# Strength of Materials. Robert Hooke. Hooke's Law Young's Modulus Stress & Strain Deformation Thermal Effects Analysis ASD vs. LRFD Modes of Failure

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1 Architecture 314 Structures I Strength of Materials Hooke's Law Young's Modulus Stress & Strain Deformation Thermal Effects Analysis ASD vs. LRFD Modes of Failure University of Michigan, TCAUP Structures I Slide 1/25 Robert Hooke Hooke, referred to as the Leonardo da Vinci of England, was a prolific engineer, architect and polymath. Barometer Microscope (Micrographia) Pocket watch Universal joint Surveyed London (after fire) Wren s engineer (St Paul s dome) Law of Springs (Hooke s Law) Optics Astronomy (gravity of bodies) Curator of experiments of the Royal Society Portrait by Rita Greer, 2009 University of Michigan, TCAUP Structures I Slide 2/25

2 Hooke's Law Ut tensio sic vis The power of any Spring is in the same proportion with the Tension 1 thereof: That is, if one power stretch or bend it one space, two will bend it two, three will bend it three, and so forward. And this is the Rule or Law of Nature, upon which all manner of Restituent or Springing motion doth proceed. Robert Hooke, De Potentia Restitutiva, 1678 With Cauchy's development of the concept of stress in 1822, Hooke s Law could be rewritten as: Strain is Proportional to Stress 1 The Seventeenth Century meaning of Tension is like the Latin, tensio or our modern word, extension or deformation. University of Michigan, TCAUP Structures I Slide 3/25 Young's Modulus material stiffness Young's Modulus, or the Modulus of Elasticity, is the material constant which generalizes Hooke s Law for any size member. It is obtained by dividing the stress by the strain present in the material. (Thomas Young, 1807) It thus represents a measure of the stiffness of the material. Thomas Young Physics - Physiology - Egyptology University of Michigan, TCAUP Structures I Slide 4/25

3 Young's Modulus Young's Modulus or the Modulus of Elasticity, is obtained by dividing the stress by the strain present in the material. (Thomas Young, 1807) When graphing stress vs strain, the slope is the stiffness of the material. E = 990 ksi E = ksi E = 3600 ksi University of Michigan, TCAUP Structures I Slide 5/25 Stress Stress is the result of some force being applied to an area of some material. Shear Stress University of Michigan, TCAUP Structures I Slide 6/25

4 Strain Strain is the amount of deformation in the material, per unit length. Deformation occurs either in stretching (tension) or in compressing (compression) but not always at the same rate. University of Michigan, TCAUP Structures I Slide 7/25 Deformation Using the stress and the Modulus of Elasticity, the total deformation of an axially loaded member can be determined. Deformation Equation University of Michigan, TCAUP Structures I Slide 8/25

5 Stiffness Deformation = Force x Stiffness Axial Matrix formulation Flexure (constant moment) University of Michigan, TCAUP Structures I Slide 9/25 Strain Calculations The amount of strain deformation is proportional to stress Cable supported span of 866 ft Jack height of 118 ft Cable length 895 ft Neckar Viaduct at Weitingen Engineer Fritz Leonhardt Completed 1978 Span 2952 ft Height 410 ft University of Michigan, TCAUP Structures I Slide 10/25

6 Strain Calculations The amount of strain deformation is proportional to stress University of Michigan, TCAUP Structures I Slide 11/25 Types of Stress Compression Tension Flexure Shear Torsion University of Michigan, TCAUP Structures I Slide 12/25

7 Stress Calculations Find the stress in each material Compression The stress equals the force spread over an area. University of Michigan, TCAUP Structures I Slide 13/25 Stress Calculations FBD reactions University of Michigan, TCAUP Structures I Slide 14/25

8 Stress Calculations The stress equals the force spread over an area. University of Michigan, TCAUP Structures I Slide 15/25 Stress Calculations The stress equals the force spread over an area. University of Michigan, TCAUP Structures I Slide 16/25

9 Stress Calculations The stress equals the force spread over an area. University of Michigan, TCAUP Structures I Slide 17/25 Stress Calculations Shear The stress equals the force spread over an area. University of Michigan, TCAUP Structures I Slide 18/25

10 Stress Calculations Tension The stress equals the force spread over an area. University of Michigan, TCAUP Structures I Slide 19/25 Stress Analysis Allowable Stress Design (ASD) use design loads (no F.S. on loads) reduce stress by a Factor of Safety F.S. Load & Resistance Factored Design (LRFD) Use loads with safety factor Use factor on ultimate strength University of Michigan, TCAUP Structures I Slide 20/25

11 Modes of Failure Strength Tension rupture Compression crushing Stability Column buckling Beam lateral torsional buckling Serviceability Beam deflection Building story drift cracking University of Michigan, TCAUP Structures I Slide 21/25 Thermal Induced Stress The amount of expansion with rising temperature or contraction with falling temperature is described by the coefficient of thermal expansion. If deformation is restrained, the result will be a thermal induced stress in the member. The build-up of thermal stress is often prevented by expansion joints. University of Michigan, TCAUP Structures I Slide 22/25

12 Thermal Induced Stress The amount of expansion with rising temperature or contraction with falling temperature is described by the coefficient of thermal expansion. If deformation is restrained, the result will be a thermal induced stress in the member. The build-up of thermal stress is often prevented by expansion joints. University of Michigan, TCAUP Structures I Slide 23/25 Thermal Induced Deformation Thermal deformation, which results in cracking, is controlled with expansion joints. Crack due to thermal stress Expansion joint in wall University of Michigan, TCAUP Structures I Slide 24/25

13 Thermal Induced Deformation The amount of expansion with rising temperature or contraction with falling temperature is described by the coefficient of thermal expansion. University of Michigan, TCAUP Structures I Slide 25/25

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Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity