = 0 3.7

푢′(퐿) = 0 3.8

푢(0) = 1 3.9

푢(퐿) = 0 3.10

By substituting our boundary conditions, equations 3.7-3.10, into the appropriate equation for slope (3.5) and displacement (3.6), the constants of integration C1 and C2 can be found.

푘11 2 (−푘21∗(0)+ ∗(0) ) 푢′(0) = 0 = 2 + 퐶 퐶 = 0 퐸∗퐼 1 1

−푘 푘 ( 11∗(0)2+ 11∗(0)3) 푢(0) = 1 = 2 6 + (0) ∗ (0) + 퐶 퐶 = 1 퐸∗퐼 2 2

Next, solve for 푘21 in terms of 푘11. Then 푘11 as a function of E, I, and L can be found. The other two stiffness values can be found by summing forces in the y-direction and moments about the beginning node.

푘11 2 (−푘21∗(퐿)+ ∗(퐿) ) 푘 푢′(퐿) = 0 = 2 + (0) 푘 = 11 ∗ 퐿 퐸∗퐼 21 2

−푘11 2 푘11 3 ( ∗(퐿) + ∗(퐿) ) 12∗퐸∗퐼 푢(퐿) = 0 = 2 6 + (0) ∗ (퐿) + 1 푘 = 퐸∗퐼 11 퐿3

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6∗퐸∗퐼 Summing forces to determine the last two stiffness values 푘 = 21 퐿2

−12∗퐸∗퐼 ∑ 퐹 = 0 = 푘 + 푘 푘 = 푦 11 31 31 퐿3

6∗퐸∗퐼 ∑ 푀 = 0 = 푘 ∗ 푥 + 푘 + 푘 푘 = 11 21 41 41 퐿2

Derivations for the other three degrees of freedom are included in more detail in

Appendix A. From these values, a local stiffness matrix can be developed to describe the member’s resistance to outside forces. Each row represents the equations of stiffness given a unit displacement of that dof and each column represents the reactions to that unit displacement.

12 6퐿 −12 6퐿 퐸퐼 6퐿 4퐿2 −6퐿 2퐿2 푘 = [ ] 3.11 푏푒푎푚 퐿3 −12 −6퐿 12 −6퐿 6퐿 2퐿2 −6퐿 4퐿2

Next, it can be observed that for any given member, the loading {F} at each node is a product of the stiffness of that member [k] and the displacement of the beginning and ending nodes {u}. For this reason, forces acting at a node in the direction of a support will not have an impact on the internal forces in that member. To simplify the mathematics, the stiffness matrix for the global system needs to be compiled only for the degrees of freedom that are unsupported as only those values of displacement have the potential to not be zero. This is explained in greater detail below.

{퐹} = [푘푏푒푎푚] ∗ {푢} 3.12

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Geometric Matrix Derivation

Slender compression members of length L are characterized by their high axial stiffness (퐴퐸 ) relative to their bending (퐸퐼 ) and shear stiffness (퐸퐼 ) values. ⁄퐿 ⁄퐿3 ⁄퐿2

Physically this is due to the way that the material interacts under different loading scenarios. Mathematically axial stiffness tends to be larger because of the contribution of the member’s length (present in the denominator in both terms), raised to a power of three for bending stiffness and raised to a power of just one for axial stiffness. This leads to the axial stiffness being orders of magnitude larger than bending stiffness in a typical beam cross-section (Cook, Malkus, and Plesha 1989).

Along with differences in axial and bending stiffness, interaction between loading and deformation are not accounted for in a standard frame section stiffness matrix.

Equation 3.13 defines the local member stiffness matrix for a frame element with three degrees of freedom at each node.

퐴퐿2 −퐴퐿2 ⁄퐼 0 0 ⁄퐼 0 0 0 12 6퐿 0 −12 6퐿 퐸퐼 2 2 푘 = 0 6퐿 4퐿 0 −6퐿 2퐿 3.13 푓푟푎푚푒 퐿3 −퐴퐿2 퐴퐿2 ⁄퐼 0 0 ⁄퐼 0 0 0 −12 −6퐿 0 12 −6퐿 [ 0 6퐿 2퐿2 0 −6퐿 4퐿2 ]

The first and third rows and columns represent the member’s axial stiffness. An index containing a zero indicates no interaction between those two degrees of freedom.

For example, the value of zero in index 푘12 indicate no interaction between axial stress and

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bending stress. In other words, given some axial force, P, no bending stress develops. For

this reason, a member that is susceptible to buckling should not be modelled as a frame

element and instead a relationship between axial force and bending stiffness must be

developed in another way.

An alternate view on overall member stiffness is to imagine a beam with some

applied axial compressive load P and transverse loading q. As the axial compressive load

increases, the beam can ultimately hold less transverse load. The opposite is true for a

tensile axial load which increases the ultimate load that can be applied transversely on the

beam. This effect is known as stress stiffening and is a function of only the applied axial

force and the length of the member.

Figure 3.4: Beam Member Bending Model Figure 3.5: Beam Member Differential Element (Cook, Malkus, and Plesha 1989) (Cook, Malkus, and Plesha 1989)

Geometric stiffness is the concept that membrane strain energy (푈푚) created by a

constant axial load transitions to bending strain energy (푈푏) and the combination of the

two work together to define the entire energy state of the member. Working with this

concept, buckling would occur when the axial compressive force acting on a member is

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large enough that it reduces the bending stiffness of that element to zero. For a member that is not loaded transversely, the energy relationship 푈푚 + 푈푏 = 0 must be satisfied.

