Structures, Fluid Mechanics, Materials, and Soil Mechanics - Math Problem Example

Structures, Fluid Mechanics, Materials, and Soil Mechanics QUESTION 1 First determine whether the structure is statically determinate or indeterminate using 2n= m + r, where n= joints, m= members, r=reactions In this case, n=11, m=19, r=3 thus 2x11=22=19+3 therefore the structure is statically determinate Let the length of each of the 5 bays be x, Determine the reactions at the supports = (1+1+1+1+1)/2 = 2.5 units, simply because the truss is symmetrical about the apex. 1 F 1 1 1 D H 1 C J A B E G I K Determination of the member forces using the method of joints Forces in question are a, b, c which according to the diagram are DE, BE and CD respectively since the truss is symmetrical Joint A C A B 2.5 units Sum of forces in the Y-axis = 0 2.5+ACsinØ=0 where Ø=the slope of the roof given by 2 vertical, 5 horizontal. AC= -2.5/sinØ Sum of forces in the x-axis=o AB+CcosØ=0, AB=-2.5/sinØXcosØ=2.5/tanØ Joint C 1 D C A B Sum of forces in the y-axis=0 -CA sinØ+ CD sinØ-1- CB cosØ=0 Sum of forces in the x-axis=o -CA cosØ+C D cosØ+ CBsinØ=0 Solving the equations we find CD &CB Joint B D C A B E Sum of forces in the y-axis=0 BC cosØ+ BD cosØ =0 Sum of forces in the x-axis=o -BA + BC sinØ + BE +BD sinØ =0 Solving we get BD and BE Joint D 1 F D C E B Sum of forces in the y-axis=0 -DC cosØ+D F cosØ- DBcos(90-Ø)-DE cos Ø -1=0 Sum of forces in the x-axis=o -DC sinØ+ DF sinØ- DBsin(90-Ø)-DE sin Ø =0 Solving we get DF QUESTION 2 Introduction Prestressing is normally accomplished in 3 ways which are: pre-tensioned concrete, followed by bonded and the unbonded and post-tensioned concrete. Pre-tensioned concrete is then cast all around the already tensioned tendons. Using this method results in a good bond in between the tendon and the concrete that both safeguards the tendon from sources of corrosion as well as allowing for direct transmission of tension. The pickled concrete sticks to and also ties itself with the bars. When the pressure is unconstrained it is relocated to the concrete by way of compression using static friction. Nonetheless, it needs stout fastening points amid which the tendon is to be stressed and the tendons are normally in a straight line. Accordingly, most pretensioned concrete essentials are assembled in a workshop and have to be transported to the building site, which confines their size. Pre-tensioned essentials may be balcony components, floor slabs, beams, lintels, or foundation piles. An inventive bridge building method by means of pre-stressing is the strained ribbon bridge design. The method I would use to prestress: I would use bonded post –tensioning method Fused post-tensioned concrete refers to the descriptive term for a process of applying firmness after pouring the concrete as well as the curing process (in situ). The concrete is placed all around a plastic, aluminum, or steel curved duct, meant to follow the region where otherwise pressure would arise in the concrete component. A collection of the tendons are fished via the duct and the concrete is drizzled. Once the concrete becomes hardened, the tendons are then tensioned by the hydraulic jacks which react (push) on the concrete associate itself. When the tendons are all stretched adequately, according to the specifications of the design (see Hooke's law), they are then lodged in position and sustain pressure after the jacks are detached, which transfers the pressure towards the concrete. The channel is then mortared to shield the tendons from decay. This method is normally used to build monolithic blocks for house construction; this normally happens in places where thr expansive soils (like adobe clay) generate difficulties for the typical foundation of the perimeter. All strains from periodic expansion as well as contraction of the core soil are reserved into the whole tensioned slab that supports the building devoid of substantial flexure. Post-tensioning is similarly used in the building of several bridges, both following the curing of concrete and after support through false work as well as by the gathering of prefabricated segments, as in the segmental bridge. Among the benefits of this system over the unbonded post-tensioning method are: The large reduction in the traditional reinforcement necessities as tendons are unable to distress in accidents. The tendons can be simply "woven" to allow a more effective design methodology. The higher definitive strength owing to bond created between the thread and the concrete. No long term problems with retaining the reliability of the anchor/dead end. This concept uses long precast concrete beams between supports and usually chorded sections say 6m are used to approximate curved geometry. Diaphragms are provided at angle points between these chorded sections. The chord length thus produces approximately 50mm offset on 92m radius curve. The beams are chorded in plan and in profile. Individual precast beams are post-tensioned together in the field to form a continuous structure. Trapezoidal box beams are used to produce a torsional rigid section that is aesthetically appealing to the eye. Post tensioning tendons are placed inside the beam void and are deflected horizontally and vertically at diaphragms between chorded sections. The tendons therefore form a string polygon that approximates a parabolic shape in profile and the curve radius in plan. Tendons are bonded to the cross section at each diaphragm but are not continually bonded along the length of the tendon Procedure 1. Step 1 (Fig. 5): Beams are fabricated full length in the plant in specially designed formwork. Beams are cast in two stages. Stage 1 includes the soffit and webs of the chorded sections, end diaphragms, and diaphragms between chorded segments. Ducts are provided by plant post-tensioning tendons and for Stage 1 and Stage 2 field post - tensioning tendons. The beam deck is cast in Stage 2. Beam casting is complete prior to remov­ing the beam from the form, Beams are lifted out of the form and transported to a yard storage/stressing area as reinforced concrete members. Plant post-tensioning tendons are stressed. 2. Step 2 (Fig. 6): Beams are transport­ed to the site and erected. Ducts for Stage I and Stage 2 field post-tensioning ten­dons are spliced over interior supports. Closure pours are made between beams over interior supports. Stage I tendons are stressed, creating continuous beams. 3. Step 3 (Fig. 7): Cross beams are cast at the midpoint or at the third points along the span at the nearest diaphragm loca­tions. The bridge deck is cast. Stage 2 ten­dons are stressed, placing the deck into compression. Traffic barriers, overlays, and expansion joints are placed, complet­ing the bridge construction. This horizontally curved prestressed precast beam concept was selected over the other concepts because it generally: 1. Improved quality 2. Reduced costs 3. Improved aesthetics Quality was enhanced using a two stage casting with removable inner forms for Stage 1. Inner surfaces and thicknesses of the beam soffit and webs can be inspect­ed and positioning of post-tensioning ten­dons can be carefully established and ver­ified. Labor costs to produce full length beams are reduced by minimizing fabrica­tion steps. Also, sloping sides delete the requirement to move back beam side forms to lift beams from the form. Material costs are reduced by eliminating costly inner void forms. Aesthetics are improved by utilizing sloping beam sides in lieu of vertical sides. Factors considered Span of the bridge that is, the bridge has along span End product that is, it’s easier to achieve the curved profile when post tensioning Machinery involved pre-tensioning requires extensive plant facilities while post tensioning does not. The thickness of the segments involved and anchorage of the tendons Loss of prestress, which is common in pre-tensioning than in post-tensioning. Economy and availability of the materials to be used. FLUID MECHANICS QUESTION 3 S=0.02,n=0.014,q=3m2/s Determine whether sub or supercritical, depth and velocity Forde number NF=v/(gyh), flow is sub critical when its less than 1,critical when equal to 1,supercritical when greater than one yc= (q2/g)1/3 =(3x3/9.81) 1/3 =0.9717 Vc=(gyc)0.5=9.81x0.9717=9.532=3.087 NF=v/(gyc),=0.917/3.087=0.297,which is less than one thus sub critical At critical depth Emin=(3/2 )yc =3/2X0.9717=1.45755 Using Manning’s formula V= (1/n) x R 2/3 S 0.5 R=hydraulic radius =A/P, but A=area==bh, P=wetted perimeter=b+2h, R=bh/b+2h), S=slope V= (1/0.014) x R 2/3 0.02 0.5 but q=Q/b=vbh/b=vh V= (1/0.014) X (bh/b+2h) 2/3X 0.02 0.5 = Q/bh MATERIALS QUESTION 5 5. (A) Cement A Advantages It has a high content of calcium oxide which provides a better bond for the required transfer of prestress Disadvantages It has a low compressive strength of 12.5mpa at 2 days yet the required compressive strength in three days is 30mpa, thus it will result into a weak prestressed element. Cement B Advantages Has a high compressive strength within 2 days thus results to strong prestressed pretensioned beam Disadvantages Cement C Disadvantages It does not give a clear specification of compressive strength within 2 days thus may result to undesirable prestressed pretensioned beam. 5. (B) Cement A Advantages It achieve a high ultimate compressive strength compressive strength in 28 days which is practically good for the elements that will be cast in deep waters. It also has a good adaption to weather and environment exposure mainly due to the good proportion of the ferric and aluminum oxides as major composition. Disadvantages It has a low compressive strength of 12.5mpa at 2 days yet the required compressive strength in three days is 30mpa, thus it will result into a weak prestressed element. Cement B Advantages Has a high compressive strength within 2 days thus results to strong prestressed pretensioned beam Disadvantages Cement C Advantages Environmentally well adapted end product would be achieve as a result of the proportion of the ferric oxides as major constituents. Disadvantages It does not give a clear specification of compressive strength within 2 days thus may result to undesirable prestressed pretensioned beam. QUESTION 6 6 a. Relating toughness to the stress strain curves is not typically true because of the regions typically achieved by the stress strain curve relationship. This is due to the varied nature of the different structural materials in place .a typical example is the concrete and the steel materials especially when in the elastic, plastic ranges. Some materials may not be stretched beyond their elastic ranges while some maybe. Due to the fact that some materials maybe tough but brittle in nature. 