Issuu

A CHAIR FOR

THE DOCTOR All Eleven of Them

Kristin Allison Paniz Moayeri

Table of Contents

Chapter 1............................................ 5 About the Doctor

Chapter 2.......................................... 11 Manifesto

Chapter 3...........................................15 Sketches

Chapter 4.......................................... 21 Precedents

Chapter 5.......................................... 27 Models | Prototype

Chapter 6.......................................... 33 Construction | Assembly

Chapter 7.......................................... 39 The Finished Chair

Chapter 8.......................................... 51 Dimensions | Weight | Cost

Chapter 9.......................................... 61 Analyses

Chapter 10...................................... 113 Conclusion

Chapter 11...................................... 117 Bibliography

CHAPTER 1 About the Doctor

Here are the BASICS: 1

The Doctorâ€™s a

They

TIMELORD, which is a race of aliens from the planet GALLIFREY.

regenerate (change their bodies into new ones when they are about to die), hence the 11

(to date) forms of the Doctor.

A) The Time Lord High Council B) The Tenth Doctor Regenerating C) Gallifrey

Heâ€™s (effectively) the last timelord. Gallifrey was thought to be completely destroyed but now we

know that the planet is only lost somewhere in space and time, AND THE DOCTOR WILL FIND IT!

Speaking of time and space, he travels

through them in the

T.A.R.D.I.S. (time and

relative dimensions in space). It looks like a

POLICE BOX from London in the 60â€™s.

Travelling through time and space, The

changed the HISTORY OF THE UNIVERSE. Doctor has

D) The T.A.R.D.I.S somewhere in space. E) The Doctor and his companion Amy Pond in the War Room with Sir Winston Churchill in WWII.

He often has a because he

HUMAN COMPANION (or companions) with him on his travels

LOVES HUMANS! He thinks we’re great.

Some of the Doctor’s Companions

The Doctor has saved earth time and time again from his ENEMIES. Some of the most notorious are:

The Daleks

The Cybermen

The Zygons

Weeping Angles

The Sontarans

The IceWarriors

The Master

Silurians

The great Intelligence

The Silence

CHAPTER 2 Manifesto

C hairs: they’re for sitting, lying down, resting, being still – not things I typically do. I mean, if you had all of space and time at the tips of your ﬁngers, why would you ever stop doing things? If I’m going to have a chair, it needs to be able to come with me. Secondly, I’m always changing. I mean physically – it’s a time-lord-regeneration thing, don’t worry about it – so I am pretty much a different person every time you

see me. That means I could be an old man who wants room to lounge one day and some springkneed young person who wants a stool the next. Point is, this chair of yours has got to be able to change as I do. Third, traveling alone is dull, so the chair should be able to seat more than one (but not all the time, everyone need a little alone time, don’t they?) There. I know that’s a lot to get out of a chair, but what the point in making a chair if you can’t have a little fun? Now hop to it.

CHAPTER 3 Sketches

Initial Ideas Sketched:

A) Modular Idea B) Sliding Panel Idea C) Final Design: Box that is Bigger on the Inside

Sketches to Solve Problems with Leg Instability:

A A) External Seat Solution with Embedded Legs B) X-leg Solution

CHAPTER 4 Precedents

Initial Precedents:

The Frame “Bigger on the Inside” Idea

Final Precedents:

A) The Gate-Leg Table Precedent B) The Paul Buckley Gate-Leg Table

CHAPTER 5 Models | Prototypes

Sketch Models:

A A) 1:100 Model B) 1: 2 Model

Prototype:

1:1 Prototype to test out construction on the:

A) Dove Tails (Bottom Left Corner Joint in the Picture) B) The Cut of the Pieces C) The CNC of the Gallifreyan D) The Piano Hinge Assembly

Important things we learned:

A) To install the legs closer to the side pieces and make them longer thus reducing the cantilever on the seats. B) To attach the piano hinge and the seats to the top piece BEFORE assembling the top piece to the frame. C) To reduce the depth and width of the Gallifreyan circles.

CHAPTER 6 Construction | Assembly

D A) CNCing the dove tails, the hole in the legs, and the Gallifreyan. B) Marking the cut-lines on pieces. C) Cutting the pieces to the right dimensions as well as cutting the mitre edges on the seats and the top piece of the frame. D) Chiseling and sanding the dove-tail pieces to create sharp edges and make them ďŹ t together. 34

H E) Testing to see if the pieces ďŹ t in together and sanding off edges to make sure they do. F) Sanding all the surfaces as well as the groves of the Gallifreyan. G) All the pieces sanded, ready for staining H) Testing out different shades, tones, and application techniques of the dye to the get the desired effect 35

A) Dying the pieces while watching Doctor Who. B) Going over the pieces once the ďŹ rst coat had dried with wet pieces of paper towel to remove the dye in certain areas. This was followed by applying highly saturated mixtures of dye (sometimes even dry powder) to other areas to obtain the desired faded effect on the pieces. C) Dying done. Having left the pieces to dry over-night, sanding the pieces. The sanding was done unevenly to further add to the faded, old look. D) Attaching the piano hinge to the top of the frame and the seats. 36

E) Testing out the piano hindge and the way the all pieces fit together. F) Hammering in the side pieces of the frame (the glue in the picture was actually NOT used) G) More hammering H) Even more hammering... and finally finishing up. Being happy on the final review day.

CHAPTER 7 The Finished Chair

“It’s bigger on the inside!” The chair we made is inspired by the T.A.R.D.I.S. in its form, and by the Doctor in its function. Flipped backwards, you can use the base as a handle to carry it arround anywhere on your journeys through time and space. It starts as a little blue box you can sit on. Folding out the embedded seats, it expands into a oneperson seat and then a bench. It ﬂips up, open, and over for as many different seating conﬁgurations as the Doctor has faces. And when you’re done, it folds up and is gone again.

Allons-y!

Configuration 1

Configuration 2

Configuration 3

Other options for when youâ€™re feeling adventurous

CHAPTER 8 Dimensions | Weight | Cost

Configuration 1

440.00 A) The Chair in Use B) Plan C) Elevations

200.00

440.00

200.00

440.00

Configuration 2

402.00

421.00

440.00

200.00

B 54

440.00

216.32

440.00

621.00

A) The Chair in Use B) Plan C) Transverse Elevation D) Longitudinal Elevation

200.00

220.23

Configuration 3

1042.00 200.00

421.00

402.00

421.00

B 56

440.00

A) The Chair in Use B) Plan C) Transverse Elevation D) Longitudinal Elevation

216.32

440.00

1042.00

220.23

200.00

220.23

19.00

402.00

421.00

402.00

421.00

440.00

19.00

200.00

440.00

19.00

200.00

329.00

19.00

50.00

8 58

389.20

414.60

389.20

329.00

50.00

19.00

Weight Wood Selection Birch Hardwood Plywood

# 1 2 3 4 5 6 7 8

Piece Name Front Seat Rear Seat Frame Top Frame Bottom Front Side Rear Side Front Leg Rear Leg

Density (kg/m3) 660-720 (assume 660)

Thickness 3/4” (19 mm)

Volume (m3) 0.003072 0.003072 0.001672 0.001125 0.001672 0.001672 0.001199 0.001199

Weight (N) 19.87 19.87 10.81 7.28 10.81 10.81 6.35 6.35

Total Volume (m3) Plywood Density (kg/m3) Chair Mass (kg) Gravity (N/kg) Weight (N) Weight (lbs)

0.01468 660 9.6908 9.8 94.97 21.35

Cost Item Birch Hardwood Plywood Piano Hinge Dowel Blue Stain

Amount 5’ x 5’ 24” 48” 1 oz.

