Studdy AI Logo
Math  /  Geometry

QuestionDraw the shear force and bending moment diagram for the beam below. Let P=800 N,a=5 m\mathrm{P}=800 \mathrm{~N}, \mathrm{a}=5 \mathrm{~m}, and L=12 m\mathrm{L}=12 \mathrm{~m}.

Studdy Solution

STEP 1

1. The beam is simply supported with a hinge at the left end and a roller at the right end.
2. Two downward forces, each of magnitude P=800 N P = 800 \, \text{N} , are acting on the beam.
3. The distance from the left support to each force is a=5 m a = 5 \, \text{m} .
4. The total length of the beam is L=12 m L = 12 \, \text{m} .

STEP 2

1. Determine the reactions at the supports.
2. Draw the shear force diagram.
3. Draw the bending moment diagram.

STEP 3

Determine the reactions at the supports.
- Sum of vertical forces: RA+RB−2P=0 R_A + R_B - 2P = 0 - Taking moments about point A A :
RB⋅L−P⋅a−P⋅(a+a)=0 R_B \cdot L - P \cdot a - P \cdot (a + a) = 0
Substitute P=800 N P = 800 \, \text{N} , a=5 m a = 5 \, \text{m} , and L=12 m L = 12 \, \text{m} :
RB⋅12−800⋅5−800⋅10=0 R_B \cdot 12 - 800 \cdot 5 - 800 \cdot 10 = 0
RB⋅12=4000+8000 R_B \cdot 12 = 4000 + 8000
RB=1200012=1000 N R_B = \frac{12000}{12} = 1000 \, \text{N}
- Substitute RB R_B back into the sum of vertical forces:
RA+1000−1600=0 R_A + 1000 - 1600 = 0
RA=600 N R_A = 600 \, \text{N}

STEP 4

Draw the shear force diagram.
- Start from the left end (x=0 x = 0 ):
V(0−)=0 V(0^-) = 0
- Just after A A (x=0+ x = 0^+ ):
V(0+)=RA=600 N V(0^+) = R_A = 600 \, \text{N}
- Just before the first force (x=5− x = 5^- ):
V(5−)=600 N V(5^-) = 600 \, \text{N}
- Just after the first force (x=5+ x = 5^+ ):
V(5+)=600−800=−200 N V(5^+) = 600 - 800 = -200 \, \text{N}
- Just before the second force (x=10− x = 10^- ):
V(10−)=−200 N V(10^-) = -200 \, \text{N}
- Just after the second force (x=10+ x = 10^+ ):
V(10+)=−200−800=−1000 N V(10^+) = -200 - 800 = -1000 \, \text{N}
- Just before B B (x=12− x = 12^- ):
V(12−)=−1000 N V(12^-) = -1000 \, \text{N}
- At B B (x=12 x = 12 ):
V(12)=−1000+1000=0 V(12) = -1000 + 1000 = 0

STEP 5

Draw the bending moment diagram.
- Start from the left end (x=0 x = 0 ):
M(0)=0 M(0) = 0
- At x=5 x = 5 :
M(5)=RA⋅5=600⋅5=3000 Nm M(5) = R_A \cdot 5 = 600 \cdot 5 = 3000 \, \text{Nm}
- At x=10 x = 10 :
M(10)=RA⋅10−P⋅(10−5)=600⋅10−800⋅5=6000−4000=2000 Nm M(10) = R_A \cdot 10 - P \cdot (10 - 5) = 600 \cdot 10 - 800 \cdot 5 = 6000 - 4000 = 2000 \, \text{Nm}
- At x=12 x = 12 :
M(12)=0 M(12) = 0
The shear force and bending moment diagrams are now complete.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord