CONSMAT-LEC.pdf

CONSMAT LEC IMPORTANCE OF CONSTRUCTION MATERIALS: Their importance lies in the following aspects: LEC 1_Construction Materials and Testing a) Strength and Durability: Constru...

CONSMAT LEC IMPORTANCE OF CONSTRUCTION MATERIALS: Their importance lies in the following aspects: LEC 1_Construction Materials and Testing a) Strength and Durability: Construction materials provide the structural integrity and longevity Construction materials required for buildings and infrastructure to - are materials or supply brought to the site by the withstand the forces of nature and use over time. Contractor or a subcontractor for incorporation into b) Safety: Proper selection and testing of the building or work. construction materials can ensure the safety of - can be traced to as early as 400BC. Civil Engineers occupants, workers, and the general public deals with the traditional and modern materials. during and after construction. c) Aesthetics: Construction materials can Traditional Construction Materials enhance the aesthetic appeal of buildings and - Are materials that have been used for centuries, structures, and contribute to the cultural and often locally sourced and based on traditional historical significance of a place. knowledge and techniques. - These materials can be natural, such as wood, TYPES OF CONSTRUCTION MATERIALS stone, adobe, cob, thatch, and bamboo, or man- Natural Materials made, such as fired bricks, terracotta tiles, and lime  Wood: A natural and renewable material, wood mortar. is commonly used in construction for its - In many parts of the world, traditional materials are durability, strength, and aesthetic appeal. It is still commonly used for construction, especially in often used for framing, flooring, and finishing, rural areas or for buildings with historical or cultural and can be treated for resistance to fire, insects, significance. and moisture. - These materials have proven to be durable, sustainable, and affordable  Stone: A natural and durable material, stone is over time, and can provide thermal and acoustic used for its aesthetic appeal and resistance to insulation. weather and fire. It is often used for walls, flooring, and decorative features, and can be cut Advance or Modern Construction Materials and polished for a smooth finish. - Modern construction materials refer to the newer materials that have emerged in the construction Man-made Materials industry, which provide innovative solutions for  Concrete: A composite material made of construction challenges, such as strength, durability, cement, aggregates, and water. Concrete is sustainability, and cost-effectiveness. used for its strength, durability, and versatility. It can be used for foundations, walls, floors, and bridges, and can be reinforced with steel to increase its strength.  Steel: A high-strength material, steel is used for PROPERTIES OF CONSTRUCTION its durability, resistance to weather and fire, and MATERIALS versatility in construction. It is often used for A. Mechanical Properties structural framing, roofing, and cladding, and i. Hardness- it is the ability to resist indentation can be recycled for sustainability. ii. Toughness- it is the ability to absorb energy and plastic deformation without fracturing  Brick: A fired clay or concrete block, brick is iii. Ductility- it is the ability to deform in plastic commonly used for its durability, insulation, and range without breaking fire resistance. It is often used for walls, iv. Stiffness- it is the ability to resist deformation chimneys, and decorative features, and can be within linear range made in various sizes and colors. v. Durability- it is the ability to last over time without wearing or significant deterioration  Glass: A transparent material, glass is used for vi. Elasticity- property of a material which makes it its aesthetic appeal, natural light, and energy return to its original dimension when the load is efficiency. It can be used for windows, doors, removed facades, and decorative features, and can be B. Chemical Properties tempered or laminated for safety. i. Corrosion Resistance- it is the ability to withstand corrosion damage caused by  Plastics: A versatile and lightweight material, oxidation plastics are used for their insulation, moisture ii. Fire Resistance- it is the ability to prevent or resistance, and flexibility. They can be used for delays the passage of excessive heat and fire piping, insulation, roofing, and structural iii. Chemical Resistance- it is the ability to elements, and can be made from recycled withstand the action of acids, alkali, seawater, materials for sustainability. and gases.  Asphalt: A petroleum-based material, asphalt is C. Physical Properties used for its durability, waterproofing, and iv. Density- ratio of mass to its volume flexibility. It is often used for paving roads, v. Specific Gravity- ratio of density of a substance parking lots, and driveways, and can be recycled to density of water for sustainability. Composition: Bitumen, vi. Dimension- measurement in one direction of a Aggregate physical object vii. Porosity- ratio of volume of voids to the volume Overall, the selection of construction materials of substance depends on their properties, suitability for specific viii. Void Ratio- ratio of volume of voids to the applications, cost-effectiveness, and environmental volume of solid sustainability. ix. Water Absorption- amount of water absorbed  TESTING OF CONSTRUCTION  Marshal Stability MATERIALS TESTS ON CEMENT Destructive Testing is the evaluation that  Finenes Test determines how a component will behave under  Soundness Test pressure.  Consistency Test Non-Destructive Testing is the evaluation that  Setting Time determines how a component will behave without permanently damaging the object TESTS ON CONCRETE  Compression Strength DESTRUCTIVE TESTING  Flexural Strength  Compression Test - It determines the  Rebound Hammer compressive strength of a specimen under  Ultrasonic Pulse Velocity compressive pressure.  Tensile Test - It determines the tensile strength, STEEL REINFORCEMENT TESTS yield strength and ductility of a metallic material  Tensile Strength by pulling up to its breaking point.  Bend Test  Flexural Test - evaluates the tensile strength of  Re-bend Test concrete indirectly. It tests the ability of  Shear Test unreinforced concrete beam or slab to withstand  Brinell Hardness Test failure in bending. The results of flexural test on  Impact Test concrete expressed as a modulus of rupture  Torsion Test which denotes as (MR) in MPa or psi. STANDARD-SETTING AGENCIES TESTS ON AGGREGATES  ASTM- American Society for Testing and  Sieve Analysis Materials  Water absorption  AASHTO- American Association of State  Aggregate Impact Value Highway and Transportation Officials  Aggregate Abrasion Value  AISC- American Institute of Steel Construction  Aggregate Crushing  ASCE- American Society of Civil Engineers  ACI- American Concrete Institute TESTS ON BITUMEN  ISO- International Organization for  Bitumen Content Standardization  Flash and Fire Point  Penetration  Softening Point  Specific Gravity  Ductility LEC 2_Concrete  Suitable for work underground, in water, or in humid environments. CONCRETE - a mixture of cement, water, aggregates (like sand and gravel) that hardens over  RAPID HARDENING CEMENT time to create a solid, strong substance.  Finer ground  Greater strength development at an early stage than OPC COMPONENTS OF A CONCRETE  Often used in prefabricated road work and ٥ Cement concrete construction ٥ Coarse aggregates ٥ Fine aggregates  EXTRA-RAPID HARDENING CEMENT ٥ Water  Sets and becomes durable faster than OPC ٥ Air and RHC ٥ Admixtures  Adding more calcium chloride to RHC ٥ Supplementary cementitious materials  Often used in repairing, for concrete drives,. and paths and steps.  LOW-HEAT CEMENT  Lower heat of hydration and needs less water to mix  High chemical corrosion resistance and wear and rupture resistance  Used in floors, and surfaces, dams, and large footings. TYPES OF CEMENT (and their uses)  ORDINARY PORTLAND CEMENT  SULFATE RESISTING CEMENT  Simply known as PORTLAND CEMENT or  Very low heat of hydration TYPE 1 CEMENT  Gains strength at a slower rate  Made of silicates of alumina and calcium  Reduces the risk of sulfate attack on carbonate concrete  Used for general construction and most  Used in constructing foundation in soil with masonry work. high sulfate content  PORTLAND POZZOLANA CEMENT  QUICK SETTING CEMENT  Ground Pozzolanic Clinker mixed with OPC  Sets faster than OPC, but maintains the  Most resistive to chemical reactions within same strength concrete   Beneficial for time-sensitive projects, such as underwater structures and in chilly and rainy climates.  BLAST FURNACE SLAG CEMENT  Ground clinker with up to 60% slag  Less expensive, but with many same properties as OPC  Used for projects where cost concerns are crucial TYPES OF CONCRETE SLUMP  TRUE SLUMP  HIGH ALUMINA CEMENT → Concrete just subsides shortly and more or  High compressive strength less maintain the mould shape  More flexible and workable than OPC → Concrete mixture is cohesive and has good  Commonly used in constructions that are workability exposed to high temperatures, such as → Good cement-to-water ratio workshops, refractories, and foundries → Most desirable  WHITE CEMENT  ZERO SLUMP  Prepared from raw materials that don’t → The concrete retains its shape completely include iron oxide → Little to no workability  