6 b. The capability of a metal to distort plastically as well as to absorb vitality in the course before breakage is called toughness. The importance of this definition ought to be placed on the capability to absorb energy before the fracture. Remember that ductility refers to a degree of how much an object distorts plastically just before fracture, but then just because an object is ductile does not, in any way, make it tough. Toughness is a good combination of the strength and the ductility of the material or object. A material with high forte and extraordinary ductility has more toughness than an object or material with little strength and a higher ductility. For that reason, one way to quantify robustness is through calculation of the area undergoing stress strain curve from the ductile test. This value is basically called “material toughness”. It has elements of energy for every unit volume. Material robustness equates to the slow absorption of the energy through the material. There are numerous variables, which have a deep influence on the robustness of a material. These factors are: Temperature Strain rate (rate of loading) Notch effect A metallic object may have reasonable toughness in static loads but then may be unsuccessful in dynamic loads and impact. As a tenet ductility and, for that reason, toughness declines as the proportion of loading rises. Temperature is the next variable that has a major impact on its robustness. As temperature is reduced, the ductility as well as, the toughness also decreases. The third variable is the notch effect, which has to do with the stress distribution. A material might show good toughness while the stress applied is uniaxial; nevertheless when a multiaxial strain state is created due to the occurrence of a notch. 6 c. Methods of increasing toughness Strength and robustness are the two most significant elements of mechanical assets for mechanical materials. They depend on more demands as a result of the gradually severe service circumstances and environments. The main idea of traditional approaches to fortify metallic materials is to constrain dislocation motion by reducing microstructural length gauges, for example spacing amid second phase, grain size, particles, and so forth. It has been corroborated empirically, that monotonically fluctuating variation microstructural length scales every so often leads to an disproportion between strength as well as robustness of materials, that is, high strength is attained at the expense of the lowered robustness. How to make an active tradeoff between forte and robustness, or find methods of increasing material strength while maintain robustness has presently developed one of the crucial hitches for the strengthening-toughening plan of metallic mechanical materials. Layered metallic resources are combinations piled alternatively by two or additional metallic constituents with manageable length scale as well as interfaces. As soon as the individual layer wideness of components falls to nanometer scale, the forte of such coated materials can stretch to the theoretical forte by a factor of 1/3, that is two to three times the strength expected by the law of mixture. The constituent length scale (dimensional as well as microstructural) of encrusted metallic resources can be modulated; additionally components with dissimilar intrinsic properties as well as varied interlayer interfaces with diverse structures can also be combined. Accordingly, it is estimated that layered metallic resources may possibly have a prospect of becoming metallic mechanical materials with high forte and good ductility. SOIL MECHANICS QUESTION 7 a. Show how in clay, peak states depend on stress level and over consolidation ratio In order to see how the peak states depend on the over consolidation ratio, a laboratory consolidation testing has to be done. The consolidation test is one dimensional, and normally, is performed with odometer cell and incremental load applications. The equipment for this test is easily available and can be easily used for the test. The CRS test (constant rate of test) has many advantages over the incremental load equipment. This is because the CRS test produces continuous deformation measurements, the vertical load, and the pore pressure, which is for the direct calculation meant for the stress strain curve, as well as the consolidation and permeability coefficients. The way soft clay behaves under one dimensional compression often changes. This happens when its load goes beyond the pre-consolidation stress. This stress is known as the yield stress and separates the small and elastic strains; normally, from the large and plastic strains. According to Jamiolkowski et al (1985), the conditions required for the pre-consolidation stress for the horizontal deposits that have geostatic stress are divided into 4 categories: Desiccation as a result of drying from freezing and evaporation Mechanical as a result of the changes in the ground water conditions, and total overburden stress Drained creep due to secondary compression of the long term Finally, the physio-chemical phenomena which lead to the cementation as well as other forms of bonding between particles The figure above indicates the different changes in soft clay when flow properties and compressibility are loaded beyond the pre-consolidation stress. S-shaped virgin solidity curves in εv-logσ'v space display constant changes in CR with strain level, with the extreme value (CRmax) situated just outside σ'p. As the loading deviates from recompression (OC) towards virgin compression (NC), the cv and Cα, similarly, undergo noticeable variations. For undisturbed clay, cv(OC) is characteristically 5 to 10 stretches the value of cv(NC) that is typically due to a lower factor of volume variation (mv = Δεv/Δσ'v) in the OC area. The degree of secondary compression surges as σ'v moves towards σ'p and every so often, touches a peak just past σ'p. This variation in Cα is distinctively associated to the gradient of the compression curve as noticeably revealed by Mesri and Castro (1987), such that Cα/CR is principally continuous for both OC as well as NC loading (Note: at this point "CR" equivalents Δεv/Δlogσ'v at all the levels of stress). For most of the cohesive soils Cα/CR = 0.04 ± 0.01 meant for the inorganic soils, and 0.05 ± 0.01 for the organic clays and silts. Read More