Baixe o app para aproveitar ainda mais
Esta é uma pré-visualização de arquivo. Entre para ver o arquivo original
Problem 12-01 An L2 steel strap having a thickness of 3 mm and a width of 50 mm is bent into a circular arc of radius 15 m. Determine the maximum bending stress in the strap. Given: b 50mm:= t 3mm:= ρ 15m:= E 200GPa:= Solution: Section Property : c t 2 := I b t 3⋅ 12 := Moment - Curvature Relationship : 1 ρ M E I⋅= M E I⋅ ρ:= Bendiug Stress : σ M c⋅ I = σ E I⋅ρ c I ⎛⎜⎝ ⎞ ⎠⋅= σ E c⋅ ρ:= σ 20 MPa= Ans Problem 12-02 A picture is taken of a man performing a pole vault, and the minimum radius of curvature of the pole is estimated by measurement to be 4.5 m. If the pole is 40 mm in diameter and it is made of a glass- reinforced plastic for which Eg = 131 GPa, determine the maximum bending stress in the pole. Given: do 40mm:= ρ 4.5m:= E 131GPa:= Solution: Section Property : c do 2 := I π do4⋅ 64 := Moment - Curvature Relationship : 1 ρ M E I⋅= M E I⋅ ρ:= Bendiug Stress : σ M c⋅ I = σ E I⋅ρ c I ⎛⎜⎝ ⎞ ⎠⋅= σ E c⋅ ρ:= σ 582.2 MPa= Ans Problem 12-03 Determine the equation of the elastic curve for the beam using the x coordinate that is valid for Specify the slope at A and the beam's maximum deflection. EI is constant.2/0 Lx <≤ Problem 12-04 Determine the equations of the elastic curve for the beam using the x1 and x2 coordinates. Specify the beam's maximum deflection. EI is constant. Problem 12-05 Determine the equations of the elastic curve using the x1 and x2 coordinates. EI is constant. Problem 12-06 Determine the equations of the elastic curve for the beam using the x1 and x3 coordinates. Specify the beam's maximum deflection. EI is constant. Problem 12-07 Determine the equations of the elastic curve for the shaft using the x1 and x2 coordinates. Specify the slope at A and the displacement at the center of the shaft. EI is constant. Problem 12-08 Determine the equations of the elastic curve for the shaft using the x1 and x3 coordinates. Specify the slope at A and the deflection at the center of the shaft. EI is constant. Problem 12-09 The beam is made of two rods and is subjected to the concentrated load P. Determine the maximum deflection of the beam if the moments of inertia of the rods are IAB and IBC, and the modulus of elasticity is E. Problem 12-10 The beam is made of two rods and is subjected to the concentrated load P. Determine the slope at C. The moments of inertia of the rods are IAB and IBC, and the modulus of elasticity is E. Problem 12-11 The bar is supported by a roller constraint at B, which allows vertical displacement but resists axial load and moment. If the bar is subjected to the loading shown, determine the slope at A and the deflection at C. EI is constant. Problem 12-12 Determine the deflection at B of the bar in Prob. 12-11. Problem 12-13 The fence board weaves between the three smooth fixed posts. If the posts remain along the same line, determine the maximum bending stress in the board. The board has a width of 150 mm. and a thickness of 12 mm. E = 12 GPa. Assume the displacement of each end of the board relative to its center is 75 mm. Given: b 150mm:= t 12mm:= L 2.4m:= E 12GPa:= δ 75mm:= Solution: Support Reactions : By symmetry, RA=RC=R + ΣFy=0; 2R P− 0= R 0.5P= Moment Function : M x( ) R x⋅= M x( ) 0.5 P⋅ x⋅= Mmax R 0.5L( )⋅= Mmax 0.25P L⋅= Section Property : c t 2 := I b t 3⋅ 12 := Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ P x⋅ 2 = E I⋅ dv dx ⋅ P x 2⋅ 4 C1+= (1) E I⋅ v⋅ P x 3⋅ 12 C1 x⋅+ C2+= (2) Boundary Conditions : Due to symmetry, dv/dx=0 at x=L/2. Also, v=0 at x=0. From Eq. (1): 0 P 4 L 2 ⎛⎜⎝ ⎞ ⎠ 2 ⋅ C1+= C1 P L2⋅ 16 −= From Eq. (2): 0 0 0+ C2+= C2 0:= The Elastic Curve : Substitute the values of C1 and C2 into Eq. (2), E I⋅ v⋅ P x 3⋅ 12 P L2⋅ 16 x⋅−= v P x⋅ 48E I⋅ 4x 2 3L2−( )⋅= (3) Require at x=L/2, v=-δ. From Eq. (3), δ− P 48E I⋅ L 2 ⎛⎜⎝ ⎞ ⎠⋅ 4 L 2 ⎛⎜⎝ ⎞ ⎠ 2 3L2−⎡⎢⎣ ⎤⎥⎦⋅= δ− P− L3⋅ 48 E⋅ I⋅= P 48 E⋅ I⋅ L3 δ⋅:= Bending Stress: Mmax 0.25P L⋅:= cmax t 2 := σmax Mmax cmax⋅ I := σmax 11.25 MPa= Ans Problem 12-14 Determine the equation of the elastic curve for the beam using the x coordinate. Specify the slope at A and the maximum deflection. EI is constant. Problem 12-15 Determine the deflection at the center of the beam and the slope at B. EI is constant. Problem 12-16 A torque wrench is used to tighten the nut on a bolt. If the dial indicates that a torque of 90 N·m is applied when the bolt is fully tightened, determine the force P acting at the handle and the distance s the needle moves along the scale. Assume only the portion AB of the beam distorts. The cross section is square having dimensions of 12 mm by 12 mm. E = 200 GPa. Given: b 12mm:= h 12mm:= L 0.45m:= E 200GPa:= δ 75mm:= R 0.3m:= Tz 90N m⋅:= Solution: Equations of Equilibrium : + ΣFy=0; Ay P− 0= (1) ΣΜB=0; Tz P L⋅− 0= (2) Solving Eqs. (1) and (2): P Tz L := Ay P:= Ay 200 N= P 200 N= Ans Moment Function : M x( ) Ay x⋅ Tz−= Section Property : I b h3⋅ 12 := Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ Ay x⋅ Tz−= E I⋅ dv dx ⋅ Ay x 2⋅ 2 Tz x⋅− C1+= (1) E I⋅ v⋅ Ay x 3⋅ 6 Tz x 2⋅ 2 − C1 x⋅+ C2+= (2) Boundary Conditions : Due to symmetry, dv/dx=0 at x=0, and v=0 at x=0. From Eq. (1): 0 0 0− C1+= C1 0:= From Eq. (2): 0 0 0− 0+ C2+= C2 0:= The Elastic Curve : Substitute the values of C1 and C2 into Eq. (2), E I⋅ v⋅ Ay x 3⋅ 6 Tz x 2⋅ 2 −= v x 2 6E I⋅ Ay x 3Tz−( )⋅= (3) At x=R, v=-s. From Eq. (3), s R2− 6E I⋅ Ay R 3Tz−( )⋅:= s 9.11 mm= Ans Problem 12-17 The shaft is supported at A by a journal bearing that exerts only vertical reactions on the shaft and at B by a thrust bearing that exerts horizontal and vertical reactions on the shaft. Draw the bending-moment diagram for the shaft and then, from this diagram, sketch the deflection or elastic curve for the shaft's centerline. Determine the equations of the elastic curve using the coordinates x1 and x2 . EI is constant. Given: h 60mm:= L1 150mm:= P 5kN:= L2 400mm:= Solution: L L1 L2+:= Support Reactions: + ΣFy=0; A B+ 0= (1) ΣΜB=0; P h⋅ A L2⋅+ 0= (2) Solving Eqs. (1) and (2): A P− h⋅ L2 := A 0.75− kN= B A−:= B 0.75 kN= Moment Function : M1 x1( ) P h⋅:= M2 x2( ) B x2⋅:= Section Property : EI kN m2⋅:= Slope and Elastic Curve : EI d2 v1⋅ dx1 2 ⋅ M1 x1( )= EI d 2 v2⋅ dx2 2 ⋅ M2 x2( )= EI d2v1 dx1 2 ⋅ P h⋅= EI d2v2 dx2 2 ⋅ B x2⋅= EI dv1 dx1 ⋅ P h⋅( ) x1⋅ C1+= (1) EI dv2 dx2 ⋅ B 2 x2 2⋅ C3+= (3) EI v1⋅ P h⋅ x12⋅ 2 C1 x1⋅+ C2+= (2) EI v2⋅ B x2 3⋅ 6 C3 x2⋅+ C4+= (4) Boundary Conditions : v1=0 at x1=0.15m, From Eq. (2): 0 P h⋅ 0.15m( )2⋅ 2 C1 0.15m( )⋅+ C2+= (5) v2=0 at x2=0, From Eq. (4): 0 0 0+ C4+= C4 0:= Ans v2=0 at x2=0.4m, From Eq. (4): 0 B 0.4m( )3⋅ 6 C3 0.4m( )⋅+ C4+= C3 B 6 − 0.4m( )2⋅:= C3 0.02− kN m2⋅= Ans Continuity Condition: dv1/dx1= - dv2/dx2 at A (x1=0.15m and x2=0.4m) From Eqs. (1) and (3), P h⋅ 0.15m( )⋅ C1+ B 2 0.4m( )2⋅ C3+= C1 B 2 0.4m( )2⋅ C3+ P h⋅ 0.15m( )⋅−:= C1 0.005− kN m2⋅= Ans From Eq. (5): C2 P h⋅ 0.15m( )2⋅ 2 − C1 0.15m( )⋅−:= C2 0.00263− kN m3⋅= Ans The Elastic Curve : Substitute the values of C1 and C2 into Eq. (2), and C3 and C4 into Eq. (4), v1 1 EI P h⋅ 2 x1 2⋅ C1 x1⋅+ C2+⎛⎜⎝ ⎞ ⎠= Ans v2 1 EI B 6 x2 3⋅ C3 x2⋅+ C4+⎛⎜⎝ ⎞ ⎠= Ans BMD : x'1 0 0.01 L1⋅, L1..:= x'2 L1 1.01 L1⋅, L..:= M'1 x'1( ) P h⋅kN m⋅:= M'2 x'2( ) P h⋅ A x'2 L1−( )⋅+⎡⎣ ⎤⎦ 1kN m⋅⋅:= 0 0.2 0.4 0 0.2 0.4 Distane (m) M om en t ( kN -m ) M'1 x'1( ) M'2 x'2( ) x'1 x'2, Problem 12-18 Determine the equations of the elastic curve using the coordinates x1 and x2 , and specify the slope and deflection at C. EI is constant. Problem 12-19 Determine the equations of the elastic curve using the coordinates x1 and x2 , and specify the slope at A. EI is constant. Problem 12-20 Determine the equations of the elastic curve using the coordinates x1 and x2 , and specify the slope and deflection at B. EI is constant. Problem 12-21 Determine the equations of the elastic curve using the coordinates x1 and x3 , and specify the slope and deflection at B. EI is constant. Problem 12-22 Determine the maximum slope and maximum deflection of the simply-supported beam which is subjected to the couple moment M0 . EI is constant. Problem 12-23 The two wooden meter sticks are separated at their centers by a smooth rigid cylinder having a diameter of 50 mm. Determine the force F that must be applied at each end in order to just make their ends touch. Each stick has a width of 20 mm and a thickness of 5 mm. Ew = 11 GPa. Given: do 50mm:= L 0.5m:= b 20mm:= t 5mm:= E 11GPa:= Solution: Section Property: I b t3⋅ 12 := Moment Function : M x( ) F− x⋅= Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ F− x⋅= E I⋅ dv dx ⋅ F x 2⋅ 2 − C1+= (1) E I⋅ v⋅ F x 3⋅ 6 − C1 x⋅+ C2+= (2) Boundary Conditions : dv/dx=0 at x=L. From Eq. (1): 0 F L2⋅ 2 − C1+= C1 F L2⋅ 2 := v=0 at x=L. From Eq. (2): 0 F L3⋅ 6 − C1 L⋅+ C2+= C2 F L3⋅ 6 C1 L⋅−= C2 F L3⋅ 3 −:= Require : v 0.5− do:= at x=0. From Eq. (2): 0.5− do E⋅ I⋅ 0− 0 F L3⋅ 3 −+= F 1.5do E⋅ I⋅ L3 := F 1.375 N= Ans Problem 12-24 The pipe can be assumed roller supported at its ends and by a rigid saddle C at its center. The saddle rests on a cable that is connected to the supports. Determine the force that should be developed in the cable if the saddle keeps the pipe from sagging or deflecting at its center. The pipe and fluid within it have a combined weight of 2 kN/m. EI is constant. Given: e 0.3m:= L 3.75m:= w 2kN m := Solution: Moment Function : M x( ) P x⋅ 1 2 w⋅ x2⋅−= Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ P x⋅ w 2 x2⋅−= E I⋅ dv dx ⋅ P x 2⋅ 2 w 6 x3⋅− C1+= (1) E I⋅ v⋅ P x 3⋅ 6 w 24 x4⋅− C1 x⋅+ C2+= (2) Boundary Conditions : v=0 at x=0 and at x=L. From Eq. (2): 0 0 0− 0+ C2+= C2 0:= 0 P L3⋅ 6 w 24 L4⋅− C1 L⋅+= (3) Also, dv/dx=0 at x=L. From Eq. (1): 0 P L2⋅ 2 w 6 L3⋅− C1+= (4) Solving Eqs. (3) and (4) for P, P L2⋅ 6 w 24 L3⋅− C1+ P L2⋅ 2 w 6 L3⋅− C1+= P L2⋅ 3 w 8 L3⋅= P 3w L⋅ 8 := P 2.813 kN= Equations of Equilibrium : + ΣFy=0; 2P F+ w 2L( )⋅− 0= F 2w L⋅ 2P−:= F 9375 N= +At C : ΣFy=0; 2Tcable e e2 L2+ ⋅ F− 0= Tcable e2 L2+ 2e F⋅:= Tcable 58.78 kN= Ans Problem 12-25 Determine the equations of the elastic curve using the coordinates x1 and x2 , and specify the slope at C and displacement at B. EI is constant. Problem 12-26 Determine the equations of the elastic curve using the coordinates x1 and x3 , and specify the slope at B and deflection at C. EI is constant. Problem 12-27 Determine the elastic curve for the simply supported beam using the x coordinate . Also, determine the slope at A and the maximum deflection of the beam. EI is constant. 2/0 Lx ≤≤ Problem 12-28 Determine the elastic curve for the cantilevered beam using the x coordinate. Also determine the maximum slope and maximum deflection. EI is constant. Problem 12-29 The beam is made of a material having a specific weight γ. Determine the displacement and slope at its end A due to its weight. The modulus of elasticity for the material is E. Problem 12-30 The beam is made of a material having a specific weight γ. Determine the displacement and slope at its end A due to its weight. The modulus of elasticity for the material is E. Problem 12-31 The leaf spring assembly is designed so that it is subjected to the same maximum stress throughout its length. If the plates of each leaf have a thickness t and can slide freely between each other, show that the spring must be in the form of a circular arc in order that the entire spring becomes flat when a large enough load P is applied. What is the maximum normal stress in the spring? Consider the spring to be made by cutting the n strips from the diamond-shaped plate of thickness t and width b. The modulus of elasticity for the material is E. Hint: Show that the radius of curvature of the spring is constant. Problem 12-32 The beam has a constant width b and is tapered as shown. If it supports a load P at its end, determine the deflection at B.The load P is applied a short distance s from the tapered end B, where s << L. EI is constant. Problem 12-33 A thin flexible 6-m-long rod having a weight of 10 N/m rests on the smooth surface. If a force of 15 N is applied at its end to lift it, determine the suspended length x and the maximum moment developed in the rod. Given: P 15N:= w 10N m := Lmax 6m:= Solution: Since the horizontal section has no curvature, the moment in the rod is zero. Hence, R acts at the end of the suspended portion and this portion acts like a simply supported beam. Thus, Equations of Equilibrium : + ΣFy=0; P R+ w x⋅− 0= (1) ΣΜ0=0; P x⋅ w2 x 2⋅− 0= (2) From Eqs. (2) : x 2P w := x 3 m= Ans Maximum moment occurs at center. Mmax P x 2 ⎛⎜⎝ ⎞ ⎠⋅ w 2 x 2 ⎛⎜⎝ ⎞ ⎠ 2 ⋅−:= Mmax 11.25 N m⋅= Ans Problem 12-34 The shaft supports the two pulley loads shown. Determine the equation of the elastic curve. The bearings at A and B exert only vertical reactions on the shaft. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a m:= P kN:= EI kN m2⋅:= EIo 1:= Solution: Support Reactions : + ΣFy=0; A B+ P− 2P− 0= (1) ΣΜB=0; A 2a( )⋅ P a⋅− 2P a⋅+ 0= (2) Solving Eqs. (1) and (2): A P− 2 := B 7P 2 := Moment Function : M x( ) A Ψ x 0−( )⋅ P Ψ x a−( )⋅− B Ψ x 2a−( )⋅+= Slope and Elastic Curve : EI d2 v⋅ dx2 ⋅ M x( )= EI d2v dx2 ⋅ A Ψ x 0−( )⋅ P Ψ x a−( )⋅− B Ψ x 2a−( )⋅+= EI dv dx ⋅ A 2 Ψ x 0−( )2⋅ P 2 Ψ x a−( )2⋅− B 2 Ψ x 2a−( )2⋅+ C1+= EI v⋅ A 6 Ψ x 0−( )3⋅ P 6 Ψ x a−( )3⋅− B 6 Ψ x 2a−( )3⋅+ C1 x⋅+ C2+= EI v⋅ A 6 x3⋅ P 6 Ψ x a−( )3⋅− B 6 Ψ x 2a−( )3⋅+ C1 x⋅+ C2+= (1) Boundary Conditions : v=0 at x=0 From Eq. (1): 0 0 0− 0+ 0+ C2+= C2 0:= Also v=0 at x=2a. From Eq. (1): 0 A 6 2a( )3⋅ P 6 Ψ 2a a−( )3⋅− 0+ C1 2a( )⋅+= C1 A 12a − 2a( )3⋅ P 12a a3⋅+:= C1 5 12 P a2⋅= Equation of Elastic Curve : v 1 EI A 6 x3⋅ P 6 Ψ x a−( )3⋅− B 6 Ψ x 2a−( )3⋅+ C1 x⋅+⎛⎜⎝ ⎞ ⎠m= v x( ) 1 EIo P 12 − x3⋅ P 6 Ψ x a−( )3⋅− 7P 12 Ψ x 2a−( )3⋅+ 5 12 P a2⋅ x⋅+⎛⎜⎝ ⎞ ⎠m:= Ans Problem 12-35 Determine the equation of the elastic curve. Specify the slopes at A and B. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a m:= w kN m := Solution: L 2a:= Support Reactions : + ΣFy=0; A B+ w a⋅− 0= (1) ΣΜA=0; w a( ) 0.5a( )⋅ B 2 a⋅( )⋅− 0= (2) Solving Eqs. (1) and (2): B 1 4 w a( )⋅:= A w a⋅ B−:= B 0.25 w a⋅= A 0.75 w a⋅= Moment Function : M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+= M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+= Slope and Elastic Curve : EI kN m2⋅:= EIo 1:= EI d2 v⋅ dx2 ⋅ M x( )= EI d2v dx2 ⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+= EI dv dx ⋅ 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ C1+= (3) EI v⋅ 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 x3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=0 and x=2a. From Eq. (4): 0 0− 0+ 0+ 0+ C2+= C2 0:= 0 1 24 − w 2a( )4⋅ 1 24 ⎛⎜⎝ ⎞ ⎠ w⋅ a 4⋅+ A 6 2a( )3⋅+ C1 2a( )⋅+= C1 1 3 w a3⋅ 1 48 w a3⋅− 2A 3 a2⋅−:= C1 3 16 − w a3⋅= Equations of Elastic Curve and Slope : v x( ) 1 EIo 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ 3w a⋅ 24 x3⋅+ 3 16 w a3⋅ x⋅−⎛⎜⎝ ⎞ ⎠:= Ans θ x( ) 1 EIo 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ C1+⎛⎜⎝ ⎞ ⎠:= Slope at A: Substitute x=0 into Eq.(3). θ 0( ) 3 16 − w a 3⋅ EIo ⎛⎜⎝ ⎞ ⎠ = Ans Slope at B: Substitute x=L into Eq.(3). θ L( ) 7 48 w a3⋅ EIo ⎛⎜⎝ ⎞ ⎠ = Ans Problem 12-36 The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 1.5m:= P 20kN:= w 6 kN m := b 3m:= Solution: L 2a b+:= Support Reactions : + ΣFy=0; A B+ P− w a⋅− 0= (1) ΣΜA=0; w a( )− 0.5a( )⋅ P a b+( )⋅+ B b⋅− 0= (2) Solving Eqs. (1) and (2): B 1 b w a( )− 0.5a( )⋅ P a b+( )⋅+[ ]⋅:= A P w a⋅+ B−:= B 27.75 kN= A 1.25 kN= Moment Function : M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+= M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+= Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+= E I⋅ dv dx ⋅ 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 Ψ x a−( )2⋅+ B 2 Ψ x a− b−( )2⋅+ C1+= (3) E I⋅ v⋅ 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 Ψ x a−( )3⋅+ B 6 Ψ x a− b−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=a and x=a+b. From Eq. (4): 0 1 24 − w a4⋅ 0+ 0+ 0+ C1 a⋅+ C2+= (5) 0 1 24 − w a b+( )4⋅ 1 24 w b4⋅+ A 6 b3⋅+ 0+ C1 a b+( )⋅+ C2+= (6) (6)-(5): C1 1 b 1 24 w a b+( )4⋅ 1 24 w b4⋅− A 6 b3⋅− 1 24 w⋅ a4⋅−⎡⎢⎣ ⎤⎥⎦:= C1 25.125 kN m 2⋅= From Eq. (5): C2 1 24 w⋅ a4⋅ C1 a⋅−:= C2 36.422− kN m3⋅= Equation of Elastic Curve : ao w 24 := a1 A 6 := a2 B 6 := a3 C1:= a4 C2:= ao 0.25 kN m = a1 0.2083 kN= a2 4.625 kN= a3 25.13 kN m2⋅= a4 36.42− kN m3⋅= Ans v 1 E I⋅ ao− x 4 ao Ψ x a−( )4⋅+ a1 Ψ x a−( )3⋅+ a2 Ψ x a− b−( )3⋅+ a3 x⋅+ a4+⎛⎝ ⎞⎠= Ans Problem 12-37 The shaft supports the two pulley loads shown. Determine the equation of the elastic curve. The bearings at A and B exert only vertical reactions on the shaft. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 0.5m:= P1 200N:= P2 300N:= Solution: Support Reactions : + ΣFy=0; A B+ P1− P2− 0= (1) ΣΜB=0; A 2a( )⋅ P1 a⋅− P2 a⋅+ 0= (2) Solving Eqs. (1) and (2): A P1 P2− 2 := B P1 3P2+ 2 := A 50− N= B 550 N= Moment Function : M x( ) A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− B Ψ x 2a−( )⋅+= Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− B Ψ x 2a−( )⋅+= E I⋅ dv dx ⋅ A 2 Ψ x 0−( )2⋅ P1 2 Ψ x a−( )2⋅− B 2 Ψ x 2a−( )2⋅+ C1+= E I⋅ v⋅ A 6 Ψ x 0−( )3⋅ P1 6 Ψ x a−( )3⋅− B 6 Ψ x 2a−( )3⋅+ C1 x⋅+ C2+= (1) Boundary Conditions : v=0 at x=0 From Eq. (1): 0 0 0− 0+ 0+ C2+= C2 0:= Also v=0 at x=2a. From Eq. (1): 0 A 6 Ψ 2a 0−( )3⋅ P1 6 Ψ 2a a−( )3⋅− 0+ C1 2a( )⋅+= C1 A 12a − 2a( )3⋅ P1 12a a3⋅+:= C1 12.50 N m2⋅= Equation of Elastic Curve : ao A 6 := a1 P1 6 −:= a2 B 6 := a3 C1:= ao 8.33− N= a1 33.33− N= a2 91.67 N= a3 12.50 N m2⋅= Ans v 1 E I⋅ ao Ψ x 0−( ) 3⋅ a1 Ψ x a−( )3⋅+ a2 Ψ x 2a−( )3⋅+ a3 x⋅+⎛⎝ ⎞⎠= Ans Problem 12-38 The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 4m:= P 50kN:= w 3 kN m := b 3m:= Solution: L a 2 b⋅+:= Support Reactions : + ΣFy=0; A B+ P− w a⋅− 0= (1) ΣΜA=0; w a( ) 0.5a( )⋅ P a b+( )⋅+ B L⋅− 0= (2) Solving Eqs. (1) and (2): B 1 L w a( ) 0.5a( )⋅ P a b+( )⋅+[ ]⋅:= A P w a⋅+ B−:= B 37.4 kN= A 24.6 kN=Moment Function : Ans M x( ) A x⋅ 0.5w x2⋅− 0.5w Ψ x a−( )2⋅+ P Ψ x a− b−( )⋅−= Slope and Elastic Curve : EI kN m2⋅:= EIo 1:= EI d2 v⋅ dx2 ⋅ M x( )= EI d2v dx2 ⋅ A x⋅ 0.5w x2⋅− 0.5w Ψ x a−( )2⋅+ P Ψ x a− b−( )⋅−= EI dv dx ⋅ A 2 x2⋅ w 6 x3⋅− w 6 Ψ x a−( )3⋅+ P 2 Ψ x a− b−( )2⋅− C1+= (3) EI v⋅ A 6 x3⋅ w 24 x4⋅− w 24 Ψ x a−( )4⋅+ P 6 Ψ x a− b−( )3⋅− C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=0. From Eq. (4): 0 0 0− 0+ 0+ 0+ C2+= C2 0:= v=0 at x=L. From Eq. (4): 0 A 6 L3⋅ w 24 L4⋅− w 24 L a−( )4⋅+ P 6 L a− b−( )3⋅− C1 L⋅+= C1 A L2⋅ 6 − w 24 L3⋅+ w 24L L a−( )4⋅− P 6L L a− b−( )3⋅+:= C1 278.70− kN m2⋅= Equation of Elastic Curve : ao A 6 := a1 w 24 := a2 P 6 := a3 C1:= a3 C2:= ao 4.10 kN= a1 0.125 kN m = a2 8.33 kN= a3 0.00 N m2⋅= Ans v 1 EI ao x 3⋅ a1 x4⋅− a1 Ψ x a−( )4⋅+ a2 Ψ x a− b−( )3⋅− a3 x⋅+ a4+⎛⎝ ⎞⎠= M x( ) A Ψ x 0−( )⋅ 0.5w Ψ x 0−( )2⋅− 0.5−( )w Ψ x a−( )2⋅− P Ψ x a− b−( )⋅−= Problem 12-39 The beam is subjected to the load shown. Determine the displacement at x = 7 m and the slope at A. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 4m:= P 50kN:= w 3 kN m := b 3m:= Solution: L a 2 b⋅+:= Support Reactions : + ΣFy=0; A B+ P− w a⋅− 0= (1) ΣΜA=0; w a( ) 0.5a( )⋅ P a b+( )⋅+ B L⋅− 0= (2) Solving Eqs. (1) and (2): B 1 L w a( ) 0.5a( )⋅ P a b+( )⋅+[ ]⋅:= A P w a⋅+ B−:= B 37.4 kN= A 24.6 kN= Moment Function : Ans M x( ) A x⋅ 0.5w x2⋅− 0.5w Ψ x a−( )2⋅+ P Ψ x a− b−( )⋅−= Slope and Elastic Curve : EI kN m2⋅:= EIo 1:= EI d2 v⋅ dx2 ⋅ M x( )= EI d2v dx2 ⋅ A x⋅ 0.5w x2⋅− 0.5w Ψ x a−( )2⋅+ P Ψ x a− b−( )⋅−= EI dv dx ⋅ A 2 x2⋅ w 6 x3⋅− w 6 Ψ x a−( )3⋅+ P 2 Ψ x a− b−( )2⋅− C1+= (3) EI v⋅ A 6 x3⋅ w 24 x4⋅− w 24 Ψ x a−( )4⋅+ P 6 Ψ x a− b−( )3⋅− C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=0. From Eq. (4): 0 0 0− 0+ 0+ 0+ C2+= C2 0:= v=0 at x=L. From Eq. (4): 0 A 6 L3⋅ w 24 L4⋅− w 24 L a−( )4⋅+ P 6 L a− b−( )3⋅− C1 L⋅+= C1 A L2⋅ 6 − w 24 L3⋅+ w 24L L a−( )4⋅− P 6L L a− b−( )3⋅+:= C1 278.70− kN m2⋅= Equations of Elastic Curve and Slope : v x( ) 1 EI A 6 x3⋅ w 24 x4⋅− w 24 Ψ x a−( )4⋅+ P 6 Ψ x a− b−( )3⋅− C1 x⋅+ C2+⎛⎜⎝ ⎞ ⎠:= Displacement at x=7m.: Substitute x=7m into Eq.(4). v 7m( ) 834.60− m EIo = Ans θ x( ) 1 EI A 2 x2⋅ w 6 x3⋅− w 6 Ψ x a−( )3⋅+ P 2 Ψ x a− b−( )2⋅− C1+⎛⎜⎝ ⎞ ⎠:= Slope at A: Substitute x=0 into Eq.(3). θ 0( ) 278.70− 1 EIo = M x( ) A Ψ x 0−( )⋅ 0.5w Ψ x 0−( )2⋅− 0.5−( )w Ψ x a−( )2⋅− P Ψ x a− b−( )⋅−= Problem 12-40 The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 2.4m:= P1 10kN:= MB 6kN m⋅:= P2 20kN:= Solution: + Support Reactions : ΣFy=0; A B+ P1− P2− 0= (1) ΣΜB=0; A 3a( )⋅ P1 2a( )⋅− P2 a⋅− MB+ 0= (2) Solving Eqs. (1) and (2): A 2P1 P2+ 3 MB 3a −:= B P1 P2+ A−:= A 12.5 kN= B 17.5 kN= Moment Function : M x( ) A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− P2 Ψ x 2a−( )⋅−= Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− P2 Ψ x 2a−( )⋅−= E I⋅ dv dx ⋅ A 2 Ψ x 0−( )2⋅ P1 2 Ψ x a−( )2⋅− P2 2 Ψ x 2a−( )2⋅− C1+= E I⋅ v⋅ A 6 Ψ x 0−( )3⋅ P1 6 Ψ x a−( )3⋅− P2 6 Ψ x 2a−( )3⋅− C1 x⋅+ C2+= (1) Boundary Conditions : v=0 at x=0 From Eq. (1): 0 0 0− 0+ C2+= C2 0:= Also v=0 at x=3a. From Eq. (1): 0 A 6 Ψ 3a 0−( )3⋅ P1 6 Ψ 3a a−( )3⋅− P2 6 Ψ 3a 2a−( )3⋅− C1 3a( )⋅+= C1 A 18a − 3a( )3⋅ P1 18a 2a( )3⋅+ P2 18a a3⋅+:= C1 76.00− kN m2⋅= Equation of Elastic Curve : ao A 6 := a1 P1 6 −:= a2 P2 6 −:= a3 C1:= ao 2.08 kN= a1 1.67− kN= a2 3.33− kN= a3 76.00− kN m2⋅= Ans v 1 E I⋅ ao Ψ x 0−( ) 3⋅ a1 Ψ x a−( )3⋅+ a2 Ψ x 2a−( )3⋅+ a3 x⋅+⎛⎝ ⎞⎠= Ans w 6 kN m := Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 1.5m:= b 3m:= P 20kN:= Problem 12-41 The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant. Solution: L 2a b+:= Support Reactions : + ΣFy=0; A B+ P− w a⋅− 0= (1) ΣΜA=0; w a( )− 0.5a( )⋅ P a b+( )⋅+ B b⋅− 0= (2) Solving Eqs. (1) and (2): B 1 b w a( )− 0.5a( )⋅ P a b+( )⋅+[ ]⋅:= A P w a⋅+ B−:= B 27.75 kN= A 1.25 kN= Moment Function : M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+= M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+= Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+= E I⋅ dv dx ⋅ 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 Ψ x a−( )2⋅+ B 2 Ψ x a− b−( )2⋅+ C1+= (3) E I⋅ v⋅ 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 Ψ x a−( )3⋅+ B 6 Ψ x a− b−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=a and x=a+b. From Eq. (4): 0 1 24 − w a4⋅ 0+ 0+ 0+ C1 a⋅+ C2+= (5) 0 1 24 − w a b+( )4⋅ 1 24 w b4⋅+ A 6 b3⋅+ 0+ C1 a b+( )⋅+ C2+= (6) (6)-(5): C1 1 b 1 24 w a b+( )4⋅ 1 24 w b4⋅− A 6 b3⋅− 1 24 w⋅ a4⋅−⎡⎢⎣ ⎤⎥⎦:= C1 25.125 kN m 2⋅= From Eq. (5): C2 1 24 w⋅ a4⋅ C1 a⋅−:= C2 36.422− kN m3⋅= Equation of Elastic Curve : ao w 24 := a1 A 6 := a2 B 6 := a3 C1:= a4 C2:= ao 0.25 kN m = a1 0.2083 kN= a2 4.625 kN= a3 25.13 kN m2⋅= a4 36.42− kN m3⋅= Ans v 1 E I⋅ ao− x 4 ao Ψ x a−( )4⋅+ a1 Ψ x a−( )3⋅+ a2 Ψ x a− b−( )3⋅+ a3 x⋅+ a4+⎛⎝ ⎞⎠= Ans w 3 kN m := Problem 12-42 The beam is subjected to the load shown. Determine the equation of the slope and elastic curve. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 5m:= b 3m:= Mo 15kN m⋅:= Solution: L a b+:= Support Reactions : + ΣFy=0; A B+ w a⋅− 0= (1) ΣΜA=0; w a( ) 0.5a( )⋅ B a⋅− Mo+ 0= (2) Solving Eqs. (1) and (2): B 1 a w a( ) 0.5a( )⋅ Mo+⎡⎣ ⎤⎦⋅:= A w a⋅ B−:= B 10.5 kN= A 4.5 kN= Moment Function : M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+ B Ψ x a−( )⋅+= M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+= Slope and Elastic Curve : E I⋅ d 2 v⋅ dx2 ⋅ M x( )= E I⋅ d 2v dx2 ⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+= E I⋅ dv dx ⋅ 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ B 2 Ψ x a−( )2⋅+ C1+= (3) E I⋅ v⋅ 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 x3⋅+ B 6 Ψ x a−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=0 and x=a. From Eq. (4): 0 0− 0+ 0+ 0+ 0+ C2+= C2 0:= 0 1 24 − w a4⋅ 0+ A 6 a3⋅+ 0+ C1 a( )⋅+= C1 1 24 w a3⋅ A 6 a2⋅−:= C1 3.125− kN m2⋅= Equation of Elastic Curve and slope : ao w 24 := a1 A 6 := a2 B 6 := a3 C1:= ao 0.125 kN m = a1 0.75 kN= a2 1.75 kN= a3 3.125− kN m2⋅= Ans v 1 E I⋅ ao− x 4 ao Ψ x a−( )4⋅+ a1 x3⋅+ a2 Ψ x a−( )3⋅+ a3 x⋅+⎛⎝ ⎞⎠= Ans θ 1 E I⋅ 4− ao⋅ x 3 4 ao⋅ Ψ x a−( )3⋅+ 3a1 x2⋅+ 3a2 Ψ x a−( )2⋅+ a3+⎛⎝ ⎞⎠= Ans Problem 12-43 Determine the equation of the elastic curve. Specify the slope at A and the displacement at C. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a m:= w kN m := Solution: L 2a:= Support Reactions : + ΣFy=0; A B+ w a⋅− 0= (1) ΣΜA=0; w a( ) 0.5a( )⋅ B 2 a⋅( )⋅− 0= (2) Solving Eqs. (1) and (2): B 1 4 w a( )⋅:= A w a⋅ B−:= B 0.25 w a⋅= A 0.75 w a⋅= Moment Function : M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+= M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+= Slope and Elastic Curve : EI kN m2⋅:= EIo 1:= EI d2 v⋅ dx2 ⋅ M x( )= EI d2v dx2 ⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+= EI dv dx ⋅ 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ C1+= (3) EI v⋅ 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 x3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=0 and x=2a. From Eq. (4): 0 0− 0+ 0+ 0+ C2+= C2 0:= 0 1 24 − w 2a( )4⋅ 1 24 ⎛⎜⎝ ⎞ ⎠ w⋅ a 4⋅+ A 6 2a( )3⋅+ C1 2a( )⋅+= C1 1 3 w a3⋅ 1 48 w a3⋅− 2A 3 a2⋅−:= C1 3 16 − w a3⋅= Equations of Elastic Curve and Slope : v x( ) 1 EIo 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ 3w a⋅ 24 x3⋅+ 3 16 w a3⋅ x⋅−⎛⎜⎝ ⎞ ⎠:= Displacement at C: Substitute x=a into Eq.(4). v a( ) 5 48 − w a 4⋅ EIo ⎛⎜⎝ ⎞ ⎠ = Ans θ x( ) 1 EIo 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ C1+⎛⎜⎝ ⎞ ⎠m:= Slope at A: Substitute x=0 into Eq.(3). θ 0( ) 3 16 − m w a 3⋅ EIo ⎛⎜⎝ ⎞ ⎠ = Ans Problem 12-44 Determine the equation of the elastic curve. Specify the slope at A and B. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a m:= w kN m := Solution: L 2a:= Support Reactions : + ΣFy=0; A B+ w a⋅− 0= (1) ΣΜA=0; w a( ) 0.5a( )⋅ B 2 a⋅( )⋅− 0= (2) Solving Eqs. (1) and (2): B 1 4 w a( )⋅:= A w a⋅ B−:= B 0.25 w a⋅= A 0.75 w a⋅= Moment Function : M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+= M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+= Slope and Elastic Curve : EI kN m2⋅:= EIo 1:= EI d2 v⋅ dx2 ⋅ M x( )= EI d2v dx2 ⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+= EI dv dx ⋅ 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ C1+= (3) EI v⋅ 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 x3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=0 and x=2a. From Eq. (4): 0 0− 0+ 0+ 0+ C2+= C2 0:= 0 1 24 − w 2a( )4⋅ 1 24 ⎛⎜⎝ ⎞ ⎠ w⋅ a 4⋅+ A 6 2a( )3⋅+ C1 2a( )⋅+= C1 1 3 w a3⋅ 1 48 w a3⋅− 2A 3 a2⋅−:= C1 3 16 − w a3⋅= Equations of Elastic Curve and Slope : v x( ) 1 EIo 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ 3w a⋅ 24 x3⋅+ 3 16 w a3⋅ x⋅−⎛⎜⎝ ⎞ ⎠:= Ans θ x( ) 1 EIo 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ C1+⎛⎜⎝ ⎞ ⎠:= Slope at A: Substitute x=0 into Eq.(3). θ 0( ) 3 16 − w a 3⋅ EIo ⎛⎜⎝ ⎞ ⎠ = Ans Slope at B: Substitute x=L into Eq.