Chapter 1 Structural Loads, Determinacy And Stability

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Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil EngineeringChapter 1Structural Loads, Determinacy and StabilityStructural LoadsIntroductory discussion points: How do you see the increasing interest and utility of computers in analysis and design ofstructures?o Matrix methodso Lengthy calculationso Accuracyo Deeper understandingo Reliance (cross checking of results) Consider the above points in comparison to the use of the classical hand calculations.Structural engineering is the science and art of planning, designing, and constructing safe andeconomical structures that will serve their intended purposes. Structural analysis is an integralpart of any structural engineering project, its function being the prediction of the performance ofthe proposed structure.LOAD ESTABLISHMENTVariation in position and timePage 1 of 16Magnitude anddistribution2014/2015 academic yearDirectionPositionPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil EngineeringLOAD CLASIFICATIONvariation inpositiondirectionstructuralresponsearea distributionvertical (gravity)dead loadsstaticconcentratedhorizontal(lateral)live loadsdynamiclinesurfacePoints of discussion: What follows from load determination? [Think of design and analysis] The purpose of design majorly entails unity of functionality and economics. What otherelements can you think of? [Consider aesthetics etc.]Following are examples of structural elements:1. Tie rods (bracing struts) slender tensile members2. BeamsPage 2 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil Engineering carry horizontal loads mainly for flexural resistance need steel reinforcement3. Columns vertical members resist axial compressive loads beam-columns: resist both axial load and bending momentTypes of Structures1. Trusses usually arranged in triangles use less material than beams2. Cables flexible tension resistance only3. Arches gain strength through material compression used in bridge structures, dome roofs and for openings in masonry walls.Page 3 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil Engineering4. Frames usually indeterminaterigid joint connections Economic benefit depends on efficiency of using smaller beams and largercolumns E.g. buildings5. Surface structures tents, air-inflated structures, plate or shell structures difficult to analyze.General building codes provide minimum loads and standards. However, detailed technicalstandards for structural design can be obtained from design codes.Dead Loads Weight of structure permanent fixturesEBCS 1o Densities of construction materials: E.g. normal concrete 24 kN/m3, cementmortar 17 kN/m3o Categories of building areaso EtcLive LoadsPage 4 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.com ASTU Civil EngineeringMovableo Stay for long periods of time. E.g. stored material in a warehouseMovingo E.g. vehicular loads on bridgeso Traffic loads for bridges according to AASHTOTime-dependento Dynamic in natureo E.g. load due to operation machineryEnvironmental Loads earthquakeWindRainLoad Combinations Load combinations for Ultimate Limit State (ULS)o E.g. Fd 1.3Gk 1.6QkWhere Fd is design loadGk is dead loadQk is live load1.3 and 1.6 are partial safety factorso Partial safety factors are provided to make up for Calculation errors Construction inaccuracies Unforeseen increases in load Load combinations for Serviceability Limit States (SLS)o E.g. Fd Gk QkDiscussion points: Is it good to combine maximum effects of all loading conditions?o Consider economic repercussions of excessively conservative designsPage 5 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.com ASTU Civil Engineeringo Discuss on general differences between structures of ancient times with those inthe era of modern civil engineering?In the load combination Fd 1.3Gk 1.6Gk, the partial safety factor for live loadexceeds that for live load. Brainstorm on why this is so.Message to the instructor:Students are encouraged to see references for a more complete understanding of loads onstructures. Tip students on how to go through multiple references.Page 6 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil EngineeringStability and Determinacy of StructuresTypes of SupportsFreely slidingguideRoller supportHinge (pin)connectionFixed or built insupportCable supportBall and socketjointRigid support inspacePage 7 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil Engineering Two-force members must have equal and opposite forces along the same line of action. Three-force members must have forces forming closed polygon.Truss: a framework composed of members joined at their ends to form a rigid structure.Plane truss: members lie in a plane.Simple truss: formed from basic triangles to form a non-collapsible system.A body is said to be internally stable or rigid if it maintains its shape and remains a rigidbody when detached from the supports.See the following two illustrations of internal stability:Will change to internallyunstable if this joint is pinnedPage 8 of 16Will change to internally unstable ifthis member is removed2014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil EngineeringBeam and Frame Structures1. External stabilitya. Static determinacy of internally stable structures:i. If (r 3), the structure is statically unstable externally.ii. If (r 3), the structure is statically determinate externally.iii. If (r 3), the structure is statically indeterminate externally.b. Static determinacy of internally unstable structures:i. If (r 3 n), the structure statically unstable externally.ii. If (r 3 n), the structure is statically determinate externally.iii. If (r 3 n), the structure is statically indeterminate externally.Where r is number of reactions; n is condition (construction) equations2. Overall