More expensive than other cement types → Best used in road construction  Often used in interior and exterior decorative → Low water-to-cement ratio projects  SHEAR SLUMP → Top half of the concrete subsides dramatically, CONCRETE SLUMP leaning to one side CONCRETE SLUMP TEST → Has workability, but low cohesion → Measures the consistency of a concrete batch to → Too much water see how easily the concrete will flow. → Ensures that the batches of the same concrete  COLLAPSE SLUMP are of constant quality and strength. → doesn’t retain its shape at all and completely collapses → Water-to-cement ratio is too high → Least desirable  AGGREGATES COARSE AGGREGATE  When the aggregate is sieved through 4.75 mm sieve, these are the aggregate that are retained Aggregate  They are the key component in concrete → are the important constituents of the concrete because they give the largest amount of → 60% - 80% of the concrete mix durability and strength → Selected for their durability, strength, and workability → Classification according to size: 1. Fine aggregate 2. Coarse aggregrate FINE AGGREGATE  when aggregate is sieved through a 4.75 mm sieve, these are the aggregate that passed through  Fill the voids in the coarse aggregate and act as a workability agent TESTS ON CONCRETE Compression Strength Test → Done by breaking cylindrical concrete specimens in a compression-testing machine → Results can be used for quality control, acceptance of concrete, strength estimation, etc. Flexural Strength Test → Evaluates the tensile strength of concrete indirectly → Tests the ability of unreinforced concrete beam or slab to resist failure in bending → Done to specify compliance with standard, as an essential requirement for concrete mix design, etc. Rebound Hammer 7000 BC - evidence of masonry construction in → A non-destructive testing apparatus Jericho, one of the world's oldest inhabited cities → The rebound of the spring-driven mass is where they use sun-dried mud bricks. measured after impact → Output is called rebound number Ancient Egyptian - the pioneers of stone masonry. → Assess the in-place uniformity, delineate poor They build structures like the Great Pyramids of Giza quality regions, and an in-place method to test that symbolize power and permanence. concrete strength Romans - introduced new materials like concrete → Strength and quality is tested through and techniques such as the arch, vault, and dome, measuring the velocity of an ultrasonic pulse enhancing the possibilities of masonry architecture. passing through the concrete → Evaluate the dynamic modulus of elasticity, Middle Ages - brick and stone masonry flourished, estimate depth of cracks, and detect internal particularly in constructing European cathedrals and flaws castles. Modern Times - the industrial revolution brought new materials like concrete blocks, providing new LEC3_Masonry opportunities for masonry. Architects like Frank Lloyd Wright used masonry not just as a structural MASONRY component but as an essential part of their aesthetic philosophy. Masonry continues to evolve, promising Masonry - is a construction technique that involves an exciting journey ahead in the architectural stacking materials, such as bricks, stone blocks or landscape. concrete blocks, on top of one another to build structures or walls. EXAMPLES OF MASONRY PROJECTS Mortar - an adhesive paste that fills the gaps and binds materials together. Masonry can make the structure of buildings safe and durable, but it's important that masonry projects EXAMPLES OF MASONRY STRUCTURES occur on solid, stable ground to ensure the materials remain intact and free of cracks or damage. HISTORY OF MASONRY PROS AND CONS OF MASONRY Pros of Masonry  Physical and Mechanical Properties  Fire protection   Diverse Design CHARACTERISTICS AND TYPES OF MORTARS  Longer Lifespan The chief properties of mortar are strength,  Durability development of good bond with building units,  Comfort resistance to weathering, mobility, placeability, and  Increase in Property Value water retention. In addition, the mortar should be cheap and durable and should not affect the Cons of Masonry durability of building units in contact. The joints made  Longer Construction Period with mortar should not develop cracks.  Limited Space  Specialized construction skills TYPES OF MORTARS  High cost ٥ Cement Mortar  Heavy materials ٥ Lime Mortar  Labor intensive repairs ٥ Lime-Cement Mortar ٥ Surkhi Mortar MORTARS ٥ Mud Mortar Mortars - is a paste (capable of setting and ٥ Special Mortars hardening) obtained by adding water to a mixture of fine aggregates such as sand and binding material TESTING MORTARS (clay, gypsum, lime or cement or their combinations).  