(3). θ L( ) 7 48 w a3⋅ EIo ⎛⎜⎝ ⎞ ⎠ = Ans Problem 12-45 The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 1.5m:= P 20kN:= Solution: L 4a:= Support Reactions : + ΣFy=0; A B+ 2P− 0= (1) ΣΜB=0; A 2a( )⋅ P 3a( )⋅− P a⋅+ 0= (2) Solving Eqs. (1) and (2): A P:= B P:= Moment Function : M x( ) P− Ψ x 0−( )⋅ A Ψ x a−( )⋅+ B Ψ x 3a−( )⋅+= Slope and Elastic Curve : EI kN m2⋅:= EI d2 v⋅ dx2 ⋅ M x( )= EI d2v dx2 ⋅ P− Ψ x 0−( )⋅ A Ψ x a−( )⋅+ B Ψ x 3a−( )⋅+= EI dv dx ⋅ P 2 − Ψ x 0−( )2⋅ A 2 Ψ x a−( )2⋅+ B 2 Ψ x 3a−( )2⋅+ C1+= (1) EI v⋅ P 6 − Ψ x 0−( )3⋅ A 6 Ψ x a−( )3⋅+ B 6 Ψ x 3a−( )3⋅+ C1 x⋅+ C2+= EI v⋅ P 6 − x3⋅ A 6 Ψ x a−( )3⋅+ B 6 Ψ x 3a−( )3⋅+ C1 x⋅+ C2+= (2) Boundary Conditions : Due to symmetry, dv/dx=0 at x=2a From Eq. (1): 0 P 2 − 2a( )2⋅ A 2 Ψ 2a a−( )2⋅+ 0+ C1+= C1 2 P⋅ A 2 −⎛⎜⎝ ⎞ ⎠ a 2⋅:= C1 67.50 kN m 2⋅=Also v=0 at x=a. From Eq. (2): 0 P 6 − a3⋅ 0+ 0+ C1 a( )⋅+ C2+= C2 P 6 a3⋅ C1 a⋅−:= C2 90− kN m3⋅= Equation of Elastic Curve : ao P 6 := a1 A 6 := a2 C1:= a3 C2:= ao 3333.33 N= a1 3.3333 kN= a2 67.5 kN m2⋅= a3 90− kN m3⋅= Ans v 1 E I⋅ ao− x 3⋅ a1 Ψ x a−( )3⋅+ a1 Ψ x 3a−( )3⋅+ a2 x⋅+ a3+⎛⎝ ⎞⎠= Ans w 2 kN m := Problem 12-46 The beam is subjected to the load shown. Determine the equation of the slope and elastic curve. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 5m:= b 3m:= P 8kN:= Solution: L a b+:= Support Reactions : + ΣFy=0; A B+ w a⋅− P− 0= (1) ΣΜA=0; w a( ) 0.5a( )⋅ B a⋅− P L⋅+ 0= (2) Solving Eqs. (1) and (2): B 1 a w a( ) 0.5a( )⋅ P L⋅+[ ]⋅:= B 17.8 kN= A w a⋅ P+ B−:= A 0.2 kN= Moment Function : M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+ B Ψ x a−( )⋅+= M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+= Slope and Elastic Curve : EI d2 v⋅ dx2 ⋅ M x( )= EI d2v dx2 ⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+= EI dv dx ⋅ 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ B 2 Ψ x a−( )2⋅+ C1+= (3) EI v⋅ 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 x3⋅+ B 6 Ψ x a−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=0 and x=a. From Eq. (4): 0 0− 0+ 0+ 0+ 0+ C2+= C2 0:= 0 1 24 − w a4⋅ 0+ A 6 a3⋅+ 0+ C1 a( )⋅+= C1 1 24 w a3⋅ A 6 a2⋅−:= C1 9.583 kN m2⋅= Equation of Elastic Curve and slope : ao w 24 := a1 A 6 := a2 B 6 := a3 C1:= ao 0.0833 kN m = a1 0.0333 kN= a2 2.9667 kN= a3 9.583 kN m2⋅= Ans v 1 EI ao− x4 ao Ψ x a−( )4⋅+ a1 x3⋅+ a2 Ψ x a−( )3⋅+ a3 x⋅+⎛⎝ ⎞⎠= Ans θ 1 EI 4− ao⋅ x3 4 ao⋅ Ψ x a−( )3⋅+ 3a1 x2⋅+ 3a2 Ψ x a−( )2⋅+ a3+⎛⎝ ⎞⎠= Ans EI d2v dx2 ⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+= w 2 kN m := Problem 12-47 The beam is subjected to the load shown. Determine the slope at A and the displacement at C. EI is constant. Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:= Given: a 5m:= b 3m:= P 8kN:= Solution: L a b+:= Support Reactions : + ΣFy=0; A B+ w a⋅− P− 0= (1) ΣΜA=0; w a( ) 0.5a( )⋅ B a⋅− P L⋅+ 0= (2) Solving Eqs. (1) and (2): B 1 a w a( ) 0.5a( )⋅ P L⋅+[ ]⋅:= A w a⋅ P+ B−:= B 17.8 kN= A 0.2 kN=Moment Function : M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+= Slope and Elastic Curve : EI kN m2⋅:= EIo 1:=EI d 2 v⋅ dx2 ⋅ M x( )= EI dv dx ⋅ 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ B 2 Ψ x a−( )2⋅+ C1+= (3) EI v⋅ 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 x3⋅+ B 6 Ψ x a−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=0 and x=a. From Eq. (4): 0 0− 0+ 0+ 0+ 0+ C2+= C2 0:= 0 1 24 − w a4⋅ 0+ A 6 a3⋅+ 0+ C1 a( )⋅+= C1 1 24 w a3⋅ A 6 a2⋅−:= C1 9.583 kN m2⋅= Equation of Elastic Curve and slope : v x( ) 1 EI 1 24 − w x4⋅ 1 24 w Ψ x a−( )4⋅+ A 6 x3⋅+ B 6 Ψ x a−( )3⋅+ C1 x⋅+⎛⎜⎝ ⎞ ⎠:= Displacement at C: Substitute x=L into Eq.(4). v L( ) 160.75− m EIo = AnsAns θ x( ) 1 EI 1 6 − w x3⋅ 1 6 w Ψ x a−( )3⋅+ A 2 x2⋅+ B 2 Ψ x a−( )2⋅+ C1+⎛⎜⎝ ⎞ ⎠:= Slope at A: Substitute x=0 into Eq.(3). θ 0( ) 9.58 1 EIo = Ans M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+ B Ψ x a−( )⋅+= Problem 12-48 The beam is subjected to the load shown. Determine the equation of the elastic curve. Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:= Given: L1 2m:= L2 3m:= wo 150 kN m := Solution: L L1 L2+:= Support Reactions : + ΣFy=0; A B+ wo L1⋅− wo 2 L2⋅− 0= (1) ΣΜA=0; wo 2 − L12⋅ wo 2 L2⋅ L2 3 ⎛⎜⎝ ⎞ ⎠⋅+ B L2⋅− 0= (2) Solving Eqs. (1) and (2): B 3− L12 L22+ 6L2 wo⋅:= A wo L1 0.5L2+( )⋅ B−:= B 25− kN= A 550 kN= Moment Function : M x( ) wo 2 − Ψ x 0−( )2⋅ wo 6L2 Ψ x L1−( )3⋅+ A Ψ x L1−( )⋅+= Slope and Elastic Curve : EI d2 v⋅ dx2 ⋅ M x( )= EI kN m2⋅:= EIo 1:= EI d2v dx2 ⋅ wo 2 − Ψ x 0−( )2⋅ wo 6L2 Ψ x L1−( )3⋅+ A Ψ x L1−( )⋅+= EI dv dx ⋅ wo 6 − Ψ x 0−( )3⋅ wo 24 L2⋅ Ψ x L1−( )4⋅+ A2 Ψ x L1−( )2⋅+ C1+= (3) EI v⋅ wo 24 − Ψ x 0−( )4⋅ wo 120 L2⋅ Ψ x L1−( )5⋅+ A6 Ψ x L1−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=L1 Given (5)From Eq. (4): 0 wo 24 − L14⋅ 0+ 0+ C1 L1⋅+ C2+= Also v=0 at x=L. From Eq. (4): 0 wo 24 − L4⋅ wo 120 L2⋅ L L1−( )5⋅+ A6 L L1−( )3⋅+ C1 L⋅+ C2+= (6) Solving Eqs. (5) and (6): Guess C1 1kN m 2⋅:= C2 1kN m3⋅:= C1 C2 ⎛⎜⎜⎝ ⎞ ⎠ Find C1 C2,( ):= C1 410 kN m2⋅= C2 720− kN m3⋅= Equation of Elastic Curve : ao wo 24 −:= a1 wo 120 L2⋅ := a2 A 6 := a3 C1:= a4 C2:= ao 6.25− 1 m kN= a1 0.42 1 m2 kN= a2 91.67 kN= Ans a3 410.00 kN m 2⋅= a4 720.00− kN m3⋅= Ans v 1 E I⋅ ao Ψ x 0−( ) 4⋅ a1 Ψ x L1−( )5⋅+ a2 Ψ x L1−( )3⋅+ a3 x⋅+ a4+⎛⎝ ⎞⎠= Ans wo 150 kN m := A B+ wo L1⋅− wo 2 L2⋅− 0= Problem 12-49 Determine the displacement at C and the slope at A of the beam. Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:= Given: L1 2m:= L2 3m:= Solution: L L1 L2+:= Support Reactions : + ΣFy=0; (1) ΣΜA=0; wo 2 − L12⋅ wo 2 L2⋅ L2 3 ⎛⎜⎝ ⎞ ⎠⋅+ B L2⋅− 0= (2) Solving Eqs. (1) and (2): B 3− L12 L22+ 6L2 wo⋅:= A wo L1 0.5L2+( )⋅ B−:= B 25− kN= A 550 kN= Moment Function : M x( ) wo 2 − Ψ x 0−( )2⋅ wo 6L2 Ψ x L1−( )3⋅+ A Ψ x L1−( )⋅+= Slope and Elastic Curve : EI d2 v⋅ dx2 ⋅ M x( )= EI kN m2⋅:= EIo 1:= EI d2v dx2 ⋅ wo 2 − Ψ x 0−( )2⋅ wo 6L2 Ψ x L1−( )3⋅+ A Ψ x L1−( )⋅+= EI dv dx ⋅ wo 6 − Ψ x 0−( )3⋅ wo 24 L2⋅ Ψ x L1−( )4⋅+ A2 Ψ x L1−( )2⋅+ C1+= (3) EI v⋅ wo 24 − Ψ x 0−( )4⋅ wo 120 L2⋅ Ψ x L1−( )5⋅+ A6 Ψ x L1−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : v=0 at x=L1 Given (5)From Eq. (4): 0 wo 24 − L14⋅ 0+ 0+ C1 L1⋅+ C2+= Also v=0 at x=L. From Eq. (4): 0 wo 24 − L4⋅ wo 120 L2⋅ L L1−( )5⋅+ A6 L L1−( )3⋅+ C1 L⋅+ C2+= (6) Solving Eqs. (5) and (6): Guess C1 1kN m 2⋅:= C2 1kN m3⋅:= C1 C2 ⎛⎜⎜⎝ ⎞ ⎠ Find C1 C2,( ):= C1 410 kN m2⋅= C2 720− kN m3⋅= Equation of Elastic Curve and Slope : v x( ) 1 EI wo 24 − Ψ x 0−( )4⋅ wo 120 L2⋅ Ψ x L1−( )5⋅+ A6 Ψ x L1−( )3⋅+ C1 x⋅+ C2+⎛⎜⎝ ⎞ ⎠ := (7) Displacement at C: Substitute x=0 into Eq.(7). v 0m( ) 720− m EIo = Ans θ x( ) 1 EI wo 6 − Ψ x 0−( )3⋅ wo 24 L2⋅ Ψ x L1−( )4⋅+ A2 Ψ x L1−( )2⋅+ C1+⎛⎜⎝ ⎞ ⎠ := Slope at A: Substitute x=L1 into Eq.(3). θ L1( ) 210 1EIo= Ans A B+ w L⋅− 0= w kN m := Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:= Given: L m:= EIo 1:= Problem 12-50 Determine the equation of the elastic curve. Specify the slope at A. EI is constant. Solution: Support Reactions : + ΣFy=0; (1) ΣΜA=0; 0.5L( )− w L⋅ B L⋅− 0= (2) Solving Eqs. (1) and (2): B 0.5− w L:= A w L⋅ B−:= A 1.5 w L⋅= Moment Function : M x( ) 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+= Slope and Elastic Curve : Ans EI kN m2⋅:= EI d2v dx2 ⋅ 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+= EI dv dx ⋅ w 6 − Ψ x 0−( )3⋅ w 6 Ψ x L−( )3⋅+ A 2 Ψ x L−( )2⋅+ C1+= (3) EI v⋅ w 24 − Ψ x 0−( )4⋅ w 24 Ψ x L−( )4⋅+ A 6 Ψ x L−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : Given v=0 at x=L From Eq. (4): 0 w 24 − L4⋅ 0+ 0+ C1 L⋅+ C2+= (5) v=0 at x=2L From Eq. (4): 0 w 24 − 2L( )4⋅ w 24 L4⋅+ A 6 L3⋅+ C1 2L( )⋅+ C2+= (6) Solving Eqs. (5) and (6): Guess C1 1kN m 2⋅:= C2 1kN m3⋅:= C1 C2 ⎛⎜⎜⎝ ⎞ ⎠ Find C1 C2,( ):= C1 13 w L3⋅= C2 724− w L4⋅= Equation of Elastic Curve and Slope : v x( ) 1 EI w 24 − x4⋅ w 24 Ψ x L−( )4⋅+ 3w L⋅ 12 Ψ x L−( )3⋅+ 1 3 w L3⋅ x⋅+ 7 24 w L4⋅−⎛⎜⎝ ⎞ ⎠:= Ans θ x( ) 1 EI w 6 − Ψ x 0−( )3⋅ w 6 Ψ x L−( )3⋅+ 9w L⋅ 12 Ψ x L−( )2⋅+ C1+⎛⎜⎝ ⎞ ⎠:= Slope at A: Substitute x=L into Eq.(3). θ L( ) 1 6 w L3⋅ EI = EI d2 v⋅ dx2 ⋅ M x( )= w kN m := Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:= Given: L m:= Problem 12-51 Determine the equation of the elastic curve. Specify the deflection at C. EI is constant. Solution: Support Reactions : + ΣFy=0; A B+ w L⋅− 0= (1) ΣΜA=0; 0.5L( )− w L⋅ B L⋅− 0= (2) Solving Eqs. (1) and (2): B 0.5− w L:= A w L⋅ B−:= A 1.5 w L⋅= Moment Function : M x( ) 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+= Slope and Elastic Curve : Ans EI kN m2⋅:= EIo 1:= EI d2v dx2 ⋅ 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+= EI dv dx ⋅ w 6 − Ψ x 0−( )3⋅ w 6 Ψ x L−( )3⋅+ A 2 Ψ x L−( )2⋅+ C1+= (3) EI v⋅ w 24 − Ψ x 0−( )4⋅ w 24 Ψ x L−( )4⋅+ A 6 Ψ x L−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : Given v=0 at x=L From Eq. (4): 0 w 24 − L4⋅ 0+ 0+ C1 L⋅+ C2+= (5) v=0 at x=2L From Eq. (4): 0 w 24 − 2L( )4⋅ w 24 L4⋅+ A 6 L3⋅+ C1 2L( )⋅+ C2+= (6) Solving Eqs. (5) and (6): Guess C1 1kN m 2⋅:= C2 1kN m3⋅:= C1 C2 ⎛⎜⎜⎝ ⎞ ⎠ Find C1 C2,( ):= C1 13 w L3⋅= C2 724− w L4⋅= Equation of Elastic Curve : v x( ) 1 EI w 24 − x4⋅ w 24 Ψ x L−( )4⋅+ 3w L⋅ 12 Ψ x L−( )3⋅+ 1 3 w L3⋅ x⋅+ 7 24 w L4⋅−⎛⎜⎝ ⎞ ⎠:= Ans v 0( ) 7 24 − w L 4⋅ EI = EI d2 v⋅ dx2 ⋅ M x( )= w kN m := Problem 12-52 Determine the equation of the elastic curve. Specify the slope at B. EI is constant. Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:= Given: L m:= Solution: Support Reactions : + ΣFy=0; A B+ w L⋅− 0= (1) ΣΜA=0; 0.5L( )− w L⋅ B L⋅− 0= (2) Solving Eqs. (1) and (2): B 0.5− w L:= A w L⋅ B−:= A 1.5 w L⋅= Moment Function : M x( ) 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+= Slope and Elastic Curve : Ans EI kN m2⋅:= EIo 1:= EI d2v dx2 ⋅ 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+= EI dv dx ⋅ w 6 − Ψ x 0−( )3⋅ w 6 Ψ x L−( )3⋅+ A 2 Ψ x L−( )2⋅+ C1+= (3) EI v⋅ w 24 − Ψ x 0−( )4⋅ w 24 Ψ x L−( )4⋅+ A 6 Ψ x L−( )3⋅+ C1 x⋅+ C2+= (4) Boundary Conditions : Given v=0 at x=L From Eq. (4): 0 w 24 − L4⋅ 0+ 0+ C1 L⋅+ C2+= (5) v=0 at x=2L From Eq. (4): 0 w 24 − 2L( )4⋅ w 24 L4⋅+ A 6 L3⋅+ C1 2L( )⋅+ C2+= (6) Solving Eqs. (5) and (6): Guess C1 1kN m 2⋅:= C2 1kN m3⋅:= C1 C2 ⎛⎜⎜⎝ ⎞ ⎠ Find C1 C2,( ):= C1 13 w L3⋅= C2 724− w L4⋅= Equation of Elastic Curve and Slope : v x( ) 1 EI w 24 − x4⋅ w 24 Ψ x L−( )4⋅+ 3w L⋅ 12 Ψ x L−( )3⋅+ 1 3 w L3⋅ x⋅+ 7 24 w L4⋅−⎛⎜⎝ ⎞ ⎠:= Ans θ x( ) 1 EI w 6 − Ψ x 0−( )3⋅ w 6 Ψ x L−( )3⋅+ 9w L⋅ 12 Ψ x L−( )2⋅+ C1+⎛⎜⎝ ⎞ ⎠:= Slope at A: Substitute x=2L into Eq.(3). θ 2L( ) 1 12 − w L 3⋅ EI = EI d2 v⋅ dx2 ⋅ M x( )= Problem 12-53 The shaft is made of steel and has a diameter of 15 mm. Determine its maximum deflection. The bearings at A and B exert only vertical reactions on the shaft. Est = 200 GPa. Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:= Given: a 200mm:= P1 250N:= b 300mm:= P2 80N:= do 15mm:= E 200GPa:= Solution: L 2a b+:= Section Property: I πdo4 64 := Support Reactions : + ΣFy=0; A B+ P1− P2− 0= (1) ΣΜB=0; A L⋅ P1 a b+( )⋅− P2 b⋅− 0= (2) Solving Eqs. (1) and (2): A P1 a b+( )⋅ P2 a⋅+ L := A 201.429 N= B P1 P2+ A−:= B 128.571 N= Moment Function : M x( ) A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− P2 Ψ x a− b−( )⋅−= Slope and Elastic Curve : E I⋅ d 2v dx2 ⋅ A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− P2 Ψ x a− b−( )⋅−= E I⋅ dv dx ⋅ A 2 Ψ x 0−( )2⋅ P1 2 Ψ x a−( )2⋅− P2 2 Ψ x a− b−( )2⋅− C1+= (1) E I⋅ v⋅ A 6 Ψ x 0−( )3⋅ P1 6 Ψ x a−( )3⋅− P2 6 Ψ x a− b−( )3⋅− C1 x⋅+ C2+= (2) Boundary Conditions : v=0 at x=0 From Eq. (2): 0 0 0− 0+ C2+= C2 0:= v=0 at x=L From Eq. (2): 0 A 6 L3⋅ P1 6 L a−( )3⋅− P2 6 a3⋅− C1 L( )⋅+= C1 A 6 − L2⋅ P1 6L L a−( )3⋅+ P2 6L a3⋅+:= C1 8.857− N m2⋅= Equation of Elastic Curve : v x( ) 1 E I⋅ A 6 x3⋅ P1 6 Ψ x a−( )3⋅− P2 6 Ψ x a− b−( )3⋅− C1 x⋅+ ⎛⎜⎝ ⎞ ⎠:= (3) Maximum Deflection: Assume vmax occurs at a < x < a+b. Given E I⋅ d 2 v⋅ dx2 ⋅ M x( )= dv/dx=0 at x', From Eq. (1): 0 A 2 x'2⋅ P1 2 x' a−( )2⋅− 0− C1+= (4) Solving Eq.(3): Guess x' 300 mm⋅:= x' Find x'( ):= x' 330.05 mm= For vmax, substitute x' into Eq.(3). v x'( ) 3.64− mm= Ans Note: The negative sign indicates downward displacement. Problem 12-54 Determine the slope and deflection at C. EI is constant. Given: L1 8m:= L2 4m:= P 75kN:= Solution: Support Reactions : L L1 L2+:= + ΣFy=0; A B+ P− 0= (1) ΣΜB=0; A L1⋅ P L2⋅+ 0= (2) Solving Eqs. (1) and (2): A P− L2 L1 ⋅:= B P L1 L2+ L1 ⋅:= A 37.5− kN= B 112.5 kN= M/EI Diagram: Set EI kN m2⋅:= EIo 1:= x1 0 0.01 L1⋅, L1..:= x2 L1 1.01 L1⋅, L..:= M'1 x1( ) A x1⋅( ) 1EI⋅:= M'2 x2( ) A x2⋅ B x2 L1−( )⋅+⎡⎣ ⎤⎦ 1EI⋅:= 0 5 10 200 0 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) x1 x2, Moment-Area Theorems : Slopes : tB.A 1 2 M'1 L1( ) L1⋅ L13⎛⎜⎝ ⎞ ⎠⋅:= θA tB.A L1 := θA 400 1 EIo = θC.A 1 2 M'1 L1( ) L1⋅ 12 M'1 L1( ) L2⋅+:= θC θC.A θA−:= θC 1400 1 EIo = Ans Deflections : tC.A 1 2 M'1 L1( ) L1⋅ L13 L2+⎛⎜⎝ ⎞ ⎠⋅ 1 2 M'1 L1( ) L2⋅ 2L23⎛⎜⎝ ⎞ ⎠⋅+:= ∆C tC.A L L1 tB.A⋅−:= ∆C 4800 m EIo = Ans Problem 12-55 Determine the slope and deflection at B. EI is constant. Problem 12-56 If the bearings exert only vertical reactions on the shaft, determine the slope at the bearings and the maximum deflection of the shaft. EI is constant. Problem 12-57 Determine the slope at B and the deflection at C. EI is constant. Problem 12-58 Determine the slope at C and the deflection at B. EI is constant. Problem 12-59 The 60-kg gymnast stands on the center of the simply supported balance beam. If the beam is made of wood and has the cross section shown, determine the maximum deflection.The supports at A and B are assumed to be rigid. Ew = 12 GPa. Given: mo 60 kg⋅:= L 2.7m:= bt 125mm:= bb 75mm:= h 150mm:= E 12GPa:= Solution: W mo g⋅:= Support Reactions : By symmetry, A=B=R + ΣFy=0; 2R W− 0= ����P ����P P ������ R 0.5W:= R 294.20 N= Section Property : ∆b bt bb−:= yc Σ yi ⎯ Ai⋅( )⋅ Σ Ai( )⋅= yc bb h⋅( ) 0.5h( )⋅ 0.5∆b h⋅ h3⎛⎜⎝ ⎞⎠⋅+ bb h⋅ 0.5∆b h⋅+ := yc 68.75 mm= I 1 12 bb⋅ h3⋅ bb h⋅( ) 0.5h yc−( )2⋅+ 136 ∆b⋅ h3⋅+ 0.5 ∆b⋅ h⋅( ) h3 yc−⎛⎜⎝ ⎞⎠ 2 ⋅+:= I 27539062.50 mm4= M/EI Diagram: x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:= M'1 x1( ) R x1⋅( ) 1E I⋅⋅⎡⎢⎣ ⎤⎥⎦:= M'2 x2( ) R x2⋅ W x2 L−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅⎡⎢⎣ ⎤⎥⎦:= 0 2 4 0 0.002 Distance (m) M /E I M'1 x1( ) M'2 x2( ) x1 x2, Moment-Area Theorems : Deflections : tA.C 1 2 M'1 L( ) L⋅ 2L 3 ⎛⎜⎝ ⎞ ⎠⋅:= ∆C tA.C:= ∆C 5.84 mm= Due to symmetry, the slope at midspan (point C) is zero. Hence, ∆max ∆C:= ∆max 5.84 mm= Ans Problem 12-60 The shaft is supported by a journal bearing at A, which exerts only vertical reactions on the shaft, and by a thrust bearing at B, which exerts both horizontal and vertical reactions on the shaft. Determine the slope of the shaft at the bearings. EI is constant. Given: L 0.3m:= a 0.1m:= P 400N:= Solution: Support Reactions : L 2L:= Due to anti-symmetry, B A−= ΣΜB=0; A 2L( )⋅ P 2a( )⋅− 0= A a L ⎛⎜⎝ ⎞ ⎠P:= B A−:= A 66.67 N= B 66.67− N= M/EI Diagram: Set EI kN m2⋅:= EIo 1:= x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:= M'1 x1( ) A x1⋅( ) 1EI⋅:= M'2 x2( ) A x2⋅ P 2a( )⋅−⎡⎣ ⎤⎦ 1EI⋅:= 0 0.5 1 0 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) x1 x2, Moment-Area Theorems : Slopes : tB.A 1− 2 M'1 L( ) L⋅ 2L 3 ⎛⎜⎝ ⎞ ⎠⋅ 1 2 M'1 L( ) L⋅ L 3 L+⎛⎜⎝ ⎞ ⎠⋅+:= θA tB.A L := θA 0.008 1 EIo = Ans Due to anti-symmetry (or in a similar manner), θB θA:= θB 0.008 1 EIo = Ans Problem 12-61 The beam is subjected to the loading shown. Determine the slope at A and the displacement at C. Assume the support at A is a pin and B is a roller. EI is constant. Problem 12-62 The rod is constructed from two shafts for which the moment of inetia of AB is I and BC is 2I. Determine the maximum slope and deflection of the rod due to the loading. The modulus of elasticity is E. Problem 12-63 Determine the deflection and slope at C. EI is constant. Problem 12-64 If the bearings at A and B exert only vertical reactions on the shaft, determine the slope at A. EI is constant. Problem 12-65 If the bearings at A and B exert only vertical reactions on the shaft, determine the slope at C. EI is constant. Problem 12-66 Determine the deflection at C and the slope of the beam at A, B, and C. EI is constant. Given: L1 6m:= L2 3m:= Mo 8kN m⋅:= Solution: L L1 L2+:= Support Reactions : + ΣFy=0; A B+ 0= (1) ΣΜB=0; A L1⋅ Mo+ 0= (2) Solving Eqs. (1) and (2): A Mo L1 −:= B A−:= A 1.333− kN= B 1.333 kN= M/EI Diagram: Set EI kN m2⋅:= EIo 1:= x1 0 0.01 L1⋅, L1..:= x2 L1 1.01 L1⋅, L..:= M'1 x1( ) A x1⋅( ) 1EI⋅:= M'2 x2( ) A x2⋅ B x2 L1−( )⋅+⎡⎣ ⎤⎦ 1EI⋅:= 0 5 10 5 0 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) x1 x2, Moment-Area Theorems : Deflections : tB.A 1 2 M'1 L1( ) L1⋅ L13⎛⎜⎝ ⎞ ⎠⋅:= tC.A 1 2 M'1 L1( ) L1⋅ L13 L2+⎛⎜⎝ ⎞ ⎠⋅ M'1 L1( ) L2⋅ L2 2 ⎛⎜⎝ ⎞ ⎠⋅+:= ∆C tC.A L L1 tB.A⋅−:= ∆C 84 m EIo = Ans Slopes : θA tB.A L1 := θA 8 1 EIo = Ans θB.A 1 2 M'1 L1( ) L1⋅:= θB θB.A θA−:= θB 16 1 EIo = Ans θC.A 1 2 M'1 L1( ) L1⋅ M'1 L1( ) L2⋅+:= θC θC.A θA−:= θC 40 1 EIo = Ans Problem 12-67 The bar is supported by the roller constraint at C, which allows vertical displacement but resists axial load and moment. If the bar is subjected to the loading shown, determine the slope and displacement at A. EI is constant. Problem 12-68 The acrobat has a weight of 750 N (~75 kg), and suspends himself uniformly from the center of the high bar. Determine the maximum bending stress in the pipe (bar) and its maximum deflection. The pipe is made of L2 steel and has an outer diameter of 25 mm and a wall thickness of 3 mm. Given: a 0.9m:= do 25mm:= W 750N:= b 0.45m:= t 3mm:= E 200GPa:= Solution: L 2a b+:= Support Reactions : By symmetry, A=B=R + ΣFy=0; 2R W− 0= R 0.5W:= R 375.00 N= Section Property : ro 0.5do:= ri ro t−:= I π 4 ro 4 ri 4−⎛⎝ ⎞⎠⋅:= Maximum Bendiug Stress : Mmax R a⋅:= σmax Mmax ro⋅ I := σmax 330.17 MPa= Ans < σY = 703 MPa (O.K. !) M/EI Diagram: x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:= M'1 x1( ) R x1⋅( ) 1E I⋅⋅:= M'2 x2( ) R x2⋅ W2 x2 a−( )⋅−⎡⎢⎣ ⎤⎥⎦ 1E I⋅⋅:= M'3 x3( ) R x3⋅ W x3 L2−⎛⎜⎝ ⎞⎠⋅−⎡⎢⎣ ⎤⎥⎦ 1E I⋅⋅:= 0 1 2 0 0.1 0.2 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) M'3 x3( ) x1 x2, x3, Moment-Area Theorems : Deflections : tA.C 1 2 M'1 a( ) a⋅ 2a 3 ⎛⎜⎝ ⎞ ⎠⋅ M'1 a( ) b 2 ⎛⎜⎝ ⎞ ⎠⋅ b 4 a+⎛⎜⎝ ⎞ ⎠⋅+:= ∆C tA.C:= ∆C 65.74 mm= Due to symmetry, the slope at midspan (point C) is zero. Hence, ∆max ∆C:= ∆max 65.74 mm= Ans Problem 12-69 Determine the value of a so that the displacement at C is equal to zero. EI is constant. Problem 12-70 The beam is made of a ceramic material. In order to obtain its modulus of elasticity, it is subjected to the elastic loading shown. If the moment of inertia is I and the beam has a measured maximum deflection ∆, determine E. The supports at A and D exert only vertical reactions on the beam. Problem 12-71 Determine the maximum deflection of the shaft. EI is constant. The bearings exert only vertical reactions on the shaft. Problem 12-72 The beam is subjected to the load P as shown. Determine the magnitude of force F that must be applied at the end of the overhang C so that the deflection at C is zero. EI is constant. Problem 12-73 At what distance a should the bearing supports at A and B be placed so that the deflection at the center of the shaft is equal to the deflection at its ends? The bearings exert only vertical reactions on the shaft. EI is constant. 