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Engineering 36 Chp 10: Beams-1 Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] Engineering-36: Engineering Mechanics - Statics 1 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Introduction In Previous Chapters We Examined • Determining External Forces Acting On A Structure – Loads & Reactions • Determining Forces Which Hold Together The Various Members Of A Structure – Trusses & Machines (at PIN Joints) Next, We Learn How to Determine The Internal Forces Which Hold Together The Various Parts Of A Given Member Engineering-36: Engineering Mechanics - Statics 2 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Introduction (2) The Four Types of INTERNAL Forces Present in Structural Members • Tension or Compression • Shear • Bending • Torsion/Twisting The Subsquent Analyses do Not Consider Torsion Loads • For More Torsion Info See ENGR45 Engineering-36: Engineering Mechanics - Statics 3 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Introduction (3) Examine in Detail Two Important Types Of Engineering Structures: 1. BEAMS - usually long, straight, prismatic (constant cross-section) members designed to support loads applied at various points along the member 2. CABLES - flexible members capable of withstanding only tension, designed to support concentrated or distributed loads Engineering-36: Engineering Mechanics - Statics 4 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Internal Forces in Members (a) Straight twoforce member AB is in equilibrium under application of F and −F. (b) Internal Forces equivalent to F and −F are required for equilibrium of free-bodies AC and CB. Engineering-36: Engineering Mechanics - Statics 5 Also Called N VIRTUAL Section Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Internal Forces in Members (2) Multiforce member ABCD is in equilibrium under the application of cable and member (pin) contact forces. FBD INTERNAL forces equivalent to a FORCECOUPLE (F/V-M) system are necessary for equilibrium of free-bodies JD and ABCJ Engineering-36: Engineering Mechanics - Statics 6 VIRTUAL Section J Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Internal Forces in Members (3) An internal FORCE-COUPLE (F/V-M or N/V-M) system is required for equilibrium of TWOFORCE members which are NOT STRAIGHT Engineering-36: Engineering Mechanics - Statics 7 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Beam in Frame Given 3-Member Structure at Left Determine the INTERNAL forces in a) member ACF at point J b) member BCD at point K Note arctan4.8m 2 2.7m 41.7 Engineering-36: Engineering Mechanics - Statics 8 Bruce Mayer, PE [email protected]edu • ENGR-36_Lec-18_Beams-1.pptx Example: Beam in Frame • Cut member ACF at J. – The internal forces at J are represented by equivalent force-couple (F/V-M or N/VM) system which is determined by considering equilibrium of either cut part. • Cut member BCD at K. Solution Plan: • Compute Rcns and Forces at connections for each member Engineering-36: Engineering Mechanics - Statics 9 – Determine force-couple (F/V-M or N/V-M) system equivalent to internal forces at K by applying equilibrium conditions to either cut part. Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Beam in Frame First Determine External Rcns & Connection-Forces Consider the ENTIRE Frame as a rigid Free Body M E 0 2400N 3.6 m F 4.8 m 0 F 1800N F y 0 2400N 1800N E y 0 E y 600N F x Engineering-36: Engineering Mechanics - Statics 10 0 Ex 0 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Beam in Frame Consider Link Member BCD as a Free Body M B 0 2400N 3.6 m C y 2.4 m 0 C y 3600N M C 0 2400N 1.2 m By 2.4 m 0 By 1200N F x Engineering-36: Engineering Mechanics - Statics 11 0 Bx C x 0 Don’t Know (yet) Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Beam in Frame Consider ABE as a Free Body M A 0: Fx 0 : Bx 2.7 m 0 Bx 0 Bx Ax 0 Ax 0 Fy 0 : Ay B y 600 N 0 Ay 1800 N Recall from Member BCD 1800 N F 0 x • But from Above Bx = 0 0 3600 N C x 0 ALL Forces on ACF are now KNOWN Engineering-36: Engineering Mechanics - Statics 12 0 Bx C x 0 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Beam