Price ($) 78.64 5.28 0.72 11.70

CNC router time

3 hours

Total Cost

120.34

CHAPTER 9 Analyses

ANALYSIS 1:

Reactions

ANALYSIS 1: Reactions

CONFIGURATION 1, Dead Load Only

R1 = R2 = 47.48 N

DL = 94.97 N

R1 = Chair Centre Line 47.48 N Load Centroid Lines

200.00

R2 = 47.48 N

Dead Load Centroid

440.00

w1 = w2 = 0.237 N/mm 200.00

440.00

Chair Centre Line

Load Centroid Lines Dead Load Centroid

Due to the symmetry of this conďŹ guration, the reactions will be equal; each will be half of the dead load of the chair. Dead Load = 94.97 N R = DL 2

R1 = 47.48 N R2 = 47.48 N

Because the reaction occurs at the middle of the reaction edge, it will be an evenly distributed reaction. R1 = 47.48 N length = 200 mm w1 = R1 l

47.48N = 0.237N/mm 200mm

w1 = 0.237 N/mm 63

R2 = 47.48 N length = 200 mm w2 = R2

l 47.48N = 0.237N/mm 200mm

w2 = 0.237 N/mm

ANALYSIS 1: Reactions

CONFIGURATION 1, Worst Case Loading

289.25

125.00

TL = 756.97 N

440.00

289.25

163 Chair Centre Line

Chair Centre Line

Live Load

Live Load Load Centroid Lines

100

Live Load Centroid Dead Load Centroid

Total Load Centroid

Load Centroid Lines

R1 = 497.62 N

Live Load Centroid Dead Load Centroid

R2 = 259.35 N

Total Load Centroid

200.00

440.00

The load in this case is positioned to give the worst case for one of the reactions at the ďŹ‚oor. Dead Load = 94.97 N Live Load = 662 N Total Load = 756.97 N

As the loading is not symmetrical, the value of each reaction is a function of its distance from the load centroid.

distance to TL = 150.75 mm

distance to TL = 289.25 mm

289.25mm (756.97N) = 497.62N 440mm

150.75mm (756.97N) = 259.35N 440mm

R1 = 497.62 N

R2 = 259.35 N

R1 = 497.62 N

R2 = 259.35 N

75.00

w1 = 1.87 N/mm

w2 = 3.11 N/mm

440.00

75.00

Chair ChairCentre CentreLine Line Load LoadCentroid CentroidLines Lines

w1 = 0.97 N/mm

w2 = 1.62 N/mm

Total TotalLoad Load

200.00

Chair ChairCentre CentreLine Line Load LoadCentroid CentroidLines Lines Total TotalLoad Load

Reaction Reaction

200.00

Reaction Reaction

Because the point load to each reaction is within the middle third of the length, but at neither the middle nor third point, the reaction will be trapezoidal in shape. R1 a b l

= = = =

497.62 N 8.33 mm 25 mm 200 mm

R2 a b l

= = = =

259.35 N 8.33 mm 25 mm 200 mm 6R R2 路 b l2

w1 = 6R1 路 b

w1 =

= 6 (497.62 N)(25 mm)/ (200 mm)2 = 74643 Nmm/ 40000 mm2 w1 = 1.87 N/mm

= 6 (259.35 N)(25 mm)/ (200 mm)2 = 38902.5 Nmm/ 40000 mm2 w1 = 0.97 N/mm

w3 =

R2 路 a w3 = 12R

12R1 路 a l2

= 12 (497.62 N)(8.33 mm)/ (200 mm)2 = 49742.1 Nmm/ 40000 mm2 w3 = 1.24 N/mm

= 12 (259.35 N)(8.33 mm)/ (200 mm)2 = 25924.63 Nmm/ 40000 mm2 w3 = 0.65 N/mm

w2 = w1 + w3 = 1.87 N/mm + 1.24 N/mm w2 = 3.11 N/mm

w2 = w1 + w3 = 0.97 N/mm + 0.65 N/mm w2 = 1.62 N/mm 65

ANALYSIS 1: Reactions

CONFIGURATION 2, Dead Load Only

The reactions of the chair at the ﬂoor are a function of the total load on the chair and the location of the center of gravity of the load(s). This conﬁguration needs to be split into two parts to solve the reactions. R2 from PART 1 will affect the loading and load centroid of PART 2. Part 1

Part 2

440.00

Chair Centre Line

Load Centroid Lines Dead Load Centroid

200.00

Part 1

Part 2

Dead Load = 94.97 N Part 1 = 26.22 N Part 2 = 68.75 N

PART 1

179.5

DL= 26.22 N

R2 = 4.85 N

Chair Centre Line

R1 = 21.36 N CL

Load Centroid Lines Dead Load Centroid Dead Load

220.23

Reaction

The value of each reaction in PART 1 is a function of its distance from the load centroid. Dead Load = 26.22 N

Part 1

distance to load centroid = 40.81 mm 179.5mm (26.22N) = 21.36N 220.23mm

R1 = 21.36 N

distance to load centroid = 179.5 mm 40.81mm (26.22N) = 4.85N 220.23mm

R2 = 4.85 N

R2 acts as an evenly distributed load on PART 2 where the seat is attached to the frame top by the hinge.

PART 2

440.00

TDL = 73.6N

Chair Centre Line Load Centroid Lines Dead Load Centroid Dead Load

R1 = 36.8 N

Reaction

R2 = 36.8 N

440.00

The value of each reaction in PART 2 is a function of its distance from the load centroid. The load of PART 2 is (220, 72.06), so the two reactions (R1 and R2) will be equal in value, Part 2 but distributed as trapezoids along their lengths.

centroid Part 1

Dead Load = 68.75 N Applied Load (R 2 from PART 1) = 4.85 N Total Dead Load = 73.6 N Total Dead Load = 73.6 N R = DL 2

R1 = 36.8 N R2 = 36.8 N

R1 = R2 = 36.8 N

440.00

72.74

Chair Centre Line Load Centroid Lines

w2 = 0.338 N/mm

w1 = 0.030 N/mm

Dead Load Centroid CL Dead Load Reaction

200.00

R1 a b l

= = = =

36.8 N 2 5.39Part mm 27.94 mm 200 mm

Chair Centre Line Load Centroid Lines Dead Load Centroid Dead Load Reaction

w1 =

6R1 路 a l2

= 6 (36.8 N)(5.39 mm)/ (200 mm)2 = 1190.11 Nmm/ 40000 mm2 w1 = 0.030 N/mm w3 =

12R1 路 b l2

= 12 (36.8 N)(27.94 mm)/ (200 mm)2 = 12338.30 Nmm/ 40000 mm2 w3 = 0.308 N/mm w2 = w1 + w3 = 0.030 N/mm + 0.308 N/mm w2 = 0.338 N/mm

ANALYSIS 1: Reactions

CONFIGURATION 2, Worst Case Loading

The reactions of the chair at the ďŹ‚oor are a function of the total load on the chair and the location of the center of gravity of the load(s). The person is sitting on the corner of the seat, so all of the live load is acting directly on PART 1.

Part 1

Part 2

Dead Load = 94.97 N Part 1 = 26.22 N Part 2 = 68.75 N

200.00

440.00

169.00

Chair Centre Line Live Load

169.00

Load Centroid Lines Live Load Centroid Dead Load Centroid

CL Part 1

Total Load Centroid

Part 2

PART 1 258.00 37.80

TL= 688.22 N

R2 = -118.14 N

Total Load Reaction

Part 2

R1 = 806.36 N 220.23

The value of each reaction in PART 1 is a function of its distance from the load centroid. R1 is a reaction at the ﬂoor, whereas R2 acts as a load on PART 2. Part 1

The reaction R2 is negative, soTotalitLoad pulls down in Reaction PART 1 and acts as an uplift force in PART 2. The reaction also acts within the middle third of the width of PART 1, so it is distributed as a trapezoid.

Dead Load = 26.22 N Live Load = 662 N Total Load = 688.22 N

R2 a b l

= = = =

-118.14 N 49.5 mm 17.5 mm 402 mm

R2 · b w1 = 6R

∑MR1 = 0 (688.22 N)(38.7 mm) = R2 (220.23 mm)

= 6 (-118.14 N)(17.5 mm)/ (402 mm)2 = -12404 Nmm/ 161604 mm2 w1 = -0.077 N/mm

R2 = 118.14 N R2 = -118.14 N

R2 · a w3 = 12R

∑MR2 = 0

(688.22 N)(258 mm) = R1 (220.23 mm)

= 12 (-118.14 N)(49.5 mm)/ (402 mm)2 = -70175.16 Nmm/ 161604 mm2 w3 = -0.434 N/mm

R1 = 806.36 N

w2 = w1 + w3 = -0.077 N/mm + (-0.434N/mm) w2 = -0.511 N/mm 71

Part 2

PART 2 177.00

TL= -49.39 N

Total Load

Tot

Reaction

440.00

Total Load

R2= -19.87 N

440.00

R1= -29.52 N

Total Load

Reaction

There is no symmetry to this loading case, so the value of each reaction is a function of its distance from the load centroid. Part 2

Dead Load = 68.75 N Applied Load (R 2 from PART 1) = -118.14 N Total Load = -49.39 N distance to load centroid = 177.00 mm 263mm (−49.39N) = −29.52N 440mm

R1 = -29.52 N

distance to load centroid = 263.00 mm 177mm (−49.39N) = −19.87N 440mm

R2 = -19.87 N

63.97

R2= -19.87 N

440.00

R1= -29.52 N

w2= -0.207 N/mm

w1= -0.307 N/mm

191.91

Total Load

Reaction

Because the load lies within the outer third of the length of each reaction side, the reaction will be distributed as a triangle that does not span the entire length of the side.

