STATICS & STRENGTH OF MATERIALS - Example 

A loaded, simply supported beam is shown below. For this beam:

A. Draw a Free Body Diagram of the beam, showing all external loads and support forces (reactions).
B. Determine expressions for the internal shear forces and bending moments in each sections of the beam.
C. Make shear force and bending moment diagrams for the beam.
Unless otherwise indicated, all joints and support points are assumed to be pinned or hinged joints.

Solution:

Part A:
STEP 1: Draw a free body diagram showing and labeling all load forces and support (reaction) forces, as well as any needed angles and dimensions.
STEP 2: Break any forces not already in x and y direction into their x and y components.
STEP 3: Apply the equilibrium conditions.

Sum F_y = (-1,000 lbs/ft)(4 ft) - (1,500 lbs/ft)(4 ft) + B_y + C_y = 0
Sum T_B = (C_y)(6 ft) + (1,000 lbs/ft)(4 ft)(2 ft) - (1,500 lbs/ft)(4 ft)(8 ft) = 0
Solving for the unknowns:
C_y = 6,670 lbs; B_y = 3,330 lbs

Part B: Determine the Shear Forces and Bending Moments expressions for each section of the loaded beam. For this process we will ‘cut’ the beam into sections, and then use Statics - Sum of Forces to determine the Shear Force expressions, and Integration to determine the Bending Moment expressions in each section of the beam.

Section 1: Cut the beam at x, where 0 < x < 4 ft. Analyze left hand section.

1. FBD. (Shown in Diagram)
2. All forces in x & y components (yes)
3. Apply translational equilibrium conditions (forces only):

Sum Fx = 0 (no net external x- forces)
Sum Fy = -1,000 lbs/ft(x) - V1 = 0

Solving: V1 = -1,000x lbs

4. Integration

M1 = -500 x² + C1

a)Boundary condition to find C1: at x=0 M=0
Apply BC: 0 = -500(0)² + C1
Solving: C1 = 0
Therefore… M1 = [-500x²] ft-lbs for 0 < x < 4 ft.

Section 2: Cut the beam at x, where 4 < x < 10 ft. Analyze left hand section.

1. FBD. (Shown in Diagram)
2. All forces in x & y components (yes)
3. Apply translational equilibrium conditions (forces only):

Sum Fx = 0 (no net external x- forces)
Sum Fy = -1,000 lbs/ft (4 ft) + 3,330 lbs - V2 = 0
Solving: V2 = -667 lbs

4. Integration

M2 = -667x + C2

a)Boundary condition to find C2: at x=4 ft M=-8000 ft-lbs (from equation M1)
Apply BC: 8000 ft-lbs = -667(4) + C2

Solving: C2 = -5,330 ft-lbs
Therefore… M2 = [-667x - 5,330] ft-lbs for 4 < x < 10

Section 3: Cut the beam at x, where 10 < x < 14 ft. Analyze left hand section.

1. FBD. (Shown in Diagram)
2. All forces in x & y components (yes)
3. Apply translational equilibrium conditions (forces only):

Sum Fx = 0 (no net external x- forces)
Sum Fy = -1,000 lbs/ft(4 ft) + 3,330 lbs + 6,670 lbs -1500lbs/ft(x-10)ft - V3 = 0

Solving: V3 = [-1,500x + 21,000] lbs

4. Integration

M3 = -750x² + 21,000x + C3

a)Boundary condition to find C3: at x=14 ft M=0 ft-lbs (end of beam, no external torque so M3=0)
Apply BC: 0 = -750(14)² + 21,000(14) + C3

Solving: C3 = -147,000 ft-lbs
Therefore… M3 = [-750x² + 21,000x - 147,000] ft-lbs for 10 < x < 14

Part C: Shear Force and Bending Moment Diagrams: Now using the expressions found in Part B above, we can draw the shear force and bending moment diagrams for our loaded beam.

V1 = -1,000x lb, V2 = -667 lb, V3 = -1,500x+21,000 lb
M1 = -500x² ft-lb, M2 = -667x-5,330 ft-lb, M3 = -750x²+21,000x-147,000 ft-lb

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