Membrane strain energy being exerted on a member can be defined by the equation

퐿 퐿 1 2 푈푚 = ∫ 푃휀푑푥 = ∫ 푃푤푥 푑푥 3.14 0 2 0

푑푠−푑푥 where 휀 = is the axial strain in the member due to load P and 푤 is the first derivative 푑푥 푥 of the lateral displacement function (푤) with respect to x. This further simplifies to

퐿 1 푇 1 푇 푈푚 = ∫ 푤푥 푃푤푥푑푥 = {푑} [푔푏푒푎푚]{푑} 3.15 2 0 2 where 푤 = [푁]{푑} and 푤푥 = [퐺]{푑}

퐿 푇 [푔푏푒푎푚] = ∫ [퐺 ]푃[퐺] 푑푥 3.16 0

Utilizing commonly known beam shape functions [N] and their derivatives [G] in equations 3.17-3.20, membrane strain energy can be calculated based on an applied axial load on a compression member.

3푥2 2푥3 6푥 6푥2 푁 = 1 − + → 퐺 = − + 3.17 1 퐿2 퐿3 1 퐿2 퐿3

2푥2 푥3 4푥 3푥2 3.18 푁 = 푥 − + → 퐺 = 1 − + 2 퐿 퐿2 2 퐿 퐿2

3푥2 2푥3 6푥 6푥2 3.19 푁 = − → 퐺 = − 3 퐿2 퐿3 3 퐿2 퐿3

푥2 푥3 2푥 3푥2 3.20 푁 = − + → 퐺 = − + 4 퐿 퐿2 4 퐿 퐿2

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Substituting [G] into equation 3.16 and integrating, 푔푏푒푎푚 can be calculated. The geometric stiffness matrix for a beam element is shown in equation 3.21. This derivation is completed in more detail in Appendix C.

36 3퐿 −36 3퐿 푇 3퐿 4퐿2 −3퐿 −퐿2 푔 = [ ] 3.21 푏푒푎푚 30L −36 −3퐿 36 −3퐿 3퐿 −퐿2 −3퐿 4퐿2

Global Stiffness Matrix Compilation

Matrix structural analysis gives designers a powerful set of tools for easily computing nodal displacements, internal forces, and can be used to calculate reaction forces at the fixed degrees of freedom. This toolset is the system of equations you get when you compile all of the rows and columns of a stiffness matrix and relate it to a load {P} as a function of the displacement vector {u} as can be seen in equation 3.12. Wholly, this set of equations describes the behavior of the entire system and the interaction of each stiffness element. This form, however, presents problems for an engineer because both the loading vector and the displacement vector contain variables. To solve this system of equations it is necessary to solve first for nodal displacements and then for reaction forces using those displacement values.

To achieve this solution methodology it is necessary to compile the global stiffness matrix [K] and [G] using only the rows and columns of the local stiffness matrices corresponding to free dofs. When completed, the displacement vector is the only unknown to be calculated. The stiffness matrix is known in its entirety based on material properties

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and axial loading, and the load vector {P} includes all of the applied loads except axial force causing stress stiffening. By removing the dofs that are restrained the unknown reaction forces in the load vector are ignored (for the moment) and the nodal displacements can be calculated using matrix multiplication.

Once the free nodal displacements are found, the system can be compiled for all degrees of freedom (including the fixed dofs which have a displacement of zero) and the load vector is calculated. This load vector includes values for the nodal loads as well as the reaction forces. The index in which they are found in the vector corresponds to their numbered dof.

To solve a buckling problem it is necessary to compile the global stiffness matrices

[K] and [G] as shown above, however, solving for nodal displacements is trivial because in a collapsed structure, nodal displacements are unbounded. Instability and collapse diverges from the normal solution process and is described in the section

Artificial Stiffness and Instability below.

Member Modelling

For a single member simply-supported beam the global stiffness matrix [K] compiled for the free dofs is shown below. Figure 3.6 depicts a single beam element, which can represent a beam of length L with homogeneous material properties throughout, or a differential beam element as part of a larger structure with similar or dissimilar material properties. Displacement in the y-direction at both nodes is restrained, therefore [K] only includes stiffness values for rotation, dof two and four. There is no displacement in the x- 34

direction because beams are never analyzed for axial displacement caused by axial loading.

Axial loading may, however, affect bending stiffness by transfering membrane strain energy to bending strain energy described in the section labelled Geometric Matrix

Derivation.

퐸퐼 2 2 퐾 = [4퐿 2퐿 ] 푚푒푚푏푒푟 퐿3 2퐿2 4퐿2

Figure 3.6: Simply Supported Beam including DOFs

From here, the model is discretized and each member is modelled with stiffness values as done above. Discretization breaks the beam into sections and each section is assigned material properties and boundary conditions. The overall goal is to choose a sufficient number of sections to balance run time with solution accuracy.

Model Discretization

The model developed for analysis of timber members in compression was used to determine the buckling loads for different numbers of sub-members. Utilizing the buckling theory described above, a 36 in. tall 2x4 SYP compression member can be analyzed for its critical load for an increasing numbers of sub-members. Using the classic Euler buckling equation it is possible to normalize this vector of critical load values to determine how accurate they are. Finally, the point where the change in calculated load by MSA is less

35

than 10-5, 15 sub-members, is highlighted by a red star and this point will become important when choosing the mesh resolution for the model.

Figure 3.7: Analysis of Model Sub-Elements Quantity

Discretization is done for two reasons: the first is so that the behavior of the beam can be analyzed throughout the member’s length instead of just at the end nodes which increases the overall accuracy of the model. Secondly, discretization allows the member to be given different material and section properties throughout its length allowing a reduction in stiffness to be analyzed along various partial member lengths. Without discretization, only displacement and rotation could be determined at the nodes; a finer

36

mesh will result in a more accurate model, however, a finer mesh will also use more computing power so a balance between solve time and accuracy must be found.

The goal of this model is to examine the buckling action for a member with reduced stiffness (EI) along a partial length from one side of the member, 15 sub-members will be used on either side of the point where stiffness degradation ends. This will provide a sufficiently accurate buckling load while allowing minimal computing power.