R1 = -29.52 N x = 63.97 mm

R2 = -19.87 N x = 63.97 mm

w1 = 2R1

R2 w2 = 2R 3x

w = 2(-29.52 N)/3(63.97 mm) = -59.04 N/ 191.91 mm w = -0.307 N/mm

w = 2(-19.87 N)/3(63.97 mm) = -39.74 N/ 191.91 mm w = -0.207 N/mm

The reactions show that in this conďŹ guration with this loading, the chair lifts off the ground at the back. This will be looked at further in Analysis 2.

ANALYSIS 1: Reactions

CONFIGURATION 3, Dead Load Only

The reactions of the chair at the ﬂoor are a function of the total load on the chair and the location of the center of gravity of the load(s). This conﬁguration is split into 3 parts to solve the reactions, but due to rotational symmetry PART 1 and PART 3 have the same reactions and effect onPart PART 2, so opposite reactions will have equal Part 1 2 Part 3 values. Dead Load = 94.97 N Part 1 = 26.22 N Part 2 = 42.53 N Part 3 = 26.22 N Part 2

Part 3

440.00

Part 1

Chair Centre Line

421.00

Part 1

200.00

Part 2

Load Centroid Lines

421.00

Dead Load Centroid

Part 3 Chair Centre Line

Load Centroid Lines Dead Load Centroid

art 2

PART 1

179.5 179.5

DL= 26.22 N

R2 = 4.85 N

R1 = 21.36 N

R2 = 4.85 N

CL 220.23

220.23

R1 = 21.36 N

Dead Load Reaction

The value of each reaction in PART 1 is a function of its distance from the load centroid. Part 3 Part 1 Part 2

Chair Centre Line Load Centroid Lines Dead Load Centroid

Dead Load = 26.22 N

Dead Load Reaction

distance to load centroid = 40.81 mm 179.5mm (26.22N) = 21.36N 220.23mm

R1 = 21.36 N

distance to load centroid = 179.5 mm 40.81mm (26.22N) = 4.85N 220.23mm

R2 = 4.85 N

R2 acts as an evenly distributed load on PART 2 where the seat is attached to the frame top by the hinge.

PART 2

TL= 52.25 N

440.00

R1 = R2 = 26.13 N

R1 = 26.13 N Chair Centre Line

R2 = 26.13 N

440.00

Load Centroid Lines

w1 = w2 = 0.131 N/mm 200.00

Dead Load Centroid Dead Load Reaction

Part 1

Due to the symmetry of this conďŹ guration, the reactions will be equal; Part 2 each will be half of the total load on this part of the chair.

Part 3

Dead Load = 42.53 N Applied Load (R 2 from PART 1 and PART 3) = 9.72 N Total Load = 52.25 N R = DL 2

R1 = 26.13 N R2 = 26.13 N

Because the reaction occurs at the middle of the reaction edge, it will be an evenly distributed reaction. R1 = 26.13 N length = 200 mm

R2 = 26.13 N length = 200 mm

w1 = R1

w2 = R2

w1 = 0.131 N/mm

w2 = 0.131 N/mm

ANALYSIS 1: Reactions

CONFIGURATION 3, Worst Case Loading The reactions of the chair at the ﬂoor are a function of the total load on the chair and the location of the center of gravity of the load(s). This conﬁguration is split into 3 parts to solve the reactions, but due to rotational symmetry PART 1 and PART 3 have the same reactions and effect on PART 2, so opposite reactions will have equal values.

Part 1

Part 2

Dead Load Part 1 Part 2 Part 3

Part 3

= = = =

94.97 N 26.22 N 42.53 N 26.22 N

CL 169.00

Part 2

169.00

Part 1

Part 3

440.00

169.00

CL 421.00

Part 1

200.00

Part 2

421.00

Part 3 CL

Chair Centre Line Live Load Load Centroid Lines Live Load Centroid Dead Load Centroid

Total Load Centroid

Chair Centre Line Live Load Load Centroid Lines Live Load Centroid Dead Load Centroid

Total Load Centroid

Part 1

PART 1 (also PART 3) 258.00

258.00

Chair Centre Line Load Centroid Lines

37.80

Total Load Reaction

TL = 688.22 N

TL = 688.22 N Part 2

Part 1

R2 = -118.14 N

R1 = 806.36 N

R1 = 806.36 CL N 220.23 220.23 Chair Centre Line Load Centroid Lines Total Load Reaction

Part 1

Part 3

Part 2

Chair Centre Line

Load Centroid Lines

Total Load

Reaction

Part 3

Part 2

The value of each reaction in PART 1 is a function of its distance from the load centroid. R1 is a reaction at the ﬂoor, whereas R2 acts as a load on PART 2.

Chair Centre Line

∑MR1 = 0 (688.22 N)(38.7 mm) = R2 (220.23 mm) R2 = 118.14 N R2 = -118.14 N

Dead Load = 26.22 N Live Load = 662 N Total Load = 688.22 N

∑MR2 = 0

(688.22 N)(258 mm) = R1 (220.23 mm) R1 = 806.36 N

The reaction R2 is negative, so it pulls down in PART 1 and 3, and acts as an uplift force in PART 2. The reaction also acts within the middle third of the width of PART 1 and 3, so it is distributed as a trapezoid. Chair Centre Line Load Centroid Lines

78Total Load Reaction

TL= 688.22 N N

w2= -0.511 N/mm

w1= -0.077 N/mm

R1= 806.36 N

R2 a b l

= = = =

-118.14 N 49.5 mm 17.5 mm 402 mm

R2 路 b w1 = 6R l2

= 6 (-118.14 N)(17.5 mm)/ (402 mm)2 = -12404 Nmm/ 161604 mm2 w1 = -0.077 N/mm R2 路 a w3 = 12R l2

= 12 (-118.14 N)(49.5 mm)/ (402 mm)2 = -70175.16 Nmm/ 161604 mm2 w3 = -0.434 N/mm w2 = w1 + w3 = -0.077 N/mm + (-0.434N/mm) w2 = -0.511 N/mm

PART 2 R1= R2 = -93.26 N

440.00

TL= -186.51 N

Chair CentreC Line L Load Centroid Lines Total Load

R1= -93.26 N

R2= -93.26 N

Chair Centre Line

Reaction

Chair Centre Line

Load Centroid Lines Total Load

w1 = w2 = -0.47 N/mm

Load Centroid Lines Total Load

440.00

Reaction

Chair Centre Line

200.00 Chair Centre Line

Load Centroid Lines

Total Load

Reaction

Chair Centre

Load Centro

Because of the rotational symmetry of this load case, the total load centroid for PART 2 is in the middle Partof 2 the frame top. Part 2 Reaction

Dead Load = Applied Load (R 2 from PART 1 and PART 3) = Total Load =

42.53 N -229.04 N -186.51 N

R = DL 2

R1 = -93.26 N R2 = -93.26 N

Because the reaction occurs at the middle of the reaction edge, it will be an evenly distributed reaction. R1 = -93.26 N length = 200 mm

Chair Centre Line Load Centroid Lines Total Load Reaction

R2 = -93.26 N length = 200 mm

w1 = R1

w2 = R2

w1 = -0.47 N/mm

w2 = -0.47 N/mm

Total Load Reaction

ANALYSIS 2:

Tipping Force and Friction

ANALYSIS 2: Tipping Force and Friction

CONFIGURATION 1, Dead Load Only

440.00

220.00

The tipping axis of the chair is the line ‘connecting’ 2 supports of the chair with the least perpendicular distance to the load centroid(s). For this conﬁguration, the easiest way to tip the chair, empty and loaded, is along the long side, so the darker line will be the tipping axis. CL

Dead Load Centroid

Critical Tipping Axis

200.00

Tipping Axis Chair Centre Line

Load Centroid Lines Dead Load Centroid

440.00

100

440.00

Choosing tipping axis Critical Tipping Axis Tipping Axis Chair Centre Line Load Centroid Lines Dead Load Centroid

100

CL 200.00

Tipping Axis Chair Centre Line

Load Centroid Lines Dead Load Centroid

The force required to tip over the chair, F, when pushed at the top perpendicular to its tipping axis is a function of the height, h, and the distance from the center of gravity to the tipping axis, x. F(h) ≥ P(x) h = 440 mm P = 94.97 N x = 100 mm F(440 mm) ≥ 94.97N (100 mm) F(440 mm) ≥ 9497 Nmm F ≥ 21.58 N The minimum force required to tip over the unloaded chair is 21.58 N.

