A long, thin beam is carrying a concentrated mass and is loaded by a time-dependent force. It is simply supported. The problem is described using the following parameters. Determine the deflections in the given test times.
Transient Response to Constant Force
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As gravity loads act on a structure, lateral displacement occurs. In turn, a secondary overturning moment is generated as the gravity load continues to act on the elements in the laterally displaced position. This effect is also known as "P-Delta (Δ)". Sec. 12.9.1.6 of the ASCE 7-16 Standard and the NBC 2015 Commentary specify when P-Delta effects should be considered during a modal response spectrum analysis.
The story drift of a building provides valuable information about its structural behavior under seismic loads. These could cause large horizontal deformations and even instabilities. Some standards, therefore, call for a check of the story drift in its center of gravity. It indicates, for example, if a second-order analysis (P-Δ effect) is necessary.
RFEM offers the option to perform a response spectrum analysis according to ASCE 7-16. This standard describes the determination of seismic loads for the American market. It might happen that the P-Delta effect has to be considered due to the stiffness of the entire structure in order to calculate the internal forces and carry out the design.
![Reduction of Building to Cantilever Structure: The individual mass points represent the floors. The deflection due to the normal compression forces shown in (a) is (b) converted into equivalent moments of displacement or shear forces [2].](/en/webimage/009762/2420261/01-en-png-12-png.png?mw=512&hash=8189361dbff525c2ff39c36a72a1ca91810208f6)
For the ultimate limit state design, EN 1998 1, Sections 2.2.2 and 4.4.2.2 [1], requires the calculation considering the second-order theory (P-Δ effect). This effect may be neglected only if the interstory drift sensitivity coefficient θ is less than 0.1. The coefficient θ is defined as follows:
$$\mathrm\theta\;=\;\frac{\displaystyle{\mathrm P}_\mathrm{tot}\;\cdot\;{\mathrm d}_\mathrm r }{{\mathrm V}_\mathrm{tot}\;\cdot\;\mathrm h}\;(1)$$
where
θ is the interstory drift sensitivity coefficient,
Ptot is the total gravity load at and above the story considered in the seismic design situation (see Expression 2),
dr is the design interstory drift, evaluated as the difference of the average lateral displacements dS at the top and bottom of the story under consideration; for this, the displacement is determined using the linear design response spectrum with q = 1.0,
Vtot is the total seismic story shear determined using the linear design response spectrum,
h is the interstory height.
$$\mathrm\theta\;=\;\frac{\displaystyle{\mathrm P}_\mathrm{tot}\;\cdot\;{\mathrm d}_\mathrm r }{{\mathrm V}_\mathrm{tot}\;\cdot\;\mathrm h}\;(1)$$
where
θ is the interstory drift sensitivity coefficient,
Ptot is the total gravity load at and above the story considered in the seismic design situation (see Expression 2),
dr is the design interstory drift, evaluated as the difference of the average lateral displacements dS at the top and bottom of the story under consideration; for this, the displacement is determined using the linear design response spectrum with q = 1.0,
Vtot is the total seismic story shear determined using the linear design response spectrum,
h is the interstory height.
![Reduction of Building to Cantilever Structure: The individual mass points represent the floors. The deflection due to the normal compression forces shown in (a) is (b) converted into equivalent moments of displacement or shear forces [2].](/en/webimage/009762/2420261/01-en-png-12-png.png?mw=350&hash=dd36dc43123116724231958668ad6cdcb13a0169)
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