Artificial Stiffness and Instability

As stated above, the beam member in our model only has the stiffness values to resist shear in the local y-direction and moment about the local z-direction. To model the column behavior it is necessary to develop a relationship between some applied axial load in the local x-direction and the native beam stiffness effects. This is done with the addition of the geometric stiffness matrix, effectively decreasing beam stiffness caused by axial compression until buckling occurs.

Eigenvalues and eigenvectors are crucial to stability analysis for a compression member. An eigenvalue (λ) is a scalar value which defines the characteristic roots of a linear system of equations (A) in the form

AX = λX 3.22

퐴11 퐴12 … 퐴1푘 푋1 푋1 3.23 퐴 퐴 … 퐴 푋 푋 [ 21 22 1푘] [ 2] = λ [ 2] ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 퐴푘1 퐴푘2 … 퐴푘푘 푋푘 푋푘

This relationship is equivalent to the form

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퐴11 − λ 퐴12 … 퐴1푘 푋1 0 퐴 퐴 − λ … 퐴 푋 0 [ 21 22 1푘 ] [ 2] = [ ] 3.24 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 퐴푘1 퐴푘2 … 퐴푘푘 − λ 푋푘 0

By Cramer’s Rule, the system of equations in this form has a solution only if the determinant of the matrix on the left is equal to zero. Using this relationship and the stiffness matrices derived above we can solve for the eigenvalues of [A]-λ[I] to determine the load at which the model buckles.

푑푒푡(퐴 − 휆퐼) = 0 3.25

푑푒푡(퐾 − 휆퐺) = 0 3.26

푑푒푡(퐺−1퐾 − 휆퐼) = 0 3.27

The solution to the eigenvalue problem results in a stiffness matrix that is not invertible, which defines instability in structural analysis.

Multiple eigenvalues exist for the same system of equations which define higher order buckling behavior. The minimum of these eigenvalues is what is of interest because it indicates first order buckling for the minimum axial load. This is the failure mode that occurs first given simple boundary conditions and normal incremental loading.

Another way to visualize column buckling in a mathematical sense is from indeterminacy of the stiffness matrix. In a typical column section, buckling would occur only when the stiffness matrix for that particular member is indeterminate (not invertible), resulting in the solution set to include unbounded nodal displacements (see below). Given the global compiled stiffness matrix [S], the applied nodal loads {P}, and the corresponding nodal displacement vector {d} this relationship is shown in equations 3.28 and 3.29. 38

[푆] ∗ {푑} = {푃} 3.28

{푑} = [푆]−1 ∗ {푃} 3.29

If the stiffness matrix [S] is not invertible, the determinant of matrix [S] is nonexistent, then the bounds on the nodal vector {d} do not exist, therefore the member is unstable and collapse may occur.

Determining a collapse scenario is done simply by determining the eigenvalues that cause the determinant of the stiffness matrices to equal zero. It is possible to capture a negative stiffening effect from axial compression by considering the geometric stiffness matrix in a subtractive manner. Using the definitions for [S] and [G]

푇 det⁡([푆] + [퐺]) = 0 3.30 30

−푇 The variable 휆푐 can be define as, 휆푐 = ⁄30, negative T for compression, and substituted into Equation 3.30. Pre-multiply both matrices inside the determinant function by the inverse of the geometric stiffness matrix [G]-1 to yield a similar form as the equation to determine eigenvalues, equations 3.25 and 3.32.

−1 det⁡([퐺] [푆] − 휆푐[퐼]) = 0 3.31

det⁡([퐴] − 휆푐[퐼]) = 0 3.32

Column Stability Modification

The most difficult part of analyzing a structure for instability is determining the load on that structure that causes buckling. For members that are both prismatic and homogeneous, this can be done using Euler’s buckling equation, however, wood that is

39

undergoing decay is mostly never homogeneous. Wood decay can affect a member along its entire length or, more often, only a portion of its length. This type of analysis must be done using the Finite Element Modelling techniques described above.

The member that we are attempting to model has some defining characteristics that must be determined. First of all, it has a stiffness (EI) and a modified stiffness (EI*). This modified stiffness is how the decayed portion of the wood is modelled. It also has an original length (L) and a modified length (L*) which is the section of the member that experiences the modified stiffness, or wood decay. The member shall represent a 2x4 framing element and thus is modelled pinned at the bottom with a roller at the top. This essentially states that the bottom shall have no horizontal or vertical displacement and the top shall only be restricted to horizontal displacement. Rotation at both ends is permitted.

The model shall be discretized, as described above, with fifteen sub-members on either side of the point where stiffness is modified. This allows for an easy running program that is sufficiently accurate. All of the nodes between the end nodes are permitted to translate horizontally, vertically, and rotate. Each sub-member gets assigned a length, geometric stiffness [g] and stiffness [k] Figure 3.8: Degraded Column Model 40

with a stiffness value depending on where it is on the member’s length. Finally, both k and g are compiled for the free degrees of freedom yielding one large square global stiffness matrix [K] and one large square global geometric stiffness matrix [G]. Using matrix algebra the two stiffness matrices are substituted into equation 3.31 and the eigenvalues are found. These eigenvalues correlate to the different modes of failure, the smallest one being the load that causes instability in the first order.

These critical buckling stresses are key to calculating an accurate column stability factor for use with traditional NDS techniques. Using constant values for EI and L with variable stiffness degradations (EI*) and critical member lengths (L*) it is possible and extremely easy to develop design aids or new factors that could assist in capturing decay effects with minimal effort and maximum efficiency.

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Chapter 4 Results

An intuitive conclusion that can be drawn from real world observation is that wood decay causes some losses in structural capacity, however, this occurrence is much more difficult to quantify. If a member is decayed uniformly throughout its entire length it could be possible to estimate some reduced moment of inertia (I) and modulus of elasticity (E), and calculate an assumed buckling load for that member. However, introducing nonhomogeneous material properties along the length of a member presents difficulties in that calculation and a more complex mathematical model must be used.