ANALYSIS 2: Tipping Force and Friction

CONFIGURATION 1, Worst Case Live Loading CL

head + neck = torso = upper arms = Total Live Load

440.00

220.00

Live Load Centroid

50 F

Critical Tipping Axis 440.00

Tipping Axis Chair Centre Line

100

Live Load

Load Centroid Lines Live Load Centroid Dead Load Centroid

200.00

90 N 420 N 92 N 602 N

The force required to tip over the chair is still a function on its height, h, but because the dead and live load centroids are in different locations, the product of the distance from the center of gravity of each load to the tipping axis, xL, and the load , L, also becomes a factor.

Choosing tipping axis Critical Tipping Axis

Tipping Axis Chair Centre Line

F(h) ≥ xDL(DL)+xLL(LL)

Live Load Load Centroid Lines Live Load Centroid Dead Load Centroid

440.00

Dead Load (DL) Live Load (LL) xDL xLL h

F(440 mm) F(440 mm) F(440 mm) F Tipping Axis

≥ ≥ ≥ ≥

= = = = =

94.97 N 602 N 100 mm 50 mm 440 mm

100 mm(94.97 N)+50 mm(602 N) 9497.02 Nmm+ 30100 Nmm 39597.02 Nmm 89.99 N

Chair Centre Line

100

CL 200.00

FRICTION Ff = µ (total load) dead load = 94.97 N live load = 602 N total load = 696.97 N

Live Load

Load Centroid Lines The minimum force required to tip over the chair is 90 N. Live Load Centroid Dead Load Centroid

The friction force, Ff, between the chair and the ﬂoor will determine if the chair will tip or slide when horizontal force is applied. Because it is more difﬁcult to tip the chair when it is loaded (someone is sitting on it), we use that situation to compare the friction force.

µ (wood-wood) = 0.4 Ff (wood-wood) = 0.4 (696.97 N) Ff (wood-wood) = 278.79 N

Ff = 278.79 N > Ftip = 90 N

The friction force is larger than the minimum required tipping force, so the chair will tip, not slide. 84

23 2. 3

ANALYSIS 2: Tipping Force and Friction

CONFIGURATION 2, Dead Load Only Choosing tipping axis Critical Tipping Axis

216.17

Tipping Axis Chair Centre Line Load Centroid Lines

440.00

Dead Load Centroid

147.1

Critical Tipping Axis Tipping Axis Chair Centre Line

200.00

Load Centroid Lines Dead Load Centroid

Dead Load Centroid

440.00

F(h) ≥ P(x) h = 440 mm P = 94.97 N x = 147 mm F(440 mm) ≥ 94.97N (147 mm) F(440 mm) ≥ 13960.59 Nmm F ≥ 31.73 N

147.1

440.00

The minimum force required to tip over the empty chair is 31.73 N.

Tipping Axis Chair Centre Line

Load Centroid Lines

200.00

Dead Load Centroid

14 3. 73 23 2. 34

103.55

220.00

ANALYSIS 2: Tipping Force and Friction

CONFIGURATION 2, Worst Case Live Loading 536.5

216.17

103.55

Choosing tipping axis Critical Tipping Axis Tipping Axis Chair Centre Line Live Load Load Centroid Lines

147.1

440.00

13 9. 71

19 8. 75

Live Load Centroid Dead Load Centroid

head + neck torso upper arms 1/2 lower arms 1/2 upper legs Total Live Load

= = = = =

90 N 420 N 92 N 20 N 100 N 722 N

Live Load Centroid

Critical Tipping Axis Tipping Axis Chair Centre Line Live Load Live Load Centroid Dead Load Centroid

147.1

198.75

F(h) ≥ xDL(DL)+xLL(LL) h = 440 mm xDL = 198.75 mm xLL = 139.71 mm Dead Load (DL) = 94.97 N Live Load (LL) = 722 N

169.00

71 13 9.

440.00

8. 19

440.00

The force required to tip over the chair, F, when pushed at the top perpendicular to its tipping axis is a function of the height, h, and the distance from the center of gravity of each load to the tipping axis, xL. The location of the live load also changes the tipping axis.

216.17

139.71

Load Centroid Lines

200.00

F(440 mm) ≥ F(440 mm) ≥ F(440 mm) ≥ F≥

198.75 mm(94.97 N)+139.71 mm(722 N) 18875.30 Nmm+100871 Nmm 119746.30 Nmm 272.15 N

Tipping Axis

The minimum force required to tip over the chair is 272.15 N.

Chair Centre Line Live Load

Load Centroid Lines

200.00

FRICTION Ff = µ (total load)

dead load = 94.97 N live load = 722 N total load = 816.97 N

Live Load Centroid Dead Load Centroid

µ (wood-wood) = 0.4 Ff (wood-wood) = 0.4 (816.97 N) Ff (wood-wood) = 326.79 N

Ff = 326.79 N > Ftip = 272.15 N The chair will tip rather than slide. 86

220.00

ANALYSIS 2: Tipping Force and Friction

440.00

220.00

23 2. 34

CONFIGURATION 3, Dead Load Only

Critical Tipping Axis Tipping Axis Chair Centre Line Load Centroid Lines

421.00

Choosing tipping axis

200.00

220.00

Dead Load Centroid

421.00

The tipping axis of the chair is the line ‘connecting’ 2 supports of the chair closest to the load centroid of the dead load. Critical Tipping Axis

Tipping Axis Chair Centre Line Load Centroid Lines Dead Load Centroid

Load Centroid Lines

440.00

220.00

Dead Load Centroid

421.00

200.00

421.00

Tipping Axis Chair Centre Line

Dead Load Centroid

F(h) ≥ P(x)

Load Centroid Lines Dead Load Centroid

h = 440 mm P = 94.97 N x = 220 mm 440.00

F(440 mm) ≥ 94.97N (220 mm) F(440 mm) ≥ 20893 Nmm F ≥ 47.48 N The minimum force required to tip over the empty chair is 47.48 N. 87

ANALYSIS 2: Tipping Force and Friction

421.00

440.00

103.55

220.00

CONFIGURATION 3, Worst Case Live Loading

200.00

Live Load Centroid

2x head + neck 2x torso 2x upper arms 2x 1/2 lower arms 2x 1/2 upper legs Total Live Load

= = = = =

180 N 840 N 184 N 40 N 200 N 1444 N

421.00

Choosing tipping axis Critical Tipping Axis

Tipping Axis

Critical Tipping Axis Tipping Axis

Chair Centre Line

Live Load

Load Centroid Lines

103.55

Load Centroid Lines

440.00

Live Load Centroid Dead Load Centroid

220.00

Live Load Centroid Dead Load Centroid

F(h) ≥ xDL(DL)+xLL(LL) Chair Centre Line

103.55

169.00

Live Load Load Centroid Lines Live Load Centroid Dead Load Centroid

440.00

220.00

169.00

Tipping Axis

421.00

200.00

421.00

FRICTION Ff = µ (total load)

dead load = 94.97 N live load = 1444 N total load = 1538.97 N µ (wood-wood) = 0.4 Ff (wood-wood) = 0.4 (1538.97 N) Ff (wood-wood) = 615.59 N Ff = 615.59 N > Ftip = 387.32 N

The chair will tip rather than slide.