Crushing Test → This test is conducted on a brickwork assembly, USES AND CLASSIFICATIONS which consists of several layers of bricks Some of the important uses of mortars are as follows: bonded together using mortar.  In brick and stone masonry, it is used in the → The brickwork assembly is placed between two vertical joints and is spread over each layer to compression plates of a compression testing give bed and a binding medium for successive machine. layers of masonry. → Gradually, a compressive load is applied until  In plastering and pointing, it is used to cover the assembly fails. exposed walls and joints to protect against weathering besides better appearance.  Tensile Strength Test  As matrix in concrete. → In this test, prepared mortar is placed in a rectangular mold with a central cross- sectional Mortars are classified on the basis of their: area of 38mm x 38mm.  Bulk Density → Once the mortar cures, a briquette is pulled  Binding Material under the grips of a tensile testing machine.  Application → The ultimate load at which the briquette fails is e) Glass Block - Glass blocks or glass bricks are recorded. wall bricks made of glass that help divide the room without blocking light transmission.  Adhesive Test → To assess the adhesive strength of mortar, two INSTALLATION PROCEDURE bricks are placed together and held in place with 1. Preparation Phase the mortar. 2. Foundation and Initial Course → The top brick is suspended from an overhead 3. Connection and Bonding support, while a board is hung from the lower 4. Course Laying and Limitations brick. 5. Special Installations and Finishing → Weights are added to the board until the bricks 6. Maintenance and Curing separate TYPES OF MASONRY 1. PREPARATION PHASE a) Brick - a block or a single unit of a kneaded  Site Inspection - Assess the site conditions, clay bearing soil, sand and lime, or concrete soil stability and any potential obstructions. material, fire hardened or air dried, used in masonry construction.  Wet Masonry Units - Wet masonry units before laying and moisten the floor under the walls. b) Concrete Block - are type of building material that is made from a mixture of cement, sand,  Material Procurement - Source quality masonry and gravel. A concrete block is typically used in materials and any necessary tools or equipment. construction to form walls, floors, and  Tools foundations.  Wire level  Plumb bob c) Stone masonry - is a traditional building  Aluminum bar technique that involves the construction of  Rubber hammer structures using carefully shaped and fitted  Brush pieces of natural stone.  Bucket  One-wheel trolley d) Composite Material - is a material which is  Mason spatula produced from two or more constituent materials. These constituent materials have notably  Wall Layout- Lay out walls for accurate spacing dissimilar chemical or physical properties and of surface bond patterns, precise location of are merged to create a material with properties openings, movement-type joints, returns, and unlike the individual elements. offsets. 2. FOUNDATION AND INITIAL COURSE 5. SPECIAL INSTALLATIONS AND FINISHING  Starting Course - Start with a full mortar bed on  Filling Cores - Fill cores in hollow masonry footings, ensuring coverage under cells. units with mortar.  Lintel Beams - Cast lintel beams in site with a  Masonry Unit Placement - Lay masonry units minimum depth of 200mm for all openings in with proper mortar thickness and maintain joint masonry assemblies, ensuring a 200mm thickness between 10mm and 12mm. minimum bearing at each jamb.  Mortar in Joints - Install mortar and fill gaps 3. CONNECTION AND BONDING between partitions and soffits, and between  Connection Between Walls - Use wall angles hollow metal frames and masonry. for bonding walls and partitions. Employ toothing  Cleaning - Clean unit masonry as work for partition connection to adjacent walls. progresses.  Work Inspection Request - Request inspection 6. MAINTENANCE AND CURING for coordination before laying the first course.  Replacement - Remove and replace loose or damaged masonry units. 4. COURSE LAYING AND LIMITATIONS  Curing - cure masonry for three days by  Course Laying - Lay courses to comply with sprinkling water twice daily. specified construction tolerances, ensuring accurate spacing and coordination with other construction. LEC4_Metals  Cutting and Shaping - When necessary, cut WROUGHT IRONS masonry units to size using tools such as a → A tough form of iron suitable for forging and hammer and chisel, masonry saw, or specialized rolling rather than casting cutting equipment. → Malleable (able to be hammered or pressed permanently without breaking)  Stopping and Resuming Work - When stopping and resuming work in each course, WHERE ARE WROUGHT IRONS ARE USED FOR? rack back one-half running bond or one-third unit  Pipe length for one-third running bond. Clean  Bars and railings exposed surfaces of set masonry and remove  Iron doors, gates, fences loose units and mortar before laying fresh  Plates masonry.  Special chains and crane hooks WROUGHT IRON PROPERTIES THE ORE EXTRACTION AND REFINING  weldable at high temperatures and ductile at low PROCESS temperatures 1. Extraction – iron ores are extracted from open pit  good forming qualities mining  resistance to corrosion 2. Refining – mined ores are crushed to remove  poor strength sand and clay away, then refined ores are  high melting point transported to blast Furnace site. 3. Manufacturing – ores goes to the blast furnace; CAST IRONS carbon monoxide reacts with the iron ore to form → A group of iron-carbon alloys, with carbon carbon dioxide and pure iron. Melted iron sinks to the content greater than 2% bottom of the furnace. → Got its name because of its excellent casting qualities Mining → Iron ore is generally extracted through open pit WHERE ARE CAST IRONS USED FOR? mining. The ground is removed from a very large  Machines area to expose the one beneath.  Automotive parts → The mined one is crushed and sorted by grade  Pots and utensils according to the percentage of iron.  Anchor for ships → The refined one is taken from the mining site ti the blast furnace where the iron uis melted. CAST IRON PROPERTIES  hard and brittle Processing  low cost → The ore enters the blast furnace from the top as  resistance to deformation hot air is blown from the bottom. Pure iron forms  prone to rusting and sinks to the bottom.  high weight to strength ratio → The melted iron is then mold cast into ingots called pigs IRON & PURE IRON STEEL → Iron is one of the most common elements of the  Steel earth → basically an alloy of iron and carbon with a → Pure iron is a soft, grayish – white metal small percentage of other metals such as → Iron is a common element, but pure iron is nickel, aluminum, tungsten etc. almost never found in nature, it only exist on → Steel may not be strong as iron but it is far fallen meteorites more resistant and does not corrode and rust like iron does. ADVANTAGES OF STEEL PROPERTIES OF FERROUS METALS  High Strength Ferrous Metals  Uniformity - are highly vulnerable to rusting and corrosion when  Elasticity exposed to moisture or an acidic or corrosive  Permanence environment due to high content of carbon molecules.  Ductility - MOST ferrous metals have good magnetic  Toughness properties and are considered to be good conductors of electricity. This property makes them suitable for METHODS FOR MAKING STEEL electronics.  OPEN-HEARTH METHOD → the specialty of open hearth is the extreme NON FERROUS METALS heat that can be obtained from them due to Metals which do NOT contain Iron their regenerative process Aluminum Lead  BASIC OXYGEN METHOD Zinc 1. Blowing of oxygen Copper 2. Remove sulfur, phosphorus and silicon Bronze 3. Heat process Brass 4. Composition of molten steel 5. Alloying to provide steel properties. Aluminum → Highly resistant to weather and corrosive  ELECTRONIC FURNACE METHOD environments → Aluminum can be economically extruded to many shapes (mouldings, edgings, window mullions) → It is very malleable, quite ductile, non-corrosive, and strong in proportion to its weight. Lead FERROUS METALS → Important physical properties include: resistance Metals that composed of IRONS to corrosion, its plasticity, and its malleability Alloy Steel → Used for waterproofing, sound and vibration Carbon Steel isolation, and radiation shield. Cast Iron → Can be combined with a tin alloy to plate iron or Wrought Iron steel → (called “terneplate”) → Use extreme care where and how lead is used b) American Standard (S) because lead vapors or dust are toxic if ingested.  They have a Sloped flange and are most often used in residential construction Zinc → Is brittle and low in strength c) HP-Section (HP) → Major use is in galvanizing (dipping hot iron or  These are generally heavier and longer than I- steel in molten zinc) beams. Appearing like the capital letter ‘H’, they → May also be used for roofing, flashing, and have longer flanges. hardware d) Channel (C) Copper  Like one half of the S-Section, these have a C- → Resistant to corrosion, impact, and fatigue; very shaped cross-section. They are used in ductile supporting lighter loads since they are not as → Primary use is electrical wiring, roofing, flashing, strong as the other beam shapes. and piping → The oxidization of copper produces what is e) Angle called a green “patina”  Angle Steel is 'L' shaped; the most common type of Steel Angles are at a 90 degree angle. The Bronze legs of the “L” can be equal