University of Michigan, TCAUP Structures I Slide 1/25

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

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Robert Hooke1635 - 1703

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

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Hooke's LawUt tensio sic vis

The power of any Spring is in the same proportion with the Tension1 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.

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Young's Modulusmaterial 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 Young1773 – 1829

Physics - Physiology - Egyptology

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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 = 29000 ksiE = 3600 ksi

E = 990 ksi

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Stress

Stress is the result of some force being applied to an area of some material.

Shear Stress

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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.

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Deformation

Using the stress and the Modulus of Elasticity, the total deformation of an axially loaded member can be determined.

Deformation Equation

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Stiffness

Deformation = Force x Stiffness

Axial

Flexure (constant moment)

Matrix formulation

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Strain Calculations

The amount of strain deformation is proportional to stress

Neckar Viaduct at WeitingenEngineer Fritz Leonhardt

Completed 1978Span 2952 ftHeight 410 ft

Cable supported span of 866 ftJack height of 118 ftCable length 895 ft

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Strain Calculations

The amount of strain deformation is proportional to stress

Types of Stress

• Compression

• Tension

• Flexure

• Shear

• Torsion

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Stress Calculations

Find the stress in each material

Compression

The stress equals the force spread over an area.

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Stress Calculations

FBD – reactions

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Stress Calculations

The stress equals the force spread over an area.

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Stress Calculations

The stress equals the force spread over an area.

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Stress Calculations

The stress equals the force spread over an area.

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Stress Calculations

Shear

The stress equals the force spread over an area.

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Stress Calculations

Tension

The stress equals the force spread over an area.

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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

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Modes of Failure

Strength• Tension rupture• Compression crushing

Stability• Column buckling• Beam lateral torsional buckling

Serviceability• Beam deflection• Building story drift• cracking

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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.

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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.

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Thermal Induced Deformation

Thermal deformation, which results in cracking, is controlled with expansion joints.

Crack due to thermal stress

Expansion joint in wall

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Thermal Induced Deformation

The amount of expansion with rising temperature or contraction with falling temperature is described by the coefficient of thermal expansion.