Tipping Axis Chair Centre Line

F(440 mm) ≥ F(440 mm) ≥ F(440 mm) ≥ F (total) ≥ F (per person) ≥

h xDL xLL Dead Load Live Load

= = = = =

440 mm 220 mm 103.55 mm 94.97 N 1444 N

220 mm(94.97 N)+103.55 mm (1444 N) 20893.44 Nmm+ 149526.2 Nmm 170419.64 Nmm 387.32 N 193.66 N

Live Load Load Centroid Lines Live Load Centroid Dead Load Centroid

The minimum force each person needs to exert to tip the chair is 193.66 N. The friction force, Ff, between the chair and the ﬂoor will determine if the chair will tip or slide when horizontal force is applied. Because it is more difﬁcult to tip the chair when it is loaded (someone is sitting on it), we use that situation to compare the friction force. 88

ANALYSIS 2: Tipping Force and Friction

Seating Safety Zones

The ‘safe seating zone’ outlines the area in which the live load centroid can be located without the chair tipping over. This can be determined by ﬁnding how far out on the cantilever the live load can be from the tipping axis while remaining in balance with the dead load. x

338.39 428.82 26.22 N

39.71 N

842 N

842 N (x mm) = 842 N (x mm) = 842 N (x mm) = x=

26.22 N(428.82 mm) + 39.71 N(338.39 mm) 11243.66 Nmm + 13254.74 Nmm 24498.4 Nmm 29.09 mm

842 N (x mm) = 842 N (x mm) = 842 N (x mm) = x=

26.22 N(438.10 mm) + 39.71 N(347.60 mm) 11486.98 Nmm + 13803.22 Nmm 25290.2 Nmm 30.04 mm

842 N (x mm) = 842 N (x mm) = 842 N (x mm) = x=

39.71 N(347.69 mm) + 26.22 N(438.22 mm) 13806.77 Nmm + 11490.13 Nmm 25296.9 Nmm 30.04 mm

pivot

347.60

438.10 26.22 N

39.71 N

842 N

pivot

347.69

438.22 842 N

39.71 N 26.22 N

pivot

143.36

Safety Zone

179.70 188.07 Safety Zone

In conclusion, the chair has a large surface area where the load centroid cannot be, due to the large cantilever on each seat (when open). That said, sitting on the chair with the load centroid outside the safety zone would probably not be very comfortable, as the area outside the safety zone does not have very large dimensions. Safety Zone

ANALYSIS 3:

Frame

ANALYSIS 3: Frame Analysis

CONFIGURATION 1, Worst Case Live Loading LL

This load case provides the worst loading for the frame side being loaded (as a column).

100

200.0

.00

440

FRAME TOP The load of the seats adds to the dead load of this piece.

Dead Load = 50.56 N Live Load = 662 N Total Load = 712.56 N

Because the load is not centrally located along the length of the frame top, the value of the reactions is a function of their distance from the total load.

Total Load Centroid = [305, 100] Live Load Reaction Dead Load

TL = 712.56 N

Total Load Live Load

w = 1.09 N/mm

TL = 712.56 N 135 mm

305mm

CL R1

w = 2.47 N/mm

distance to load centroid = 135.00 mm 305.00mm (712.56N) = 493.93N 440mm

Live Load Reaction

R1 = 493.93 N

Dead Load

TL = 712.56 N

Total Load

w = 1.09 N/mm

135

CL w = 2.47 N/mm

Live Load Reaction

Live Load

TL = 712.56 N

distance to load centroid = 305.00 mm 135mm (712.56N) = 218.63N 440mm

305

R2 = 218.63 N

The load occurs at the midpoint of the reaction R1 R2 edge length, so each reaction will be an evenly distributed rectangle. R1 = 493.93 N length = 200 mm

Dead Load Total Load

w1 = R1

w2 = R2

w1 = 2.47 N/mm

w2 = 1.09 N/mm

Live Load

R2 = 218.63 N length = 200 mm l

LEGS Dead Load = 6.35 N (0.019 N/mm) DL = 6.35 N 161.75 mm

w DL= 0.019

161.75

167.26 mm

N/mm

DL = 6.35 N

DL = 6.35 N R1=6.76N

019 N/mm

w DL= 0.019

R2=6.67N

N/mm

distance to load centroid = 161.75 mm R1=6.76N

167.25mm R2=6.67N (6.35N) = 3.23N 329.00mm

R2=6.67N R1=6.76N

R1 = 3.23 N

Applied Load Reaction

distance to load centroid = 167.25 mm Dead Load Total Load Live Load

R1 = 3.22 N To Floor

161.75mm (6.35N) = 3.12N 329.00mm

R2 = 3.12 N To Frame Bottom

Applied Load

R2 = 3.12 N

Applied Load

Reaction Dead Load

Reaction

Total Load

Dead Load

Live Load

Total Load

R2 from each leg acts as a point load on FRAME BOTTOM, while R1 acts on the ďŹ‚oor (but we assume this to have no practical impact).

Live Load

R1 = 3.22 N R2 = 3.12 N To Floor To Frame Bottom

R2 = 3.12 N To Frame Bottom

13.52 N

FRAME BOTTOM 13.52 N

Dead Load = 7.28 N Applied Load = 6.24 N Total Load R2 = 13.52 N

3.12 N

13.52 N

w= 3.12 N

7.28 N

3.12 N

0.0

R1 N/m

m R1

0.0

3.12 N

N/m

w= 7.28 N

3.12 N

0.0

Each reaction will be evenly distributed across its length.

N/m

Rotational Symmetry 3.12 N

N/m

Total = 13.52 N R = 6.76 N

Applied Load

.03

N/m

0.0

Rotational Symmetry N/m

Reaction Rotational Symmetry Dead Load Total = 13.52 N RTotal = 6.76Load N

Total = 13.52 N R = 6.76 N

Live Load Applied Load Applied Load Reaction Dead Load Total Load

Total Load = 13.52 N R = TL 2 R1 = 6.76 N R2 = 6.76 N

Reaction

Dead Load Total Load Live Load

R = 6.76 N length = 200 mm w= R l w = 0.034 N/mm

FRONT SIDE

wdead =

Dead Load = 0.054 N/mm Applied Load = 2.47 N/mm = 0.034 N/mm Total Load = 2.56 N/mm

0.054 N /mm

w2 = 0.03

4 N/mm

w1 = 2.47

N/mm

The reaction on this component is an evenly distributed rectangle equal to the total load. Total Load = 2.56 N/mm Reaction = 2.56 N/mm This column will be further analysed in Analysis 4.

Applied Load

Reaction

Dead Load

Total Load

Live Load

w = 2.56 N/mm

BACK SIDE wdead =

0.054 N/m

w2 = 0.03

4 N/mm

w1 = 1.09

N/mm

Dead Load = 0.054 N/mm Applied Load = 1.09 N/mm = 0.034 N/mm Total Load = 1.18 N/mm The reaction on this component is an evenly distributed rectangle equal to the total load. Total Load = 1.18 N/mm Reaction = 1.18 N/mm

Applied Load

Reaction

Dead Load

Total Load

Live Load

w = 1.178 N/mm

54 N/mm

ANALYSIS 3: Frame Analysis

CONFIGURATION 2, Worst Case Live Loading Z DL

.00

440

169

TL = 681.87 N

169

.00

402

421.0 440.00

Live Load Reaction Dead Load

200.0

Total Load Live Load

OPEN SEAT Live Load

Total Load Live Load

This load case provides the worst loading for the open seat (as a beam) and the open leg (as a column). R2 R1 Even though it was determined in Analysis 2 that this position is outside the SAFE SEATING ZONE, we use it to ﬁnd the limits theNchair components TL =of 681.87 (with regards toN/mm strength). w=0.77

Dead Load = 19.87 N w=0.054 N/mm Live Load = 662 N Total Load = 681.87 N

Reaction Dead Load

Y X

R1=846.80 N

Y X

.87 N

87 N

TL = 681.87 N

237.5

147.5

273.5

53.27

147.5

2.7

200.77

9.2

220.23

R2=-164.93 N

R1=846.80 N

The value of the reactions is dependent on their distance from the total load. TL = 681.87 N

w=0.77 N/mm

∑MR1 = 0 (681.87 N)(53.27 mm) = R2 (220.23 mm) R2 = -164.93 N

∑MR2 = 0 (681.87 N)(237.5 mm) = R1 (220.23 mm) R1 = 846.80 N

R2 is a negative (upward) reaction, so it will act as an upward force on the frame top. Because it acts within the middle third of the x-length of the seat, the reaction will be distributed as a trapezoid.

