Find maximum positive shear and bending moment at point B due to the series of four moving concentrated loads

Explanation of Solution
Calculation:
Apply a 1 k unit moving load at a distance of x from left end A.
Sketch the free body diagram of beam as shown in Figure 1.
Refer Figure 1.
Find the equation of support reaction
Take moment about point C.
Consider moment equilibrium at point C.
Consider clockwise moment as positive and anticlockwise moment as negative.
Sum of moment at point C is zero.
Find the equation of support reaction
Apply vertical equilibrium equation of forces.
Consider upward force as positive
Substitute
Influence line for shear at B.
Find the equation of shear force at B of portion AB
Sketch the free body diagram of the section AB as shown in Figure 2.
Refer Figure 2.
Apply equilibrium equation of forces.
Consider upward force as positive
Substitute
Find the equation of shear force at B of portion BC
Sketch the free body diagram of the section BC as shown in Figure 3.
Refer Figure 3.
Apply equilibrium equation of forces.
Consider upward force as positive
Substitute
Thus, the equations of the influence line for
Find the value of influence line ordinate of shear force at various points of x using the Equations (3) and (4) and summarize the value as in Table 1.
x | |
0 | 1 |
30 | 0 |
Draw the influence lines for the shear force at point B using Table 1 as shown in Figure 4.
Refer Figure 4.
Find the slope
Here,
Substitute 15 ft for
Find the slope
Here,
Substitute 15 ft for
Sketch the loading position as shown in Figure 5.
Find the maximum positive shear force at B.
Sketch the loading position on the beam when the load 1 placed at just right of B as shown in Figure 6.
Refer Figure 6.
Find the shear force at B when the load 1 placed at just right of B.
Substitute 15 ft for
Sketch the loading position on the beam when the load 2 placed at just right of B as shown in Figure 7.
Refer Figure 7.
Find the shear force at B when the load 2 placed at just right of B.
Substitute 15 ft for
Sketch the loading position on the beam when the load 3 placed at just right of B as shown in Figure 8.
Refer Figure 8.
Find the shear force at B when the load 3 placed at just right of B.
Substitute 15 ft for
Sketch the loading position on the beam when the load 4 placed at just right of B as shown in Figure 9.
Refer Figure 9.
Find the shear force at B when the load 3 placed at just right of B.
Substitute 15 ft for
Maximum positive shear force at B as follows.
The maximum positive shear at B is the maximum of
Therefore, the maximum positive shear at point B is
Influence line for moment at B.
Refer Figure 2.
Consider clockwise moment as positive and anticlockwise moment as negative.
Find the equation of moment at B of portion AB
Substitute
Refer Figure 3.
Consider clockwise moment as negative and anticlockwise moment as positive.
Find the equation of moment at B of portion BC
Substitute
Thus, the equations of the influence line for
Find the value of influence line ordinate of moment at various points of x using the Equations (5) and (6) and summarize the value as in Table 2.
x | |
0 | 0 |
30 | 0 |
Draw the influence lines for the moment at point B using Table 2 as shown in Figure 10.
Refer Figure 9.
The slope of portion AB and BC is same.
Find the slope
Here,
Substitute 15 ft for
Find the maximum positive bending moment at B.
Refer Figure 6.
Find the bending moment at B when the load 1 placed at just right of B.
Substitute 15 ft for
Refer Figure 7.
Find the bending moment at B when the load 2 placed at just right of B.
Substitute 15 ft for
Refer Figure 8.
Find the bending moment at B when the load 3 placed at just right of B.
Substitute 15 ft for
Refer Figure 9.
Find the bending moment at B when the load 1 placed at just right of B.
Substitute 15 ft for
Maximum positive bending moment at B as follows.
The maximum positive bending moment at B is the maximum of
Therefore, the maximum positive bending moment at point B is
Want to see more full solutions like this?