During the course of design, resistance factors and conservative assumptions from situational uncertainties can lead to a structural model that is highly conservative.

Alternatively, if our understanding of a particular subject is minimal, assumptions that seem to be cautious may not be and structures could be susceptible to premature failure.

That is, misunderstanding of the effect of environmental stimuli on a member’s mechanism of failure can lead to unintentionally under-designed members. This can be the case in certain geographic areas where wood rot and decay is widespread.

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Original Column Stability Factor

As discussed in Chapter 2, compression members fail due to a combination of crushing and buckling. The pure crushing load in a column member is based strictly on material mechanics altered by some assumed outside factors; wet service, temperature, size, and incising are the factors that the wood design experts have deemed significant.

The difficulty in determining the critical buckling load occurs when a member’s properties change over the member’s length and thus, the buckling component of the interaction cannot be easily calculated. When this occurs, finite element modelling becomes useful to determine the complete picture of interaction.

The column stability factor is intended to represent a reduction in the capacity of a column member based on that member’s slenderness. The factor, as prescribed by the

NDS, is designed to determine member capacity based on an assumed interaction between pure crushing and pure buckling action. Analyzing the value of the factor as a function of length indicates a rapid decrease in capacity of the model 2x4, seen in Figure 4.1. It is more convenient to consider the column stability factor as the percentage of the crushing stress, 퐹′′푐, that the column can resist. The value 퐹′′푐 is equal to the reference design value for compression parallel to the grain multiplied by all applicable adjustment factors from

Table 2.1 except the column stability factor (American Wood Council 2015).

퐹′′푐 = 푓푐 × 퐶푀 × 퐶푡 × 퐶퐹 × 퐶푖 × 퐾퐹 × 휙 × 휆 4.1

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As an example, a 10-foot tall column constructed of Southern Yellow Pine retains approximately 3.5% of the crushing capacity, 퐶푃 = 0.035, as shown in Figure 4.1.

Similarly, a 36-in. tall column’s capacity reduces to roughly 35%, 퐶푃 = 0.35, as predicted by buckling-crushing interaction without any modification to stiffness from wood decay.

This rapid reduction in capacity indicates that thin members that are not protected from weak axis buckling are susceptible to weak axis buckling at increasing lengths for compression acting parallel to the grain.

Figure 4.1: Column Stability Factor for a 2x4 SYP as Prescribed by the NDS Additional analysis can be performed for varying levels of stiffness reduction relative to the member’s cross-section. Plotting CP in Figure 4.2 and varying the critical 44

buckling stress that is assumed to act on the section along its entire length, it is possible to

see how drastic the capacity decreases as the section stiffness is degraded for specific

member lengths. This is useful initially because it allows analysis of the failure action for

a column assumed to decay to a certain degree without needing to solve the problem using

structural analysis due to heterogeneous material properties. Later, a more in-depth

analysis can be accomplished by varying the stiffness over a partial member length.

Figure 4.2: Column Stability Factor Varied with Member Length and Critical Stress Observing Figure 4.2 for a 36-in. tall member, the column capacity drops from

35.5% for a situation with no wood decay a total of 16.7% for a situation where decay

causes the buckling stress to drop by one-half, resulting in an overall capacity percentage 45

of 18.8%. The information in Table 4.1 shows the percentage of capacity reduction (CP) along the entire length of a 36-in. tall Southern Yellow Pine 2x4 for different Fcr values, where Fcr is the Euler buckling load defined above.

Table 4.1: CP Factor for Different Values of Fcr, 36-in. Tall 2x4 SYP

Fcr 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

CP .0391 .0775 .1152 .1522 .1883 .2235 .2579 .2913 .3237 .3550 As an additional example, Table 4.2 shows the variation in column stability factor for a 20-in. tall SYP column for varying stiffness reduction. This data indicates a loss of approximately 25% in capacity between a member that has no degradation and a member with half of the original Euler buckling strength. That is, the difference in percent capacity

(CP) of a member in which the critical buckling stress is equal to that predicted by Euler’s equation (76.8%) and one where wood rot has reduced the critical buckling stress to 50% of Euler (52.34%) is approximately 25% (exactly 24.46%).

Table 4.2: CP Factor for Different Values of Fcr, 20-in. Tall 2x4 SYP

Fcr 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

CP 0.1241 0.2402 0.3463 0.4411 0.5234 0.5931 0.6509 0.6982 0.7367 0.7680 There is an apparent bounded region of lengths where stiffness reduction is particularly harmful to the column capacity. A drastic drop in column capacity can be observed in Figure 4.2 for short member lengths from 5 inches to 40 inches. A 2x4 column without any stiffness reduction, represented by the curve labelled 100% Fcr in Figure 4.2, is expected to lose approximately 70% of its capacity for a 40-in. tall specimen due to buckling interaction. After the member reaches 40-in. tall the capacity reduction starts to tail off and further column capacity reduction is minimal. Similarly, a column that has 46

been affected by decay, represented by the other curves in Figure 4.2, can be expected to lose capacity at a greater rate over the same length; the slope of the curves increases as the critical buckling stress reduces due to increased decay. This indicates that for slender members prone to a more buckling driven failure mechanism (i.e.- short columns), capacity is lost at a greater rate than both blocks and tall columns.

The information presented in the tables above would be considered a worst-case scenario for each value of Fcr as decay will only occur along the entire length of a member in rare situations. In a low to moderate level flood event, this would most likely not be the case. Floodwater would be expected to contact, and possibly degrade, the lower portion of the column leaving the upper, unaffected portion with the original section and material properties. Assuming the water-contacted wood will have some level of degradation, the column capacity will be negatively affected.

When calculating the critical buckling stress of a member due to variable stiffness reduction over the member length using Finite Element Modelling, a similar result to what is seen in Figure 4.2 should emerge. In the preliminary analysis above, the column capacity was modified given an assumed percent reduction in overall Euler buckling stress. The following discussion investigates how the overall buckling stress changes as stiffness is varied over a differential portion of the member length leading to decreased column capacity. Additionally, whether that reduced column capacity is a concern to engineers designing with timber members.

47

In the following discussion the percent change in stiffness is referred to as 퐸퐼푝 such

∗ that 퐸퐼 = 퐸퐼 × 퐸퐼푝 from Figure 3.8.

Overall Effect of Stiffness Reduction

Analyzing the percent reduction in capacity due to column failure interaction may be observed by inspecting the column stability factor for any particular member length and critical buckling stress. Alternatively, another method is to calculate the capacity of a column using FEM given some level of decay over a section of the member’s length. This value normalized by the capacity of a member with no decay can show a more complete picture of how decay and stiffness degradation affects column stability in real world

푃∗ construction. This ratio is where 푃∗ is the engineered capacity of a column of length 푃

∗ (H) degraded by some percentage (퐸퐼푝) over length (퐻 ) using FEM, and P is the engineered capacity of a column of similar length with no stiffness degradation.

Leaving all other NDS prescribed factors constant, Figure 4.3 shows the column

∗ capacity ratio for variable 퐻 and 퐸퐼푝. The critical buckling load ratio is useful because it illustrates a percent reduction in column capacity given some set of variables similar to what CP represented. For example, a 36-in. tall member with half of its length affected by decay would have varying levels of capacity reduction. For an assumed stiffness reduction of 50% it would lose about 35% of its capacity. For decay that was unchecked for many years and was able to do extensive damage to the member’s cross-section and modulus, the capacity could be reduced by 85% which corresponds to a value of 퐸퐼푝 = 0.1.

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∗ Figure 4.3: Critical Buckling Load Ratio for Variable 퐻 and 퐸퐼푝, 36-in. Member

The new question that an engineer must ask given the data above is the point in which a certain degree of degradation in stiffness becomes critical. Given a certain percent stiffness reduction, how can it be determined if the member designed is still adequate? This can be done by revisiting the interaction diagrams described in Chapter 2 and is the basis for both the column stability factor developed by the NDS and the modified interaction curves proposed below.

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Modified Interaction Diagrams

The most natural way to deal with decay and reduced capacity is to revisit the column stability factor and investigate further what it was comprised of and how it changes based on outside environmental influences. The NDS created the column stability factor as a curve fit to the experimentally comprised compression failure data from Figure 2.1.

This fit illustrated the interaction between column crushing and column buckling for a pure compression member with no lateral load. Similar to the interaction curves from Figure

2.2, it is possible to modify them for reduced values of buckling that correspond to reduced values of stiffness. By modifying the interaction diagram, engineering design aids can be produced to be used to check ultimate capacities.

The type of lumber that is of particular importance is Southern Yellow Pine sawn lumber because of its use in residential construction. Therefore, modifying the curve

‘c=0.8 Sawn Lumber’ for stiffness reductions over a length of 20% of the member’s overall length, Figure 4.4 can be produced. The variable in these plots is the column capacity ‘P’, which would decrease as the assumed percent stiffness, 퐸퐼푝, decreased. On the y-axis, the column capacity is normalized by the crushing strength of wood (Pc) as defined by equation

4.1. The capacity is normalized by the Euler buckling force (Pcr) on the x-axis.

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Figure 4.4: 2x4 Column Interaction Diagram, 20% Affected Member Length

Data points with a higher buckling ratio indicate taller specimens because the

buckling stress, which is found in the denominator of the ratio, for a tall specimen is

reduced relative to a similar, yet shorter specimen. For a member with only 20% of its

length affected by decay, the overall effect will be minimal compared to members that have

larger affected lengths as seen in Figure 4.5. Furthermore, shorter members would be less

affected overall because the controlling failure mechanism is crushing which is not

modified in this model. It is reasonable to think that if a member is decayed, crushing

strength would also be affected as well, however, this action is not captured in the model.

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Figure 4.5: 2x4 Column Interaction Diagram, 60% Affected Member Length

Analysis of existing column structures can be accomplished using the developed aids such as Figure 4.4 and Figure 4.5. When powerful mathematical structural analysis tools are not available, checking for premature failure due to material degradation can be easily accomplished with supplemental information such as these plots.

When placed side-by-side it is easier to understand the effects of stiffness reduction on a column member (Figure 4.6). Visual inspection of the changes in positioning of any

52

one colored line between the four plots can help understand how affected member length plays a part in overall member capacity.

Figure 4.6: Wood Decay Design Aids

Overall, it is extremely important to understand all of the environmental factors that act on a particular structure. These factors currently include wind, rain, snow, earthquake, and now they should include wood decay. Particularly prevalent in areas of intense flooding where certain types of wood consuming micro-organisms are abundant, the

53

analysis for wood decay is no longer too difficult to check for as long as engineers are given appropriate tools to do those checks.

Capacity Reduction

The capacity ratios presented in the interaction diagrams in Figure 4.6 and the critical buckling load ratios from Figure 4.3 are comparable. This is evident by presenting the capacity ratio data for a 36-in. column member in a more logical way. The two tables below, Table 4.3 and Table 4.4, tabulate the data from the interaction curves for each point corresponding to a 36-in. tall member for a certain value of stiffness degradation over a variable affected length. The difference between the capacity ratios for variable lengths can be calculated by the ∆ 푃⁄ = 푃⁄ (0%) − 푃⁄ (퐿푀퐹%) which indicates the actual 푃푐 푃푐 푃푐 difference in capacities between a column that is not degraded and one that is. More importantly, dividing either 푃⁄ or 푃⁄ by its unmodified value given no decayed length 푃푐 푃푒

∆푃 results in the capacity reduction ratio of that point ⁄푃. These values, when graphed against the length modification percentage yields the same results found in Figure 4.3.