R1=846.80 N

TL = 681.87 N

R2=-164.93 N R2=-164.93 N R1=846.80 N

R2 R1

TL = 681.87 N w=0.77 N/mm 147.5

Z Y

2.7

w=0.054 N/mm

.87 N

R2 a b l

= = = =

R1=846.80 N

-164.93 N 58.25 mm 8.75 mm 402 mm

N w2 = w1TL+=w681.87 3 = -0.054 N/mm + (-0.71N/mm) w2 = -0.76 N/mm

R2 路 b w1 = 6R l2

87 N

= 6 (-164.93 N)(8.75mm)/ (402 mm)2 = -8658.83 Nmm/ 161604 mm2 w1 = -0.054 N/mm

The distributed load R2 is stopped from rotating the R2=-164.93 N open seat in the x-direction by the hinge connecting the seat to the R1=846.80 frame top. This is analysed in N Analysis 5.

R2 路 a w3 = 12R l2

= 12 (-164.93 N)(58.25 mm)/ (402 mm)2 = -115286.07 Nmm/ 161604 mm2 w3 = -0.71 N/mm OPEN LEG

R2=-164.93 N

Dead Load = 0.019 N/mm Applied Load = 846.80 N Total Load = 853.15 N AL = 846.80 N

Because the applied load acts directly in line with R1, all of the load is taken by R1. R2 is the same as if there were no applied load.

w DL= 0.019 N/mm

R1 = 846.80 N + 3.23 N = 850.03 N

R2 = 3.12 N

Applied Load Reaction Dead Load Total Load Live Load

R2 = 850 N

R1 = 3.12 N

ANALYSIS 3: Frame Analysis

CONFIGURATION 3, Worst Uplift Case in Frame Top LL LL .00

169

169.00

169.0

9.0

This load case causes the worst uplift in the frame top (as a beam). Because the loading is identical (albeit doubled) to that of Conﬁguration 2, we’ve skipped ahead to analyze the frame top.

FRAME TOP

Dead Load = 10.81 N Applied Load = -164.93 N = -164.93 N Total Load = -319.05 N

w= -0.77 N/mm

w= -0.054 N/mm

Live Load Reaction Dead Load Total Load

w= 0.79 N/mm

Live Load

w= 0.79 N/mm

CL CL

CL TL = 319.05 N

TL = 319.05 N

Due to the rotational symmetry of this loading conﬁguration, the total dead load will act at the center of the frame top, and so the reactions will be equal and evenly distributed rectangles. Total Load = -319.05 N R = TL 2 R1 = -159.53 N R2 = -159.53 N

R = -159.53 N length = 200 mm w= R l w = -0.79 N/mm

ANALYSIS 3: Frame Analysis

CONFIGURATION 3, Worst Case Loading for Frame Top Upper Legs ......... 200N

Head + Neck ..... 90N 1/2 Torso ......... 210N Upper Arms ....... 92N --------392 N

1/2 Torso ......... 210N Lower Arms ....... 40N --------250 N

200 N

392 N

This load doesnâ€™t cause the worst possible loading for the frame top (that case is analysed in Analysis 4), but it does create an interesting case for the amount of live and applied load on theZ frame top.

0.0

20 TL

Y X 411.87 N

OPEN SEAT (HEAD) CL

CL Z Y X

Dead Load = 19.87 N Live Load = 392 N CL N Total Load = 411.87

R2 = 18.20 N

R1 = 393.67 N

Live Load Reaction

411.87 N

Dead Load

Total Load

411.87 N

Live Load

201

C L N 411.87

R2 = 18.20 N w2 = 0.045 N/mm

411.87 N

CL 393.67 N

The value of the reactions is dependent on Y their distance from the total load. X CL distance to load centroid = 9.73 mm

411.87 N

R1 = 393.67 N

distance to load centroid = 210.50 mm

393.67 CL N

(9.73mm/220.23mm)(411.87N) = 18.20 N R1 = 393.67 N R2 = 18.20 N R2 = 18.20 N R2 occurs midspan of the x-length of the seat, so it will be an evenly distributed reaction.

R2 = 18.20 N length = 402 mm

CL w2 = 0.045 N/mm

R1 = 393.67 N

411.87 N

9.73 mm

(210.50mm/220.23mm)(411.87N) = 393.67 N

210.5 m

R1 = 393.67 N

411.87 N 210.5 mm

w2 = R2 l

w2 = 0.045 N/mm

393.67 N

OPEN SEAT (LEGS) Z Y

TL = 219.87 N

Y X

TL = 219.87 N

210.5

TL = 219.87 N 200.75 mm 9.73 mm

R1 = 210.16 N

R1TL = 219.87 N

R1 = 210.16 N

TL = 219.87 N

The value ofTLthe reactions is dependent on = 219.87 N CL C L CL load. their distance from the total

R1 = 210.16N

N/ L= 9.73 mm distance to 024 load centroidC = 0. mm

w CL CL (210.50mm/220.23mm)(219.87N) = /mm210.16N

R1 = 210.16N

distance R1 = 210.16N

R2 = 9.71 N

0.02

R1 = 210.16 N

/mm = 210.50 mm to load centroid 24 N w=

210.5 mm

m 1m

R1 = 210.16 N

Dead Load = 19.87 N Live Load = 200 N Z Total Load = 219.87 N

0.0

(9.73mm/220.23mm)(219.87N) = 9.71N

R2 = 9.71 N R2 occurs midspan of the x-length of the seat, so it wil be an evenly distributed reaction. R2 = 9.71 N length = 402 mm w2 = R2 l

w2 = 0.024 N/mm

R2 = 9.71 N

FRAME TOP Dead Load = 10.81 N Live Load = 200 N Applied Load = 18.20 N = 9.71 N Total Load = 238.73 N TL = 238.73 N

TL = 238.73 N

TL = 238.73 N 411.87 N

w= 0.024 N/mm

N/mm

TL = 238.73 N

w= 0.045 N/mm

CLCL w= 0.045 N/mm

CL w= 0.024 N/mm

411.87 N

210.5 mm

411.87 N 9.73 mm

w= 0.045 N/mm

R2 = 119.36 N

R1 = 119.36 N

R2 = 119.36 N

R1 = 119.36 N

R2 = 119.36 N

R1 = 119.36 N

R2 TL = 238.73 N

TL = 238.73 N

R1 Due to the rotational symmetry of this loading conďŹ guration, the total dead load will act at the center of the frame top, and CL so the reactions will be equal and evenly distributed rectangles.

Total Load

TL = 238.73 N

= 238.73 w2= 0.59N N/mm C CL L TL R CL = CL 2 0.59 N/mm N R1 = w1=119.36 w= 0.024 N/mm R2 = 119.36 N CL CL

R = 119.36 N CLlength = 200 mm CL w= 0.045 N/mm w= R w2= 0.59 N/mm l CL CL w = 0.59 N/mm w2= 0.59 N/mm

R1 = 119.36 N

w1= 0.59 N/mm R1

TL = 238.73 N

CL CL

w2= 0.59 N/mm

w1= 0.59 N/mm

411.87 N

R2 = 119.36 N

ANALYSIS 4:

Beams and Columns

100

ANALYSIS 4: Beams and Columns

Frame Top, Worst Loading Case Dead Load Live Load Total Load length

= = = =

0.025 N/mm 2.27 N/mm 2.29 N/mm 440 mm

Load Centroid Lines

BENDING

Live Load Reaction

|Mmax| (from moment diagram) = 37063.11 Nmm S (section modulus) = 12033.33 mm3

Live Load

200.00

Beam Diagram Section

.00

402

fb = S = 3.08 N/mm2 fb = 3080.04 kPa fb= 3080.04 kPa < Fb = 11376.75 kPa The beam is adequate in bending.