in Frame Cut member ACF at J • The internal forces at J are represented by an equivalent force-couple system Consider Free Body AJ MJ 0: 1800 N 1.2 m M 0 Fx 0 : F 1800 N cos 41.7 0 M 2160 N m F N 1344N Fy 0 : V 1800 N sin 41.7 0 Engineering-36: Engineering Mechanics - Statics 13 V 1197 N Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Beam in Frame Cut member BCD at K • Determine a force-couple system equivalent to internal forces at K Consider Free Body BK MK 0: 1200 N1.5 m M 0 M 1800 N m Fx 0 : F N 0 Fy 0 : 1200 N V 0 Engineering-36: Engineering Mechanics - Statics 14 V 1200 N Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx 3D Internal Forces Full 3D Loading: • Axial Force, Ny • Torsional Moment, My • TOTAL Shear Magnitude Vtot Vx2 Vz2 For Structural Analysis (ENGR45) need: Engineering-36: Engineering Mechanics - Statics 15 • TOTAL Bending Moment Magnitude M tot M x2 M z2 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Beam – What is it? Beam Structural member designed to support loads applied at various points along its length Beams can be subjected to CONCENTRATED loads or DISTRIBUTED loads or a COMBINATION of both. Beam Design is 2-Step Process 1. Determine Axial & Shearing Forces and Bending Moments Produced By Applied Loads 2. Select Structural Cross-section & Material Best Suited To Resist SHEARING-Forces and BENDING-Moments Engineering-36: Engineering Mechanics - Statics 16 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Beam Loading and Supports Beams are classified according to the Support Method(s); e.g., Simply-Supported, Cantilever Reactions at beam supports are Determinate if they involve exactly THREE unknowns. • Otherwise, they are Statically INdeterminate Engineering-36: Engineering Mechanics - Statics 17 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Shear & Bending-Moment Engineering-36: Engineering Mechanics - Statics 18 Goal = determine bending moment and shearing force at any point, say C, in a beam subjected to concentrated and distributed loads Determine reactions at supports by treating whole beam as a free-body. Cut beam at C and draw freebody diagrams for AC and CB exposing V-M System From equilibrium considerations, determine M & V or M’ & V’. Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx 2D V & M Sign Conventions Consider a TypicalCase (Gravity) Loaded SimplySupported Beam with the X-Axis Origin Conventionally Located at the LEFT C P x Engineering-36: Engineering Mechanics - Statics 19 Next Consider a Virtual Section Located at C DEFINE this Case as POSITIVE • Shear, V – The Virtual Member LEFT of the Cut is pushed DOWN by the Right Virtual Member • Moment, M – The Beam takes BOWL Shape Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx 2D V & M Sign Conventions (2) Positive Shear • Right Member Pushes DOWN on Left Member Positive Bending • Beam Concave UP POSITIVE Internal Forces, V & M • Note that at a Virtual Section the V’s & M’s MUST Balance Engineering-36: Engineering Mechanics - Statics 20 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx V & M Diagrams With the Signs of V&M Defined we Can now Determine the MAGNITUDE and SENSE for V&M at ANY arbitrary Virtual-Cut Location PLOTTING V&M vs. x Yields the Stacked Load-Shear-Moment (LVM) Diagram Engineering-36: Engineering Mechanics - Statics 21 LOAD Diagram “Kinks” at LoadApplication Points SHEAR Diagram MOMENT Diagram Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Build V&M Diagram a) Determine reactions at supports b) Cut beam at C (dist x) and consider left member AC V P 2 M P 2x c) Cut beam at E and consider right member EB V ' P 2 M ' P 2L x d) Plot V vs x e) Plot M vs x Note: For a beam subjected to CONCENTRATED LOADS, shear is CONSTANT between loading points and moment Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics varies LINEARLY 22 [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Torqued Beam Solution Plan For the Given Load & Geometry, Draw the shear and bending moment diagrams for the beam AB Engineering-36: Engineering Mechanics - Statics 23 • Taking entire beam as free-body, calculate