54

Table 4.3: Capacity Ratios for a 36-in. Tall Column, EIp=20%

80% Stiffness Reduction (EIp=20%), 36-in. Tall Member Length Modification 0% 20% 40% 60% 80% 100% 푃⁄ (y-axis) 푃푐 0.3548 0.2853 0.1485 0.0975 0.0803 0.0775 ∆푃⁄ 푃푐 0.0000 0.0695 0.2063 0.2573 0.2745 0.2773 푃⁄ (x-axis) 푃푒 0.9004 0.7240 0.3767 0.2475 0.2039 0.1967 ∆푃⁄ 푃푒 0.0000 0.1764 0.5237 0.6529 0.6965 0.7037 Percent Capacity 100% 80.4% 41.8% 27.5% 22.6% 21.8%

The information in Table 4.3 show the capacity ratios for each axis of the interaction diagrams presented above. These data points represent a particularly serious case of decay, but illustrate the severity in the reduction of structural capacity that is possible. For a member that is affected over 80% of its length, the change in crushing capacity ratio (∆푃⁄푃푐) drops by 27.45% whereas, the change in buckling capacity ratio

(∆푃⁄푃푒) drops by 69.65%. Using the relationship shown in equation 4.2 this indicates an overall capacity reduction of approximately 23% of the original capacity given by the NDS.

(푃⁄푃 − ∆푃⁄푃 ) 1 − ∆푃⁄ = 푒 푒 ⁄ 4.2 푃 푃⁄푃푒

Similarly, although not as critical, the capacity reduction of a member that experiences a stiffness reduction of only 20% (EIp=80%) can be compared and analyzed.

The data in Table 4.4 show that between a column with decay over 0% of its length and decay over 80% of its length, the change in crushing capacity ratio (∆푃⁄푃푐) drops by 6.1% and the change in buckling capacity ratio (∆푃⁄푃푒) drops by 15.47%. Again, using equation

4.2, the reduction in capacity ratios correlates with an overall capacity reduction of just 55

82.8% of an unaffected column. The difference in percent capacities for the two decay scenarios vary widely, however, even a modest amount of decay can affect a short column by a large degree.

Table 4.4: Capacity Ratios for a 36-in. Tall Column, EIp=80%

20% Stiffness Reduction (EIp=80%), 36-in. Tall Member Length Modification 0% 20% 40% 60% 80% 100% 푃⁄ (y-axis) 0.3548 0.3512 0.332 0.3077 0.2938 0.2913 푃푐 ∆푃⁄ 0.0036 0.0228 0.0471 0.061 0.0635 푃푐 푃⁄ (x-axis) 0.9004 0.8912 0.8426 0.7809 0.7457 0.7392 푃푒 ∆푃⁄ 0.0092 0.0578 0.1195 0.1547 0.1612 푃푒 Reduction Percent 100% 99.0% 93.6% 86.7% 82.8% 82.1%

The reduction in structural capacity seems to be most critical when decay occurs between 10% and 40% of a member’s length. That is, the Capacity Reduction from Figure

4.3 curves are steepest between these points. Furthermore, a larger rate of decay will reduce the column capacity of a member to a greater extent. These two points should be used to develop code limits to determine the seriousness of a design decay issue.

56

Chapter 5 Discussion

Determining the adequacy of an in-service column member affected by decay is a significant concern for wood engineered structures. This concern is magnified for column elements since they are key components in structural load path. Additionally, weakened columns can lead to collapse of the structure, often catastrophic, and should be protected against to a significant degree.

The easiest way to determine whether a column has the capacity to resist the applied loads is to modify the interaction diagram discussed in Chapter 2. This requires minimal calculations which is advantageous because accurately calculating the capacity of a member with nonhomogeneous material properties is difficult without computer software.

Similar to concrete column design capacity interaction diagrams, these diagrams would allow the engineer to calculate the members crushing stress, 퐹′′푐, using NDS methods, the

Euler buckling stress, and the factored ultimate loads that must be resisted and plot the ratio on a prepared interaction diagram. Any point that falls below the interaction line corresponding to the assumed degree of decay would indicate a safe design.

Wood Rot Capacity

Intuitively, taller members that are affected by an increasing amount of decay have a decreasing level of capacity. That is, as member length and decay increase, overall 57

column capacity decreases. The interaction diagrams in Figure 5.1 support this hypothesis; the failure of taller members is controlled to a larger extent by buckling fall closer to the x-axis on the interaction curve and those points are affected drastically as stiffness is decreased.

Figure 5.1: 2x4 Column Interaction Diagram, 60% Affected Member Length

Members that are taller and more prone to buckling failure are more susceptible to premature failure initiated by wood decay. This can be observed from the data in Table

4.3 and Table 4.4 where the loss in buckling capacity, ∆푃⁄푃푒, increases at a greater rate than the loss in crushing capacity, ∆푃⁄푃푐, as more of the member’s length is affected by wood rot. This is due to a limitation of this model, that decay is assumed to affect the 58

buckling component while leaving the cross-section unaffected and able to carry the fully expected crushing load.

Normally, short members less than a few feet in length, typically known as blocks, are used in limited quantities in wood construction other than for bracing and other special framing scenarios. In these scenarios, crushing perpendicular to the grain loading conditions would govern for compression. Moreover, tall column members that are subjected to moisture and decay are numerous in typical wood construction and stiffness degradation could become a real factor if not prevented or mitigated. Most column members in residential construction that are affected by decay causing organisms will have their capacity greatly affected due to their buckling failure mode.