stabilityThe following situations can be observed for a planar beam-type structure:a. There are (3ma ra) unknown quantities since each existing member isdefined by an axial force, shear force, and bending moment. Also all reactionforces at the existing supports must be determined.b. There are (3j n) available equations since each joint must satisfy Fx 0, Fy 0, and Mz 0. Additionally, there are n condition equations.If (3ma ra) (3j n), then structure is statically unstableIf (3ma ra) (3j n), then structure is statically determinateIf (3ma ra) (3j n), then structure is statically indeterminateWhere ra is number of existing reactions; j is number of joints, ma number ofexisting membersRemark: Internal degree of indeterminacy can be calculated by subtracting external degree ofindeterminacy from the overall degree of indeterminacy.Page 9 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comExample StructureASTU Civil EngineeringStructureCharacteristicsj n ma ra4 1 34r 3 nr 3 1 4Classificationr raDeterminate;stable(3ma ra) (3j n)9 4 13 12 1 13ClassificationDeterminateStable25246r 3 2 5r raIndeterminate(1st degree);stable12 6 1815 2 17Indeterminate(1st degree);stable34133r 3 1 4r raUnstable9 3 1212 1 13Unstable480103r 3r raDeterminate30 3 3324 0 24Indeterminate(9th degree);stable1External ClassificationOverall ClassificationFree end of a cantilever beam isconsidered as a joint.Page 10 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: risticsj n ma ra8 3 8965External Classification155Classificationr raIndeterminate(4th degree)(3ma ra)24 9 33(3j n)24 3 27ClassificationIndeterminate(9th degree);stabler 3 0r raIndeterminate(2nd degree);stable15 5 2015 1 16Indeterminate(4th degree);stableIf there are p members that frame into acommon pin support, (p – 1) memberend conditions must be introduced.Page 11 of 162014/2015 academic yearOverall Classificationr 3 nr 3 2 5N.B. Only 2 are externalExample StructureASTU Civil EngineeringPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil EngineeringExample 1: Determine the reactions at the supports5 kN/m3 kN/mBECA20 m20 mD50 mF20 m 20 mSolutionFree Body Diagram5 kN/m3 kN/myxAxBAyEDyCy20 m 20 m50 mFy20 m 20 mTake a section to the left of hinge BTake the entire beam MB 0, CCW ve MC 0, CCW ve(5*20*10) – (Ay*20) 0(5*40*20) – (Ay*40) – (3*90*45) (Dy*50) (Fy*90) 0Ay 50 kNBut Ay 50 kN, Fy 30 kNDy 149 kNTake a section to the right of hinge E ME 0, CCW ve Fy 0, upward ve(Fy*20) – (3*20*10) 0- (5*40) – (3*90) Ay Cy Dy Fy 0Fy 30 kNBut Ay 50 kN, Fy 30 kN, and Dy 149 kNCy 241 kNPage 12 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil EngineeringExample 2: Find the reaction at A and C40 kN/mRight of hinge MH 0, CCW ve15 kN/mH– (40*10*5) (Cy*10) – (Cy*10) 0– 2000 10Cy 10Cx 0 . ②AC10 m10 mEntire archSolution Fx 0, right veFree body diagram(15*10) Ax Cx 0 . ③40 kN/mEntire arch Fy 0, upward ve15 kN/mH– (40*20) Ay Cy 0 . ④AxCxAyyCy10 m10 mMoment at C MC 0, CCW ve(40*20*10) – (15*10*5) – (Ay*20) 0x– 2000 10Cy 10Cx 0Left of hingeAy 362.5 kN . ⑤ MH 0, CCW ve(40*10*5) (15*10*5) (Ax*10) – (Ay*10) 02750 10Ax – 10Ay 0 . ①Insert ⑤ in ④, Cy 437.5 kN . ⑥Insert ⑥ in ②, Cx – 237.5 kN . ⑦Insert ⑦ in ③, Ax 87.5 kNPage 13 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil EngineeringTrusses1. External stability:The analysis is the same as in beam and frame structures discussed above.2. Internal stability:There are (m ra) unknown quantities where m is the number of members and ra is thenumber of existing reaction forces.There are 2j available equations for plannar trusses, and 3j available equations for spacetrusses where j is the number of joints and n is the number of condition equations.m ra 2j or m 2j – ra for statically determinate stable structures.If (ma m), then structure is statically unstableIf (ma m), then structure is statically determinateIf (ma m), then structure is statically indeterminateExample Structure123Page 14 of 16ExternalClassificationj 8, ma 13, ra 3, r 3ra rDeterminate;stablej 8, ma 15, ra 3, r 3ra rDeterminate;stableInternalClassificationm 2j – ra 16– 3 13ma mDeterminate;stablem 2j – ra 16– 3 13ma mIndeterminate(2nd degree);stablej 8, ma 13, ra m 2j – ra 16 3, r 3 1 4 – 3 13ra rma mUnstableDeterminate;unstable2014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comExample Structure45678ASTU Civil EngineeringExternalClassificationj 7, ma 9,ra 4, r 3ra rIndeterminate(1st degree);stablej 14, ma 24, ra 4, r 3 1 4ra rDeterminate;stableInternalClassificationm 2j – ra 14 – 3 11ma mDeterminate;stablej 12, ma 23, ra 3, r 3ra rIndeterminate(1st degree);stablej 7, ma 11,ra 3, r 3ra rDeterminate;stablej 16, ma 26, ra 6, r 3 3 6ra rDeterminate;stablem 2j – ra 14 – 3 11ma mDeterminate;stablem 2j – ra 28 – 4 24ma mDeterminate;stablem 2j – ra 14 – 3 11ma mDeterminate;stablem 2j – ra 32 – 6 26ma mDeterminate;stableAnalysis of TrussesPage 15 of 162014/2015 academic yearPrepared by Iskinder Yacob

Theory of Structures I Lecture Note Chapter 1Course website: theoryofstructures.wordpress.comASTU Civil EngineeringPoints of discussion:1. The method of joints and the method of sections for truss analysis have been dealt with inengineering mechanics I (statics). Make quick revision if necessary.2. What are the major differences between the method of joints and the method of sections?3. Discuss qualitatively and quantitatively how to go about solving example 8 above usingboth methods.Page 16 of 162014/2015 academic yearPrepared by Iskinder Yacob

1. External stability: The analysis is the same as in beam and frame structures discussed above. 2. Internal stability: There are (m r a) unknown quantities where m is the number of members and r a is the number of existing reaction forces. There are 2j available equations for plannar trusses, and 3j available equations for space 1

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