1) Several loads are applied to a cantilevered beam as shown. A WT305 X 70 standard steel shape is used for the beam and the dimensions are shown below. a) Plot the shear force and bending moment diagram of the beam using the graphical method. Be sure to track calculations and clearly specify the maximum positive and negative values for shear and moment. [Ans. to Check: Vmax-positive = 40 kN; Mmax-negative = -70 kN-m] b) Locate the neutral axis of the cross-section relative to the bottom of the shape in mm. [Ans. to Check: y = 233 mm from bottom] c) d) Compute the moment of inertia in mm* about the z-axis. [Ans. to Check: I₂ = 78,338,100 mm¹] Calculate the bending stress in the top and bottom of the beam at the point of maximum positive moment (in MPa). Be sure to specify which fiber is in tension and which fiber is in compression. Plot the normal stress distribution for this location using the rules we learned in class. [Ans. to Check: Otop=49.15 MPa (C)] e) Calculate the bending stress in the top and bottom of the beam at the point of maximum negative moment (in MPa). Be sure to specify which fiber is in tension and which fiber is in compression. Plot the normal stress distribution for this location using the rules we learned in class. [Ans. to Check: 0 top = 68.8 MPa (T)] f) Use your answers from parts (d) and (e) to determine the location, magnitude, and sign of the maximum normal tensile and compressive stresses due to bending (in MPa). Note: Specify where these stresses happen along the beam (e.g. x = 2m) and where in the section (e.g. top or bottom). 50 kN-m 20 kN 2 m 60 KN 287.8 mm B Beam loading along its length 230 mm An 3 m 1 40 kN/m -13.1 mm 22.2 mm 1 m. D Cross-section of the beam (view looking down the beam from A to D)
1) Several loads are applied to a cantilevered beam as shown. A WT305 X 70 standard steel shape is used for the beam and the dimensions are shown below. a) Plot the shear force and bending moment diagram of the beam using the graphical method. Be sure to track calculations and clearly specify the maximum positive and negative values for shear and moment. [Ans. to Check: Vmax-positive = 40 kN; Mmax-negative = -70 kN-m] b) Locate the neutral axis of the cross-section relative to the bottom of the shape in mm. [Ans. to Check: y = 233 mm from bottom] c) d) Compute the moment of inertia in mm* about the z-axis. [Ans. to Check: I₂ = 78,338,100 mm¹] Calculate the bending stress in the top and bottom of the beam at the point of maximum positive moment (in MPa). Be sure to specify which fiber is in tension and which fiber is in compression. Plot the normal stress distribution for this location using the rules we learned in class. [Ans. to Check: Otop=49.15 MPa (C)] e) Calculate the bending stress in the top and bottom of the beam at the point of maximum negative moment (in MPa). Be sure to specify which fiber is in tension and which fiber is in compression. Plot the normal stress distribution for this location using the rules we learned in class. [Ans. to Check: 0 top = 68.8 MPa (T)] f) Use your answers from parts (d) and (e) to determine the location, magnitude, and sign of the maximum normal tensile and compressive stresses due to bending (in MPa). Note: Specify where these stresses happen along the beam (e.g. x = 2m) and where in the section (e.g. top or bottom). 50 kN-m 20 kN 2 m 60 KN 287.8 mm B Beam loading along its length 230 mm An 3 m 1 40 kN/m -13.1 mm 22.2 mm 1 m. D Cross-section of the beam (view looking down the beam from A to D)
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:## Analysis of a Cantilevered Beam under Load
1) **Problem Overview:**
Several loads are applied to a cantilevered beam as indicated in the diagram below. A WT305 X 70 standard steel shape is employed for the beam, with the stated dimensions.
### Tasks:
a) **Shear Force and Bending Moment Diagram:**
- Plot the shear force and bending moment diagram of the beam using the graphical method.
- Track calculations and clearly specify the maximum positive and negative values for shear and moment.
- **Values to Check:**
- \( V_{\text{max-positive}} = 40 \, \text{kN} \)
- \( M_{\text{max-negative}} = -70 \, \text{kN-m} \)
b) **Neutral Axis Location:**
- Locate the neutral axis of the cross-section relative to the bottom of the shape in millimeters.
- **Value to Check:** \( \bar{y} = 233 \, \text{mm from bottom} \)
c) **Moment of Inertia:**
- Compute the moment of inertia in mm\(^4\) about the z-axis.
- **Value to Check:** \( I_z = 78,338,100 \, \text{mm}^4 \)
d) **Bending Stress at Maximum Positive Moment:**
- Calculate the bending stress in the top and bottom of the beam at the point of maximum positive moment (in MPa).
- Specify which fiber is in tension and which fiber is in compression.
- Plot the normal stress distribution for this location using the class rules.
- **Value to Check:** \( \sigma_{\text{top}} = 49.15 \, \text{MPa (C)} \)
e) **Bending Stress at Maximum Negative Moment:**
- Calculate the bending stress in the top and bottom of the beam at the point of maximum negative moment (in MPa).
- Specify which fiber is in tension and which fiber is in compression.
- Plot the normal stress distribution for this location using the class rules.
- **Value to Check:** \( \sigma_{\text{top}} = 68.8 \, \text{MPa (T)} \)
f) **Maximum Stresses Evaluation:**
- Use answers from parts (d) and (e) to determine the location, magnitude
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