SHEAR

Vmax (from shear diagram) = 505.41 N A (cross-sectional area) = 3800 mm2

440.00

fv =

wTL = 2.29 N/mm

Load (N)

Load Centroid Lines Live Load Reaction

Live Load 505.41 N

3V = 0.19 N/mm2 2A

fv = 199.50 kPa

fv= 199.50 kPa < Fv = 1310.1 kPa

The beam is adequate in shear.

505.41 N

DEFLECTION 505.41 N

Shear (N)

w L E I

18531.60 Nmm Moment (Nmm)

= = = =

Total Load

0.025 N/mm 440Reaction mm 12411 MPa 44952.99 mm4

∆ = 0.402 mm ∆actual = 0.402 mm < L/240 = 1.83 mm

-37063.11 Nmm

-37063.11 Nmm 101

Line

Load Centroid Line

4 ∆ = wL 384EI

-505.41 N

Because the dead and live loads are equally distributed across the entire length of the beam, we treat them as one evenly distribued Chairload. Centre

The beam is adequate in deﬂection.

oid Lines

ANALYSIS 4: Beams and Columns

Open Seat, Worst Loading Case Dead Load Live Load Total Load length

= = = =

0.047 N/mm 3.61 N/mm 3.65 N/mm 421 mm

BENDING |Mmax| (from moment diagram) = 73710.98 Nmm S (section modulus) = 24187 mm3 M

fb = S = 3.04 N/mm2 fb = 3047.54 kPa

Load Centroid Lines Live Load

fb= 3047. kPa < Fb = 11376.75 kPa

Reaction Live Load Beam Diagram Section

200.00

The beam is adequate in bending.

SHEAR Vmax (from shear diagram) = 731.45 N A (cross-sectional area) = 7638 mm2

.00

200

fv =

3V = 0.14 N/mm2 2A

fv = 143.65 kPa

fv= 143.65 kPa < Fv = 1310.1 kPa The beam is adequate in shear.

421.00

DEFLECTION ∆dead =

200.00

wLL = 3.61 N/mm

wDL = 0.047 N/mm

-329.49 N

Load (N) 1071.36 N

220.23

∆dead

731.45 N

Shear (N) -339.91 N

Moment (Nmm)

w L E I a x

= = = = = = =

∆live = w L E I a

= = = = =

wx (4a2 l − l 3 + 6a2 x − 4ax 2 + x 3 ) 24EI

0.047 N/mm 421 mm 12411 MPa 44952.99 mm4 200 mm 200 mm 0.012 mm wa3 (4l + 3a) 24EI

Chair Centre Line

Load Centroid Lines

3.61 N/mm 221 mm Total Load 12411 MPa 44952.99 mm4 Reaction 200 mm

∆live = 3.20 mm

∆total = 3.21 mm < L/240 = 1.75 mm -73710.98 Nmm

The beam fails in deﬂection with this loading, but as we determined in Analysis 2, this is ouside the safe seating zone, so the chair would already have tipped over. 102

ANALYSIS 4: Beams and Columns

Handle, Worst Loading Case Total Load = 94.97 N length = 440 mm

BENDING |Mmax| (from moment diagram) = 7835.02 Nmm S (section modulus) = 1925.33 mm3 M

fb = S = 4.07 N/mm2 fb = 4069.44 kPa

Beam Diagram Section

Load Centroid Lines

200.00

Dead Load

fb= 4069.44 kPa < Fb = 11376.75 kPa

Reaction

The beam is adequate in bending.

Live Load

0.0

SHEAR

Vmax (from shear diagram) = 47.49 N A (cross-sectional area) = 607.81 mm2 fv =

3V = 0.12 N/mm2 2A

fv = 117.19 kPa

fv= 117.19 kPa < Fv = 1310.1 kPa The beam is adequate in shear.

DEFLECTION

440.00 TL = 94.97 N 2611.68 Nmm

2611.68 Nmm

Load (N)

47.48 N

∆= P L E I

= = = =

PL3 192EI

94.97 N 440 mm 12411 MPa 44952.99 mm4

∆ = 0.076 mm

47.48 N

Shear (N)

-47.48 N 2611.68 Nmm

The beam is adequate in deﬂection. Load

Centroid Li

Total Load

2611.68 Nmm

Chair ∆ = 0.076 mm < L/240 = 1.83 mmCentre Line

Moment (Nmm)

-7835.02 Nmm 103

Reaction

ANALYSIS 4: Beams and Columns

Open Leg, Worst Loading Case AXIAL LOAD (COMPRESSION) P (from Analysis 3) = 846.80 N A (cross-sectional area) = 950 mm2 P

fa = A = 0.89 N/mm2 fa = 891.37 kPa

Load Centroid Lines

fa= 891.37 kPa < Fa = 10618.3 kPa

Live Load Reaction

The column is adequate in compres-

Live Load

BUCKLING 200.00

π E k r L

00 00.

414.60

fcr =

= = = = =

3.14 12411 MPa 0.8 9.41 414.6 mm

π2E = 98.54 kPa 2 ( kL ) r

fcr = Pcr = 0.033 N/mm2 A

COLUMN CROSS-SECTION

19.00 Load Centroid Lines Reaction

= = = <

(0.098 N/mm2)(950 mm2) 93.62 N 936.17 Pcr x 10

The actual load is less than 10 times the critical buckling load, so the column probably won’t buckle.

I = 84078.75 mm4

Live Load

50.00

Live Load

Pcr= fcr x A Pcr Pcr x 10 Pactual= 846.80 N

VERTICAL DEFORMATION

A = 950 mm2 r2 = I/A = 84078.75 mm4/950 mm2 r2 = 88.50 mm2 r = √88.50 mm2 r = 9.41 mm

∆ = PL P L E A

= = = =

846.80 N 414.6 mm 12411 MPa 950 mm2

∆ = 0.029 mm

This column is rigidly connected at one end and pinned at the other, so it has a k value of 0.8.

104

ANALYSIS 4: Beams and Columns

Frame Side, Worst Loading Case AXIAL LOAD (COMPRESSION) P (from Analysis 3) = 493.93 N A (cross-sectional area) = 3800 mm2 P

fa = A = 0.13 N/mm2 fa = 129.98 kPa

Load Centroid Lines

fa= 129.98 kPa < Fa = 10618.3 kPa

Live Load

The column is adequate in compres-

Reaction Live Load

BUCKLING

200.00

π E k r L

.0 200

fcr =

= = = = =

3.14 12411 MPa 0.65 4.704 440 mm

π2E = 33.13 kPa 2 ( kL r )

fcr = Pcr = 0.033 N/mm2 A

Pcr= fcr x A Pcr Pcr x 10 Pactual= 493.93 N

COLUMN CROSS-SECTION

19.00

VERTICAL DEFORMATION

A = 3800 mm2

Reaction

200.00

Live Load

r = I/A = 84078.75 mm4/3800 mm2 r2 = 22.13 mm2 2

Live Load

(0.033 N/mm2)(3800 mm2) 125.9 N 1259.10 Pcr x 10

The actual load is less than 10 times the critical buckling load, so the column probably won’t buckle.

I = 84078.75 mm4

Load Centroid Lines

= = = <

r = √22.13 mm2 r = 4.704 mm

∆ = PL P L E A

= = = =

493.93 N 440 mm 12411 MPa 3800 mm2

∆ = 0.0046 mm

This column is rigidly connected at both ends, so it has a k value of 0.65.

105

ANALYSIS 5:

Joints

106

ANALYSIS 5: Joint Analysis

DOVE TAILS

BENDING

21.64

19.00

CROSS-SECTIONAL AREA

12 .92

12.92

‘TAB’ PLAN VIEW

bh2 6

= 12.92(19)2/6

S = 777.35 mm3 The critical dovetail joints are those where the frame top and sides meet. The worst loading case for bending in these joints occurs when the frame top is loaded with the total load in the middle of the beam span.

2 Mmax wL =

M = 4632.89 Nmm S (section modulus) = 777.35 mm3

(2.2973N/mm)(440mm)2 24

Mmax = 18531.60 Nmm

fb = S = 5.95 N/mm2 fb = 5959.84 kPa

Since there are 4 ‘tabs’ in each dovetail joint, the amount of moment each must bear is:

fb= 5959.84 kPa < Fb = 11376.75 kPa The tabs are strong enough to resist failure by bending.

