Question

Q1. Figure 1 shows a cantilever I-beam subjected to a distributed load whose maximum intensity is $ q $ at the free end. The dimensions of the I-section of the beam are indicated in Figure 2. Figure 1 Figure 2 Considering the $ x $-axis to be parallel to the longitudinal axis of the beam, the $ z $-axis to be perpendicular to the plane of loading and the origin of the $ x-y-z $ axes to coincide with the centroid of the I-section at the fixed end, (a) Evaluate the shear stress $ \tau_{x z} $ in the top flange of the I-section at $ x=L / 2 $ and at a variable distance $ z \in\left[-b / 2,-t_{w} / 2\right] \cup\left[t_{w} / 2, b / 2\right] $ along its width. Plot the variation of this stress with $ z $. Assume that this stress does not vary across the thickness $ t_{f} $ of the flange. (b) Evaluate the shear stress $ \tau_{x y} $ in the web of the I-section at $ x=L / 2 $ and at variable distance $ y \in[-h / 2, h / 2] $ along its depth. Plot the variation of this stress with $ y $. Assume that this stress does not vary across the thickness $ t_{w} $ of the web. (c) Using the shear stress obtained in (b), determine the fraction of the total shear force at $ x=L / 2 $ that is borne by the web. (d) Consider a horizontal plane (parallel to $ x-\mathrm{z} $ plane) passing through the beam's web at $ y=h / 4 $. Evaluate and plot the variation of the shear stress $ \tau_{x y} $ on this plane as a function of the distance $ x \in[0, L] $ from the fixed-end of the beam.

          Q1. Figure 1 shows a cantilever I-beam subjected to a distributed load whose maximum intensity is \( q \) at the free end. The dimensions of the I-section of the beam are indicated in Figure 2.
Figure 1
Figure 2

Considering the \( x \)-axis to be parallel to the longitudinal axis of the beam, the \( z \)-axis to be perpendicular to the plane of loading and the origin of the \( x-y-z \) axes to coincide with the centroid of the I-section at the fixed end,
(a) Evaluate the shear stress \( \tau_{x z} \) in the top flange of the I-section at \( x=L / 2 \) and at a variable distance \( z \in\left[-b / 2,-t_{w} / 2\right] \cup\left[t_{w} / 2, b / 2\right] \) along its width. Plot the variation of this stress with \( z \). Assume that this stress does not vary across the thickness \( t_{f} \) of the flange.
(b) Evaluate the shear stress \( \tau_{x y} \) in the web of the I-section at \( x=L / 2 \) and at variable distance \( y \in[-h / 2, h / 2] \) along its depth. Plot the variation of this stress with \( y \). Assume that this stress does not vary across the thickness \( t_{w} \) of the web.
(c) Using the shear stress obtained in (b), determine the fraction of the total shear force at \( x=L / 2 \) that is borne by the web.
(d) Consider a horizontal plane (parallel to \( x-\mathrm{z} \) plane) passing through the beam's web at \( y=h / 4 \). Evaluate and plot the variation of the shear stress \( \tau_{x y} \) on this plane as a function of the distance \( x \in[0, L] \) from the fixed-end of the beam.

$Q1. Figure 1 shows a cantilever I-beam subjected to a distributed load whose maximum intensity is q at the free end. The dimensions of the I-section of the beam are indicated in Figure 2. Figure 1 Figure 2 Considering the x-axis to be parallel to the longitudinal axis of the beam, the z-axis to be perpendicular to the plane of loading and the origin of the x-y-z axes to coincide with the centroid of the I-section at the fixed end, (a) Evaluate the shear stress τx z in the top flange of the I-section at x=L / 2 and at a variable distance z ∈[-b / 2,-tw / 2] ∪[tw / 2, b / 2] along its width. Plot the variation of this stress with z. Assume that this stress does not vary across the thickness tf of the flange. (b) Evaluate the shear stress τx y in the web of the I-section at x=L / 2 and at variable distance y ∈[-h / 2, h / 2] along its depth. Plot the variation of this stress with y. Assume that this stress does not vary across the thickness tw of the web. (c) Using the shear stress obtained in (b), determine the fraction of the total shear force at x=L / 2 that is borne by the web. (d) Consider a horizontal plane (parallel to x-z plane) passing through the beam's web at y=h / 4. Evaluate and plot the variation of the shear stress τx y on this plane as a function of the distance x ∈[0, L] from the fixed-end of the beam.$

Added by Marcus W.

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