0 1 2 0 0.01 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) M'3 x3( ) x1 x2, x3, Given: a 0.5m:= b 0.8m:= c 1.2m:= do 50mm:= P1 600N:= P2 1200N:= E 200GPa:= Solution: L a b+ c+:= Support Reactions : + ΣFy=0; A B+ P1− P2− 0= (1) ΣΜB=0; A L⋅ P1 b c+( )⋅− P2 c⋅− 0= (2) Solving Eqs. (1) and (2): A P1 b c+( )⋅ P2 c⋅+ L := B P1 P2+ A−:= A 1.056 kN= B 0.744 kN= Section Property : I π do4⋅ 64 := M/EI Diagram: x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:= M'1 x1( ) A x1⋅( ) 1E I⋅⋅:= M'2 x2( ) A x2⋅ P1 x2 a−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅:= M'3 x3( ) A x3⋅ P1 x3 a−( )⋅− P2 x3 a− b−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅:= Problem 12-74 Determine the slope of the 50-mm-diameter A-36 steel shaft at the bearings at A and B. The bearings exert only vertical reactions on the shaft. Moment-Area Theorems : Slopes : tB.A 1 2 M'3 L c−( ) c⋅ 2c 3 ⋅ 1 2 M'2 a b+( ) M'1 a( )−( ) b⋅ b3 c+⎛⎜⎝ ⎞⎠⋅+ ... M'1 a( ) b⋅ b 2 c+⎛⎜⎝ ⎞ ⎠⋅ 1 2 M'1 a( ) a⋅ L 2a 3 −⎛⎜⎝ ⎞ ⎠⋅++ ... := θA tB.A L := θA 0.01046 rad= Ans θB tA.B L := θB 0.00968 rad= Ans tA.B 1 2 M'3 L c−( ) c⋅ L 2c 3 −⎛⎜⎝ ⎞ ⎠⋅ 1 2 M'2 a b+( ) M'1 a( )−( ) b⋅ 2 b⋅3 a+⎛⎜⎝ ⎞⎠⋅+ ... M'1 a( ) b( )⋅ b 2 a+⎛⎜⎝ ⎞ ⎠⋅ 1 2 M'1 a( ) a⋅ 2a 3 ⎛⎜⎝ ⎞ ⎠⋅++ ... := 0 1 2 0 0.01 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) M'3 x3( ) x1 x2, x3, Given: a 0.5m:= b 0.8m:= c 1.2m:= do 50mm:= P1 600N:= P2 1200N:= E 200GPa:= Solution: L a b+ c+:= Support Reactions : + ΣFy=0; A B+ P1− P2− 0= (1) ΣΜB=0; A L⋅ P1 b c+( )⋅− P2 c⋅− 0= (2) Solving Eqs. (1) and (2): A P1 b c+( )⋅ P2 c⋅+ L := B P1 P2+ A−:= A 1.056 kN= B 0.744 kN= Section Property : I π do4⋅ 64 := M/EI Diagram: x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:= M'1 x1( ) A x1⋅( ) 1E I⋅⋅:= M'2 x2( ) A x2⋅ P1 x2 a−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅:= M'3 x3( ) A x3⋅ P1 x3 a−( )⋅− P2 x3 a− b−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅:= Problem 12-75 Determine the maximum deflection of the 50-mm diameter A-36 steel shaft. It is supported by bearings at its ends A and B which only exert vertical reactions on the shaft. Moment-Area Theorems : Deflections : tB.A 1 2 M'3 L c−( ) c⋅ 2c 3 ⋅ 1 2 M'2 a b+( ) M'1 a( )−( ) b⋅ b3 c+⎛⎜⎝ ⎞⎠⋅+ ... M'1 a( ) b⋅ b 2 c+⎛⎜⎝ ⎞ ⎠⋅ 1 2 M'1 a( ) a⋅ L 2a 3 −⎛⎜⎝ ⎞ ⎠⋅++ ... := θA tB.A L := θA 0.01046 rad= Ans Maximum Deflection: Assume vmax occurs at point E ( a < x' < a+b ). Slopes : θE.A 1 2 M'1 a( ) a⋅ M'1 a( ) x'⋅+ 1 2 M'2 a b+( ) M'1 a( )−( ) x'b⋅ x'⋅+= θE.A 1 2 A a⋅ E I⋅ a⋅ A a⋅ E I⋅ x'⋅+ 1 2E I⋅ A a b+( )⋅ P1 b⋅− A a⋅−⎡⎣ ⎤⎦ x' b ⋅ x'⋅+= θE.A 1 2E I⋅ A a 2⋅ 2A a⋅ x'⋅+ A P1−( ) x'2⋅+⎡⎣ ⎤⎦= θE θA− θE.A+= θE θA− 1 2E I⋅ A a 2⋅ 2A a⋅ x'⋅+ A P1−( ) x'2⋅+⎡⎣ ⎤⎦+= At point E, where θE=0. Given 0 θA− 2E I⋅( )⋅ A a2⋅+ 2A a⋅ x'⋅+ A P1−( ) x'2⋅+= (3) Solving Eq.(3): Guess x' 0.7 m⋅:= x' Find x'( ):= x' 733.27 mm= < 0.8m (O.K. !) Deflection: tA.E 1 2 M'1 a( ) a⋅ 2a 3 ⎛⎜⎝ ⎞ ⎠⋅ M'1 a( ) x'⋅ x' 2 a+⎛⎜⎝ ⎞ ⎠⋅+ 1 2 M'2 a b+( ) M'1 a( )−( ) x'b⋅ x'⋅ 2 x'⋅3 a+⎛⎜⎝ ⎞⎠⋅+⎡⎢⎣ ⎤⎥⎦:= ∆max tA.E:= ∆max 8.161 mm= Ans 0 0.5 1 0.1 0.05 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) M'3 x3( ) x1 x2, x3, a 0.2m:= b 0.3m:= c 0.5m:= do 20mm:= Problem 12-76 Determine the slope of the 20-mm-diameter A-36 steel shaft at the bearings at A and B. The bearings exert only vertical forces on the shaft. P1 800N:= P2 350N:= E 200GPa:= Solution: L a b+ c+:= Support Reactions : + ΣFy=0; A B+ P1− P2− 0= (1) ΣΜB=0; P1− L⋅ A b c+( )⋅+ P2 c⋅− 0= (2) Solving Eqs. (1) and (2): A P1 L⋅ P2 c⋅+ b c+:= B P1 P2+ A−:= A 1218.75 N= B 68.75− N= Section Property : I π do4⋅ 64 := M/EI Diagram: x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:= M'1 x1( ) P1− x1⋅( ) 1E I⋅⋅:= M'2 x2( ) P1− x2⋅ A x2 a−( )⋅+⎡⎣ ⎤⎦ 1E I⋅⋅:= M'3 x3( ) P1− x3⋅ A x3 a−( )⋅+ P2 x3 a− b−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅:= Given: Moment-Area Theorems : Slopes : tB.A 1 2 M'3 a b+( ) c⋅ 2c 3 ⋅ M'3 a b+( ) b⋅ b 2 c+⎛⎜⎝ ⎞ ⎠⋅+ 1 2 M'2 a( ) M'3 a b+( )−( ) b⋅ 2 b⋅3 c+⎛⎜⎝ ⎞⎠⋅+ ...:= θA tB.A b c+:= θA 0.01811 rad= Ans tA.B 1 2 M'3 a b+( ) c⋅ c 3 b+⎛⎜⎝ ⎞ ⎠⋅ M'3 a b+( ) b⋅ b 2 ⎛⎜⎝ ⎞ ⎠⋅+ 1 2 M'2 a( ) M'3 a b+( )−( ) b⋅ b3⎛⎜⎝ ⎞⎠⋅+ ...:= θB tA.B b c+:= θB 0.00592 rad= Ans 0 0.5 1 0.1 0.05 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) M'3 x3( ) x1 x2, x3, Given: a 0.2m:= b 0.3m:= c 0.5m:= do 20mm:= P1 800N:= P2 350N:= E 200GPa:= Solution: L a b+ c+:= Support Reactions : + ΣFy=0; A B+ P1− P2− 0= (1) ΣΜB=0; P1− L⋅ A b c+( )⋅+ P2 c⋅− 0= (2) Solving Eqs. (1) and (2): A P1 L⋅ P2 c⋅+ b c+:= B P1 P2+ A−:= A 1218.75 N= B 68.75− N= Section Property : I π do4⋅ 64 := M/EI Diagram: x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:= M'1 x1( ) P1− x1⋅( ) 1E I⋅⋅:= M'2 x2( ) P1− x2⋅ A x2 a−( )⋅+⎡⎣ ⎤⎦ 1E I⋅⋅:= M'3 x3( ) P1− x3⋅ A x3 a−( )⋅+ P2 x3 a− b−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅:= Problem 12-77 Determine the displacement of the 20-mm-diameter A-36 steel shaft at D. The bearings at A and B exert only vertical reactions on the shaft. Moment-Area Theorems : Deflection : tD.B 1 2 M'3 a b+( ) c⋅ L 2c 3 −⎛⎜⎝ ⎞ ⎠⋅ M'3 a b+( ) b⋅ b 2 a+⎛⎜⎝ ⎞ ⎠⋅+ 1 2 M'2 a( ) M'3 a b+( )−( ) b⋅ b3 a+⎛⎜⎝ ⎞⎠⋅ 12 M'1 a( ) a⋅ 2a3⋅++ ...:= tA.B 1 2 M'3 a b+( ) c⋅ c 3 b+⎛⎜⎝ ⎞ ⎠⋅ M'3 a b+( ) b⋅ b 2 ⎛⎜⎝ ⎞ ⎠⋅+ 1 2 M'2 a( ) M'3 a b+( )−( ) b⋅ b3⎛⎜⎝ ⎞⎠⋅+ ...:= ∆D tD.B L b c+ tA.B−:= ∆D 4.98 mm= Ans Problem 12-78 The beam is subjected to the loading shown. Determine the slope at B and deflection at C. EI is constant. Problem 12-79 Determine the slope at B and the displacement at C. The bearings at A and B exert only vertical reactions on the shaft. EI is constant. Problem 12-80 Determine the displacement at D and the slope at C. The bearings at A and B exert only vertical reactions on the shaft. EI is constant. Fh 1kN:= ∆max ∆C:= Given: a 50mm:= do 32mm:= Fv 4.5kN:= Support Reactions : b 650mm:= h 450mm:= E 200GPa:= Solution: L a b+:= ∆max 1.78 mm= + ΣFy=0; Fv A− B+ 0= (1) ΣΜA=0; Fv a⋅ Fh h⋅+ B b⋅− 0= (2) Solving Eqs. (1) and (2): B Fv a⋅ Fh h⋅+ b := A Fv B+:= Ans Section Property : I π 64 do 4⋅:= M/EI Diagram: x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, L..:= M'1 x1( ) Fh h⋅ Fv x1⋅+( ) 1E I⋅⋅:= 0 0.2 0.4 0.6 0 0.05 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) x1 x2, Moment-Area Theorems : Deflections : tA.B 1 2 M'1 a( ) b⋅ b 3 ⎛⎜⎝ ⎞ ⎠⋅:= θB tA.B b = θB M'1 a( ) b⋅ 6 := The maximum displacement occurs at point C, where θC = 0. Let x' = L - x θC.B 1 2 M'1 a( ) x' b ⎛⎜⎝ ⎞ ⎠⋅ ⎡⎢⎣ ⎤⎥⎦⋅ x'⋅= θC θB θC.B+= 0 M'1 a( ) b⋅ 6 − 1 2 M'1 a( )⋅ x'2 b ⋅+= x' b 1 3 ⋅:= x' 375.28 mm= < b = 650 mm (O.K. !) ∆C tB.C= tB.C θB.C 2x' 3 ⎛⎜⎝ ⎞ ⎠⋅= θB.C θC.B= ∆C 1 2 M'1 a( ) x' b ⎛⎜⎝ ⎞ ⎠⋅ ⎡⎢⎣ ⎤⎥⎦⋅ x'⋅ 2x' 3 ⎛⎜⎝ ⎞ ⎠⋅:= Problem 12-81 The two force components act on the tire of the automobile as shown. The tire is fixed to the axle, which is supported by bearings at A and B. Determine the maximum deflection of the axle. Assume that the bearings resist only vertical loads. The thrust on the axle is resisted at C. The axle has a diameter of 32 mm and is made of A-36 steel. Neglect the effect of axial load on deflection. M'2 x2( ) Fh h⋅ Fv x2⋅+ A x2 a−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅:= EI kN m2⋅:= Given: a 0.3m:= P 20kN:= b 0.9m:= Solution: L 2a b+:= Support Reactions : By symmetry, A=B=R + ΣFy=0; 2R 2P− 0= R P:= Maximum Bendiug Stress : Mmax R a⋅:= M/EI Diagram: Set Ans EIo 1:= x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:= M'1 x1( ) R x1⋅( ) 1EI⋅:= M'2 x2( ) R x2⋅ P x2 a−( )⋅−⎡⎣ ⎤⎦ 1EI⋅:= M'3 x3( ) R x3⋅ P x3 a−( )⋅− P x3 a− b−( )⋅−⎡⎣ ⎤⎦ 1EI⋅:= 0 0.5 1 1.5 0 5 10 Distance (m) M /E I ( 1/ m ) M'1 x1( ) M'2 x2( ) M'3 x3( ) x1 