18531.6N · mm = 4632.89N · mm 4

Using this M value, we can check whether the ‘tabs’ will fail in bending at the narrowest part of their width.

107

COMPRESSION

19.00

21.64 b

a 12 .92 ‘TAB’ PLAN VIEW

(a + b)h 2 (12.92mm + 21.64mm)19mm 2

A = 328.32 mm2 We can use the same loading for bending to ﬁnd the worst compression in the dovetail joint. Total Load =

1010.81 N

Because the total load is centrally located, each dovetail joint gets 1/2 the total load, and each ‘tab’ gets 1/4 of that (1/8 of the total load). R = DL 2

R = 505.41 N 505.41N = 126.35N 4

The load on each tab is spread across the area of the tab, creating a compression stress in N/mm2 (MPa). 126.35N = 0.385N/mm2 328.32mm2

fc = 0.385 MPa fc = 384.84 kPa

fc= 384.84 kPa < allowable = 4500 kPa The actual compression on each tab is less than the allowable stress, so the tabs will not fail in compression. 108

ANALYSIS 5: Hinge Analysis

PIANO HINGE

The other crucial joints in the chair are the piano hinges connecting the seats to the frame top. Under worst case loading, they are subject to torque, which puts varying stresses on the screws connecting the hinges to the wood. Beam Section

1000 N 3P

-P

-2P

-3P

pivot

The piano hinge has 7 screws in it; the outermost three exert a downward force on the seat, the innermost three exert an upward force, and the central screw does no work.

When the load acting on the open seat is is not at the seats’ center of gravity, the hinge (more speciﬁcally the screws connecting the hinge to the wood) has to resist this rotational force.

Looking at the downward acting screws, we can solve for the vertical resistance in the critical screw, 3P.

For this system to balance:

402.00 mm a 3P

a 2P

1000 N

(1000 N)(197.32 mm) =

3P(3a) + 2P(2a) + P(a)

(1000 N)(197.32 mm) =

14P(a)

(1000N)(197.32mm) =P 14a (1000N)(197.32mm) =P 14(68.23mm)

197.32 mm

pivot

P =

206.57 N

The vertical resistance in the critical screw is 3P = 619.71 N

(402.00mm − 197.32mm) = 68.23mm 3 109

45° The hinge may fail in one of two ways: 1. Withdrawal of the screw from the wood (parallel to the screws’ length) or, 2. Shearing through the wood (perpendicular to the screws’ length) Therefore, the vertical resistance in the screw has a parallel withdrawal component and a perpendcular shear component.

At this angle, the withdrawal component is 0, so the entire vertical resistance is made up of the shear force component.

45°

The force and component create a 1-1-√2 triangle, so we can use similar triangles to calculate the shear value.

√2

shear

vertical resistance

1 vertical resistance

shear √ 2

shear =

vertical x √2

shear =

619.71 N x √2

shear =

876.40 N

At maximum loading, the critical screw is bearing 876.4 N of shear force. 110

111

CHAPTER 10 Conclusion

113

Right. So, we made a chair. Good. And what did we learn? Well, for starters, making a chair is harder than it looks. Someone should have told us that, but we got it done and it’s pretty solid for a ﬁrst try. How about the actual chair? Here are a few things we found out:

1. The joints are critical. Though our chair is constructed out of simple pieces, the connections between those pieces turned out to be much more important than we initially though, in terms of how strong they’d need to be and how much stress they were taking. As one of our early concerns was about whether the frame would be stable enough, it was a huge relief to realize how sturdy and rigid the dovetail joints were. Not only did they solve the racking issue, they also made the frame top a lot stronger as a beam. Just as crucial was the piano hinge (connecting the seats to the frame top), which turned out to be the component resisting moment in the seats when someone site on the edge. During our starting analyses and models, we really hadn’t considered what exactly was happening in that hinge to counteract the load on the seat, so it wasn’t until the prototype (chapter 5) that the hinge was actually really important. Because the screws in the piano hinge are resisting so much stress from the load on the seat, they are the most vulnerable point in the chair, being most likely to fail by the screws pulling or breaking out of the wood (seat or frame top).

2. This chair is a bit…wibbly-wobbly. We thought, what with it having so many legs, that our chair would be very stable ALL THE TIME! As it turns out, the fact that the seats cantilever almost half their length over the legs makes the chair very easy to tip over when the load centroid is on the cantilever. This necessitated that we outline the safety zone (analysis 2), but anyone who sits on the chair will ﬁgure out pretty quickly where it’s actually possible to comfortably sit.

3. The bottom is a handle. We didn’t even ﬁgure this one out, and it blew our minds. Full credit to Heinz [Koller] for realizing that, when the chair is closed up and ﬂipped upside-down, the bottom of the frame can be used as a handle to pick up and carry the chair like a briefcase. If we were to do this over, we would probably use something more solid in place of the dowels that the legs pivot on, because they kept breaking, and replacing them was very troublesome. That, or we just wouldn’t do this again at all.

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CHAPTER 11 Bibliography

117

Information Sources Chapter 1: Doctor Who Wiki. Accessed September 21, 2013. http://tardis.wikia.com/wiki/Doctor_Who_Wiki

Chapter 8: See See Plywood. “Plywood density” Accessed October 7, 2013. http://www.plywood.cc/2009/03/27/ plywood-density/

Chapter 9: The Engineering Toolbox. “Friction and Coefficients of Friction.” Accessed October 29, 2013. http:// www.engineeringtoolbox.com/friction-coefficients-d_778.html Georgia-Pacific Building Products. “Plywood Design Specification.” Accessed November 3, 2013. https://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&ved=0CF4QFjAF&url=http% 3A%2F%2Fwww.buildgp.com%2FDocumentViewer.aspx%3Frepository%3Dbp%26elementid%3D3809&ei=09qkUoGFI6ag2gX4_4CwCA&usg=AFQjCNEGUEFyguu1J2RWyQMUHl-wlmQ1HQ&sig2=l aB02-556eoTqiSCc6fWyg&bvm=bv.57752919,d.b2I

Images Table of contents: Shirt Design by ‘Zerobriant’. Accessed December 15, 2013. http://shirtoid.com/38990/eleven-doctors/ Chapter title images by ﬂickr user ‘Under The Name’. Accessed December 15, 2013. http://www. ﬂickr.com/photos/underthename/sets/72157625523683528/ 118

Chapter 2: Jordan, G. “Doctor Who 50th Anniversary Airs Today.” Accessed November 23, 2013. http://guardianlv.com/wp-content/uploads/2013/11/The-Doctors-Who.png

Chapter 4: 1 + 4. The Furniture Home. “Drop Leaf Table Types.” Accessed September 20, 2013. http://thefurniturehome.com/1552-table-design-for-small-dining-room/drop-leaf-table-types/ 2 + 5. IKEA. “Norden.” Accessed September 20, 2013. http://www.ikea.com/ca/en/catalog/products/20104718/ 3. The Village Woodworker. “Restoring A Gateleg Table – Part 1.” Accessed September 20, 2013. http://thevillagewoodworker.blogspot.ca/2012/10/restoring-gateleg-table-part-1.html 6. Big Tree Turnings. Photograph. Accessed September 20, 2013. http://www.bigtreeturnings.com/ images/articles/gateleg-table/tn_gateleg.jpg Paul Buckley, Gate-Leg Table. 1975. White Oak. 76.4 x 155.6 x 81.6 cm. Renwick Gallery, Boston. From: The Smithsonian American Art Museum, http://americanart.si.edu/collections/search/ artwork/?id=76496 Villareal, Alberto. “Silla Guarda / InsideOut Collection.” Renderings. Accessed September 20, 2013. http://alberto_villarreal.prosite.com/74406/658997/projects/silla-guarda-insideout-collection Tran, Long. “Chair Inside A Chair.” Photographs. 2011. Accessed September 20, 2013. http://www. yankodesign.com/2011/03/30/chair-inside-a-chair/ Peter Capaldi Eye Shot Appreciation Fanclub. Accessed December 15, 2013. http://petercapaldieyeshotfanclub.tumblr.com/

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