or unequal in length. → Originally a copper-tin alloy, but now aluminum or silicon added to copper f) Tee (WT or ST) → Now may be “phosphor bronze”, “aluminum  WT shapes are cut from a wide flange. ST bronze” or “silicon bronze” shapes are cut from American Standard Beams → Widely used for casting delicate mold impressions (Cathedral doors) TESTING OF METALS Two categories: Brass 1. NON-DESTRUCTIVE → Copper with zinc to form an alloy 2. DESTRUCTIVE → Used for doors, windows, railings, trim, grilles and for finish hardware NON-DESTRUCTIVE - Test performed without breaking the sample STEEL FRAME CONSTRUCTION a) Wide-Flange (W) HARDNESS TESTING  They are commonly used in industrial  used to determine hardness of metal applications. → capacity to resist wear and deformation  They have a straight flange and are most  can be used to predict properties and often used in residential construction performance of the metal Two types of testing process: Vickers Hardness Test 1. Measure depth of penetration → Based on an optical measurement system  Rockwell hardness test → Specifies a range of light loads using diamond  Brinell Hardness test indenter  Vickers Hardness test → Used for very thin material  Meyer Hardness test → Typical Loads : 10 g to 1 kg 2. Measure height of rebound  Scleroscope Hardness test Scleroscope Hardness Test → The Scleroscope test consists of dropping a diamond tipped hammer, which falls inside a NON-DESTRUCTIVE TESTING glass tube under the force of its own weight from Rockwell Hardness Test a fixed height, onto the test specimen. The → Indicates hardness value by depth that the height of the rebound travel of the hammer is penetrator advances into metal under known measured on a graduated scale. pressure ○ Height of rebound is converted to hardness → Designed by various letters and numbers (A, B, reading C, D) Brinell Hardness Test → Indenting the metal surface with a 10-mm DESTRUCTIVE diameter steel ball at a load of 3000kg for 30 - Test performed by breaking the material to seconds and the impression is measure by a determine the quality of material low-power microscope Standard load of 500kg is TENSILE TEST used for nonferrous metal. IMPACT TEST → Brinell Hardness Number (BHN) is expressed as a. Charpy Test the load P divided by a surface area of the b. Izod Test indentation. TORQUE TEST Meyer Hardness Test DESTRUCTIVE TESTING → Less sensitive to the applied load than that of Tensile Test Brinell Hardness → Maximum amount of pull a material can → Cold-worked material: relatively constant and withstand independent of load → Indicates elastic limit, yield point, percent of area → Annealed materials: increases continuously with reduction and percentage elongation load because of strain → Tensile Strength - maximum load divided by the original cross-sectional area → Tensile Strength of metal increases as hardness  increases, and the ductility decreases as Impact Test hardness increases → Measures toughness of metal or ability to withstand sudden shock or impact → TWO TEST: a. Charpy Test b. Izod Test I. 10-mm-square specimen II. Swinging Pendulum of fixed mass raised to standard height ٥ Proportional Limit - Point the needle stop moving ٥ Yield Point - Beyond PL and start to stretch Charpy Test ٥ Necking - Reduction in diameter ٥ Specimen mounted in fixture and supported ٥ Ultimate Strength - Highest travel of at both ends with V or notch placed on the needle/Maximum pull before breaking side opposite direction of pendulum’s swing ٥ Breaking Stress ٥ Pendulum strikes, knife edge strikes sample ٥ Difference in height of pendulum at beginning and end indicates amount of energy used to fracture specimen Izod Test ٥ One end of specimen is gripped in clamp with notched side toward the direction of pendulums swing ٥ Amount of energy required to break SAMPLE PROBLEM specimen on scale A steel alloy bar 100mm long with a rectangular cross Torsion Test sectional area of 10mm x 50mm is subjected to → Determine the behaviour of a material exhibits tension with a load of 200kN and experiences an when twisted or under torsional forces as a increase in length of 0.10mm. If the length is entirely result of applied moments that cause shear elastic, calculate the modulus of elasticity of the steel stresses about the axis alloy. → Indicates torsional shear stress, maximum torque, shear modulus and the angle of twist SAMPLE PROBLEM A rod with a length of 1 m and a radius of 20 mm is made of high-strength steel. The rod is subjected to a torque T, which produces a shear stress below the proportional limit. If the cross section at one end is rotated 45 degrees in relation to the other end, and the shear modulus G of the material is 90 GPa, what is the amount of applied torque?