x2, x3, Moment-Area Theorems : Deflections : tA.C 1 2 M'1 a( ) a⋅ 2a 3 ⎛⎜⎝ ⎞ ⎠⋅ M'1 a( ) b 2 ⎛⎜⎝ ⎞ ⎠⋅ b 4 a+⎛⎜⎝ ⎞ ⎠⋅+:= ∆C tA.C:= ∆C 1.5975 m EIo = Due to symmetry, the slope at midspan (point C) is zero. Hence, ∆max ∆C:= ∆max 1.5975 m EIo = Problem 12-82 Determine the displacement at D and the slope at C. The bearings at A and B exert only vertical reactions on the shaft. EI is constant. Problem 12-83 Beams made of fiber-reinforced plastic may one day replace many of those made of A-36 steel since they are one-fourth the weight of steel and are corrosion resistant. Using the table in Appendix B, with σallow = 160 MPa and τallow = 84 MPa, select the lightest-weight steel wide-flange beam that will safely support the 25-kN load, then compute its maximum deflection. What would be the maximum deflection of this beam if it were made of a fiber-reinforced plastic with Ep = 126 GPa and had the same moment of inertia as the steel beam? Given: σallow 160MPa:= P 25kN:= L 3m:= τallow 84MPa:= E 126GPa:= Solution: Support Reactions : By symmetry, A=B=R + ΣFy=0; 2R P− 0= R 0.5P:= R 12.50 kN= Maximum Moment and Shear: Vmax R:= Vmax 12.5 kN= Mmax R L⋅:= Mmax 37.5 kN m⋅= Bending Stress: Sreq'd Mmax σallow := Sreq'd 234375.00 mm3= Select W 310x21 : Sx 244 10 3( )⋅ mm3:= d 303mm:= tw 5.08mm:= I Sx 0.5d( )⋅:= Shear Stress : Provide a shear stress check. τmax Vmax tw d⋅ := τmax 8.12 MPa= < τallow =84 MPa (O.K.!) Hence, Use W 310x21 Ans M/EI Diagram: x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:= M'1 x1( ) R x1⋅( ) 1E I⋅⋅⎡⎢⎣ ⎤⎥⎦:= M'2 x2( ) R x2⋅ P x2 L−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅⎡⎢⎣ ⎤⎥⎦:= 0 2 4 6 0 0.005 Distance (m) M /E I M'1 x1( ) M'2 x2( ) x1 x2, Moment-Area Theorems : Deflections : tA.C 1 2 M'1 L( ) L⋅ 2L 3 ⎛⎜⎝ ⎞ ⎠⋅:= ∆C tA.C:= ∆C 24.15 mm= Due to symmetry, the slope at midspan (point C) is zero. Hence, ∆max ∆C:= ∆max 24.15 mm= Ans Problem 12-84 The simply supported shaft has a moment of inertia of 2I for region BC and a moment of inertia I for regions AB and CD. Determine the maximum deflection of the shaft due to the load P. The modulus of elasticity is E. Problem 12-85 The A-36 steel shaft is used to support a rotor that exerts a uniform load of 5 kN/m within the region CD of the shaft. Determine the slope of the shaft at the bearings A and B. The bearings exert only vertical reactions on the shaft. Problem 12-86 The beam is subjected to the loading shown. Determine the slope at B and deflection at C. EI is constant. Problem 12-87 Determine the slope of the shaft at A and the displacement at D. The bearings at A and B exert only vertical reactions on the shaft. EI is constant. Problem 12-88 Determine the slope at B and the displacement at C. The member is an A-36 steel structural tee for which I = 30(106) mm4. Given: P 25kN:= L 1m:= w 25 kN m := E 200GPa:= I 30 106( ) mm4:= Solution: Support Reactions : By symmetry, A=B=R + ΣFy=0; 2R' P− 0= 2R'' w 2L( )⋅− 0= R' 0.5P:= R'' w L⋅:= R' 12.50 kN= R'' 25.00 kN= M/EI Diagram: x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:= For point load P: M'1 x1( ) R' x1⋅( ) 1E I⋅⋅:= M'2 x2( ) R' x2⋅ P x2 L−( )⋅−⎡⎣ ⎤⎦ 1E I⋅⋅:= For uniform load w : M''1 x1( ) R'' x1⋅ 0.5w x12⋅−⎛⎝ ⎞⎠ 1E I⋅⋅:= M''2 x2( ) R'' x2⋅ 0.5 w⋅ x22⋅−⎛⎝ ⎞⎠ 1E I⋅⋅:= 0 1 2 0 0.002 Distance (m) M /E I M'1 x1( ) M'2 x2( ) x1 x2, 0 1 2 0 0.002 Distance (m) M /E I M''1 x1( ) M''2 x2( ) x1 x2, Moment-Area Theorems : Slope : Due to symmetry, the slope at midspan (point C) is zero. Hence, θB = θB/C θB.C 1 2 M'1 L( )⋅ L⋅ 2 3 M''1 L( )⋅ L⋅+:= θB θB.C:= θB 0.00243 rad= Ans Deflection : tA.C 1 2 M'1 L( )⋅ L⋅ 2L 3 ⎛⎜⎝ ⎞ ⎠⋅ 2 3 M''1 L( )⋅ L⋅ 5L 8 ⎛⎜⎝ ⎞ ⎠⋅+ ⎡⎢⎣ ⎤⎥⎦:= ∆C tA.C:= ∆max ∆C:= ∆max 1.56 mm= Ans Problem 12-89 The W200 X 71 cantilevered beam is made of A-36 steel and is subjected to the loading shown. Determine the displacement at its end A. Given: P 6kN:= Mo 3kN m⋅:= E 200GPa:= a 2.4m:= Solution: L 2a:= Use W 200x71 : I 76.6 106( )⋅ mm4:= Elsastic Curve : The elastic curves for the concentrated load and couple moment are drawn separately as shown. Method of Superposition : Using the table in Appendix C, the required slope and displacement are, ∆A1 P L3⋅ 3E I⋅:= ∆C2 Mo a 2⋅ 2E I⋅:= θC2 Mo a⋅ E I⋅:= ∆A2 ∆C2 θC2( ) a⋅+:= Hence, the displacement at A is ∆A ∆A1 ∆A2+:= ∆A 16.13 mm= Ans Problem 12-90 The W200 X 71 cantilevered beam is made of A-36 steel and is subjected to the loading shown. Determine the displacement at C and the slope at A. Given: P 6kN:= Mo 3kN m⋅:= E 200GPa:= a 2.4m:= Solution: L 2a:= Use W 200x71 : I 76.6 106( )⋅ mm4:= Elsastic Curve : The elastic curves for the concentrated load and couple moment are drawn separately as shown. Method of Superposition : Using the table in Appendix C, the required slope and displacement are, ∆C1 P a2⋅ 6E I⋅ 3L a−( )⋅:= ∆C2 Mo a 2⋅ 2E I⋅:= θA1 P L2⋅ 2E I⋅:= θC2 Mo a⋅ E I⋅:= θA2 θC2:= Hence, the slope at A is θA θA1 θA2+:= θA 0.00498 rad= Ans Hence, the displacement at C is ∆C ∆C1 ∆C2+:= ∆C 5.08 mm= Ans Problem 12-91 The W360 X 64 simply supported beam is made of A-36 steel and is subjected to the loading shown. Determine the deflection at its center C. Given: a 3m:= Mo 60kN m⋅:= E 200GPa:= w 30 kN m := Solution: L 2a:= Use W 360x64: I 179 106( )⋅ mm4:= Elsastic Curve : The elastic curves for the uniform load and couple moment are drawn separately as shown. Method of Superposition : Using the table in Appendix C, the required displacements are, ∆C1 5w L4⋅ 768E I⋅:= ∆C2 Mo 6 E⋅ I⋅ a L ⎛⎜⎝ ⎞ ⎠⋅ a 2 3 L⋅ a⋅− 2 L2⋅+( )⋅:= Hence, the displacement at C is ∆C ∆C1 ∆C2+:= ∆C 10.84 mm= Ans Problem 12-92 The W360 X 64 simply supported beam is made of A-36 steel and is subjected to the loading shown. Determine the slope at A and B. Given: a 3m:= Mo 60kN m⋅:= E 200GPa:= w 30 kN m := Solution: L 2a:= Use W 360x64: I 179 106( )⋅ mm4:= Elsastic Curve : The elastic curves for the uniform load and couple moment are drawn separately as shown. Method of Superposition : Using the table in Appendix C, the required slopes are, θA1 7w L3⋅ 384E I⋅:= θA2 Mo L⋅ 6 E⋅ I⋅:= Hence, the slope at A is θA θA1 θA2+:= θA 0.00498 rad= θA 0.285 deg= Ans θB1 3w L3⋅ 128E I⋅:= θB2 Mo L⋅ 3 E⋅ I⋅:= Hence, the slope at B is θB θB1 θB2+:= θB 0.00759 rad= θB 0.435 deg= Ans Problem 12-93 Determine the moment M0 in terms of the load P and dimension a so that the deflection at the center of the beam is zero. EI is constant. Problem 12-94 The beam supports the loading shown. Code restrictions, due to a plaster ceiling, require the maximum deflection not to exceed 1/360 of the span length. Select the lightest-weight A-36 steel wide-flange beam from Appendix B that will satisfy this requirement and safely support the load. The allowable bending stress is σallow = 168 MPa and the allowable shear stress is τallow = 100 MPa. Assume A is a roller and B is a pin. Given: σallow 168MPa:= a 3.6m:= τallow 100MPa:= w 60 kN m := E 200GPa:= Solution: L 2a:= + ΣFy=0; A B+ w a⋅− 0= (1) ΣΜB=0; A L⋅ 0.5−⋅ w a2⋅ 0= (2) Solving Eqs. (1) and (2): A w a2 2L := B w a⋅ A−:= A 54 kN= B 162 kN= Maximum Moment and Shear: Vmax max A B,( ):= Vmax 162 kN= Mmax occurs at x where V(x) = 0: V x( ) A w Φ x a−( )⋅−= 0 A w Φ x a−( )⋅−= x A w a+:= Mmax A x⋅ 0.5w x a−( )2⋅−:= Mmax 218.7 kN m⋅= Strength criterion: Sreq'd Mmax σallow := Sreq'd 1301785.71 mm3= Select W 410x74 : Sx 1330 10 3( )mm3:= d 413mm:= tw 9.65mm:= I Sx 0.5d( )⋅:= Shear Stress : Provide a shear stress check. τmax Vmax tw d⋅ := τmax 40.65 MPa= < τallow =100 MPa (O.K.!) Deflection criterion: ∆allow L 360 := ∆allow 20 mm= Using the table in Appendix C, ∆max 0.006563 w L4⋅ E I⋅:= ∆max 19.27 mm= < ∆allow =20mm (O.K.!) Hence, Use W 410x74 Ans x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, L..:= V1 x1( ) A( ) 1kN:= V2 x2( ) A w x2 a−( )⋅−⎡⎣ ⎤⎦ 1kN⋅:= M1 x1( ) A x1⋅( ) 1kN m⋅:= M2 x2( ) A x2⋅ 0.5 w⋅ x2 a−( )2⋅−⎡⎣ ⎤⎦ 1kN m⋅⋅:= 0 5 200 0 Distance (m) Sh ea r ( kN ) V1 x1( ) V2 x2( ) x1 x2, 0 5 0 200 Distance (m) M (k N -m ) M1 x1( ) M2 x2( ) x1 x2, Problem 12-95 The simply supported beam carries a uniform load of 29 kN/m. Code restrictions, due to a plaster ceiling, require the maximum deflection not to exceed 1/360 of the span length. Select the lightest- weight A-36 steel wide-flange beam from Appendix B that will satisfy this requirement and safely support the load. The allowable bending stress is σallow = 168 MPa and the allowable shear stress is τallow = 100 MPa. Assume A is a pin and B a roller support. Given: σallow 168MPa:= a 1.2m:= b 2.4m:= τallow 100MPa:= P 40kN:= E 200GPa:= w 29 kN m := Solution: L 2a b+:= Support Reactions : By symmetry, A=B=R + ΣFy=0; 2R 2P− w L⋅− 0= R 0.5 2P w L⋅+( ):= R 109.60 kN= Maximum Moment and Shear: Vmax R:= Vmax 109.6 kN= Mmax occurs at midspan. Mmax R 0.5L( )⋅ P 0.5L a−( )⋅− 0.5w 0.5L( )2⋅−:= Mmax 131.52 kN m⋅= Strength criterion: Sreq'd Mmax σallow := Sreq'd 782857.14 mm3= Select W 360x51 : Sx 794 10 3( )mm3:= d 355mm:= tw 7.24mm:= I