Progression of Pyramid DesignStep PyramidBent PyramidKhufu's Pyramid: 147m in height, 51.90.Built in stages,. Poor foundation,subsidenceexpanded later: Granite beamsInward leaningblocks.Internal structural: Facing stones fittedcollapse·Graniteroofbeamwith a tolerance of less· Below ground chambers · Reduced angle by 10°than 1 mm: Stress relieving: Corbelled ceilingschambers6
Step Pyramid 6 Progression of Pyramid Design • Built in stages, expanded later • Inward leaning blocks • Granite roof beam • Below ground chambers Bent Pyramid • Poor foundation, subsidence • Internal structural collapse • Reduced angle by 10° • Corbelled ceilings • 147 m in height, 51.90 , • Granite beams • Facing stones fitted with a tolerance of less than 1 mm • Stress relieving chambers Khufu’s Pyramid
Lessons Learned by Trial and Error? Site must be solid rock (no shifting sand): Foundation must be strong enough to support thestructure above it, but not be crushed or dislodged: Precise Measurements: to maintain shape and verticalplumb: Masonry Courses: placed horizontally to distribute loads. Strength of Materials Knowledge: Limestone vs. Granite> Limestone: readily available, soft - easy to work with, cut byhand, limited strength (< 3 m beams)> Granite: used for large beams in interior chambers because ofits superior strength7
Lessons Learned by Trial and Error • Site must be solid rock (no shifting sand) • Foundation must be strong enough to support the structure above it, but not be crushed or dislodged. • Precise Measurements: to maintain shape and vertical plumb • Masonry Courses: placed horizontally to distribute loads • Strength of Materials Knowledge: Limestone vs. Granite Limestone: readily available, soft – easy to work with, cut by hand, limited strength (< 3 m beams) Granite: used for large beams in interior chambers because of its superior strength 7
Archimedes (287-212 BC)·唯一与现代科学相通的天才. The Law ofLever: “Magnitudes are in equilibriumat distances reciprocally proportional to theirweights." Said to have discovered Archimedes'principle"while bathing and subsequently ran naked throughthe streets of Syracuse shouting ‘Eureka!"EUREKAI8
8 • 唯一与现代科学相通的天才 • The Law of Lever: “Magnitudes are in equilibrium at distances reciprocally proportional to their weights.” • Said to have discovered “Archimedes’ principle” while bathing and subsequently ran naked through the streets of Syracuse shouting “Eureka!”. Archimedes (287-212 BC)
Archimedes (287-212 BC): Most known for geometric proofs andmay have invented calculus: Invented the pulley, screw, and catapult;definedthe law oflevers. Outlined methods for finding the center ofArchimedes portrait ingravity of bodies.Fields Medal0SunMirrotArchimedesHeatMirrorRayMirrorArchimedes'Heat Ray“On the Sphere &Cylinder""Archimedes'Screw
Archimedes (287-212 BC) “On the Sphere & Cylinder” Archimedes’ Screw Archimedes’ Heat Ray Archimedes portrait in Fields Medal • Most known for geometric proofs and may have invented calculus. • Invented the pulley, screw, and catapult; defined the law of levers. • Outlined methods for finding the center of gravity of bodies
Roman Empire. Archimedes killed by Romansoldiers while doing research. The Romans did not understandstrength of materials, so theybuilt semi-circular arches ofsmall span.· E is “Flexural Rigidity":Romans used large E and I.. Modern materials: when E islarge I does not have to be large10
Roman Empire 10 • Archimedes killed by Roman soldiers while doing research. • The Romans did not understand strength of materials, so they built semi-circular arches of small span. • EI is “Flexural Rigidity”: Romans used large E and I. • Modern materials: when E is large I does not have to be large