Critical Member Lengths

From the data it can be concluded that not all members are effected equally by decay. Short blocks and sufficiently slender members lose a smaller percentage of their ultimate capacity compared to members of intermediate slenderness. Bracketed in Figure

5.2, the critical member height that is affected by wood decay are members with a height between 5 inches and 40 inches.

59

Figure 5.2: Critical Member Height for Various Levels of Wood Decay

The percentage of capacity reduction for members at heights of 5, 20, and 96 inches are highlighted in Table 5.1 corresponding to short blocks, intermediate column studs, and tall slender columns. The last row in the table indicate the maximum difference in capacity between a member with no decay and a member that retains only 10% of the original Euler buckling capacity. Intermediate stud members have the potential to lose close to 65% of their crushing capacity purely due to decay. Alternatively, short blocks and tall studs lose

12% and 5% respectively. This indicates that studs between 5 inches and 40 inches are most critically affected by wood decay. 60

Table 5.1: Critical Member Height Capacity

Fcr H = 5in. H = 20in. H = 96in.

10% 0.8706 0.1241 0.0055

20% 0.9434 0.2402 0.0111

30% 0.9640 0.3463 0.0166

40% 0.9737 0.4411 0.0221

50% 0.9792 0.5234 0.0275

60% 0.9829 0.5931 0.0330

70% 0.9854 0.6509 0.0385

80% 0.9873 0.6982 0.0439

90% 0.9888 0.7367 0.0494

100% 0.9899 0.7680 0.0548

Difference (100%-10%) 11.93% 64.39% 4.93%

61

Additions to NDS

The necessity for more robust data on the criticality of wood decay is imperative.

Research into how wood rot affects column stiffness as raw capacity changes needs to be investigated further. This data will give engineers and code-writers a better idea of where on these charts typical column members fall given actual levels of stiffness degradation and decay. Those actual degradation data paired with easily developed interaction diagrams are an essential addition to the NDS code and would provide an easy check for column members where flooding is a real threat.

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Figure 5.3: Wood Decay Design Aids A design aid such as the ones seen in Figure 5.3 provide an engineer the ability to

check a wood column’s capacity given an assumed level of decay with very little work on

their part. Given 푃푒 and 푃푐 easily calculated using current NDS procedures, the actual

factored column capacity, P, can be determined such that this design point falls inside the

interaction diagram corresponding to an assumed level of decay.

63

Chapter 6 Conclusion

Limitations

The preceding work hinges on a few key assumptions that need to be researched further for relevance. These assumptions do not take away from the fact that wood rot and decay is a real threat to timber framed structures that are subjected to large amounts of flooding and where wood consuming micro-organisms are present and therefore, stiffness should assumed to be degraded. Furthermore, column members taller than a few feet in length are affected to the largest extent because their capacity is mostly governed by buckling failure whereas, shorter members are mostly affected by crushing failure.

First of all, the largest assumption that this paper makes is the rate at which wood decay affects structural capacity. Decay is assumed to critically affect the stiffness of wood column members. Also, wood that is affected by decay causing micro-organisms for extended periods of time are assumed to be affected to a larger extent than members where decay has been mitigated or avoided. If experimental testing shows that decay and stiffness degradation are only slightly affected in wood design then EIp is close to 1.0 and the modified interaction diagrams shift only slightly from original interaction diagram.

64

Future Work

As stated above, the applicability of this project ultimately depends on real world column member capacity data as well as the sensitivity of that data falling within design specifications after adequate design factors are applied. Large-scale testing to better understand wood decay and its effects are needed to determine the pervasiveness of this type of structural degradation.

A useful course of action would be a series of controlled experimentation utilizing micro-organism that are known to consume wood followed by a break test measuring the amount of force required for failure. This type of experiment would allow researchers to observe an overall trend in the effect of decay on a member’s capacity and therefore allow us to infer where a typical member would fall on the interaction diagrams above.

Additionally, determining probabilistic failure of wood members due to degradation is needed. Analyzing how the probability curve would shift from this environmental stimulus provides another way to look at overall structural capacity to determine if decay is a problem that needs to be researched further.

65

Bibliography

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Paper Association.

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Appendix A: Beam Stiffness Derivation

To derive the last three stiffness relationships we need to utilize the moment- curvature relationship just as we did in Chapter 3 to derive stiffness for degree of freedom

1.

푀 푑2푢 A.1 = 2 퐸 ∗ 퐼 푑푥

1 A.2 푢′′ = 푀 ∗ 퐸 ∗ 퐼

To find the equation for moment with respect to x we can use Figure A.1. To do this, cut the member at a location x from the left node and sum moments about the cut.

Figure A.1: Beam Member with Internal Forces

푀 = −푘21 + 푘11 ∗ 푥 A.3

Substitute the equation for M into the moment-curvature equation and integrating once to arrive at the equation for rotation. Integrate again to find the equation for displacement. 68

(−푘 + 푘 푥) 푢′′ = 21 11 A.4 퐸 ∗ 퐼

푘11 2 (−푘21푥 + 푥 ) ∫ 푢′′푑푥 = 푢′ = 2 + 퐶 A.5 퐸 ∗ 퐼 1

−푘 푘 ( 21 푥2 + 11 푥3) ∫ 푢′푑푥 = 푢 = 2 6 + 퐶 푥 + 퐶 A.6 퐸 ∗ 퐼 1 2

We can then substitute the boundary conditions from virtual unit displacements into these equations to solve for the constants of integration. Start out with a unit rotation at the degree of freedom 2 location as shown in Figure A.2.

y 푘24 푢2 = 1 x 푘22 푘21 푘23 Figure A.2: Degree of Freedom 2 Unit Rotation This beam member has nodal displacements shown below.

푢′(0) = 1 A.7

푢′(퐿) = 0 A.8

푢(0) = 0 A.9

푢(퐿) = 0 A.10

Systematically substituting equations A.7 through A.10 into equations A.5 and A.6 first to find the constants of integration and then to find the stiffness values 푘21 and 푘22.

