
Concept explainers
Find the equations for slope and deflection of the beam using direct integration method.

Answer to Problem 6P
For segment AB:
The equation for slope is
The equation for deflection is
For segment BC:
The equation for slope is
The equation for deflection is
Explanation of Solution
Calculation:
Consider flexural rigidity EI of the beam is constant.
Draw the free body diagram of the beam as in Figure (1).
Refer Figure (1),
Consider upward force is positive and downward force is negative.
Consider clockwise is negative and counterclowise is positive.
Determine the support reaction at A using the relation;
Determine the support reaction at B using the relation;
Show the reaction values as in Figure (2).
Take a section at a distance of x.
Show the section as in Figure (2).
Consider the segment AB:
Refer Figure (2),
Write the equation for bending moment at x distance.
Write the equation for
Write the equation for slope as follows:
Integrate Equation (1) with respect to x.
Write the equation for deflection as follows:
Integrate Equation (2) with respect to x.
Find the integration constants
Write the boundary conditions as follows:
Apply the above boundary conditions in Equation (3):
Write the boundary conditions as follows:
Apply the above boundary conditions in Equation (3):
Find the equation for slope.
Substitute
Thus, the equation for slope is
Find the equation for deflection.
Substitute
Thus, the equation for deflection is
Consider segment BC;
Show the distance at a distance of x as in Figure (4).
Refer Figure (2),
For segment BC the limit should be
Write the equation for bending moment at x distance.
Write the equation for
Write the equation for slope as follows:
Integrate Equation (4) with respect to x.
Write the equation for deflection as follows:
Integrate Equation (5) with respect to x.
Write the boundary conditions as follows:
The slope at left and right of support B is equal.
Substitute
Apply the above boundary conditions in the above Equation
Substitute
Hence, the Equation for slope is
Write the boundary conditions as follows:
Apply the above boundary conditions in Equation (6):
Substitute
Hence, the Equation for deflection is
Want to see more full solutions like this?
Chapter 6 Solutions
Structural Analysis
- 6000 units have been installed to date with 9,000 units to install. Labor costs are $23,300.00 to date. What is the unit cost for labor to date?arrow_forwardThe base rate for labor is $15/hr. The labor burden is 35% and 3% for small tools for the labor. There are 1000 units to install. Records indicate that trade workers can install 10 units per hour, per trade worker. The owners need 15% overhead and profit to pay bills, pay interest on loan and provide some profit to the partners. What is the minimum bid assuming no risk avoidance factor?arrow_forwardCan you show me how to obtain these answers thanks, will rate!arrow_forward
- I have the answers for part a just need help with b mostly thanksarrow_forwardPlease explain step by step and show formulasarrow_forward5. (20 Points) Consider a channel width change in the same 7-foot wide rectangular in Problem 4. The horizontal channel narrows as depicted below. The flow rate is 90 cfs, and the energy loss (headloss) through the transition is 0.05 feet. The water depth at the entrance to the transition is initially 4'. 1 b₁ TOTAL ENERGY LINE V² 129 У1 I b₂ TOP VIEW 2 PROFILE VIEW h₁ = 0.05 EGL Y₂ = ? a) b) c) 2 Determine the width, b₂ that will cause a choke at location 2. Determine the water depth at the downstream end of the channel transition (y₂) section if b₂ = 5 feet. Calculate the change in water level after the transition. Plot the specific energy diagram showing all key points. Provide printout in homework. d) What will occur if b₂ = = 1.5 ft.?arrow_forward
- 4. (20 Points) A transition section has been proposed to raise the bed level a height Dz in a 7-foot wide rectangular channel. The design flow rate in the channel is 90 cfs, and the energy loss (headloss) through the transition is 0.05 feet. The water depth at the entrance to the transition section is initially 4 feet. b₁ = b = b2 1 TOTAL ENERGY LINE V² 129 Ут TOP VIEW 2 hloss = 0.05 " EGL Y₂ = ? PROFILE VIEW a) Determine the minimum bed level rise, Dz, which will choke the flow. b) If the step height, Dz = 1 ft, determine the water depth (y2) at the downstream end of the channel transition section. Calculate the amount the water level drops or rises over the step. c) Plot the specific energy diagram showing all key points. Provide printout in Bework. d) What will occur if Dz = 3.0 ft.?. Crest Front Viewarrow_forward1. (20 Points) Determine the critical depth in the trapezoidal drainage ditch shown below. The slope of the ditch is 0.0016, the side slopes are 1V:2.5H, the bottom width is b = 14', and the design discharge is 500 cfs. At this discharge the depth is y = 4.25'. Also, determine the flow regime and calculate the Froude number. Ye= ? Z barrow_forward3. (20 Points) A broad crested weir, 10 feet high, will be constructed in a rectangular channel B feet wide. The weir crest extends a length of B = 120 feet between the banks with 2 - 4 foot wide, round nosed piers in the channel. The width of the weir crest is 8 feet. If H = 6', determine the design discharge for the weir.arrow_forward
- Parking Needs vs. Alternative Transportation Methods for presentation slides include images and graphsarrow_forwardPlease explain step by step and show formulararrow_forwardBeam ABD is supported and loaded as shown. The cross-section of the beam is also shown. The modulus of elasticity of the beam is 200 GPa. 6.0 kN/m Cross-section: 330 mm 4.5 kN 8.0 kNm 40 mm 2.5 m 1.5 m 20 mm Set up the discontinuity moment function in terms of x. List all the appropriate boundary conditions. Determine the slope function in terms of x. Determine the deflection function in terms of x. Determine the support reactions. Determine the maximum deflection. 290 mmarrow_forward
