# five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture

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1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture five mechanics of materials Mechanics of Materials 1

2 Mechanics of Materials MECHANICS MATERIALS Mechanics of Materials 2

3 Mechanics of Materials external loads and their effect on deformable bodies use it to answer question if structure meets requirements of stability and equilibrium strength and stiffness other principle building requirements economy, functionality and aesthetics Mechanics of Materials 3

4 Knowledge Required material properties member cross sections ability of a material to resist breaking structural elements that resist excessive deflection deformation Mechanics of Materials 4

5 Problem Solving 1. STATICS: equilibrium of external forces, internal forces, stresses 2. GEOMETRY: cross section properties, deformations and conditions of geometric fit, strains 3. MATERIAL PROPERTIES: stress-strain relationship for each material obtained from testing Mechanics of Materials 5

6 Stress stress is a term for the intensity of a force, like a pressure internal or applied force per unit area P stress f A Mechanics of Materials 6

7 Design materials have a critical stress value where they could break or yield ultimate stress yield stress compressive stress fatigue strength (creep & temperature) acceptance vs. failure Mechanics of Materials 7

8 Design (cont) we d like f actual F allowable stress distribution may vary: average uniform distribution exists IF the member is loaded axially (concentric) Mechanics of Materials 8

9 Scale Effect model scale material weights by volume, small section areas structural scale much more material weight, bigger section areas scale for strength is not proportional: 3 L L 2 L Mechanics of Materials 9

10 Normal Stress (direct) normal stress is normal to the cross section stressed area is perpendicular to the load f t or c P A Mechanics of Materials 10

11 Shear Stress stress parallel to a surface P P f v ave A td Mechanics of Materials 11

12 Bearing Stress stress on a surface by contact in compression P P f p A td Mechanics of Materials 12

13 Bending Stress normal stress caused by bending Mc M f b I S Mechanics of Materials 13

14 Torsional Stress shear stress caused by twisting f v T J Mechanics of Materials 14

15 Structures and Shear what structural elements see shear? beams bolts splices slabs footings walls wind seismic loads connections V Mechanics of Materials 15

16 Bolts connected members in tension cause shear stress connected members in compression cause bearing stress Mechanics of Materials 16

17 Single Shear seen when 2 members are connected P P Mechanics of Materials 17 f v A d 2 4

18 Double Shear seen when 3 members are connected two areas F= f v P P P 2 2 2A A 2 d 4 Mechanics of Materials 18

19 Bolt Bearing Stress compression & contact projected area F= P P f p A projected td Mechanics of Materials 19

20 Strain materials deform axially loaded materials change length bending materials deflect STRAIN: change in length over length + UNITLESS Mechanics of Materials 20 strain L L

21 Shearing Strain deformations with shear parallelogram change in angles stress: strain: unitless (radians) s L s L tan Mechanics of Materials 21

22 Shearing Strain deformations with torsion twist change in angle of line stress: strain: unitless (radians) L Mechanics of Materials 22

23 Load and Deformation for stress, need P & A for strain, need & L how? TEST with load and measure plot P/A vs. Mechanics of Materials 23

24 Material Behavior every material has its own response 10,000 psi L = 10 in Douglas Fir vs. steel? Mechanics of Materials 24

25 Behavior Types ductile - necking true stress f engineering stress (simplified) f P A P A o Mechanics of Materials 25

26 Behavior Types brittle semi-brittle Mechanics of Materials 26

27 Stress to Strain important to us in - diagrams: straight section LINEAR-ELASTIC recovers shape (no permanent deformation) f Mechanics of Materials 27

28 Hooke s Law straight line has constant slope Hooke s Law f f E E E Modulus of elasticity Young s modulus units just like stress 1 Mechanics of Materials 28

29 Stiffness ability to resist strain f u steels same E different yield points different ultimate strength Mechanics of Materials 29

30 Isotropy & Anisotropy ISOTROPIC materials with E same at any direction of loading ex. steel ANISOTROPIC materials with different E at any direction of loading ex. wood is orthotropic Mechanics of Materials 30

31 Elastic, Plastic, Fatigue elastic springs back plastic has permanent deformation fatigue caused by reversed loading cycles Mechanics of Materials 31

32 Plastic Behavior ductile at yield stress Mechanics of Materials 32

33 Lateral Strain or what happens to the cross section with axial stress x f f y z f x E 0 strain in lateral direction negative equal for isometric materials y z Mechanics of Materials 33

34 Poisson s Ratio constant relationship between longitudinal strain and lateral strain lateral strain axial strain f x y z E y x z x sign! Mechanics of Materials 34

35 Calculating Strain from Hooke s law substitute f E P A E L get PL AE Mechanics of Materials 35

36 Orthotropic Materials non-isometric directional values of E and ex: plywood laminates polymer composites Mechanics of Materials 36

37 Stress Concentrations why we use f ave increase in stress at changes in geometry sharp notches holes corners Mechanics of Materials 37

38 Maximum Stresses if we need to know where max f and f v happen: F P 0 cos 1 f max A o 45 cos sin 0.5 Mechanics of Materials 38 f vmax P 2A o f 2 max

39 Maximum Stresses Mechanics of Materials 39

40 Deformation Relationships physical movement axially (same or zero) rotations from axial changes 20 kn aluminum steel relates to P PL AE Mechanics of Materials 40

41 Deformations from Temperature atomic chemistry reacts to changes in energy solid materials can contract with decrease in temperature can expand with increase in temperature linear change can be measured per degree Mechanics of Materials 41

42 Thermal Deformation - the rate of strain per degree UNITS :, F C length change: thermal strain: T T T T L no stress when movement allowed Mechanics of Materials 42

43 Coefficients of Thermal Expansion Material Mechanics of Materials 43 Coefficients () [in./in./f] Wood 3.0 x 10-6 Glass 4.4 x 10-6 Concrete 6.0 x 10-6 Cast Iron 6.1 x 10-6 Steel 6.5 x 10-6 Wrought Iron 6.7 x 10-6 Copper 9.3 x 10-6 Bronze 10.0 x 10-6 Brass 10.4 x 10-6 Aluminum 12.8 x 10-6

44 Stresses and Thermal Strains if thermal movement is restrained stresses are induced 1. bar pushes on supports 2. support pushes back 3. reaction causes internal stress P f E A L Mechanics of Materials 44

45 Superposition Method can remove a support to make it look determinant replace the support with a reaction enforce the geometry constraint Mechanics of Materials 45

46 Superposition Method total length change restrained to zero p sub: f PL AE PL AE constraint: P A P T T L 0 T T T E L 0 Mechanics of Materials 46

47 Design of Members beyond allowable stress... materials aren t uniform 100% of the time ultimate strength or capacity to failure may be different and some strengths hard to test for RISK & UNCERTAINTY f u P u A Mechanics of Materials 47

48 Factor of Safety accommodate uncertainty with a safety factor: allowable load ultimate load F. S with linear relation between load and stress: F. S ultimate load allowable load ultimate stress allowable stress Mechanics of Materials 48

49 Load and Resistance Factor Design loads on structures are not constant can be more influential on failure happen more or less often UNCERTAINTY R u D R D L R L R n - resistance factor - load factor for (D)ead & (L)ive load Mechanics of Materials 49

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