Finally, sum the forces in the y-direction to solve for 푘23 and sum the moments about either

69

node to solve for 푘24. This same process can be repeated, starting with solving for the constants of integration, for degrees of freedom 3 and 4.

퐶1 = 1 A.11

퐶2 = 0 A.12

6퐸퐼 A.13 푘 = 12 퐿2

4퐸퐼 A.14 푘 = 22 퐿

6퐸퐼 A.15 푘 = − 32 퐿2

2퐸퐼 A.16 푘 = 42 퐿

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For degree of freedom 3:

Figure A.3: Degree of Freedom 3 Unit Displacement

퐶1 = 0 A.17

−푘 퐿2 −푘 퐿3 A.18 13 + 23 퐶 = 1 − 2 6 2 퐸퐼

12퐸퐼 A.19 푘 = − 13 퐿3

6퐸퐼 A.20 푘 = − 23 퐿2

12퐸퐼 A.21 푘 = 33 퐿3

6퐸퐼 A.22 푘 = − 43 퐿2

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For degree of freedom 4: y

푘44 x 푢4 = 1 푘24 푘14 푘34 Figure A.4: Degree of Freedom 4 Unit Rotation

2 −푘24퐿 −푘14퐿 + A.23 퐶 = 1 − 2 1 퐸퐼

퐶2 = 0 A.24

6퐸퐼 A.25 푘 = 14 퐿2

2퐸퐼 A.26 푘 = 24 퐿

6퐸퐼 A.27 푘 = − 34 퐿2

4퐸퐼 A.28 푘 = 44 퐿

The final stiffness matrix takes the form

12 6퐿 −12 6퐿 퐸퐼 6퐿 4퐿2 −6퐿 2퐿2 푘 = [ ] 퐿3 −12 −6퐿 12 −6퐿 6퐿 2퐿2 −6퐿 4퐿2

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Appendix B: Column Stability Factor Derivation

Begin with a straight line interaction (B.29) and modifying the equation to fit a particular data set (B.30)

푃 푃 + = 1 B.29 푃푐 푃푒

푃 푃 푃 푃 B.30 + = 1 + × × 푐 푃푐 푃푒 푃푐 푃푒

Simplifying both sides with a common denominator yields

푃(푃 + 푃 ) 푃2 푒 푐 = 1 + 푐 B.31 푃푐푃푒 푃푐푃푒

Multiplying both sides by the inverse of what is on the left

푃 푃 푃(푃 + 푃 ) 푃2 푃 푃 푐 푒 × 푒 푐 = (1 + 푐 ) × 푐 푒 B.32 (푃푒 + 푃푐) 푃푐푃푒 푃푐푃푒 (푃푒 + 푃푐) and simplifying

푃2 푃 푃 푃 = 푐 + 푐 푒 B.33 (푃푒 + 푃푐) (푃푒 + 푃푐)

Rearranging this equation to arrive at the form for a quadratic

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푐 푃 푃 0 = 푃2 − 푃 + 푐 푒 B.34 (푃푒 + 푃푐) (푃푒 + 푃푐)

Solve for the roots of the equation above using the quadratic equation

푐 푃 푃 1 ± √1 − 4 푐 푒 (푃푒 + 푃푐) (푃푒 + 푃푐) B.35 푃 = 푐 2 (푃푒 + 푃푐)

Multiplying the denominator to both sides of the numerator term a form similar to what is seen in the column stability formula starts to take shape.

(푃 + 푃 ) (푃 + 푃 )2 푐푃 푃 (푃 + 푃 )2 푒 푐 푒 푐 푐 푒 푒 푐 B.36 푃 = ± √ 2 − 4 2 2 2푐 (2푐) (푃푒 + 푃푐) 4푐

Simplifying what is inside the root to its most basic form yields and multiplying by -1

(푃 + 푃 ) (푃 + 푃 )2 푃 푃 푃 = 푒 푐 ± √ 푒 푐 − 푐 푒 B.37 2푐 (2푐)2 푐

Finally, dividing both sides by the crushing strength and recognizing that the force and

푃 퐹 stress ratios are equal 푒⁄ = 푒⁄ , and 푃⁄ = 퐹⁄ B.38 and B.39 are formulated. 푃푐 퐹푐 푃푐 퐹푐

푃 푃 푃 (1 + 푒⁄ ) (1 + 푒⁄ )2 푒⁄ 푃푐 √ 푃푐 푃푐 B.38 푃⁄ = ± − 푃푐 2푐 (2푐)2 푐

퐹 퐹 퐹 (1 + 푒⁄ ) (1 + 푒⁄ )2 푒⁄ 퐹푐 √ 퐹푐 퐹푐 B.39 퐹⁄ = ± − 퐹푐 2푐 (2푐)2 푐

These two equations represent column stability based on either applied force or stress.

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Appendix C: Geometric Stiffness Matrix Derivation

This derivative is picked up where Chapter 3 ends.

N represents the shape functions of a beam given the four unit displacement scenarios. G is the derivative of each shape function with respect to x.

Geometric stiffness is based on bending strain energy transferring to

membrane strain energy, that is, an

axial force creating axial displacement causing an increase or decrease in membrane strain energy.

Positive axial force increases membrane energy effectively stiffening the member, negative axial force does the opposite.

Now we just need to substitute the shape function derivatives into the equation for strain and factor out the terms that represent geometric stiffness.

75

Strain energy is represented by

U= =

U= {d}'[ ]{d}

where [ ]= d because {d} and {d} note: is the derivative of w.r.t. ,

is the solution to the integral for the strain energy equation from above.

Factoring out leaves us with a

matrix which is visually friendlier and easier to program. Using this matrix with global stiffness , the buckling force occurs when the determinant of + is equal to zero.

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