Sx 0.5d( )⋅:= Shear Stress : Provide a shear stress check. τmax Vmax tw d⋅ := τmax 42.64 MPa= < τallow =100 MPa (O.K.!) Deflection criterion: ∆allow L 360 := ∆allow 13.33 mm= Using the table in Appendix C, ∆max 5w L4⋅ 384E I⋅ 2 P a⋅ b⋅ 6E I⋅ L⋅ ⎛⎜⎝ ⎞ ⎠⋅ L 2 a2− 0.5L( )2−⎡⎣ ⎤⎦⋅+:= ∆max 11.61 mm= < ∆allow =20mm (O.K.!) Hence, Use W 360x51 Ans x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:= V1 x1( ) R w x1⋅−( ) 1kN:= V2 x2( ) R w x2⋅− P−( ) 1kN⋅:= V3 x3( ) R w x3⋅− 2P−( ) 1kN⋅:= M1 x1( ) R x1⋅ 0.5 w⋅ x12−⎛⎝ ⎞⎠ 1kN m⋅:= M2 x2( ) R x2⋅ 0.5 w⋅ x22− P x2 a−( )⋅−⎡⎣ ⎤⎦ 1kN m⋅⋅:= M3 x3( ) R x3⋅ 0.5 w⋅ x32− P x3 a−( )⋅− P x3 a− b−( )⋅−⎡⎣ ⎤⎦ 1kN m⋅⋅:= 0 2 4 100 0 100 Distance (m) Sh ea r ( kN ) V1 x1( ) V2 x2( ) V3 x3( ) x1 x2, x3, 0 2 4 0 100 Distance (m) M (k N -m ) M1 x1( ) M2 x2( ) M3 x3( ) x1 x2, x3, Problem 12-96 The W250 X 45 cantilevered beam is made of A-36 steel and is subjected to unsymmetrical bending caused by the applied moment. Determine the deflection of the centroid at its end A due to the loading. Hint: Resolve the moment into components and use superposition. Given: L 4.5m:= Mo 2.5kN m⋅:= E 200GPa:= θ 30deg:= Solution: Use W 250x45: Ix 71.1 10 6( )⋅ mm4:= Iy 7.03 10 6( )⋅ mm4:= Displacements : Using the table in Appendix C, the required displacements are, ∆x_max Mo sin θ( )⋅ L2⋅ 2E Iy⋅ := ∆y_max Mo cos θ( )⋅ L2⋅ 2E Ix⋅ := Hence, the displacement at A is ∆A ∆x_max ∆y_max+:= ∆A 10.54 mm= Ans eliz Rectangle Problem 12-97 The assembly consists of a cantilevered beam CB and a simply supported beam AB. If each beam is made of A-36 steel and has a moment of inertia about its principal axis of Ix = 46(106) mm4, determine the displacement at the center D of beam BA. Given: P 75kN:= I 46 106( )⋅ mm4:= E 200GPa:= LDA 2.4m:= LBC 4.8m:= LBA 4.8m:= Solution: Consider beam AB : Support Reactions : By symmetry, A=B=R + ΣFy=0; 2R P− 0= R 0.5P:= Method of Superposition : Using the table in Appendix C, the required slope and displacement are, Consider beam BC : ∆B R LBC 3⋅ 3E I⋅:= Consider beam AB : ∆D1 P LBA 3⋅ 48E I⋅:= ∆D2 LDA LBA ∆B⋅:= Hence, the displacement at D is ∆D ∆D1 ∆D2+:= ∆D 93.91 mm= Ans Problem 12-98 The rod is pinned at its end A and attached to a torsional spring having a stiffness k, which measures the torque per radian of rotation of the spring. If a force P is always applied perpendicular to the end of the rod, determine the displacement of the force. EI is constant. Problem 12-99 The relay switch consists of a thin metal strip or armature AB that is made of red brass C83400 and is attracted to the solenoid S by a magnetic field. Determine the smallest force F required to attract the armature at C in order that contact is made at the free end B. Also, what should the distance a be for this to occur? The armature is fixed at A and has a moment of inertia of I = 0.18(10-12) m4. Given: ∆B 2mm:= L 50mm:= E 101GPa:= I 0.18 10 12−( )m4:= Solution: LAC L:= LCB L:= Elsastic Curve : The elastic curves for the concentrated load and couple moment are drawn separately as shown. Method of Superposition : Using the table in Appendix C, the required slope and displacement are, θC F LAC 2⋅ 2E I⋅= ∆C F LAC 3⋅ 3E I⋅= (1) Hence, the displacement at B is ∆B ∆C θC LCB⋅+= ∆B F LAC 3⋅ 3E I⋅ F LAC 2⋅ 2E I⋅ LCB⋅+= F 6∆B E⋅ I⋅ 2 LAC 3⋅ 3 LCB⋅ LAC2⋅+ := F 0.349 N= Ans From Eq.(1), ∆C F LAC 3⋅ 3E I⋅:= ∆C 0.800 mm= Ans Problem 12-100 Determine the vertical deflection and slope at the end A of the bracket. Assume that the bracket is fixed supported at its base, and neglect the axial deformation of segment AB. EI is constant. Given: h 75mm:= L 100mm:= P 400N:= w 4 kN m := Solution: Set EI 1kN m2:= Elsastic Curve : The elastic curves for the concentrated load, uniform distributed load and couple moment are drawn separately as shown. MB 0.5w L 2⋅:= Method of Superposition : Using the table in Appendix C, the required slope and displacement are, θA1 w L3⋅ 6EI := θB2 MB h⋅ EI := θA2 θB2:= θB3 P h2⋅ 2EI := θA3 θB3:= ∆A1 w L4⋅ 8EI := ∆A2 θB2 L( )⋅:= ∆A3 θB3 L( )⋅:= Hence, the slope at A is θA θA1 θA2+ θB3+:= θA 0.00329 rad= θA 0.00329 EI = Ans The displacement at A is ∆A ∆A1 ∆A2+ ∆A3+:= ∆A 0.313 mm= ∆A 0.313 EI = Ans Problem 12-101 The W610 X 155 A-36 steel beam is used to support the uniform distributed load and a concentrated force which is applied at its end. If the force acts at an angle with the vertical as shown, determine the horizontal and vertical displacement at point A. Given: L 3m:= P 25kN:= E 200GPa:= cx 3 5 := cy 4 5 := w 30 kN m := Solution: Use W 610x155: Ix 1290 10 6( )⋅ mm4:= Iy 108 10 6( )⋅ mm4:= Elsastic Curve : The elastic curves for the concentrated load and uniform load are drawn separately as shown. Py P cy⋅:= Py 20 kN= Px P cx⋅:= Px 15 kN= Method of Superposition : Using the table in Appendix C, the required displacement are, ∆A1 w L4⋅ 8E Ix⋅ := ∆A2 Py L 3⋅ 3E Ix⋅ := The vertical displacement at A is ∆A_y ∆A1 ∆A2+:= ∆A_y 1.875 mm= Ans The horizontal displacement at A is ∆A_x Px L 3⋅ 3E Iy⋅ := ∆A_x 6.250 mm= Ans Problem 12-102 The framework consists of two A-36 steel cantilevered beams CD and BA and a simply supported beam CB. If each beam is made of steel and has a moment of inertia about its principal axis of Ix = 46(106) mm4, determine the deflection at the center G of beam CB. Given: P 75kN:= I 46 106( )⋅ mm4:= E 200GPa:= LDC 4.8m:= LBC 4.8m:= LAB 4.8m:= Solution: Consider beam BC : Support Reactions : By symmetry, B=C=R + ΣFy=0; 2R P− 0= R 0.5P:= Method of Superposition : Using the table in Appendix C, the required slope and displacement are, Consider beam DC : ∆C R LDC 3⋅ 3E I⋅:= Due to symmetry, ∆B ∆C:= Consider beam BC : ∆'G P LBC 3⋅ 48E I⋅:= Hence, the displacement at G is ∆G ∆C ∆'G+:= ∆G 169.04 mm= Ans Problem 12-103 Determine the reactions at the supports A and B, then draw the moment diagram. EI is constant. Problem 12-104 Determine the reactions at the supports A and B, then draw the shear and moment diagrams. EI is constant. Neglect the effect of axial load. Problem 12-105 Determine the reactions at the supports A, B, and C; then draw the shear and moment diagrams. EI is constant. Problem 12-106 Determine the reactions at the supports, then draw the shear and moment diagram. EI is constant. Problem 12-107 Determine the moment reactions at the supports A and B. EI is constant. Problem 12-108 Determine the value of a for which the maximum positive moment has the same magnitude as the maximum negative moment. EI is constant. Problem 12-109 Determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant. Problem 12-110 The beam has a constant E1I1 and is supported by the fixed wall at B and the rod AC. If the rod has a cross-sectional area A2 and the material has a modulus of elasticity E2 , determine the force in the rod. Problem 12-111 Determine the moment reactions at the supports A and B, and then draw the shear and moment diagrams. Solve by expressing the internal moment in the beam in terms of Ay and MA. EI is constant. Problem 12-112 Determine the moment reactions at the supports A and B. EI is constant. Problem 12-113 Determine the moment reactions at the supports A and B, then draw the shear and moment diagrams. EI is constant. Problem 12-114 Determine the moment reactions at the supports A and B, then draw the shear and moment diagrams. EI is constant. Problem 12-115 Determine the reactions at the supports. EI is constant. b 1.8m:=Given: P 25kN:= a 1.2m:= Solution: L a b+:= Due to symmetry, the slope at support B is zero. Hence, θB = 0 and tanθB = 0. Moment-Area Theorems : Ans Set EI 1kN m2:= A 1kN:= x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, L..:= For point load P: M'1 x1( ) 0( ) 1EI⋅:= M'2 x2( ) P− x2 a−( )⋅⎡⎣ ⎤⎦ 1EI⋅:= For reaction A : M''1 x1( ) A x1⋅( ) 1EI⋅:= M''2 x2( ) A x2⋅( ) 1EI⋅:= 0 1 2 3 50 0 Distance (m) M /E I M'1 x1( ) M'2 x2( ) x1 x2, 0 1 2 3 0 5 Distance (m) M /E I M''1 x1( ) M''2 x2( ) x1 x2, Deflection : tA.B_1 1 2 M'2 L( )⋅ b⋅ a 2 b⋅ 3 +⎛⎜⎝ ⎞ ⎠⋅:= tA.B_2 1 2 M''2 L( )⋅ L⋅ 2L 3 ⎛⎜⎝ ⎞ ⎠⋅:= tA.B tA.B_1 tA.B_2 Ay A ⎛⎜⎝ ⎞ ⎠⋅+= Compatibility : tA.B 0:= 0 tA.B_1 tA.B_2 Ay A ⎛⎜⎝ ⎞ ⎠⋅+= Ay A− tA.B_1 tA.B_2 ⎛⎜⎝ ⎞ ⎠ ⋅:= Ay 10.80 kN= Ans Support Reactions : Due to symmetry, Cy Ay:= Cy 10.80 kN= Ans + ΣFy=0; Ay By+ Cy+ 2P− 0= By 2P Ay− Cy−:= By 28.40 kN= Problem 12-116 Determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant. x1 0 0.01 a⋅, a..:= x3 L 1.01 L⋅, L b+( )..:= x2 a 1.01 a⋅, L..:= x4 L b+( ) 1.01 L b+( )⋅, 2L( )..:= V1 x1( ) Ay( ) 1kN:= V3 x3( ) Ay P− By+( ) 1kN⋅:= V2 x2( ) Ay P−( ) 1kN⋅:= V4 x3( ) Ay P− By+ P−( ) 1kN⋅:= M1 x1( ) Ay x1⋅( ) 1kN m⋅:= M2 x2( ) Ay x2⋅ P x2 a−( )⋅−⎡⎣ ⎤⎦ 1kN m⋅⋅:= M3 x3( ) Ay x3⋅ P x3 a−( )⋅− By x3 L−( )⋅+⎡⎣ ⎤⎦ 1kN m⋅⋅:= M4 x4( ) Ay x4⋅ P x4 a−( )⋅− By x4 L−( )⋅+ P x4 L− b−( )⋅−⎡⎣ ⎤⎦ 1kN m⋅⋅:= 0 1 2 3 4 5 6 20 0 20 Distance (m) Sh ea r ( kN ) V1 x1( ) V2 x2( ) V3 x3( ) V4 x4( ) x1 x2, x3, x4, 0 1 2 3 4 5 6 20 0 20 Distance (m) M (k N -m
Compartilhar