Chapter 9 Solutions
EBK STRUCTURAL ANALYSIS
- A traffic signal has a 60-second cycle length (Red time + Green time). For the travel direction of interest, the red and green times are 30 seconds each, the arrival rate is constant at 20 [veh/min] and the saturation flow (i.e., the departure rate) is 1 [veh/sec]. a. Calculate the average delay (for all vehicles) for the travel direction of interest. b. Assume a work zone on the street downstream of the intersection so that only 25 [veh/min] (in the direction of interest) can pass. Calculate the average delay caused by the work zone to a vehicle leaving the intersection. Assume that the queue at the work zone never backs- up into the intersection. c. Discuss qualitatively the implications of queue spillback from the work zone on the delay of the system. Traffic Direction (a) Traffic Direction (b)arrow_forwardAttached pics is a sample problem, can you compute it for me, I just want to compare my answer. Thank you.arrow_forwardProblem 2: The Douglas fir beam below supports uniform live (WL) and dead loads (WD) as shown below. Assume the total distributed load is 700 lb/ft. WD=300#/PT. W₁ = 400# W₁ = 400#/FT- J J J J I J J J L=161 a) Assuming an alllowable deflection of L/360, compute the magnitude of the allowable deflection. b) Using an 8"x12" timber beam (see Table A1-b on page 567 of your text for properties) compute the actual deflection. Assume E = 1.6 x 100 psi. c) Based on your answers for parts a and b, determine if an 8"x12" timber beam is safe for this applicationarrow_forward
- Please use relationship method as I do not understand that one.arrow_forwardA freeway study resulted in a speed-density relationship: v=60(1-0.008k). Determine: a. The free-flow speed. b. The jam density. c. The speed-flow relationship. d. The flow-density relationship. e. The critical density. f. The capacity.arrow_forwardAverage Shear Stress BASIC FORMULAS General Shear Stress fv lb Shear Flow Flexural or Bending Stress VQ q Fb Mc M 1 Sx dtw fy-shear stress V = shear capacity d depth of the section tw = web thickness Radius of Gyration Q-statical moment of area Imoment of inertia of whole section b value of width where you cut Note: can be used to solve maximum shear stress Note: Maximum Shear Stress is located at the neutral axis (smallest width) Section Modulus Sx=x NOTE: If not given Fy 248 MPa for A36 Steel Fu-400 MPa for A36 Steel Es-200,000 MPa fb flexural or bending stress M-moment capacity I-Moment of Inertia of the whole section Sx-section modulus fy actual shear stress (from loads) Fv allowable shear stress (from NSCP) fv-Fv; safe and economical Vmax= P2 P Mmax = COMMON LOADINGS AND SUPPORTS PPP L/4 4↓1 44 L/4 PL PL Ymax = 48E1 Vmax = 1.5P Mmax = 22 PL 19PL Ymax = 384E1 P P [[ P P WL Vmax= 2 Mmax = P P L/3 ↓ L/3 ↓ L/3 WL' Swi Ymax = 384 ΕΙ PL 23PL Vmax = P Mmax = 이 Ymax =. 648E1arrow_forward
- Consider, M people (aka pax) who want to travel by car from O to D. They all start working at D at Q (e.g., Q-8am). If a person departs at time t, assume the time needed to go from O to D is given by c(t)=A+Bx(t), where x(t) is the flow of people departing at time t [car/unit of time]. In addition, a is the penalty for being early at work (E(t) is how early the person arrived when departing at time t), and ẞ is the penalty for being late at work (L(t) is how late the person arrived when departing at time t). Assume 0 < a < 1 < ß. Further assume the departure time choice problem under the equilibrium conditions. Prove that the arrival time of people who depart when most of the M people start their trips is equal to Q.arrow_forwardConstruction Methodarrow_forwardConstruction Method Ps. not graded, just a reviewerarrow_forward
- If you could help me answer these questions in matlab that would be great, I provided an additional picture detailing what the outcome should look like.arrow_forwardA Water at A flows out of the 1-in.-diameter nozzle at 8 ft s and strikes the 0.5 -lb-plate.Determine the height h above the nozzle at which the plate can be supported by the water jetarrow_forward= = Q1/A cantilever sheet-pile wall penetrating a granular soil. Here, L₁= 3 m, L2 = 6 m, y 17.3kN/m³, Ysat 19.4 kN/m², and 0= 30. a. What is the theoretical depth of embedment, D? b. For a 30% increase in D, what should be the total length of the sheet piles? c. What should be the minimum section modulus of the sheet piles? Use σall = 172 MN/m².arrow_forward