reactions at A and B. • Determine equivalent internal force-couple systems at sections cut within segments AC, CD, and DB • Plot Results Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Torqued Beam Taking entire beam as a free-body, calculate reactions at A and B. M A 0: By 32 in. 480 lb 6 in. 400 lb 22 in. 0 B y 365 lb MB 0: 480 lb 26 in. 400 lb 10 in. A32 in. 0 A 515 lb Fx 0 : Bx 0 • Note that the 400 lb load at E may be REPLACED by a 400 lb force and 1600 in-lb couple at D Engineering-36: Engineering Mechanics - Statics 24 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Torqued Beam Evaluate equivalent internal force-couple systems at sections cut within segments AC, CD, and DB • For AC use Cut-1 Fy 0 : 515 40x V 0 V 515 40x M1 0 : 515x 40x12 x M 0 M 515x 20x 2 Engineering-36: Engineering Mechanics - Statics 25 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Torqued Beam Evaluate equivalent internal force-couple systems at sections cut within segments AC, CD, and DB • For CD use Cut-2 Fy 0 : 515 480 V 0 V 35 lb M 2 0 : 515x 480x 6 M 0 M 35 lb x 2880 lb in Engineering-36: Engineering Mechanics - Statics 26 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Torqued Beam Evaluate equivalent internal force-couple systems at sections cut within segments AC, CD, and DB • For DB Use Cut-3 Fy 0 : M 3 0: 515 480 400 V 0 V 365 lb 515x 480 x 6 1600 400 x 18 M 0 M 11,680lb in 365lbx Engineering-36: Engineering Mechanics - Statics 27 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Example: Torqued Beam Plot Results • From A to C: V 515 40x M 515x 20x 2 • From C to D V 35 lb M 35 x 2880 • From D to B V 365 lb M 11,680 365 x – Note that over A-C The Bending-Moment Equation is SECOND Order, and Concave DOWN Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 28 [email protected] • ENGR-36_Lec-18_Beams-1.pptx Axial Forces In Civil Engineering Most Beams are loaded Transversely Relative to the Beam Axis • Most Beams do NOT have AXIAL Loads In ME however, many beam-like structures (e.g., Shafts) have significant axial loads that accompany the transverse Shear-Force and Bending-Moment Engineering-36: Engineering Mechanics - Statics 29 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Axial Force Diagram When Significant Axial Loads are present in a Beam, An AXIAL-Force Diagram is added to the typical V&M Diagrams The “N” Diagram is typically placed ABOVE the V-diagram Engineering-36: Engineering Mechanics - Statics 30 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Axial-Force Diagram Engineering-36: Engineering Mechanics - Statics 31 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx WhiteBoard Work Let’s Work These Nice Problems Engineering-36: Engineering Mechanics - Statics 32 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Engineering 36 Appendix Bruce Mayer, PE Registered Electrical & Mechanical Engineer [email protected] Engineering-36: Engineering Mechanics - Statics 33 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Bent Bar Prob Determine at Pt-C the magnitudes of • Axial Force • Shear Force • Bending Moment • Torsional Moment – The Moment on the AC Axis Engineering-36: Engineering Mechanics - Statics 34 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Beam Prob For the Beam with Loading a shown: • Draw the SHEAR and BENDINGMOMENT Diagrams • Determine the Largest-Magnitude BendingMoment and its Location Engineering-36: Engineering Mechanics - Statics 35 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Engineering-36: Engineering Mechanics - Statics 36 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Engineering-36: Engineering Mechanics - Statics 37 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Engineering-36: Engineering Mechanics - Statics 38 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Engineering-36: Engineering Mechanics - Statics 39 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Engineering-36: Engineering Mechanics - Statics 40 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx Engineering-36: Engineering Mechanics - Statics 41 Bruce Mayer, PE [email protected] • ENGR-36_Lec-18_Beams-1.pptx